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VOLUME 3

Participants in

Mathematics Teacher Education

Individuals, Teams,
Communities and Networks

Konrad Krainer and Terry Wood (Eds.)

The International Handbook of Mathematics
Teacher Education

Series Editor:

Terry Wood
Purdue University
West Lafayette
USA

This Handbook of Mathematics Teacher Education, the first of its kind, addresses the learning of
mathematics teachers at all levels of schooling to teach mathematics, and the provision of activity and
programmes in which this learning can take place. It consists of four volumes.

VOLUME 1:

Knowledge and Beliefs in Mathematics Teaching and Teaching Development

Peter Sullivan, Monash University, Clayton, Australia and Terry Wood, Purdue University, West

Lafayette, USA (eds.)

This volume addresses the "what" of mathematics teacher education, meaning knowledge for

mathematics teaching and teaching development and consideration of associated beliefs. As well as

synthesizing research and practice over various dimensions of these issues, it offers advice on best

practice for teacher educators, university decision makers, and those involved in systemic policy

development on teacher education.

paperback: 978-90-8790-541-5, hardback: 978-90-8790-542-2, ebook: 978-90-8790-543-9

VOLUME 2:

Tools and Processes in Mathematics Teacher Education

Dina Tirosh, Tel Aviv University, Israel and Terry Wood, Purdue University, West Lafayette, USA

(eds.)

This volume focuses on the "how" of mathematics teacher education. Authors share with the readers

their invaluable experience in employing different tools in mathematics teacher education. This

accumulated experience will assist teacher educators, researchers in mathematics education and those

involved in policy decisions on teacher education in making decisions about both the tools and the

processes to be used for various purposes in mathematics teacher education.

paperback: 978-90-8790-544-6, hardback: 978-90-8790-545-3, ebook: 978-90-8790-546-0

VOLUME 3:

Participants in Mathematics Teacher Education: Individuals, Teams, Communities and Networks

Konrad Krainer, University of Klagenfurt, Austria and Terry Wood, Purdue University, West Lafayette,

USA (eds.)

This volume addresses the "who" question of mathematics teacher education. The authors focus on the

various kinds of participants in mathematics teacher education, professional development and reform

initiatives. The chapters deal with prospective and practising teachers as well as with teacher educators

as learners, and with schools, districts and nations as learning systems.

paperback: 978-90-8790-547-7, hardback: 978-90-8790-548-4, ebook: 978-90-8790-549-1

VOLUME 4:

The Mathematics Teacher Educator as a Developing Professional

Barbara Jaworski, Loughborough University, UK and Terry Wood, Purdue University, West Lafayette,

USA (eds.)

This volume focuses on knowledge and roles of teacher educators working with teachers in teacher

education processes and practices. In this respect it is unique. Chapter authors represent a community

of teacher educators world wide who can speak from practical, professional and theoretical viewpoints

about what it means to promote teacher education practice.

paperback: 978-90-8790-550-7, hardback: 978-90-8790-551-4, ebook: 978-90-8790-552-1

Participants in Mathematics
Teacher Education

Individuals, Teams, Communities and Networks

Edited by

Konrad Krainer

University of Klagenfurt, Austria

and

Terry Wood

Purdue University, West Lafayette, USA

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BIBLIOTECA - LEGARIA

SENSE PUBLISHERS
ROTTERDAM / TAIPEI

A C.I. P. record for this book is available from the Library of Congress.

ISBN 978-90-8790-547-7 (paperback)
ISBN 978-90-8790-548-4 (hardback)
ISBN 978-90-8790-549-1 (e-book)

Published by: Sense Publishers,

P.O. Box 21858, 3001 AW Rotterdam, The Netherlands

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Badell river, Tavernes de la Valldigna, Valencia, Spain
© Pepa and Ana Llinares 2007

One drop of water does not make a river, yet each drop bears in itself the full fluidity and power of
water. A river is more than millions of drops, it is a large and vital system. It represents an astonishing
journey from the source to its mouth at the sea, from the micro to the macro level. A river marks a
steady coming together and growing but also finding different branches and maybe courses. It depends
on and is influenced by external elements like rain and rocks, but also by pollution. It is formed by its
environment, but is in turn a force upon its environment. A river is a journey of necessary collaboration
in a joint process.
© Salvador Llinares and Konrad Krainer 2008

Printed on acid-free paper

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by
any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written
permission from the Publisher, with the exception of any material supplied specifically for the purpose
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TABLE OF CONTENTS

Preface ix

Individuals, Teams, Communities and Networks: Participants and Ways

of Participation in Mathematics Teacher Education: An Introduction 1

Konrad Krainer

Section 1: Individual Mathematics Teachers as Learners

Chapter 1 : Individual Prospective Mathematics Teachers: Studies on

Their Professional Growth 13

Helia Oliveira and Markku S. Hannula

Chapter 2: Individual Practising Mathematics Teachers: Studies on

Their Professional Growth 35

Marie-Jeanne Perrin-Glorian, Lucie DeBlois, and Aline Robert

Section 2: Teams of Mathematics Teachers as Learners

Chapter 3: Teams of Prospective Mathematics Teachers: Multiple Problems

and Multiple Solutions 63

Roza Leikin

Chapter 4: Teams of Practising Teachers: Developing Teacher Professionals 89
Susan D. Nickerson

Chapter 5: Face-to-Face Learning Communities of Prospective Mathematics
Teachers: Studies on Their Professional Growth 1 1 1

Fou-Lai Lin andJoao Pedro da Ponte

Section 3: Communities and Networks of Mathematics Teachers as Learners

Chapter 6: Face-to-Face Communities and Networks of Practising

Mathematics Teachers: Studies on Their Professional Growth 133

Stephen Lerman and Stefan Zehetmeier

TABLE OF CONTENTS

Chapter 7: Virtual Communities and Networks of Prospective Mathematics
Teachers: Technologies, Interactions and New Forms of Discourse 1 55

Salvador Llinares and Federica Olivero

Chapter 8: Virtual Communities and Networks of Practising Mathematics
Teachers: The Role of Technology in Collaboration 1 8 1

Marcelo Borba and George Gadanidis

Section 4: Schools, Regions and Nations as Mathematics Learners

Chapter 9: School Development as a Means of Improving Mathematics
Teaching and Learning: Towards Multidirectional Analyses of Learning
across Contexts 209

Elham Kazemi

Chapter 10: District Development as a Means of Improving Mathematics
Teaching and Learning at Scale 23 1

Paul Cobb and Thomas Smith

Chapter 1 1 : Studies on Regional and National Reform Initiatives as a Means

to Improve Mathematics Teaching and Learning at Scale 255

John Peggand Konrad Krainer

Section 5: Teachers and Teacher Educators as Key Players in the
Further Development of the Mathematics Teaching Profession

Chapter 12: The Use of Action Research in Teacher Education 283

Gertraud Benke, Alena Hospesovd, and Marie Tichd

Chapter 13: Building and Sustaining Inquiry Communities in Mathematics
Teaching Development: Teachers and Didacticians in Collaboration 309

Barbara Jaworski

Chapter 14: Mathematics Teaching Profession 33 1

Nanette Seago

Section 6: Critical Respondants

Chapter 15: Pathways in Mathematics Teacher Education: Individual

Teachers and Beyond 355

Gilah C. Leder

TABLE OF CONTENTS

Chapter 16: Individuals, Teams and Networks: Fundamental Constraints of
Professional Communication Processes of Teachers and Scientists about
Teaching and Learning Mathematics 369

Heinz Steinbring

PREFACE

It is my honor to introduce the first International Handbook of Mathematics
Teacher Education to the mathematics education community and to the field of
teacher education in general. For those of us who over the years have worked to
establish mathematics teacher education as an important and legitimate area of
research and scholarship, the publication of this handbook provides a sense of
success and a source of pride. Historically, this process began in 1987 when
Barbara Jaworski initiated and maintained the first Working Group on mathematics
teacher education at PME. After the Working Group meeting in 1994, Barbara,
Sandy Dawson and I initiated the book, Mathematics Teacher Education: Critical
International Perspectives, which was a compilation of the work accomplished by
this Working Group. Following this, Peter de Liefde who, while at Kluwer
Academic Publishers, proposed and advocated for the Journal of Mathematics
Teacher Education and in 1998 the first issue of the journal was printed with
Thomas Cooney as editor of the journal who set the tone for quality of manuscripts
published. From these events, mathematics teacher education flourished and
evolved as an important area for investigation as evidenced by the extension of
JMTE from four to six issues per year in 2005 and the recent 15 th ICMI Study, The
professional education and development of teachers of mathematics. In preparing
this handbook it was a great pleasure to work with the four volume editors, Peter
Sullivan, Dina Tirosh, Konrad Krainer and Barbara Jaworski and all of the authors
of the various chapters found throughout the handbook.

Volume 3, Participants in mathematics teacher education: Individuals, teams,
communities and networks, edited by Konrad Krainer, focuses not only on
individual prospective and practicing teachers as learners but also on teams,
learning communities, networks of teachers, schools and on the teaching profession
as a whole. In this volume, the emphasis is on describing and critically analysing
participants' organizational contexts. This is the third volume of the handbook.

Terry Wood
West Lafayette, IN
USA

REFERENCES

Jaworski, B., Wood, T., & Dawson, S. (Eds). (1999). Mathematics teacher education: Critical

international perspectives. London: Falmer Press.
Krainer, K., & Wood, T. (Eds.). (2008). International handbook of mathematics teacher education: Vol.

3. Participants in mathematics teacher education: Individuals, teams, communities and networks.

Rotterdam, the Netherlands: Sense Publishers.
Wood, T. (Series Ed), Jaworski, B., Krainer, K., Sullivan, P., & Tirosh, D. (Vol. Eds.). (2008).

International handbook of mathematics teacher education. Rotterdam, the Netherlands: Sense

Publishers

KONRAD KRAINER

INDIVIDUALS, TEAMS, COMMUNITIES AND

NETWORKS: PARTICIPANTS AND WAYS OF

PARTICIPATION IN MATHEMATICS TEACHER

EDUCATION

An Introduction

This chapter outlines the main idea of the third volume of this handbook. The focus
is to make transparent the large diversity of experiences evident in the field of
mathematics teacher education and to describe and learn from these differences.
This is achieved by focusing on the practices of various teacher education
participants and the environments in which they work. Notions like individuals,
teams, communities and networks are introduced. In addition, school development
as well as regional and national teacher education initiatives are regarded. Each
chapter considers differences within and among these distinctions.

THE MAIN IDEA OF THIS VOLUME

What is (mathematics) teacher education and who are its participants! The teacher
educator can be regarded as the teacher of prospective or practising teachers.
Although, traditionally, the latter are regarded mainly as the participants and
learners, also the teacher educator is a part of the learning process and grows
professionally from participating in the process. Therefore, it makes sense to regard
student teachers, teachers and teacher educators as teachers and as (lifelong)
learners at the same time (see also Llinares & Krainer, 2006). Teachers are active
constructors of their knowledge, embedded in a variety of social environments.
These environments influence and shape teachers' beliefs, knowledge and practice;
similarly, teachers themselves influence and shape their environments. Therefore,
teachers should be expected to continuously reflect in and on their practice and the
environments in which they work; teachers should also be prepared to make
changes where it is appropriate. Whereas teachers and students in a classroom (at a
school or in a course for prospective teachers) mostly show considerable
differences in age and experience, the relation in the context of professional
development activities might be more diverse (e.g., having a young researcher
working with experienced teachers).

Teacher education is a goal-directed intervention in order to promote teachers'
learning, including all formal kinds of teacher preparation and professional
development as well as informal (self-organized) activities. Mathematics teacher

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 1-10.
© 2008 Sense Publishers. All rights reserved.

KONRAD KRAINER

education can aim at improving teachers' beliefs, their knowledge and their
practice, at increasing their motivation, their self-confidence and their identity as
mathematics teachers and, most importantly, at contributing to their students'
affective and cognitive growth.

Although the education of prospective mathematics teachers is organized in a
wide variety of ways in different countries and at different teacher education
institutes, teacher education activities for practicing mathematics teachers are even
more diverse. They include, for example: formal activities led by externals (e.g., by
mathematics educators and mathematicians) or informal and self-organized ones
(e.g., by a group of mathematics teachers themselves); single events or continuous
and long-lasting programmes; small-group courses or nation-wide mathematics
initiatives (eventually with hundreds of participants); heterogeneous groups of
participants (e.g., mathematics and science teachers from all parts of a country) or
a mathematics-focused school development programme at one single school;
obligatory participation in courses or voluntary engagement in teacher networks;
focus on specific contents (e.g., early algebra) or on more general issues (e.g., new
modes of instruction); theory-driven seminars at universities or teaching
experiments at schools; focus on primary or secondary schooling; teacher
education accompanied by extensive research or confined to minimal evaluation;
activities that aim at promoting teachers' (different kinds of) knowledge, or beliefs,
or practice etc.

It is a challenge to find answers to the questions of where, under which
conditions, how and why mathematics teachers learn and how important the
domain-specific character of mathematics is. It is important to take into account
that teachers' learning is a complex process and is to a large extent influenced by
personal, social, organisational, cultural and political factors.

Discussions about "effective", "good", "successful" etc. mathematics teaching
or teacher education indicate different aspects which need to be considered (see
e.g., Sowder, 2007). However, in any case the dimensions contents, communities
and contexts (see e.g., Lachance & Confrey, 2003; Krainer, 2006) are addressed at
least implicitly:

- Contents are needed that are relevant for all people who are involved (e.g.,
contents that are interesting mathematical activities for the students, challenging
experiments, observations and reflections for teachers, constructive initiatives
and discussions for mathematics departments at schools);

- Communities (including small teams, communities of practice and loosely-
coupled networks) are needed that allow people to collaborate with each other
in order to learn autonomously but also to support others' and the whole
system's content-related learning;

- Contexts (within a professional development program, at teachers' schools, in
their school district, etc.) are needed that provide conducive general conditions
(resources, structures, commitment, etc.).

The "community" aspect does not mean that initiatives by individual teachers
are not essential. In contrast, the single teacher is of great importance, in particular

REFLECTING THE DEVELOPMENT

at the classroom level. However, in order to bring about change at the level of a
whole department, school, district or nation, thinking only in terms of individual
teachers is not sufficient ("one swallow does not make a summer"). Research on
"successful" schools shows that such schools are more likely to have teachers who
have continual substantive interactions (Little, 1982) or that inter-staff relations are
seen as an important dimension of school quality (Reynolds et al., 2002). The latter
study illustrates, among others, examples of potentially useful practices, of which
the first (illustrated by a US researcher who reflects on observations in other
countries) relates to teacher collaboration and community building (p. 281):

Seeing excellent instruction in an Asian context, one can appreciate the
lesson, but also understand that the lesson did not arrive magically. It was
planned, often in conjunction with an entire grade-level-team (or, for a first-
year teacher, with a master teacher) in the teachers' shared office and work
area. [Referring to observed schools in Norway, Taiwan and Hong Kong: ...]
if one wants more thoughtful, more collaborative instruction, we need to
structure our schools so that teachers have the time and a place to plan, share
and think.

This example underlines the interconnectedness of the three dimensions content,
community and context. Most research papers in mathematics teacher education
put a major focus on the content dimension, much less attention is paid to the other
two dimensions. In order to fill this gap, the major focus and specific feature of this
volume is the emphasis on the "community" dimension. This dimension raises the
"who" question and asks where and how teachers participate and collaborate in
teacher education. It puts the participants and their ways of participation into the
foreground.

How can a volume of a handbook that focuses on participants in teacher
education deal with this diversity and offer a viable structure? One option is to pick
one of the manifold differences (formal versus informal; single events versus
continuous and sustained programmes etc., see above). A very relevant difference
is the number of participants as, for example, shown (see Table 1) in the case of
supporting practising teachers to improve their practice in mathematics classrooms
and related research on their professional growth. Of course, entities like teams,
communities, networks, schools, districts, regions and nations are only assigned
provisionally to one of the levels. There might, for example, be teams with ten or
more people or small schools (e.g., in rural primary schools) with less than ten
people that teach mathematics (and also most other subjects).

Table I. Levels of teacher education

Number of
M teachers

Relevant environments (in addition
to mathematics teacher educators)

Major mathematics education
research focus on \. . . ]

Micro level

Students, Parents, . . .

Individual teachers, Teams

Meso level

10s

Colleagues, Leaders, ...

Communities, Networks, Schools

Macro level

100s

Superintendents, Policy makers, ...

Districts, Regions, Nations

KONRAD KRAINER

However, in principle, it makes a difference whether we regard the professional
growth of one or a few teachers in a mathematics department (micro level), or of
tens of mathematics teachers at a larger school (meso level) or of hundreds or even
thousands of mathematics teachers in a district or a whole nation (macro level).

Concerning the three levels, quite different people are interested in the impacts
of teacher education initiatives: in the case of single classrooms, the students and
their parents are the most concerned environments; in contrast, superintendents and
(above all) policy makers are more interested to get a whole picture over all
classrooms in a country. For example, PISA plays a major role for nations' system
monitoring of mathematics teaching, but not so much for individual teachers and
parents. They are more interested in the learning progress of their own students.
Schools as organizations or networks of dedicated teachers lay somewhat in
between. On the one hand, a school is important for teachers and parents since this
organizational entity forms a crucial basis and environment for students' learning;
for example, this includes important feelings of being accepted, autonomous,
cognitively supported, a member of a community, safe, taken serious etc. On the
other hand, reformers need to see schools as units of educational change since they
cannot reach teachers and students directly. All in all, each of the three levels is
important and the three should be regarded as closely interconnected.

However, our knowledge on teachers' learning is not equally distributed among
these three levels. A survey by Adler, Ball, Krainer, Lin, and Novotna (2005) of
recent research in mathematics teacher education culminated in three claims:

- Claim 1: Small-scale qualitative research predominates. Most studies
investigate the beliefs, knowledge, or practice (and often also the professional
growth) of individual or of a few teachers. So the focus is primarily laid on the
micro (and partially on the meso) level, more emphasis is needed on the meso
and the macro level.

- Claim 2: Most teacher education research is conducted by teacher educators
studying the teachers with whom they are working. Also here, the focus is more
on the micro (and partially on the meso) level since the context of the research is
prospective or practising teachers' classrooms. Again, more research is needed
on the meso and the macro level.

- Claim 3: Research in countries where English is the national language
dominates the literature. Therefore, we do not know enough about research
projects in countries where the dominant language is not English. It is more
likely to get studies from individual researchers from these countries than
reports about national reforms. There is a big need for comparison of cases of
reform initiatives (in international publications).

Since the recent research focus in mathematics teacher education is still on the
macro level - with a tendency to spread more and more also to the meso level,
most chapters in this volume also deal with these two levels. This makes it
necessary to offer a framework that allows us to differentiate between different
kinds of groups in which teachers participate. A definition by Allee (2000), which

REFLECTING THE DEVELOPMENT

distinguishes between the notions "teams", "communities" and "networks", is
helpful here:

- Teams (and project groups) are regarded as mostly selected by the management,
have pre-determined goals and therefore have rather tight and . formal
connections within the team.

- Communities are regarded as self-selecting, their members negotiating goals and
tasks. People participate because they personally identify with the topic.

- Networks are loose and informal because there is no joint enterprise that holds
them together. Their primary purpose is to collect and pass along information.
Relationships are always shifting and changing as people have the need to
connect.

I regard these distinctions (of course, others might be possible, too) as crucial
since, for example, it makes a difference whether initiatives in teacher education
are predominantly planned from top-down (e.g., by installing teams, task forces
etc.) or whether they support bottom-up-approaches (e.g., by funding networks of
teachers that establish their own plans and actions); and, of course, there are
several approaches in between. The extent to which autonomy is given to
participants is a crucial social aspect of the community dimension. Of great
importance is also the extent to which participants feel that they are supported in
their growth of a competent mathematics teacher and to which extent they belong
to and might have an influence on a teacher education initiative. This is the kernel
of the self-regulation theory of Deci and Ryan (2002) which proposes that
perceived support of basic psychological needs (support of autonomy, support of
competence and social relatedness) are associated with intrinsic motivation or self-
determined forms of extrinsic motivation. Since I have become increasingly
familiar with that approach, which seems to be particularly worthwhile for use in
teacher education research, only in the last few months it was not possible for this
volume to take this theory more into account when writing the chapters. However,
the reader might focus on these ideas when reading the texts.

The differentiation between teams, communities and networks is used to
structure this volume. In addition, where appropriate, different chapters focus on
prospective or practising teachers specifically. In two cases, also a distinction is
made whether teacher education is held (primarily) in face-to-face or in virtual
settings. The idea behind working with these differences is that this format could
facilitate the reader making comparisons between the various strands. Each of
these chapters gives an overview of our recent knowledge on this particular issue
and illustrates it through one or a few examples.

THE CHAPTERS OF THIS VOLUME

In the remainder of this chapter, each of the six sections and sixteen chapters will
be introduced very briefly. Since both chapters of the critical respondents in
Section 6 refer to all texts in the former five sections - and thus necessarily also
provide descriptions and views on these chapters - I restrict myself here to sketch

KONRAD K.RAINER

the main topic and to pick out a few issues that could focus readers' attention to
potentially interesting commonalities and differences.

Section 1 is on individual teachers as learners. Whereas the chapter by Helia
Oliveirq and Markku S. Hannula focuses on prospective mathematics teachers,
Marie-Jeanne Perrin-Glorian, Lucie DeBlois and Aline Robert put practising
mathematics teachers in the foreground. In both chapters, although focusing on
individual teachers, not only the importance of the individual but also that of the
social aspect of teacher learning is highlighted. Similarly, both chapters stress that
teachers' beliefs, knowledge and practices cannot be separated (e.g., when aiming
to understand teachers' practice or growth). This makes sense since the focus on
the individual (and thus on the micro level of teacher education), in general, means
closer proximity to teachers' practice than in the case of studies at the macro level;
the nearness and the smaller sample provides the opportunity to go deeper with
various qualitative methods and thus to see more commonalities and differences
between teachers' beliefs, knowledge and practices. In contrast, at the meso and (in
particular at) the macro level, in order to gather a larger data base, it is not easy to
focus on teachers' beliefs, knowledge and practices within one study and also to
combine them. In many cases (e.g., where quantitative approaches with larger
numbers of questionnaires are taken and this is most easily done with regard to
beliefs), teachers' beliefs (and sometimes also, or instead, their knowledge is
tested) are measured.

Section 2 puts an emphasis on teams of mathematics teachers as learners. The
chapter by Roza Leikin focuses on prospective mathematics teachers. In addition to
that, Susan D. Nickerson deals with teams of practising teachers. The chapters on
prospective teachers in Section I and Section 2 have in common that management
skills and organizational issues are not given special attention. In contrast, both
chapters on practising teachers deal intensively with issues to do with school
context. There are a few reasons for this difference. For example, prospective
teachers are only in schools for relatively short periods during their practicum and
hence the full impact of school contextual issues might not be evident. Practising
teachers are in a better position to both identify school contextual issues and to
address these issues over an extended period. The four chapters in these two
sections all have in common that the complexity of (mathematics) teaching is
stressed in various ways. This leads to notions like (teaching) challenges,
dilemmas, ethical questions, paradoxes, problems, tensions etc. I assume that this
phenomenon (which cannot be found as much in the other sections) has to do with
the specific focus of the micro dimension of teacher education. The chapters
support the view that the complexity of teaching can be seen (at least) as a function
of the diversity and richness of the mathematics (content), the varying relationship
between the individual and the groups in which s/he is involved (community) and
the resources and general conditions (context), which, however, play a bigger role
in the case of practising teachers than prospective ones.

Section 3 puts an emphasis on communities and networks of mathematics
teachers as learners. Fou-Lai Lin and Joao Pedro da Ponte focus on face-to-face
learning communities of prospective mathematics teachers, whereas Stephen

REFLECTING THE DEVELOPMENT

Lerman and Stefan Zehetmeier do the same for practising mathematics teachers. In
contrast to that, Salvador Llinares and Federica Olivero and Marcelo C. Borba
and George Gadanidis deal with virtual communities and networks of prospective
and practising mathematics teachers, respectively. Although the four chapters
represent four different domains where communities (and partially also networks)
can be initiated or emerge in a self-organized way, they have in common that each
of them contains various communities representing a diversity of forms, goals and
purposes. In all four cases, in particular in the face-to-face communities, the
question of (internal and/or external) expertise or leadership (mathematical
competent prospective teachers, teacher leaders, principals, qualified experts, more
knowledgeable others, facilitators, steering group) is raised. This seems to
highlight that the self-selective nature of communities and networks and the
corresponding negotiation of goals and activities does not exclude issues of
expertise or leadership, but in contrast, brings them to the fore. Although in all four
cases - due to the issue of communities and networks - an important focus is on
the social aspect, in particular the contents (and to some extent also the contexts)
are in most cases well described and reflected. It is interesting that also in the
chapters about virtual communities the integration of face-to-face periods (same
room, same time) or at least synchronous periods of online-participation (same
time, e.g., working interactively on the same mathematical problem using the same
software) are suggested and realized. On the other hand, the flexibility concerning
time and thus autonomy that asynchronous forms of online teacher education give
to (in particular practising) teachers is so attractive that about 95% of continuing
teacher education programmes of a particular university are organized online. One
major difference between face-to-face and virtual learning environments seems to
be generated by the tools: new technologies and tools seem to change the form of
communication. Partially, the tools are not only regarded as mediators but also as
co-actors in the communication process. The new technical options in the web
allow a transformation from read-only to read-and- write communication (e.g.,
weblogs and wikis). Probably, it is not by chance that the chapters and many
references on virtual forms of teacher education stem from countries with large
distances to cover like Australia, Brazil, Canada, or Spain. It is worth noticing that
beginning with Section 3 (and not in the sections before that which primarily
belong to the micro level) the question of sustainability of teacher education is
raised in several chapters.

Section 4 shows a shift of focus to the development of schools, regions and
nations as a means of improving mathematics teaching and learning. Elham
Kazemi puts an emphasis on school development and thus relates to the meso level
most extensively. Paul Cobb and Thomas Smith deal with district development as a
means of improving mathematics teaching and learning at scale, the same focus is
taken by John Pegg and Konrad Krainer regarding regional and national reform
initiatives. These two chapters put their primary focus on the macro level, the
former one more extensively on the theoretical foundation and on planned
activities, the latter more on reform initiatives (carried out or still continuing)
reflecting and comparing cases from four countries concerning their genesis, goals,

KONRAD KRAINER

results of evaluation and research etc. Entering the meso and macro level, thus
focusing on the improvement of a larger number of mathematics classrooms, the
organizational aspects of change (development of schools, districts etc.), new
relevant environments for mathematics teaching and learning like parents,
principals, other kinds of leaders, but also organizations like universities and
ministries come into play. They are regarded as crucial co-players of bringing
about change, although the teachers and the students (and their interaction) are seen
as the key for systemic change. Given the complexity of the different actors (from
individuals, teams, communities, networks, organizations, media etc.), places,
resources, goals to be defined or negotiated, reflection and evaluation, ways of
communication and decision making, the content - at least at a first view -
becomes less important. However, it is naive to assume that bringing about change
in mathematics teaching at a large scale can be reduced to defining central
regulations, working out national mathematics standards, writing textbooks and
research papers, golden rules about good teaching and to transfer it to individual
teachers. Thinking that way, the autonomous culture established over years at
schools, districts etc. are underestimated and not taken seriously. Systemic change
needs to take into account the participants of initiatives at the micro, the meso and
the macro level in order to avoid "system resistance" at one or more of these levels.
The three chapters have in common that they deal with notions like (inter)national
assessments and benchmarks, brokers, change agents, differences between
designed and lived organizations, economic development, intervention strategies,
key boundaries objects, leadership content knowledge, regional networks,
stakeholders, support structures, systemic reform etc. which indicate that
interconnections between different levels, social systems etc. need to be balanced.
On the one hand, it is understandable that the three chapters deal to a large extent
with practising teachers. On the other hand, the lack of putting an emphasis on
prospective teachers might reveal the problem that schools and districts usually do
not have strategies for integrating novice teachers in a way so that both sides can
profit (or even do not have strategies for personal development at all). The journey
from the micro to the macro level shows that not only the focus of attention is
shifting from one class or a few to maybe thousands, but that also larger entities of
teachers and their professional growth are investigated, using more qualitative
studies at the micro and more quantitative studies at the macro level. Of course, the
participants of teacher education and professional development initiatives (and not
the students) form the focus of this volume, nevertheless, there is a tendency to
look at students' growth (affective or cognitive) rather at the macro level, in
particular with regard to (inter)national assessments that often are the starting point
of national reform initiatives.

Section 5 focuses on teachers and teacher educators as key players in the further
development of the mathematics teaching profession. Gertraud Benke, Alena
Hospesovd and Marie Tichd analyse the use of action research in mathematics
teacher education. Barbara Jaworski discusses a specific way of collaborating
between teachers and didacticians. Finally, Nanette Seago reflects on the
mathematics teaching profession in general. These three chapters form a kind of

REFLECTING THE DEVELOPMENT

reflection on teacher education that does not go along the levels introduced in this
volume, but along particular thematic issues relevant to the participants and
organizers of teacher education. Often underestimated, however of particular
importance, are forms of teacher education where prospective or practising
teachers are investigating their own practice. Therefore, (critical) reflection is a key
notion in the first of these chapters. It is interesting to follow the different ways
action research is used and the issues educational researchers have to attend to
when supporting teachers engaged in such projects. This is the bridge to the second
chapter in this section where the focus is on the collaboration between teacher
educators (didacticians) and practising teachers as partners and co-researchers in
inquiry communities. Whereas the first chapter combines general considerations on
characteristics and aspects of action research with examples from two countries,
the second chapter explains the theoretical background of building and sustaining
environments for co-learning inquiry and demonstrates this by giving an insight
into a specific project, illuminating its issues and tensions. The third chapter
addresses the professional ization of mathematics teaching, the specialized
knowledge mathematics teachers need to have and design principles for
professional development. This is illustrated by a specific mathematics teacher
development programme that uses video-records of classroom practice. Like all
other chapters, these three chapters aim at addressing general insights and research
results as well as particular views on a few cases in order to make the general
aspects more concrete and authentic.

Section 6 contains two chapters written by Gilah C. Leder and Heinz Steinbring
who look in a kind of "critical response" to the whole volume from two specific
perspectives. The first chapter follows the structure of this volume, refers to all
sections, describes the chapters in the order they are published, sifts out interesting
issues and indicates the complexity and diversity of the field and the variety of
contributions, approaches, theoretical and practical stances in this volume. In
contrast, the last chapter offers a theoretical perspective on fundamental problems
in the context of investigating the communication issues raised in this volume. In
particular, the theory-practice-problem within mathematics education and the
importance of the subject (mathematics) and of teaching activity are raised. Doing
that, links to all previous sections and chapters are made. These two chapters form
a reflective closure of the whole volume.

ACKNO WLEDGEM ENTS

Together with Terry Wood, I wish to gratefully thank all authors of this volume for
their great efforts and efficient work on this challenging joint endeavour of putting
the participants and community dimension into the foreground. We appreciate very
much that Gilah C. Leder and Heinz Steinbring not only wrote the final chapters
but also reviewed all chapters. In particular, we also want to highlight that the
whole review process (each chapter was also peer-reviewed by a first author or a
colleague of another chapter) was done in a critical but always constructive way.
Furthermore, I thank Dagmar Zois for her helpful correction and layout work

K.ONRAD KRAINER

concerning all chapters. We hope that you, as the readers of this volume, will gain
new and interesting insights into various aspects of mathematics teacher education
that partially gorge new pathways; they are not paved, they are tentative in nature
and work in progress. You are invited to collaborate and to be a participant in this
joint process.

REFERENCES

Adler, J., Ball, D., Kramer, K., Lin, F.-L„ & Novotna, J. (2005). Mirror images of an emerging field:

Researching mathematics teacher education. Educational Studies in Mathematics, 60, 359-38 1 .
Allee, V. (2000). Knowledge networks and communities of practice. OD Practitioner, Journal of the

Organization Development Network, 32(4), 4-13.
Deci, E. L., & Ryan, R. M. (Eds.). (2002). Handbook on self-determination research Rochester:

University of Rochester Press.
Krainer, K. (2006). How can schools put mathematics in their centre? Improvement = content +

community + context. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlkova (Eds.), Proceedings of

the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol.

1 , pp. 84-89). Prague, Czech Republic: Charles University.
Lachance, A., & Confrey, J. (2003). Interconnecting content and community: A qualitative study of

secondary mathematics teachers . Journal of Mathematics Teacher Education, 6, 107-137.
Little, J. W. (1982). Norms of collegiality and experimentation: Workplace conditions of school

success. American Education Research Journal, 19, 325-340.
Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In

A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education.

Past, present and future (pp. 429-459). Rotterdam, the Netherlands: Sense Publishers.
Reynolds, D., Creemers, B., Stringfield, S., Teddlie, C, & Schaffer, G. (Eds.). (2002). World class

schools. International perspectives on school effectiveness. London: Routledge Falmer.
Sowder, J. (2007). The Mathematical Education and Development of Teachers. In F. Lester (Ed),

Second handbook of research on mathematics teaching and learning (pp. 1 57-223) Greenwich, CT:

NCTM.

Konrad Krainer

Institut fur Unterrichts- und Schulentwicklung

University of Klagenfurt

Austria

SECTION 1

INDIVIDUAL MATHEMATICS
TEACHERS AS LEARNERS

HELIA OLIVEIRA AND MARKKU S. HANNULA

1. INDIVIDUAL PROSPECTIVE MATHEMATICS
TEACHERS

Studies on Their Professional Growth

There are two types of goals for those who learn to become mathematics teachers.
Firstly, they need to learn mathematics, and secondly, they need to learn how to
teach it. On the one hand, those who specialize in order to become secondary level
subject teachers usually have a strong mathematical background but they may
have weaker identities as teachers. Those who become elementary education
teachers, on the other hand often have problems with the mathematical content
themselves, but they have stronger identities as teachers. In this chapter we will
take three perspectives to mathematics teachers' learning during their teacher
education: 1) the development of teachers ' knowledge and beliefs during that
period; 2) the development of necessary skills for the teaching profession, such as,
observing and interpreting classroom incidents, and reflecting on them, as well as
different modes of interaction with the class; and, 3) the adoption of a productive
disposition and identity as a teacher who is a reflective and collaborative
professional and is willing to engage in future professional development. The last
one will be illustrated with a case study focusing on the development of the identity
of beginning secondary mathematics teachers. We then discuss the challenges to
initial teacher education of a perspective centred on the idea of the individual
prospective teacher as learner.

INTRODUCTION

When we speak about becoming a (mathematics) teacher, we are talking about a
learning experience that can be seen as an individual one, but also as a social one.
Our focus in this chapter is on research on teacher education, mainly work
published over the last ten years, which centres on the development of the
individual prospective mathematics teacher and embraces both the uniformity and
the diversity of individuals in this process. This would seem to be an easy task
since for many years studies have centred on a small number of participants and
focused on the individual. Adler, Ball, Krainer, Lin, and Novotna (2005) provided
a detailed table which revealed that out of the 160 studies on teachers that were
reported in the proceedings for the Psychology of Mathematics Education (PME),
Journal of Mathematics Teacher Education (JMTE) and Journal of Research in
Mathematics Education (JRME) over the years (1999-2003), 21 were reports on a
single teacher, while only 10 had one hundred or more participants. In the review
of research presented in PME about teacher education, from 1998 to 2003, the

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 13-34.
© 2008 Sense Publishers. AH rights reserved.

HELIA OLIVEIRA AND MARKKU S. HANNULA

majority of studies also had less than 20 participants (Llinares & Kramer, 2006).
Nevertheless, concentrating on the learning of the individual prospective teacher
proved to be a very complex task because there are many studies but these quite
often do not target the individual with great depth.

Since the launch of JMTE in 1998, this journal is the main forum for publishing
studies on mathematics teacher education, and also in our review this journal has
been a very important source. We also searched for papers in Educational Studies
in Mathematics (ESM), JRME and PME conference proceedings, mainly between
1998 and 2007. We also included a few studies from other publications that
seemed particularly relevant for the purposes of this chapter.

When we study prospective teachers' learning to become a teacher, we asked,
what kind of learning are we expecting to see? To attain our goal, we have
identified three perspectives on learning that we shall use to structure this chapter:
1 ) one view perceives learning as acquiring knowledge or beliefs, 2) another view
perceives learning as mastering a skill, for example, the ability to observe and
reflect, and 3) the third perspective perceives learning as adoption of a certain
disposition, for example, an identity or an orientation.

COMPETENCE AS "MATTER": BELIEFS AND KNOWLEDGE

For the last two decades, research on prospective teachers' knowledge and beliefs
has received enormous attention from the community of mathematics teacher
educators. The recent work of Llinares and Krainer (2006), and Ponte and
Chapman (in press) provide an extensive review of the literature on these areas and
offer a good overview of the themes that are being addressed and of the knowledge
that has been accumulated. In the present chapter we want to stress the individual
dimension of learning to teach as the development of beliefs and knowledge.

Prospective Teachers ' Beliefs

Research on teacher's beliefs (see also Leikin, this volume) has a long tradition in
research on teacher education and has been "perhaps the predominant orientation in
research on teachers and teacher education" (Lerman, 2001, p. 35). Changes in
beliefs are assumed to reflect development. Experiences and reflection are two
basic sources of influence that are considered to be important in the formation,
development and change of beliefs. Llinares and Krainer (2006) highlight the
importance of early experiences as a learner of mathematics in the formation of
teachers' beliefs. Once beliefs have been formed, they are not easy to change.
Liljedahl, Rolka, and Rosken (2007) have identified three different methods used
to change elementary education teachers' mathematics-related beliefs. The first
method is to challenge their beliefs. Many beliefs are held implicitly, but when
they are challenged, they can become explicit and subject to reflection, creating the
opportunity for change. The second method is to involve them as learners of
mathematics, usually in a constructivist setting. A third method for producing
changes in belief structures is to provide prospective teachers with experiences of

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

mathematical discovery, which seems to have a profound, and immediate,
transformative effect on their beliefs regarding the nature of mathematics, as well
as the teaching and learning of mathematics (Liljedahl, 2005). Llinares and Kramer
(2006) note that it is often assumed, not always correctly, that learning
mathematics through inquiry will influence prospective teachers' beliefs and
attitudes in a positive way, namely in regarding the implementation of those
methodologies in their practice as in the case described by Langford and Huntley
(1999).

In some cases it seems that the use of "reform-oriented" teaching material in
teacher education has been useful in influencing prospective teachers' beliefs about
the nature of mathematics. In Lloyd (2006, p. 77), one prospective teacher
explained: "I am learning about how to look for reasons and explanations as
opposed to simply believing 'the rules' that some really ancient dead guy came up
with. I prefer being able to use my own mind in solving problems". However,
according to Szydlik, Szydlik, and Benson (2003, p. 254), research has shown that
prospective teachers tend to "see mathematics as an authoritarian discipline, and
that they believe that doing mathematics means applying memorized formulas and
procedures to textbook exercises".

One of the big issues in research on teachers' beliefs is the mismatch between
teachers' personal theories of learning and their actual teaching practices that seem
not to support the theory. This is a special case of a more general mismatch
between espoused and enacted beliefs. It has been suggested that theory and
practice can be connected if integrated in authentic contexts. Bobis and Aldridge
(2002) have designed a continuum of authentic contexts for their master level
elementary teacher education course. Their students experienced mathematics
teaching and reflected on their experiences in typical student workshops and
practice schools, but they also used university based clinics and school-based small
group teaching. Somewhat paradoxically, they only were able to report changed
beliefs of their students during the course, as they have not followed the students
after they moved to their positions as teachers. The possibility of a dynamic
interaction between coursework and fieldwork in the development of alternative
perspectives on teaching and learning is illustrated by Ebby (2000). This research
shows that the three prospective teachers learned from their teacher education
coursework and fieldwork in quite different ways, influenced by their beliefs,
dispositions, and individual experiences.

In many studies, the development of prospective elementary education teachers'
learning of content knowledge is intertwined with their changing beliefs about the
nature of mathematics (e.g., Amato, 2006; Nicol, Gooya, & Martin, 2002; Olsen,
Colasanti, & Trujillo, 2006). One example of change in beliefs as a result of
learning content is the case of the prospective elementary education teacher,
Jennifer (in Davis & McGoven, 2001), in the context of a mathematics content
course that was built around problem solving and included class discussions and
reflective writing. Jennifer begun the course holding a typical belief that
mathematics is about applying the right formula to get the right answer. She
learned to see and value the mathematical connections between different tasks and

HELIA OLIVEIRA AND MARKK.U S. HANNULA

her test performance improved from not satisfactory to excellent. One
characteristic of Jennifer was her extensive and elaborated reflective writing. A
similar bond between content knowledge and beliefs was identified by Kinach
(2002), when she was challenging her prospective secondary mathematics teachers
to provide instructional explanations for subtraction of negative numbers. She and
her prospective teachers found themselves "to be in disagreement about what
counted as an explanation, and even more fundamentally, about what counted as
knowing, and therefore learning mathematics" (Kinach, 2002, p. 179). Hence, the
problem of teaching mathematical content was essentially also a problem of
changing the prospective teachers' beliefs about the nature of mathematics.

Prospective Teachers ' Knowledge

The above indicated interdependence of knowledge and beliefs in learning to teach
becomes more explicit from the growing body of empirical research dealing with
both concepts (e.g., Llinares, 2003) and how theory is being built around them
(Pehkonen & Pietila, 2003). However, the analysis of the knowledge that
prospective teachers should develop to teach is part of a long enterprise in research
on teacher education (Ponte & Chapman, in press).

It is well recognized that research on teachers' knowledge is strongly influenced
by the work of Shulman (1987), focusing mainly on the mathematical knowledge
of teachers and on pedagogical content knowledge. Regarding the first aspect,
studies have confirmed prospective teacher misconceptions (or lack of conceptual
understanding) in different branches of subject matter knowledge (Llinares &
Krainer, 2006; Ponte & Chapman, in press). This problem is more frequently
addressed among prospective elementary education teachers, but also prospective
secondary mathematics teachers at times struggle to attain a deep understanding of
mathematics (Kinach, 2002). As one prospective teacher wrote in her journal:
"Over the past semester, there have been a few times where 1 have stopped and
realized that there are some math concepts that 1 thought 1 knew, but actually
didn't" (Kinach, 2002, p. 176). This reflection about what one knows and how one
is learning is fundamental in the process of becoming a mathematics teacher. In
many countries, secondary mathematics teachers have a solid mathematic
education, with three or more years of mathematics courses in several domains.
However, the failure in some courses is very high, even for prospective teachers
who see themselves as good at mathematics in secondary school. One of the
reasons for this may be the way mathematics is taught to prospective teachers, and
not only because of the content to which they are exposed.

One important question that arises concerning individual prospective teachers as
learners is how they evaluate the mathematical knowledge they are developing in
terms of the development of subject knowledge for teaching. For example, some of
the beginning mathematics teachers studied by Oliveira (2004) did not see much
relation between the mathematics they learnt in the mathematics courses they
attended for three years and the way they are to teach. But they still considered that
the study of "hard mathematics" enhanced their mathematical reasoning, giving

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

them confidence to answer their students' questions and to solve mathematical
problems. Curiously, the beginning teacher who scored the highest marks in the
mathematics courses, and who deliberated between becoming a secondary
mathematics or a mathematician, was the one who recognized that sometimes he
had great difficulties in solving the mathematical problems given in methods
courses. This made him question the "hard mathematics" he had been studying
during that period in mathematics courses.

Certainly, prospective mathematics teachers also develop mathematical
knowledge in different ways, as a consequence of their past experiences with
school mathematics and in higher education. Sanchez and Llinares (2003, p. 21)
describe four university graduates, prospective secondary mathematics teachers,
who expressed different ways of knowing the function concept; this, according to
the authors, "influenced what they considered important for the learner and
affected their use of the modes of representation in teaching that were considered
as teachers' tools to obtain his/her teaching goals". This raises another important
question concerning prospective teachers' subject matter knowledge and its
relationship with their emerging pedagogical content knowledge. Tirosh (2000, p.
329) argued that teacher education needs to take into account the prospective
teachers' knowledge of students' common responses to given tasks, considering
that "such knowledge is strongly related to prospective teachers' SMK [subject
matter knowledge]". From a situated perspective on learning, the research
developed by Llinares (2003, p. 205) showed that the case analysis that prospective
teachers developed in interaction provided a context for them to attempt to
understand the students' way of thinking and simultaneously "to think about the
meanings of some elements of their subject matter knowledge and pedagogical
content knowledge". With respect to the process of learning in a small group of
four prospective primary teachers as learning in a community of practice, the
author explained that they engaged differently in the task and assumed different
responsibilities in exploring new domains. Despite their individual engagement, the
interaction among them was very important for the reification of beliefs, something
that Llinares contends could not be attained if reflection would stand merely at a
discursive level.

Summary

The studies reviewed provide evidence, to a certain extent, of the attention paid to
the individual prospective teacher. Collectively they show that there are important
differences that can be relevant influences on how they are going to experience
teacher education and develop as prospective teachers. Therefore, those differences
should be taken into account, when designing and developing teacher education
programmes. For teacher educators this means to further develop two competences:
to notice such differences (see e.g., Krainer, 2005, referring to Willke, 1999, who
writes about the art of precise observation being typical for experts in contrast to
laymen); and to produce relevant differences (which Willke, 1999, in Krainer,
2005, defines as interventions). Such an intervention in teacher education could

HELIA OLIVEIRA AND MARKKU S. HANNULA

be, for example, to make differences among prospective teachers' views fruitful for
their joint reflection on these differences.

COMPETENCE AS SKILL: LEARNING IN AND FROM PRACTICE

The practicum experience is generally considered to be an integral part of teacher
preparation. However, it has been argued that prospective teachers need to develop
a critical stance towards their field experiences and a "certain distance" from the
ongoing actions (Jaworski & Gellert, 2003) in order to reflect about the events of
the classroom, otherwise "field experiences perpetuate apprenticeship and trial-and
error views of teaching" (Mewborn, 1999, p. 318), and oversimplify its nature.

In this section, we are looking at the role of practice in learning to teach from
three different - although in some cases complementary - approaches: 1 ) observing
teachers and students engaged in mathematical activity, 2) prospective teachers'
interaction with students, and 3) prospective teachers' reflection. In our opinion,
these three ways of seeing how prospective teachers learn to teach entail different
activities that are nuclear for the teaching profession (see Seago, this volume).
There are additional important issues regarding the PCK prospective teachers are
developing, namely when planning and selecting appropriate tasks and resources
for students.

Observing Teachers and Students Engaged in Mathematical Activity

When prospective teachers come in contact with pedagogical situations in
elementary and secondary schools, they have already spent many hours in
classrooms as students. Therefore, the classroom environment and pedagogical
situations may seem quite familiar for them. It is necessary for them to produce a
shift of perspective from one of a prospective teacher to that of a teacher: "teaching
is fundamentally about attention; producing shifts in the locus, focus, and structure
of attention, and these can be enhanced for others by working on one's own
awareness" (Mason, 1998, p. 244). It is not surprising that novice and expert
teachers have different competencies in observing mathematics video taped
teaching episodes. In a study that used eye-tracking technology, it was found that
prospective elementary teachers attended more to mathematics content, while
experienced teachers and mathematics educators focused more on the activities of
the teacher and student (Philipp & Sowder, 2002). Some teacher education
programmes prepare prospective teachers to conduct clinical interviews,
considering that those are useful tools for learning to understand how students
think mathematically and to "develop an increased awareness of the ways in which
people learn mathematics" (Schorr, 2001, p. 159). Ambrose (2004) argues that by
having prospective elementary teachers interviewing children, namely on specific
difficult mathematical concepts, they realise that the mathematics they were
supposed to teach was not so simply and required conceptual understanding.

Morris (2006) focused on the prospective teachers' ability to collect evidence
about student learning to analyse the effects of instruction and to use the analysis to

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

revise instruction. The results of this study suggested that prospective teachers
possessed some initial diagnostic and revision skills, and that the quality of
analysis was influenced by the instruction given prior to the task, namely that when
a lesson was perceived as problematic it encouraged them to look more closely at
students and to ask themselves questions about the situation.

In the case of a programme that used a multimedia resource, elementary and
secondary prospective teachers investigated assessment and teaching strategies in a
situated learning environment (Herrington, Herrington, Sparrow, & Oliver, 1998).
They could observe video clips of teachers who were using different assessment
and teaching strategies, and at the same time they had access to teachers'
discussions about their approaches. Prospective teachers recognized the influence
of this environment in the assessment strategies they adopted later in their teaching
practice.

These initial experiences with the classroom reality and with students can have
an impact on the development of certain attitudes and knowledge that are central
for teaching, namely a disposition to listen to students and to try to understand how
they think and reason mathematically. In all these studies, prospective teachers'
practice is very limited in time and scope, but these experiences constitute also an
opportunity for the prospective teacher to start feeling what it is like to be a
teacher. Assuming that the teacher's role in classroom is much more complex than
working with one student or a small group of students on a specific theme or
concept, and that the teaching situations are so diverse, multimedia explorations
can assist prospective teachers in grasping that reality.

Prospective Teachers ' Interaction with Students

In spite of being, by definition, an interactive profession, teaching through
meaningful interactions with students is one of the most demanding aspects of that
practice. Research shows that prospective teachers often tend to focus their
attention on issues concerning class management and pedagogy (Van Zoest &
Bohl, 2002), considering subject matter learning less problematic.

Moyer and Milewicz's (2002) study on prospective teachers' questioning skills
looked at the developing interaction skills of practicing teachers. They argued, that
"when open-ended questioning is used and there are many right answers, the
learning environment becomes complex and less predictable as teachers attempt to
interpret and understand children's responses" (p. 296). In their research they gave
prospective elementary teachers the task to interview individual students on
mathematical concepts and later to reflect on the audio-recorded interview. They
concluded that the prospective teachers did not yet have good questioning skills,
but that the experience of interviewing "is a first step towards developing the
questioning strategies that will be used in the multi-dimensional, simultaneous,
unpredictable environment of the classroom" (p. 311).

The nature of discourse in the mathematics classroom is another way of looking
at teachers' learning. Blanton (2002) describes the experiences of eleven secondary
mathematics teachers in a course that attempted to challenge their notions about

HELIA OLIVEIRA AND MARKKU S. HANNULA

discourse as univocal or dialogical. Throughout the course, prospective teachers
developed a positive stance toward dialogic discourse but they felt there were
many obstacles to developing that kind of discourse in their practice. They
recognized that their discourse was predominately univocal as, for example, they
felt the need to structure and control the class. Blanton suggests that it is difficult
for prospective teachers to develop a dialogic discourse due to aspects such as:
sharing authority with students; focusing on students' thinking by acknowledging
and incorporating their ideas; and in balancing the need for dialogic discourse with
the time constraints of "covering the curriculum" (p. 149).

There is evidence from research that many prospective teachers begin their
practice trying authentically to assume a new role as teacher, one that is attuned
with reform ideas presented in university courses, but lack the skills to implement
them. For example, Lloyd (2005) described the internship of one prospective
secondary mathematics teacher who started his student teaching with an approach
that was student-centred, relying on group work and the use of manipulatives and
technology but that left unchanged the mathematical nature and content of
students' activity. As a consequence of the students' complaint that they "were not
being taught" (p. 457), he recognized that he had been emphasizing "how to solve
problems rather than why certain ideas and methods are related" (p. 457). The
prospective teacher was learning through interaction with students by attending to
the students' feedback and then reflecting on his own actions as a teacher.

Questioning students, and communicational aspects, in a more general way, are
central issues in learning to teach, but research shows that these are quite
demanding for prospective teachers. Implementing reform ideas or changing
approaches to mathematics teaching is challenging in many of the diverse
classroom contexts (see also Leikin, this volume). The professional growth of
prospective teachers depends on the opportunities to effectively experiment with
teaching and in a context where they are stimulated to interact with students in
meaningful ways, with the support of those responsible for their education as
teachers.

Teachers ' Reflection

Reflection has become a popular concept in teacher education and it is assumed to
be a critical element in belief change in the process of becoming a teacher (Llinares
& Krainer, 2006; Hannula, Liljedahl, Kaasila, & RSsken, 2007; Liljedahl, Rolka, &
RSsken, 2007). Elements that have been found to support reflection are
collaboration (Kaasila, Hannula, Laine, & Pehkonen, 2006) and use of multimedia
(Masingila & Doerr, 2002; Goffree & Oonk, 2001).

Reflection is seen as an indispensable element in the process of learning from
and in practice. Many teacher education programmes have implicitly addressed the
idea of promoting reflection about prospective teachers' practice by creating
moments for discussing their classroom lessons with tutors or/and mentors and
helping them "to analyse in more detail their own teaching practices" (Jaworski &

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

Gellert, 2003, p. 843). This is done in different programmes in a more or less
structured way.

In a study conducted by Artzt (1999), prospective secondary teachers were
engaged in structured reflection on their teaching. The course built on prospective
teachers' existing knowledge and beliefs and used both pre-lesson and post-lesson
reflective activities. Prospective teachers were encouraged to think about the
decisions they made in light of their goals for students. This research presents two
contrasting cases. Mrs. Carol was revealed to be insecure about her teaching and
mathematical abilities and about becoming a teacher. Through her writing
assignment the supervisor could detect a low self-esteem. But these feelings
constituted a motivation for her to be open-minded about learning new approaches
and she started to plan and implement different lessons. By the end of the semester,
she had understood the importance of reflecting about her underlying beliefs about
students and how they learn. In contrast, the other prospective teacher, Mr. Wong,
had strong beliefs about the "right" way to teach that left him inflexible and
unmotivated to learn new instructional approaches. So the course did not cater
equally for all prospective teachers.

Another study, about the teaching practicum, examines the role of reflection in
prospective teachers' practice by looking at how they use their pedagogical content
knowledge in solving problems identified from the classroom (McDuffie, 2004).
The elementary prospective teachers approached teaching as a problem solving
endeavour, since their internship included completing a classroom-based action
research project on their own teaching (see also Benke, Hospesova, & Ticha, this
volume), and focused on facilitating their understanding and anticipating problems
in teaching and learning.

As prospective teachers' observation skills are often limited, new technology
has been introduced as new means for reflection. Multimedia allows the same
episode to be watched several times and also to hear interviews of the people
involved in the episode more than once. In the programme Multimedia Interactive
Learning Environment (MILE), video clips of actual teaching situations are used,
and elementary education students are invited into reflective discussions about the
situations in order to enhance their practical knowledge (Goffree & Oonk, 2001).
Nevertheless, the authors consider that it is still necessary to understand better
"how to establish a link between the fieldwork of teacher education students and
their investigation and discourses in MILE" (p. 143).

The connections established by prospective teachers between multimedia case
studies and their practice was the focus of one study presented by Masingila and
Doerr (2002), involving grade 7-12 prospective mathematics teachers. The cases
were created in a way "that would reflect the complexities of classroom
interactions, teacher decisions, and students' mathematical thinking" (p. 244), in
order to promote investigation, analysis and reflection. Prospective teachers were
asked to select a specific issue from their own practice that they saw addressed in
the teacher's practice there and discuss it. They were able to discuss the case,
taking into account their own practice, and focused on complex issues instead of

HELIA OLIVE1RA AND MARK.KU S. HANNULA

the usual management concerns. However, the authors did not investigate if the
prospective teachers subsequently changed their teaching practices.

Mewborn (1999), studying prospective teachers who participated in a field
experience during a mathematics methods course, claimed that the ways they went
about making sense of what they observed in a mathematics classroom "can be
characterized as reflective thinking in the manner described by Dewey" (p. 324).
This happened when the locus of authority was internal to the prospective teachers,
and was promoted by the teacher educator and the classroom teacher who tried to
remove themselves "from positions of authority" (p. 337). They tried not to answer
directly to prospective teachers' questions, instead they "turn questions back to
them and encouraged them to rely on their peers for evaluation of their ideas" (p.
337).

It is not surprising that reflection on teaching practice is such a difficult task for
prospective teachers since learning in practice is so demanding for them. Even
more challenging is to reflect about the events and to change rapidly the course of
the lesson, adapting the plan that they made. Sometimes, they are supposed to
reflect "on mathematical activity while participating in the discourse" (Lloyd,
2006, p. 462). This cannot be seen exclusively as a skill or disposition to reflect on
the moment but as contingent action that relies on the (prospective) teacher's
confidence and willingness to assume risks (Rowland, Huckstep, & Thwaites,
2005), in a situation where, at the same time, she or he is under the scrutiny of
experts.

Summary

One of the classical problems of teacher education has been the bridging between
theory and practice. Observing and practice teaching has been used as a general
solution to this problem. Looking at the studies reported above it is possible to
recognize the important role of university tutors and teacher mentors in helping
prospective teachers to observe learning incidents in their complexity, to generate
fruitful interaction with students, and to reflect upon them, even when that role is
not specifically targeted. New multimedia learning environments hold a lot of
promise to develop prospective teachers' understanding about the complexity of
mathematics teaching, but educational environments that are supportive and
flexible, taking into account their unicity and individuality, are also very important
for their professional growth.

COMPETENCE AS DISPOSITION AND IDENTITY

Our last approach to prospective teachers' development through teacher education
looks at their disposition as a teacher and at what they want to do as professionals
(and as persons). Often the professional and personal aspects are intertwined and
the development of a skill is necessary to promote the inclination to use it (Peretz,
2006). Learning to teach is also a "process of becoming" (Jaworski, 2006, p. 189),
developing a new identity, one that integrates a professional side.

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

Disposition towards Mathematics and Its Learning

Disposition and identity issues have their specificities in the case of elementary
teachers, considered as generalist teachers, and in the case of secondary teachers,
usually regarded as specialized in mathematics. Especially among prospective
elementary education teachers, there are several whose attitude towards
mathematics is negative; sometimes they develop even a more serious condition -
mathematics anxiety. As such, it is important that elementary teacher education
programmes help students overcome this problem (e.g., Kaasila, 2006; Liljedahl et
al., 2007; Pietila, 2002; Uusimaki & Nason, 2004). These examples collectively
use what is often referred to as a therapeutic approach (Hannula et al., 2007). One
of the elements of a therapeutic approach is the effort to provide students with
positive experiences with mathematics. This can be achieved within the context of
elementary mathematics, where hands-on material is used to give prospective
teachers an example of teaching in a constructivist way, whilst at the same time
providing an opportunity for many of them to really understand mathematics for
the first time in their life (Pietila, 2002; Amato, 2006). In a supportive classroom
climate, anxious prospective teachers may express their thoughts and feelings and
ask for advice without fear of stigmatisation (Pietila, 2002). However, experience
alone is not sufficient for a major change in students' mathematical self-concept -
it needs to be supported by reflection and peer collaboration (Hannula et al.,
2007).

On the one hand, a negative view can seriously influence students' becoming
good mathematics teachers (Uusimaki & Kidman, 2004), on the other hand,
prospective teachers who have experienced only success in school mathematics
may find it hard to understand students for whom learning is not so easy (Kaasila,
2000), and that happens quite often with secondary mathematics teachers. These
teachers very often identify themselves very strongly with the subject they teach,
but develop a stronger frame of reference to certain forms of teaching (Sowder,
2007), which constitutes a challenge to teacher education programmes. In fact,
during their school years, prospective teachers begin to develop personal beliefs
about teaching and perspectives about teacher's and student's roles, and the nature
of the subject they are going to teach, through which they will interpret teacher
education programmes.

Assuming Different Roles in Teacher Education

Another issue in research concerning the development of prospective teachers'
professional identity is whether they identify themselves as learners of
mathematics or if they already identify themselves as (prospective) teachers of
mathematics. Although they are learning to become teachers, at the beginning of
their studies they are primarily identifying themselves as students. This duality has
been explicitly expressed, for example, in Bowers and Doerr (2001). Sometimes
there is a multiplicity of roles to perform as documented in one case study
presented by Stehlikova (2002). Molly, a prospective secondary mathematics

HELIA OLIVEIRA AND MARKKU S. HANNULA

teacher, over the course of her five-year education developed from "a role of a
pupil [who] expected to be taught [...] into an independent problem solver,
autonomous learner, 'mathematician' at times, teacher and teacher researcher" (p.
245). Although Molly was an exceptional case of an enthusiastic learner, she
exemplifies the variety of different roles a prospective teacher is expected to unite
into a coherent identity.

Teachers ' Interactions with Students

Interactions with students constitute one of the main configurative elements in the
process of identity construction. Quite often prospective teachers meet with
students who do not match their expectations (Munby, Russell, & Martin, 2001).
However, these interactions can promote a change in their perspectives. For
example, Skott (2001) documented that learning to be a mathematics teacher
involved more than "merely" teaching mathematics, and that it was important to
(re)evaluate their priorities concerning teaching practice, namely the necessity to
attend to students' problems. Oliveira (2004) also found that some secondary
mathematics teachers started to realise, early in their career, that they had an
important role as educators in spite of having to teach a socially "strong" subject.

Developing an Identity in Different Contexts

Sfard and Prusak (2005) define identities as collections of those narratives that are
reifying, enforceable and significant. According to these authors, different
identities may emerge in different situations and that might happen with
prospective teachers as they emerge in different contexts. In a longitudinal study
with four elementary prospective teachers, Steele (2001) illustrated "some of the
problems and realities of the workplace that interfere with teachers sustaining a
change in conceptions" (p. 168). Two of them remained attached to the
conceptions developed in teacher education while the other two did not. One of
these prospective teachers felt specially pressured by the school administration and
implemented a "teacher-proof curriculum" (p. 169). This was also the prospective
teacher who showed less change in her conceptions during the period of teacher
education and simultaneously was more confident about mathematics. Steele
conjectures that "perhaps her past experiences of learning mathematics were in the
end more influential in her approach to teaching" (p. 168).

Opportunities for learning during teaching practice are shaped by the
characteristics of the contexts in which teacher education occurs, in some instances
by the strong evaluative flavour (Mewborn, 1999; Johnsen Heines & Lode, 2006)
and they are also fused with "networks of power" (Walshaw, 2004). Using insights
from the post-structural ideas of Foucault, Walshaw discusses what it means to
engage in pedagogical work in the context of elementary mathematics classrooms.
Teaching practice is regarded as a strategic and interested activity since the
practice of prospective teachers "in schools always works through vested interests,
both of their own and others' rhetoric of opinions and arguments" (Walshaw, 2004,

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

p. 78). In some cases, the institutional practices of the university course and of the
school involve painful negotiations to produce individual subjectivity: "I wanted to
introduce new ideas but did not have enough confidence. I just followed my
associate's plans. I felt I could not try new things as my associate was set in the
way things were done" (p. 78). Walshaw concluded that the best intentions of each
prospective teacher can be prevented by "a history of response to local discursive
classroom codes and wider educational discourse and practices" (p. 78).

Ensor (2001) analysed the case of Mary, a prospective teacher who expressed
alignment with teacher educator's ideas, centred on the notion of innovation, but
who begun to change her opinions during teaching practice in school. Mary started
to identify herself with some classroom teachers and "to raise questions about the
applicability of what she had learned in the mathematics method course" (p. 306),
since some of them argued that their methods achieved results. Working for results
is a central aim for many teachers and is part of a certain professional culture with
which prospective teachers are acquainted.

In contrast to Ensor' s observations, the case of a prospective teacher
documented by Van Zoest and Bohl (2002) developed in a context of alignment
between the university programme and the school internship site. The social
context of the school was supportive of reform curricular materials and teachers
indicated that they wanted to change their teaching practices. The prospective
teacher had a profound conviction that the particular curriculum (CPMP) at this
school contributed to the mathematical understanding of students. According to
this, when she assumed a new position as teacher in a school with a traditional
single-subject curriculum, she put a great effort in adapting it "so that students
would do more of the types of thinking that she believed CPMP demanded" (p.
281). This shows the importance of communication and negotiation of
philosophies between teacher education institutes and schools during the
practicum.

Looking for Their Professional Development

One important aspect concerning the development of teachers' professional
identity is how they assume responsibility for their own professional development.
There are some studies which start to address this perspective. Olson, Colasanti,
and Trujillo (2006) described two prospective teachers who accepted leadership
roles when they began their teaching career. The researchers hypothesised that the
transformative experiences (cognitive and affective ones) during university
education promoted their self-efficacy and thus positioned them to assume
leadership roles. The above mentioned study by Van Zoest and Bohl (2002) also
described the first year in the career of a teacher who developed an important role
as a reformer teacher at her new school, and it seems that "the fact that there was a
social network of jointly-engaged educators working towards the same goals [...]
had great impact" (p. 284).

Goos (2005) also presented evidence to illustrate how beginning teachers used
their (technology-related) expertise "to act as catalysts for technology in schools"

HELIA OLIVEIRA AND MARKKU S. HANNULA

(p. 56). The beginning teachers in the study showed initiative in trying to develop
themselves as technology users. Particularly in the case of one of these, the author
argued that he was "not simply reproducing the practices he observed nor yielding
to environmental constraints, but instead re-interpreting these social conditions in
the light of his professional goals and beliefs" (p. 55). Goos emphasized that this
study shows it is possible that teachers implement innovative approaches from the
very beginning of their careers.

Summary

The development of a professional identity as (mathematics) teachers is a process
that is intrinsically connected with their participation in different communities. In
some studies, attention is given to the activity of the prospective teachers as part of
one or more communities as they practice teaching in the classroom. For example,
reflection, as a central process in learning to teach, as presented in the studies
above, does not exist in a social vacuum. Some of these studies illustrate conflicts
between the perspectives of different communities involved in initial teacher
education. For Lerman, "reflective practice takes place in communities of practice
[...] and learning can be seen as increasing participation in that practice" (2001, p.
4 1 ), and that involves very strongly the development of a certain professional
identity.

CASE STUDY: DEVELOPING PROFESSIONAL IDENTITY AS
(MATHEMATICS) TEACHER

In a longitudinal study, Oliveira (2004) followed four secondary mathematics
teachers, Cheila, Guilherme, Rita, and Susana, for three years after they finished
their five year teacher education programme and she tried to characterize the
development of their professional identity and the role of different contexts and
processes. Adopting a psycho-sociological model of the person (Gohier, Anadon,
Bouchard, Charbonneau, & Chevrier, 200 1 ), professional identity is regarded as a
process that starts from the time the first ideas of becoming a teacher appear.
Assuming that professional identity develops through a complex and dynamic
process that faces many constraints and threats with respect to continuity,
congruence, self-esteem and personal and professional orientation (Oliveira, 2004),
we choose to illustrate a tension that is pointed in this research as continuity versus
rupture. The first significant rupture identified in the development of these
beginning teachers' identities occurred during their teacher education programme.
All of them recognized that the courses on the didactics of mathematics contributed
strongly to a change in their perspectives about the teaching and learning of
mathematics and the mathematics teacher's role. For example, they stressed the
importance of promoting student-centred teaching methodologies and the use of
several strategies and resources, in contrast to the teaching style they were used to
when they were secondary students. Nevertheless, the study revealed different
levels of rupture and of focus among these four beginning teachers.

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

Guilherme expressed a deep change in his perspectives about the nature of
mathematics as a consequence of the readings and discussions that took place in
the methods classes. He had been a very successful mathematics student in
secondary school and at university, but began to see "another type" of mathematics
and to rethink what it means to teach mathematics. This was a turning point for
him and he started to see the teaching profession as a stimulating intellectual job,
one that is constantly changing.

Rita and Susana changed their own visions about what it means to be a good
mathematics teacher, one who developed good pedagogical content knowledge that
focused not only on the students' success but also on achieving more significant
mathematics learning. They came to see the teaching of mathematics as a much
more demanding and complex profession than they thought it would be, namely
that the teacher has to give attention to a lot of aspects beyond teaching the content.
Cheila also recognized that her vision had been transformed and now considered
that it is necessary to change old methodologies if students are to be more
motivated to learn.

The teaching practicum that the four teachers experienced occupied a full year at
a school and involved teaching two different mathematics classes. This was a time
in which the teachers predominantly continued with the ideas they developed
before, except for Cheila who reached a new turning point. Her expectations that
the ideas she developed concerning "new methodologies" would have an impact on
students' motivation remained unfulfilled. This situation caused many doubts about
the possibility of putting in practice what she learned. It is worth noticing that
among the four teachers, Cheila was the only one who felt that she did not have the
support she needed from the mentor to work with her very low achieving students.

The main changes for these four teachers began when they had to face the first
year of teaching on their own, in basic schools that did not have any induction
programme for new teachers. Now they had to regard themselves as autonomous
teachers. Rita and Guilherme came to understand that they could have an important
role as teachers and not "merely" as mathematics teachers. Guilherme made every
effort to know the students well and to attend to their various needs. Rita also
clearly assumed the role of an educator, who wants to contribute to the social and
personal development of students.

Cheila, at this point, expressed a great rupture with the ideas she associated with
the university programme. She questioned the applicability of the programme's
ideas in practice and developed a teaching style consistent with the one she
experienced as a student and in which she succeeded. From there on, her major
concern was for having professional stability and, consequently, maintaining a
continuity in her perspectives and practices.

In Susana's case, there was an initial moment of continuity in the first year;
however, as time went by she began to question her own perspectives through
confrontation with other teachers' perspectives and through reflection on her
incapacity to deal with some difficult situations in the classroom. Since she looked
at the profession as her natural vocation, this tension created deep doubts about her
professional and personal projects.

HELIA OLIVEIRA AND MARKKU S. HANNULA

It is important to stress that most of these beginning teachers had very difficult
positions in schools that were labelled as "unwanted". However, some of them
were creative in terms of their identity development, one that was positive and
congruent from their point of view.

Besides the teacher education programme, there are many conditions that appear
to have contributed to the development of these different professional identities. It
is interesting to note that Rita and Guilherme immediately became involved with
continuing teacher education and, especially, Rita participated in a research group
located at the Mathematics Teacher Association (APM, in Portuguese). In contrast,
Cheila and Susana only participated in some short sessions in schools about
specific aspects of the teachers' roles. Additionally, Cheila had no reference group
and Susana was far way from those who constituted her reference group (her
colleagues from the teaching practice).

These beginning teachers participated in the same teacher education programme
but developed very contrasting professional identities (Oliveira identifies four
different identity configurations). When these beginning teachers talked about the
perspectives they developed in the programme, superficially they sounded quite
similar. However, their teaching practices differed markedly. Teacher education
discourse does not affect all prospective teachers in the same way, as they are
different people, with diverse expectations, experiences and origins. This study
also showed that their biographies can reveal much about their beliefs, values and
knowledge about mathematics and its teaching.

NURTURING INDIVIDUAL PROSPECTIVE TEACHERS' GROWTH

Learning to be a mathematics teacher is a single trajectory, through multiple
contexts (Perresini, Borko, Romagno, Knuth, & Willis, 2004), and involving many
characters. However, in the research we analysed we focused on the learning of
individual prospective (mathematics) teachers. Programmes that focus on teacher
education and research on teachers' beliefs show sensitiveness to the individual
prospective teacher. Usually in this research, "the student teacher is recognized as a
learner and an active processor of knowledge", one who develops "systems of
constructs through which they interpret their undergraduate experiences" (Llinares
& Krainer, 2006, p. 430). However, in the need to theorize about these, research
sometimes does not attend to the prospective teacher holistically as a person and to
the social origins of his or her beliefs. It seems that research is now beginning to
incorporate the fact that beliefs are also contextual i zed (Llinares & Krainer, 2006).
Studies on prospective teachers' knowledge focus on the individual, but quite
often these do not show the prospective teachers ' perspectives about what they are
learning and how, neither do they explain the development of that knowledge,
taking into account past and present personal experiences of prospective teachers.
As Ponte and Chapman (in press) notice, it is rare for reflection on self to be
addressed in research on the development of knowledge of mathematics teaching;
it occurs mainly as "a by-product". Making tacit knowledge explicit would be an
important element in their development as future mathematics teachers. In the case

INDIVIDUAL PROSPECTIVE MATHEMATICS TEACHERS

of prospective secondary mathematics teachers, with a strong mathematical
background, it is common for them to "value mathematics as important and
beautiful but lack a critical attitude to mathematics and to teaching itself.
Elementary teachers often value educational goals but just use "mathematical weak
conceptions" (Jaworski & Gellert, 2003, p. 843).

The recognition of the importance of linking theory and practice led many
programmes to develop frameworks for promoting learning in context. There is
now a growing body of research aiming to understand how prospective teachers
"make sense of their beliefs, reflect, and learn while participating in field
experiences" (Lloyd, 2005, p. 443).

Research has shown that many prospective teachers are receptive to the reform
ideas that are presented in teacher education programmes at the universities, but
interpret these differently. It seems that they develop a professional argot,
phraseology typical of mathematics education, and some myths about classroom
reform. When they try to change their role as teachers, they realize that this may
involve taking on practices that differ the ones they were used to observe. But they
often do not know how to develop new communication patterns in the classroom or
to teach for conceptual understanding. Teacher educators clearly have a demanding
task to help prospective teachers to assume their role in a new classroom setting
and informed by new theories on learning.

The process of learning to teach in practice has to do with teachers' perceptions
of their own knowledge, of what they are able to do or not do, of what in particular
they think will benefit their students, and so on; and this can be seen as evolving
their beliefs and their knowledge, as well as themselves as persons. Prospective
and beginning teachers' dispositions and identities are receiving increasing
research attention. There is a growing debate about the dispositions for teaching
and efforts to define and assess them (Borko, Liston, & Whitcomb, 2007).
Although, if we want to attend to the individuality of the prospective teacher
regarded as a whole person, it is not fruitful to try to match his or her development
with a list of dispositions. What some of the studies we analysed suggest is that the
enormous difficulty involved in learning to teach in the current, challenging times,
needs to be balanced by a deep consciousness about what it means to be a teacher.

The paradox for the prospective teacher (and consequently for teachers
educators) is that teaching is a long term enterprise, meaning that the teacher's
decisions depend upon other long term decisions and not just of those they can
make during a limited period of time. Issues of conflict between prospective
teachers' perspective and those of their mentors raise important ethical questions
for teacher education, namely, how is it possible to create a good balance between
promoting the development of a professional identity that is attuned to the
educational aims of the institution and respecting the professional autonomy of the
mentors and of the schools were the practicum occurs!

Research on prospective and beginning mathematics teachers reveals that they
do not simple reproduce the institutionalized practices in which they teach and that
it is possible for them to become active agents of their own development (Goos,
2005); this shows that they are re-interpreting the social conditions of their

HELIA OLIVEIRA AND MARKKU S HANNULA

professional context "in the light of [their] own professional goals and beliefs" (p.
55). We have to recognize that initial teacher education is just the beginning of a
long journey of professional development for teachers, and that the constitution of
a professional identity is subject to multiple influences (personal and contextual),
many of which teacher education can not fully anticipate. Becoming a mathematics
teacher is not "a sudden move from novice to experienced practitioner on the
completion of a module or the passing of a test" (Jaworski, 2006, p. 1 89).

We add some final words about some challenges to teacher education. A focus
on the individual prospective teacher as learner has a strong parallel with the need
for the teacher to make an effort to know individually the students in front of him
or her. But this can be very difficult when teacher educators are responsible for
dozens of prospective teachers (as also happens with school teachers and their
students). The advancement of technology has opened new opportunities in teacher
education. The use of multimedia can facilitate learning from practice. However,
using these resources gives only a certain picture of mathematics teaching and is no
substitute for real interaction with students in the classroom. The use of the Internet
to promote virtual interaction is another possibility for teacher education but there
is still much to understand about how people "perform" in these scenarios. At the
same time, we observe in educational research an increasing interest in looking at
learning as participating in communities of practice, and it becomes clear that it is
not possible to study the individual prospective teacher's learning without
considering the contexts where it takes place. However, in this chapter, we
intended to illustrate how learning to teach is also an idiosyncratic process and to
review research providing evidence for this. As the studies analysed in this chapter
show, much of the research on initial teacher educator is done by the teacher
educators themselves (see also Adler et al., 2005). Therefore, there is a strong
perception that the knowledge accumulated can be used readily in the design and
development of new programmes and courses and (most importantly) in the
interactions between teacher educators and prospective teachers.

ACKNOWLEDGEMENTS

We wish to gratefully acknowledge the helpful comments and feedback on former
versions of this chapter provided by Gilah Leder, Heinz Steinbring and Terry
Wood, and the continued encouragement and feedback from Konrad Kramer.

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Helia Oliveira

Centra de /nvestigacao em Educaqao da FCUL

University of Lisbon

Portugal

Markku S. Hannula

Department of Applied Sciences of Education

University of Helsinki

Finland

MARIE- JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND
ALINE ROBERT

2. INDIVIDUAL PRACTISING MATHEMATICS
TEACHERS

Studies on Their Professional Growth

This chapter focuses on the professional growth of practising teachers. In the first
part, two examples illustrating major trends of current research allow to launch
the discussion. We then present our methodology and the main results from our
review of literature. Finally, we conclude on three main issues and perspectives: 1)
the change of paradigm proposed in practising teacher education poses new
teaching problems, thus it seems important to study teaching in its context; 2) when
teachers have to change because of an external constraint, some changes may
occur, but they are not always the "wanted" ones; we need better understanding of
deep components of practice; perhaps the study of the stabilization of practice
during the first years of career may help progress; 3) it seems important to
construct concepts or systems capable of taking into consideration the variety of
teachers ' work (from planning to classroom interactions).

INTRODUCTION

This chapter focuses on the professional growth of practising teachers. We are
interested in the evolution of teachers' practice, knowledge and beliefs, as well as
the constraints, dilemmas and difficulties they face in adapting their practice to
students or to a new curriculum. Though the constructs and theoretical frameworks
are often different, we distinguish three main issues: 1) How are teachers'
conceptual, belief and knowledge systems organized in relation to their practice? 2)
How are these systems to be changed in order to improve teaching? 3) How do
teachers' practices change "naturally" and in what ways do teachers learn from
their own practice? The first two of these questions are close to issues raised by
Adler, Ball, Krainer, Lin, and Novotna (2005) in their survey on mathematics
teacher education, whereas the third issue, at the core of our chapter, is scarcely
addressed in the literature.

Part one of this chapter presents our general point of view regarding the
problem. Two examples from our own recent research are included as a point of
departure. We then specify the methodology we used in conducting our review of
the literature. In the third and fourth parts, we state our research questions and
present our results. We then move on to provide a more in-depth discussion of
those issues with more recent research about actual practices and their

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, $5-59.
© 2008 Sense Publishers. All rights reserved.

MARIE-JEANNE PERRIN-GLOR1AN, LUCIE DEBLOIS, AND ALINE ROBERT

development. Finally, we draw some provisional conclusions as to what new
questions, concepts and methodologies might be considered.

THE PROBLEM

Our Point of View

One's learning as a mathematics teacher is a lifelong learning process which starts
with one's own experiences as a learner of mathematics, later supported by both
prospective and practising teacher training. In this chapter, we view professional
growth as a progressive transformation of mathematics teachers' actual practice in
relationship to their individual and professional experience, their knowledge and
their beliefs or conceptions about mathematics and mathematics teaching. In the
literature, terms such as teacher development, changes in teachers' practice or
teachers' learning are often used indiscriminately, though they are not equivalent.
For practising teachers, teacher development can be seen through the changes that
occur in their practice. Nonetheless, recent research seems to indicate that teachers
may learn about teaching, for example about student's logical development or
challenging problems but not change their practices, or they may change their
practices without really new thinking.

Behind the three research issues considered above, important questions arise:

1) Is it possible to study only beliefs, or knowledge, or practice? What
relationships between these concepts are most relevant? For example, beliefs can
influence practice, but practice can also influence beliefs. How can we study
systems of beliefs and knowledge and their relationship to practice? Research
issues depend on these implicit or explicit assumptions. Many models have been
constructed to explain interactions between different views of teachers. In
reviewing the literature, we see that during the last decade a shift has occurred
from research describing one issue (beliefs, or knowledge or practice) to research
taking greater account of complexity and the relationship of these constructs to
context.

2) In what sense can we speak of professional growth? What constitutes good
teaching? Very little research goes so far as to study the effect of teaching on
students' learning. It looks like as if an implicit agreement exists about what is
good teaching (see e.g., Wilson, Cooney, & Stinson, 2005) and such teaching
produce students' best learning. In many studies, particularly in US, the implicit
reference is to the NCTM standards without questioning its effect on students'
learning, particularly in the case of socio-cultural differences between students.
Another possible reference for professional development among research is the
participation in a community of practice (and development of this participation).
Otherwise, the reference may only be the researcher's conception of good teaching.

3) What is meant by "natural" change? The difficulty in accessing natural
teacher growth without modifying it from outside may explain why there are so
few studies on this topic (e.g., self-study, often from teacher educators or teachers
involved in research projects). Teachers may decide themselves to register for a

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

practising teacher education course. This decision is an indication that they wish to
improve their practice but the changes are influenced by the teacher education
program and are not "natural".

4) The relevance of making a synthesis of questions and results achieved from
very different theoretical perspectives and with different aims may be questioned.
Theory may be a theorisation of practice: assumptions on learning and on effective
teaching; in such case, research consists in elaborating and testing such a theory of
practice. In other research, theory is a construct in order to analyse, understand and
explain practice and relationships between teaching and learning, for any style of
teaching. Sometimes, both positions co-exist in the same research, in particular in
the case of collaborative research (e.g., Scherer & Steinbring, 2006). In this
chapter, we consider mainly research the aim of which is to study professional
practice in order to understand it and thus what may make it evolve.

Two Examples to Launch Our Discussion

Example 1: Enhancing our understanding of stability of practice. We refer here to
a research which was done in 2004-2006 (Paries, Robert, & Rogalski, in press).
This work aims at identifying specific difficulties that experienced mathematics
teachers' face when changing their own practices. In the following, we discuss one
of the two cases presented in the article mentioned above.

This case investigates two lessons given by the same teacher at grades eight and
nine in "ordinary" classes 1 . The lessons focused on the same kind of geometrical
activities for both groups of students (the resolution of problems just after a lesson
on a new theorem). To exhibit stability of practice both mathematical tasks and
lesson management were analysed.

There was a real difference in how the tasks were proposed in these classes. In
the first class, students had to conceive of some steps on their own before applying
corresponding theorems (once the Pythagoras theorem, once the converse, and then
a theorem on the sum of angles). In the second class, the problem applying
Thales's theorem 2 was more direct. The difficulty, however, was to write "x"
instead of a formal length "EM" in an equality. There were few differences in
terms of classroom management. The activities explicitly organised by the teacher
for her students and the order in which the activities are carried out are the same in
both groups (drawing the figure, looking for a strategy and then for the resolution
and correction). We note the similarity in the length of each activity (including the
total), even if there are slight differences. At that point, it is important to assess the
previous comment: this stability of classroom management has been confirmed by
the teacher as typical for this kind of problem when she was asked about the
representativeness of these videos. So we can assume that they represent the usual
way of working for this teacher on analogous problems.

1 Students work in class, at home and there are no special problems of discipline.

2 The theorem of proportional lengths made by a parallel to one side of a triangle.

MARIE-JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

Table I. Lesson plan (with time in minutes)

On Pythagoras'
theorem

grade 8

On Thales'
theorem

grade 9

Work organised
by teacher for
students

First question

(1) beginning, (2)

end

Second question

Total for
the grade 8
class

Draw the

geometrical

figure

Drawing
2'20

More than
2'20

Looking for a
strategy to
solve the
problem

Individual then

collective

8'30

Collective (I)
5'30

Collective
2'30

Looking for the
resolution

Individual
4'30

In two steps
2'

Individual
2'10

4' 10

Correcting and
recopying the
correction

The teacher

writes on the

blackboard at

students'

dictation

Two students chosen
by the teacher write
successively on the
blackboard
2'10and2'40

One student
writes on the
blackboard
9'50

Almost
15'

Total

28'

30'

The relationship between tasks and the students' activity are different in the two
lessons, not only in terms of the tasks themselves but also in terms of the teacher's
management 3 . As the first problem, which can be more difficult than the second
problem, required more steps, it was impossible for many grade 8 students to come
up with something during the 8 minutes they had to look for a strategy. The teacher
relied on some isolated students' proposals and finally explained the whole strategy
to the students who, for the most part, did not find anything. She then wrote the
three steps on the blackboard. The students then had to work on three easier tasks.
Perhaps, if the teacher had allowed more time or if the students had had the
possibility of working in small groups, more students would have begun to move in
the direction of a resolution.

One can argue that the teacher does not mind changing the task in that way. It is
not clear because the teacher's last sentence in grade 8 is "You see [...] you are able
to solve a problem without intermediate questions." So, even if the teacher does not
mind this reduction of students' activities, it does not change the fact that her
(stable) management style is incompatible with letting students work by
themselves on such complex activities. On the contrary, in grade 9, more students
find the strategy. The difficulty of calculating with x is widely anticipated by the
teacher and it seems that a lot of students get involved in it. Maybe the teacher's

5 We use here the term "management" to precise the part of practice by which the teacher manages the
students' mathematical activity.

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

management is well adapted to such a problem because this exercise was an old
one for the teacher and the other problem was more recent, selected for the purpose
of providing complex activities. What can be deduced from these analyses?

The stability of experienced mathematics teachers' practice concerns, first of all,
the management of the session at the scale of lessons. In other words, tasks are
easier to change than management (see e.g., Cogan & Schmidt, 1999). Even if
tasks adopted by teachers for students change, due to, for example, new curricula
or new standards, this does not ensure that the expected consequences for students'
work occur, because of no variation in the way they choose to manage their
classroom. In the case where students have to work in a precise way on some (new)
tasks to benefit from new activities, as finding steps by themselves before applying
a theorem, we can guess that nothing will occur in the class if the teacher's usual
management does not let students work in a compatible way. Consequently, it may
explain why it is so difficult for teachers to change their own practice. Maybe, the
management of their practice is so stable that in order to change it has to be
revealed by somebody else, for example, by mutual observation, and explicitly
discussed to be questioned (see also Nickerson, this volume).

Example 2: Conditions leading to the transformations of the interpretation and the
intervention. Previous research of DeBlois and Squalli (2001) leads to the study of
how discussions among teachers may help them transform their understanding of
their students and develop their practice.

A first study (DeBlois, 2006) was done with elementary school teachers
concerning the errors contained in students' written products. It showed, for
example, that when teachers compare the task to all the tasks usually proposed,
they evaluate the influence of the student's work habits. This type of reflection
makes the error "logical". It is no longer synonymous with a lack of attention on
the part of the student. The error becomes an extension of the procedures known by
students. This reflection then leads the teacher to choose a form of intervention that
allows students to break with their usual work habits. This study was continued
with teachers at the early high school level. Seminars allowed teachers to discuss
the errors contained in students' work. At the beginning, teachers expressed
themselves in an affirmative way when they described the students' production or
when they talked about the teaching offered and other parts of environment to
which they are sensitive. During the discussion, they were entering a process of
dissociation from the teaching offered. This dissociation allowed them to review
the tasks performed with and by the students in the classroom.

For example, four teachers (D, K, M, J) discussed a 13-year-old student's
production (Figure 1). At first, one teacher could not find any justification to
explain the strip diagram. Another teacher explained that/how he had worked on
the vertical and horizontal diagrams with the students. He added that he informed
the students about what could be asked. The problem exposed in the student's
production became afterwards proof of confusion between the two elements of
knowledge.

MARIE-JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

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Figure I. Student's production.

Time and a collective decision became the means of the student's understanding
as shown in this excerpt:

D - 1 can impose you, [precise] what I want. I can say: trace the one that is

appropriate and you will have to choose the one that is appropriate.

K - If you had shown the two of them, they could have mixed them up [...].

D - Because different situations required two different diagrams.

D - Give a class showing the possibility of making a horizontal or vertical

diagram and take a decision in group. In two groups, the choice of the strip's

form was different, one chose the vertical diagram and the other chose the

horizontal.

Other teachers considered the error as a way for students to regulate how to get
good answers that, by itself, is part of the student's learning process.

M - Maybe they wanted to do a horizontal strip first? The zero should have

been there.

M - It is possible. He could have changed his mind at the last minute.

The fact that a teacher brings out the notion of regulation weakens for the group
the hypothesis of an automatism suggested at the beginning of the discussion. The

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

research method undoubtedly influenced the discussion. In fact, asking teachers to
describe their students' written products was above all an invitation that freed them
from concerns they had. Indeed, the teachers' interpretations seemed to change as
other possible clues concerning the origin of the error were explored. Thus, in this
case, among the possible interventions, we find the wish of knowing the student's
representation of a graphic and a diagram. This way, we can consider the
discussion between colleagues as an environment for re-examining students'
production.

In the excerpt studied, it was the moments of intense interaction that modified
the teachers' interpretation and allowed the understanding of the student's
production, resulting in an effect on the planned interventions. Their sensitivity
appears mostly in regard to the particular learning conditions, to students'
uncommon procedures and to institutional knowledge. The teachers agree that,
despite the confusion that led to an inversion of the representation, the final result
makes sense. The analysis shows that, from time to time, the researcher, while also
playing her role as a mathematics teacher educator, uses teachers' affirmations as a
starting point to call attention to other hypotheses or observable facts in the
student's production which were ignored until then. This way, the interaction
produced between teachers and the researcher played an important role in the
process of transformation.

These two examples illustrate two major trends of current research: On the one
hand, there are diagnoses of difficulties for transforming practice, they involve
beliefs, classroom management, or cultural and institutional constraints. On the
other hand, research presents some ways of overcoming these difficulties,
including new concepts for analysing teachers' appropriation of teacher education.

OUR METHODOLOGY

In order to define the three areas of research mentioned at the beginning of this
chapter, we started a literature review with the following interrelated questions in
mind:

1. What is studied (practice, knowledge, beliefs, relations among them, how to
change them, how they can change)? We also tried as much as possible to specify
the cultural and political context as main factors of practice.

2. What factors are taken into account for professional growth and their effects
(institutional or cultural factors: reforms, new technologies; later effects of
previous training; research participation; individual factors: learning from
practice)?

3. What theoretical framework, implicit or explicit hypotheses, and what
methodologies are used to gain access to individual professional growth?

4. What are the findings? We will distinguish the results concerning primary and
secondary school teachers. Indeed, their conditions are not the same, neither
concerning mathematics (their learning of mathematics as students which may

MARIE-JEANNE PERRJN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

influence their practice as teachers) nor with regard to teaching (the former teach
many subjects and not only mathematics).

The key words used in our search of literature were: teachers' experience,
teachers' beliefs, teachers' practice, teachers' learning, professional knowledge,
and practices of mathematics teachers. We used also handbooks and synthesis
articles as well as their bibliographies in selecting research concerning the
professional development of practising teachers. Moreover, a variety of reviews
were systematically studied. For the selected papers, we drew up a summary taking
into account issues I to 4 above. This allowed us to define categories used to
present the third and fourth parts.

RESEARCH QUESTIONS AND FIRST RESULTS

A Recent Emergence

The interest of scholars for teachers as object of study is recent. Fifteen years ago,
Hoyles (1992) deplored the scarcity of teacher- focused research and appealed to
develop it. Her request was granted: For example, Sfard (2005), Ponte and
Chapman (2006) noticed a growing interest in this subject and a surprising growth
of papers on teachers' practice since 1995 (see also Krainer, volume 4). Similarly,
Margolinas and Perrin-Glorian (1997) noticed that research concerning teachers'
action has developed in France since 1990 and a book on Teacher Education
emerged from the first European meeting on Mathematics Education (Krainer,
Goffree, & Berger, 1999).

Several reasons may explain this growing interest in teachers' practices and the
way they can evolve: results that did not answer the teachers' questions but the
researchers' questions; research did not consider enough constraints of the class.
Thus, mathematics education researchers felt the need to better understand
teachers' conceptions of mathematics teaching and their various constraints to
conduct their teaching. These new questions required an extension of the
theoretical frameworks. For example, in France where the question of the teacher
was posed early, the theory of didactic situations in mathematics (Brousseau,
1997), particularly the notion of milieu, was refined to study "ordinary" classes.
The teacher's action on 4 the milieu of a didactic situation is a key issue to
understand his or her role in class (Bloch, 2002; Hersant & Perrin-Glorian, 2005;
Margolinas, 2002; Margolinas, Coulange, & Bessot, 2005; Perrin-Glorian, 1999;
Salin, 2002). Extending the notion of didactic transposition, a new framework
emerged in order to help describe the organisation of study at school (Barbe,

In the theory of didactical situations (Brousseau, 1997), the milieu of a didactic situation is the part of
the context that can bring a feedback to student's actions to solve a problem. The teacher can act on the
milieu bringing some new information or new equipment, for example, asking a question or giving a
compass; acting on the milieu, he changes the knowledge needed to solve the problem. One can refer to
Warfield (2006) for an introduction to this theoretical framework

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

Bosch, Espinoza, & Gascon, 2005; Bosch & Gascon, 2002; Chevallard, 1999;
Chevallard, 2002). Other scholars endeavoured to model teachers' didactic action
using a combination of these two theories (Sensevy, Mercier, & Schubauer-Leoni,
2000; Sensevy, Schubauer-Leoni, Mercier, Ligozat, & Perrot, 2005). The didactical
and ergonomical twofold approach of Robert and Rogalski (2002, 2005) copes in
another way with the complexity of practice.

In Quebec, different program reforms have led researchers to develop new
contexts for practising teacher education. The theoretical frameworks of SchOn
(1983), Lave (1991), Erickson (1991), and Bauersfeld (1994), contributed to the
development, experimentation and analysis of collaborative research (Bednarz,
2000). In this perspective, the teachers' practice is the starting point of the training.
The didactic content is more a tool for training than a corpus of knowledge to be
transmitted. In this way, teachers could develop a deep understanding of
mathematical knowledge. Recently, Bednarz (2007) observed that research was
suffering from an absence of a teacher-oriented database which allows for a
thorough consideration of classroom situations.

At the same time, at the international level, sociocultural theories (D'Ambrosio,
1999) and references to Vygotsky's work were spreading as well as a new interest
for teacher and students' interactions. Several models appeared; theoretical efforts
were done to control the teacher's role or action inside theory in order to
understand teacher's work. However, it is not sufficient to understand how to
change practice; for that, it is necessary to understand teacher's work in and of
itself. Thus, research concerning teacher change is even more recent, involving
various questions and methods referring to psychological or sociological
perspectives (Richardson & Placier, 2001). The recent 15th ICMI Study
Conference on The Professional Education and Development of Teachers of
Mathematics (2005) was a crucial moment to discuss this theme.

Understand Teachers ' Practice and Teachers ' Growth

Previous experience of learning and teaching strongly influences current teaching.
Though we cannot separate beliefs, knowledge and practice to understand teachers'
practice or teachers' growth, the aim of this section is to clarify these notions and
their use to better understand the way practising teachers may evolve.

Beliefs. Research on beliefs was first carried out from psychological perspectives
and beliefs were treated as cognitive phenomena (see e.g., Thompson, 1992). Some
studies used questionnaires, others focused on describing teachers' beliefs or
conceptions in relation to a particular aspect of teaching or learning mathematics,
problem solving, students' errors or technology for many of those involving
practising teachers. The results of these studies led researchers to appreciate the
complexity of the notion of beliefs (Ernest, 1989; Jaworski, 1994; Mura, 1995).

More recently, their contextual ised nature and their social origin were
considered (see e.g., Gates, 2006; Leder, Pehkonen, & TOrner, 2003; Llinares &

MARIE-JEANNE PERRJN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

Krainer, 2006). Nevertheless, some scholars, for example in France (Robert &
Robinet, 1996) referred earlier to social representations. Be that as it may, beliefs
and attitudes about mathematics, mathematics teaching and the role of the teacher
were regarded as a main factor influencing teachers' teaching and their learning
processes about mathematics teaching (see also Oliveira and Hannula, this
volume).

Knowledge. Many studies have attempted to know more about teachers' knowledge
for teaching mathematics, and its effect on students' learning.

Early research showed that there was no clear relationship between the number
and the level of mathematics courses completed by teachers and students'
achievement. Then research focused on a specific mathematical content, identified
deficiencies and misconceptions in teachers' mathematical content knowledge on
many topics, generally for prospective teachers or primary school teachers. A
number of papers point out explicitly implications for teacher education but the
relationship with teaching was not evidenced. Thus researchers grappled with the
question of what would constitute conceptual understanding for teachers and felt
the need to extend the theorisation of teachers' mathematics knowledge by
including teachers' knowledge of mathematics for teaching (Ball, Lubienski, &
Mewborn, 2001 ; Ponte & Chapman, 2006).

Shulman (1986) was the first to call attention to a special kind of teacher
knowledge that linked content and pedagogy: pedagogical content knowledge
(PCK). Research into PCK identified the necessity of teachers' awareness on
students' difficulties in specific subjects. Later, the notion of PCK was often
combined with other theoretical constructs. Studies involving PCK showed an
effort in establishing a critical perspective. For example, Ma's study (1999)
described what she called "profound understanding of fundamental mathematics"
to explain the difference between American and Chinese teachers. That said, Ball,
Lubienski, and Mewborn (2001) claim that the descriptions of teachers' knowledge
do not necessarily illuminate the knowledge that is critical to good practice.
Moreover, distance remains between studies on teachers' knowledge and on
teaching itself. Ball, Bass, Sleep, and Thames (2005) developed an extension of the
notion of PCK. They identified four domains of which two are close to the
Shulman's work: knowledge of students and content, knowledge of teaching and
content, and two are new: common content knowledge and specialized content
knowledge.

For Suurtamm (2004), professional development necessarily passes through a
deepening of one's mathematical knowledge in order to answer the students'
questions and to guide them in their exploration process. Many authors confirm
this vision, by linking risk taking, self confidence, teamwork and the use of
appropriate resources with teachers' professional development (Brunner et al.,
2006; Carpenter & Fennema, 1989). Even (2003) recalls that insofar as learning is
a personal construction, the construction of mathematical knowledge does not
necessarily reflect instruction. Some common experiences (subjective and
sociocultural) will favour a common signification contributing to deepen reflection.

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

Thus, the research has revealed the complexity of mathematical knowledge for
teaching, its links with mathematical knowledge thus indicating "a shift away from
regarding mathematical knowledge independent of context to regarding teachers'
mathematical knowledge situated in the practice of teaching" (Llinares & Krainer,
2006, p. 432).

The research approach developed mainly in France around the basic notions of
the Theory of Didactic Situations does not focus on actors but on teaching and
relationships to mathematical knowledge. Rather than pedagogical content
knowledge or knowledge on mathematics teaching, Margolinas et al. (2005) use
the notion of "teacher's didactic knowledge". This notion is defined as part of
teacher's knowledge "which is related to the mathematical knowledge to be
taught". They prefer this notion because, as Steinbring (1998) has also stressed,
mathematical knowledge and pedagogical knowledge cannot be separated for
teaching. Indeed, crucial questions for teacher education now are 1) what special
form of mathematical knowledge is fundamental for teaching? and 2) how can
teachers not only acquire such knowledge in order to use it effectively in the
classroom but go on developing it over the course of their teaching careers? We
will see in part 4 how recent research addresses this question.

Practices. There is a large variety of studies on teachers' practices referring to
various theoretical frameworks and methods. Some of them describe practice in
terms of indicators such as amount of time spent on lesson development, types of
problems selected during development, teacher's types of questioning and so on.

Other studies focus on the relationship between the structure of the lesson and
the teachers' understanding of a specific mathematical content and others are
biographical studies. Among them, some studies are linked to education reform
efforts or to the quest for effective practices. There are also many studies at a
microdidactic level which study classroom interactions, with an interest in the
language used, the nature of classroom discourse, the role that the teacher plays in
classroom discussions, the identification and characterisation of interaction
patterns. We can notice that teachers' practices and research on teachers' practices
strongly depend on the cultural and political context (e.g., Cogan & Schmidt,
1999). We also must consider results coming from international evaluations like
TIMSS and PISA. These evaluations may have some influence on practices. For
example, some countries may develop students' training in problem situations like
PISA to have best performance. Thus, the teachers' practice, the curriculum and
the studies could nowadays be influenced by international evaluation (Bodin, 2006;
DeBlois, Freiman, & Rousseau, 2007).

Ponte and Chapman (2006) point out that the notion of practice used in research
has evolved. Mostly regarded as actions or behaviours in early studies, practice
includes later what the teacher does, knows, believes and intends. Boaler (2003)
and Saxe (1999) consider the notion of stability and recurrence of practices. Saxe
emphasizes the socially organised nature of these practices; Boaler considers not
only activities but also norms. We consider with Ponte and Chapman (2006, p.
483) that "teachers' practices can be viewed as the activities they regularly

MARIE-JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

conduct, taking in consideration their working context and their meanings and
intentions".

Relations between Beliefs and Practices

Initially, it seemed that one way to improve practice was to improve beliefs and
knowledge. But research has shown that a change in beliefs does not necessarily
entail a change in practices especially for practising teachers. Many studies
focused, directly or indirectly, on the relationship between beliefs or conceptions
and practices. These studies found inconsistencies between them, particularly when
teachers were faced with innovation, notably involving computers. Nevertheless,
these inconsistencies may be apparent only. For example, Skott (2004) explains
these inconsistencies in terms of the existence of the multiple motives of teachers'
activity, experienced as incompatible. These inconsistencies may thus be seen as
situations in which the teachers' priorities are dominated by other motives, maybe
not immediately related to school mathematics, for example, developing students'
self-confidence. Vincent (2001) and DeBlois and Squalli (2001) talked about
preoccupations about the space and the time of learning. Moreover, Lerman (2001)
criticises past research on beliefs and comparison with practice, including his own,
because it does not take sufficiently contexts into account: for example, the
interview context is different from class context. He suggests that "whilst there is a
family resemblance between concepts, beliefs, and actions in one context and those
in another, they are qualitatively different by virtue of these contexts" (Lerman,
2001, p. 36), and that "contexts in which research on teachers' beliefs and practices
is carried out should be seen as a whole". More recently, Herbel-Eisenmann,
Lubienski, and Id-Deen (2006) distinguish local and global changes, taking into
account the importance of curricular context for local changes. Thus, students' and
parents' expectations and desires as well as the curriculum materials may influence
the teacher adopt local adaptations not really compatible with his global beliefs.

GROWTH AND LEARNING

What Actually Changes, What Resists? What Means Seem Effective?

Two main questions on practices strongly concern teacher education: (I) what
constitutes good practice for students' effective learning? and (2) how can
teachers' practice be improved? For the first question, research often gives an
implicit answer supported by constructivist or socio-constructivist theories
supporting practice that let a large place to students' action. For the second
question, the research to better understand teachers' practice concludes on the
complexity of teachers' practice so that it is now widely recognized that
professional development programs that attempt to achieve real changes in
classroom practices must address teachers' knowledge, beliefs and practice.
However, the nature and genuineness of changes further complicates this question.
The question of efficiency is very difficult to evaluate; actual and deep changes

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

require time. Moreover, to find efficiency in students' achievement, one has to be
sure that the planned changes in practices were achieved (Bobis, 2004; Sullivan,
Mousley, & Zevenbergen, 2004). Other reasons could explain these difficulties: the
methodology used, the paradigm adopted, the cultural factors and the context, and
teachers' engagement.

Thus, an important methodological problem occurs to know what changes and
what is resistant to change. How can we compare or measure changes that occur in
teachers' beliefs, knowledge or practices? Teachers' answers to questionnaires are
not easily interpretable into actual changes made. Class observation, more often
used in recent research, is very time-consuming such that only a few cases can be
studied. Some researchers adopt a mixed approach, asking teachers to react to class
contexts, for example using video clips of classroom excerpts.

A common feature of reforms in mathematics education carried out in the last
decades is the change in relative emphasis from mathematical products to
processes, with a greater importance given to individual processes. The student is
expected not only to learn predetermined concepts and procedures but also to
become involved in genuinely creative individual and collective processes of
investigating, experimenting, generalising, naming and formalising. This
adaptation supposes a rather different and more difficult role for the teacher. The
teacher has to adopt a certain interpretative stance; to engage in reflexive activity
enabling him to a flexible use of interactions with the students, what Skott (2004)
named "forced autonomy". He observed some novice teachers claiming teaching
priorities inspired by the reform. Their classroom practices were in line with these
priorities most of time but not at other moments, identified as critical incidents,
where teachers are "playing a very different game than one of teaching
mathematics". In this type of research, some recent work tries to specify new tasks
for students or/and new tasks for teachers, to get students to become more engaged
and more effective problem solvers (Doerr, 2006). Recent research also tries to
describe the diversity of ways of adoption of this new "job", sometimes in relation
with an appropriate (or hoped so) teachers' training program (Herbst, 2003). Even
though teacher education programs have gained some positive results, much
research highlights the difficulties for teachers to change their practices in a deep
way.

Other recent work tries to understand the difficulty to change mathematics
teachers' practice, and how their beliefs are implicated in a complex way in
practices, at different levels, combined with social or cultural considerations. For
example, Arbaugh, Lannin, Jones, and Park-Rogers (2006) studied 26 secondary
teachers using a problems-based mathematics textbook Core-Plus. They conclude
that adopting a problem-based textbook series and using it in a classroom is not
enough, in itself, to have an effect on teacher instructional practices - to get them
to teach in a more reform-oriented manner, according to the strength of previous
habits and beliefs. Wilson, Cooney, and Stinson (2005) reveal some subtle aspects
of these beliefs in what constitutes good mathematics teaching and how it develops
for teachers: they find considerable overlap between the teachers' espoused beliefs
and the writing in NCTM standards documents as well as important differences,

MARIE-JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

linked to these teacher-centred/student-centred classroom conceptions. Moreover,
the teacher's view and the researcher's view of change may be different. For
example, Sztajn (2003) shows the way in which elementary school teachers adapt
their practices to their students' needs, that is to what they think their students'
needs are, and at the same time being certain that they are acting according to
current reform visions adapted as much as possible to their specific students.

Thus, professional development efforts often result only in surface changes. We
can notice with Tirosh and Graeber (2003) the importance of cultural factors and
the environment of teachers. Some research reports success of individual teachers
changing their practices though their colleagues do not (e.g., Koch, 1997). But
successful change is more likely to occur when simultaneous attention is given to
changing the system in which teachers work. Many studies have noted the value
derived from discussions with colleagues who are experiencing similar concerns
and can provide ideas for solving problems encountered in change (e.g., Kramer,
2001). Students themselves may resist (Brodie, 2000; Ponte, Matos, Guimaraes,
Leal, & Canavarro, 1994). Thus it may be easier to change practices with a
program involving all the teachers in a given school (e.g., Sztajn, Alex-Saht,
White, & Hackenberg 2004); nevertheless, volunteer teachers will change more
easily; research projects often balance between these two options.

Indeed, student achievement is the main goal of the vast majority of efforts to
change classroom practice and the main motive for teachers to try to do it,
especially at a large scale (see e.g., large programs for primary schools in
Australia: Bobis, 2004; Sullivan et al., 2004) or with low-achieving students in
South Africa (Graven, 2004).

Taking into account the fact that deep changes take a long time, Franke,
Carpenter, Levi, and Fennema (2001) also addressed the question why some
teachers continue to develop their practices when teacher education programs are
over. Their study, a follow-up of the Cognitive Guided Instruction (CGI) program
for practising teachers (Carpenter & Fennema, 1989), shows that some teachers of
this program were engaged in a generative growth. They suggest that focusing on
students' thinking is a means for engaging teachers in continued learning and that
helping teachers' collaboration with their colleagues can support it. Graven (2004),
in a study in the Lave and Wenger's theoretical framework of community of
practice, adds the notion of confidence as both a product and a process of learning,
in relation to teacher learning as "learning as mastery". It seems that the notion of
confidence could help particularly in this case, involving teachers with a low
(sometimes even absent) mathematical background in their teaching preparation
and students coming from low socio-cultural backgrounds. However, engaging
teachers in learning to examine mathematical tasks using the Level of Cognitive
Demand criteria supports both a growth in pedagogical content knowledge (ways
of thinking about mathematical tasks) and a change in practice (choosing
mathematical tasks) proved to be a non-threatening way to start teachers thinking
more deeply about their practices (Arbaugh & Brown, 2005).

Nevertheless, an important condition is the teacher' involvement in the program
during its unfolding and after it. Doerr and English (2006) show that the modelling

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

tasks were actually a means used to bring about change in teachers' practices, even
if that change is not the same for any two teachers. Wood (2001, p. 432) attempted
to find out if teachers were actually learning by investigating their process of
reflection, namely "how teachers use reflective thinking in their pedagogical
reasoning and how their thinking relates to changes in teaching". She confirms that
teachers develop differently while giving some insight into how this difference
continues. But the challenge remains to explain why this development is different
among teachers or to know if another choice in approach to teacher education
would produce another effect. The study of Empson and Junk (2004) is devoted to
an assessment of a training program for elementary teachers, novice and
experienced teachers alike, based on knowledge of students' mathematics. It shows
that teachers' knowledge of students' non-standard strategies is broadest and
sometimes deepest in the chosen topic, even if it depends on the teachers; the
teachers use information given in their training, some of them even extended their
knowledge.

To change practices (in a broad sense), the importance of reflection on practice
as well as collaboration and discussion among colleagues about concrete cases (for
example excerpts of videos, particularly critical classrooms incidents) is nowadays
stressed (e.g., Nickerson, this volume). Nevertheless, as Llinares and Krainer
(2006, p. 444) comment:

At the present time, we need to understand better the relationship between
these instruments [reflection and collaboration] and teachers' different levels
of development, as well as the changes in teachers' practice.

Spontaneous Changes. How Do Teachers Learn through Teaching?

Interest of scholars in the individual and spontaneous changes in teachers' practice
is quite recent. Moreover, questions and methods used are different: Searching why
and how teachers change without questioning if it is in a desirable direction
(change is not always desirable, especially at the end of their career), studies have
found that teachers are always changing. They refer mainly to psychological
background and methods are often based on case studies looking for relationship
between biography, professional experience and professional knowledge
development. Results show mainly stability of style with certain flexibility. Large
differences between teachers are observed, some of them being more able than
others to frame puzzles stemming from practice (Richardson & Placier, 2001). For
example, Sztajn (2003) analyses the case of Helen, an elementary school teacher
with 31 years of teaching experience, who thinks that her mathematics teaching has
greatly improved during her career. Nevertheless, the same fact may be seen as
change for the teacher and as stability by the researcher if actually the students'
activity is not really changed. Thus, Sztajn stresses the difference of scale between
teachers' and researchers' views about change and the necessity for researchers to
understand teachers from the teachers' point of view.

MARIE-JEANNE PERRIN-GLOR1AN, LUCIE DEBLOIS, AND ALINE ROBERT

Most scholars share the view that teachers increase their understanding of
learning and teaching mathematics indirectly from their practice, through years of
participating in classroom life (Stigler & Hiebert, 1999). However, even though
"direct and indirect learning are interrelated and depend on each other" (Zaslavsky,
Chapman, & Leikin, 2003), little research concerns indirect practising teacher
learning (embedded in practice). Such research may consist in self-analysis carried
out by teacher educators of their own practice as teachers (e.g., Tzur, 2002) or joint
analysis of a teacher and a researcher (e.g., Rota & Leikin, 2002) or analysis by a
researcher of teacher's learning from class observation (Margolinas et al., 2005).
Tzur (2002) conducted, as researcher-teacher, a teaching experiment in a third-
grade classroom observing his own improvement of practice. His reflection is
similar to Ma's (1999) and raises the question if and how a teacher (not researcher)
can lead this kind of reflection in his class. Rota and Leikin (2002) studied the
development of one elementary school beginning teacher's proficiency in
managing a whole class mathematics discussion in an inquiry-based learning
environment. They found a large growth inflexibility for the teacher, much more
attentive to the students, without any professional development intervention, except
the existence of their research. They also stress the difficulty of teaching a teacher
when to apply a particular teaching action.

More research is needed on teachers' learning through teaching, especially for
beginning teachers. However, some models explain how such learning may occur,
and enhance in some way previous results, as they suggest that a teacher cannot
"see" what he is not prepared to see. This explains that a deep professional growth
cannot occur in some cases without some external interventions. For example,
Zaslavsky et al. (2003, p. 880) refer to Steinbring's model (1998):

According to this model, the teacher offers a learning environment for his or
her students in which the students operate and construct knowledge of school
mathematics in a rather autonomous way. This occurs by subjective
interpretations of the tasks in which they engage and by ongoing reflection on
their work. The teacher, by observing the students' work and reflecting on
their learning processes, constructs and understanding, which enables him or
her to vary the learning environment in ways that are more appropriate for the
students. Although both the students' learning processes and the interactive
teaching process are autonomous, the two systems are nevertheless
interdependent. This interdependence can explain how teachers learn through
their teaching.

Zaslavsky et al. (2003) extend this model to teacher educators' learning with
one more layer.

Another model of teachers and teacher educators' development, strongly
emphasising collaboration, is the co-learning partnership described by Jaworski
(1997) and Bednarz (2000), in which "teachers and educators learn together in a
reciprocal relationship of a reflexive nature". Margolinas et al. (2005), using a
model extending Brousseau's work, carry out two case studies of lessons in which

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

the teacher's didactical knowledge seemed insufficient for dealing with students'
solutions and identified two kinds of teacher's learning.

Let us notice that the three quoted models also implicitly suggest that teachers
learn more from their practice when they provide their students with a rich
environment allowing for autonomous reflection. There is in some way a dialectic
process with actions reciprocal to the usual ones: Students' learning Teachers'
learning Teacher educators' learning. However, Margolinas et al. (2005)
suggest that it may not be sufficient for general and stable learning.

CONCEPTS TO CONSIDER PROFESSIONAL GROWTH IN THE SYSTEM OF
TEACHERS' WORK

As early as 1993, Ball recognized that the tensions (Cohen, 1990) and the
dilemmas (Lampert, 1985) with which teachers are confronted, are the
distinguishing features of teaching and must be treated throughout the teacher's
career. The importance of the practice context and of the contextual ised
pedagogical contents has been acknowledged in relation to the restructuring of a
repertoire of interventions (Brodie, 2000; Bednarz, 2000). Grossman,
Smagorinsky, and Valencia (1999) use the concept of appropriation to determine
the process by which a person adopts a pedagogical tool available in a particular
social environment. The degree of appropriation depended on the congruence
between the values, the experiences and the goals of the members of this culture.
The American Institutes for Research recognize the influence of the length of the
training in the transformation of teaching practices (Garet et al., 2001). This
component permitted the emphasis on the mathematical and scientific contents and
also to perceive a greater coherence between the teacher's objectives, the standards
and the other teachers, and offered the opportunity for active learning. Finally, a
direct relationship is established between what people know and the way they
learned the knowledge and the practices.

To consider these components, a variety of concepts were used: dilemmas and
tensions (Herbst, 2003; Suurtamm, 2004; Herbst & Chazan, 2003), constraints and
conditions (Barbe et al., 2005; Robert & Rogalski, 2002; Roditi, 2006); the concept
of practical rationality (Herbst & Chazan, 2003), and the concept of teachers'
sensibility to a milieu (DeBlois, 2006).

Tensions and Dilemmas

Practising teacher education programs usually require a change of paradigm.
Consequently, the teachers are often confronted with dilemmas that many studies
have identified. Brodie (2000), Herbst (2003), and Suurtamm (2004) describe how
a constructivist approach could create tensions and dilemmas. For example,
questioning students makes their weaknesses more apparent. Experimenting new
tasks with students requires paying attention to the explicitly expected product or
to the new ideas which students may develop while engaged in the activity. The
research of coherent evaluation methods according to the teaching practices, the

MARIE-JEANNE PERRIN-GLOR1AN, LUCIE DEBLOIS, AND ALINE ROBERT

time to explore, develop and find an appropriate curriculum and evaluation
resources add to the tensions lived by the teachers.

These dilemmas concern the coordination to be accomplished between the new
teaching practices and the contents to cover, the students' rhythm, the type of
questions formulated by the teacher, the conciliation of norms, and "authentic"
evaluation. Some conditions make it easier to cope with these dilemmas and
tensions: collaboration between teachers (collegiality), adequate resources (e.g.,
time), school administration support.

Practical Rationality

The concept of practical rationality allows practitioners to use their acquired
knowledge (knowledge, personal engagement) in a similar or different manner than
the others and to describe the action in its context (Herbst & Chazan, 2003). This
way, it becomes possible to predict the dilemmas which the teachers will be
confronted with when they will engage themselves into actions activating these
dispositions. Like the disposition network, activated in particular situations, it
becomes possible to justify the presence of very different teaching styles according
to the objective characteristics of the position of "mathematics teacher" compared
with connected practices, but still different (e.g., teaching history, doing
mathematics).

The concept of didactical and ergonomical twofold approach, developed in
France by Robert and Rogalski (2002), seems to get closer to the concept of
practical rationality. However, these authors seem to give more consideration to
non premeditated actions than to those derived from practical rationality. Different
components are identified and defined: cognitive, mediative, institutional, social,
personal (Robert & Rogalski, 2002), and collective (Roditi, 2005, 2006).

Teachers ' Sensibility

Rene de Cotret (1999) introduces the following distinction between milieu
(Brousseau, 1997) and environment (Maturana & Varela, 1994): the environment
relates to the observer's description of a situation and the milieu relates to the
student's sensibility to this environment. DeBlois (2006) uses this distinction to
develop the concept of "sensibility" of teachers to speak of their interpretation of
this milieu (for the student). She observed teachers that were asked to interpret
their students' errors in mathematics in order to examine the interpretative process
and its influence on the choice of teaching strategies in a mathematics class. Four
types of teachers' sensibilities (teaching, familiarity of the student with the task,
students' understanding, curriculum and characteristics of the task) were identified.
At that point, it was possible to recognize a relationship between a variety of
interpretations of students' productions (attention, extension of students'
procedures, students' ability, product of an interaction between student and the
task) and kinds of teaching strategies (method of working, creation of a gap with
the habits, reconsider exercises or manipulative, play with didactic variables).

INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

When she analyses her data, Leikin (2006) prefers the notion of awareness to
explain teachers' choices when they teach. She includes this notion in one of the
two types of factors: momentary factors (reasoning, noticing and awareness) and
preliminary factors (knowledge, abilities, and beliefs). These two concepts
(sensibility and awareness) could help to understand the choice of teachers.

CONCLUSION

This review of literature shows how difficult it is to organize the variety of results
and develop some conceptualisation. Nonetheless, we have identified three main
issues. First, the change of paradigm proposed in practising teacher education
poses new teaching problems. Research on mathematics teacher education shows
the importance of flexibility, depth and connectedness of teachers' mathematical
knowledge. It also shows how difficult it is for teachers to acquire such flexible
knowledge and use it to manage students' learning in challenging classroom
mathematics activities. Thus, it seems important to study teaching in its context
(e.g., in relation with the students or within the larger social or political context).

Second, when teachers have to change because of an external constraint, reform,
or new syllabus, it is often difficult for them, and research shows that whereas
some changes may occur, they are not always the "wanted" ones. The difficulties
often stem from the very stable imbrications between teachers' beliefs and
knowledge, and social and institutional expectations as well as cultural context.
Previous research had already shown that "isolated" changes are not sufficient to
guarantee real improvement in practice. Some studies show that there are actually
teachers who learn alone through their teaching, but these studies and inferences
from other research lead us to believe that these improvements occur inside the
teachers' previous "teaching style", for example, their beliefs and choices
regarding contents and classroom management. Perhaps some progress will come
from better understanding of the differences between this kind of improvement and
a deeper transformation of practice: what indicators may help researchers assess
this difference? A way to enlighten this issue may be to study more deeply how
novice teachers' practice stabilizes during the first five years of their career. We
meet here one of perspectives by Leikin (this volume) drawn from her analysis of
research on education of prospective mathematics teachers. Such studies might
give us evidence of different ways of improving practice according to the level of
teaching (primary or secondary) and mixing different kinds of knowledge.
Moreover, recent studies enable us to think that professional growth takes a long
time and requires a collective investment and the implementation of specific new
tasks to be used by students and/or teachers. However, due to the complexity of
these elements of change and the length of time needed, it becomes even more
difficult for research to assess progress.

Third, it seems important to construct concepts or systems capable of taking into
consideration the variety of teachers' work (planning, analysis, classroom
interactions, including relations with parents etc.). Perhaps some further
development of research will occur when researchers are able to elaborate models

MARIE-JEANNE PERRIN-GLORIAN, LUCIE DEBLOIS, AND ALINE ROBERT

of teachers' professional growth which involve mathematical development blended
with teaching issues and are adaptable to individual specificity.

ACKNOWLEDGMENTS

The contribution of Lucie DeBlois was partly supported by Social Sciences and
Humanities Research Council of Canada 410-2005-0406 and the revision of
English language by DIDIREM-EA 1547 (Paris).

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INDIVIDUAL PRACTISING MATHEMATICS TEACHERS

i Marie-Jeanne Perrin-Glorian
■ Equipe DIDIREM

Ins ti tut Universitaire de Formation des Maitres, Universite d'Artois

France

Lucie DeBlois
\ Centre de recherche sur V intervention et la reussite scolaire (CRIRES)
\ Faculte des sciences de {'education, Universite Laval

Canada

\ Aline Robert

\ Equipe DIDIREM

\ Institut Universitaire de Formation des Maitres, Universite de Cergy-Pontoise

France

SECTION 2

TEAMS OF MATHEMATICS TEACHERS
AS LEARNERS

ROZA LEIKIN

3. TEAMS OF PROSPECTIVE MATHEMATICS
TEACHERS

Multiple Problems and Multiple Solutions

In this chapter, I first address four interrelated problems that mathematics teacher
educators (MTEs) are currently facing in the education of prospective mathematics
teachers (PMTs): (1) The importance of challenging mathematics and PMTs '
limited experience in challenging mathematics; (2) Changing approaches to
mathematics teaching and the resistance to change in teaching; (3) The need to
intermingle the different components of teachers' knowledge in teacher education;
and (4) The difficulty of becoming a member of a community of practice. To
analyse a variety of solutions that MTEs suggest for these problems when working
with teams of PMTs, I provide in a second part a comparative analysis of 30
studies focusing on diverse issues integrated in PMT education programmes.
Finally, I make connections between the problems outlined in the chapter and the
solutions discussed in the observed studies. The argument presented in the chapter
is that even though many solutions are suggested and proven to be effective in
solving some of the outlined problems, we are still lack of evidence that those
solutions are ample for preparing PMTs to become members of the communities of
practice they join. I suggest that combining the various solutions proposed for the
education of teams of teachers in various studies is necessary in order to prepare
PMTs to be effective mathematics teachers.

PROBLEMS IN THE EDUCATION OF PMTS

This chapter is based on two main positions: first, teaching is a complex system that
includes interrelated and mutually dependent elements such as teachers, students,
curriculum, textbooks, school management, students' families, local settings, and
other factors that effect classroom procedures; second, teaching is a cultural
activity, reflected in knowledge and beliefs that guide behaviour and determine the
expectations of the participants. The chapter also assumes that prospective teachers
should be prepared to teach in ways that would allow students to fulfil their
learning potential, and that the mathematical challenge is an irreducible element of
mathematics education. Based on these assumptions and on the analysis of the
research literature, I outlineybwr interrelated problematic issues in the education of
prospective mathematics teachers.

K. Kramer and T. Wood(eds.), Participants in Mathematics Teacher Education, 63-88.
© 2008 Sense Publishers. All rights reserved.

ROZA LEI KIN

The Importance of Challenging Mathematics and PMTs ' Limited Experience in
Challenging Mathematics

The learners' intellectual potential is a multivariable function of ability,
motivation, belief, and learning experiences (National Council of Teachers of
Mathematics, 1995). Principles of "developing education" (Davydov, 1996), which
integrate Vygotsky's (1978) notion of ZPD (Zone of Proximal Development) and
Leontiev's (1983) theory of activity, claim that to fulfil the learners' mathematical
potential the leaning environment must involve challenging mathematics. A
mathematical challenge is an interesting mathematical difficulty that a person can
overcome (Leikin, 2007). Mathematical challenge is subjective because it depends
on the learner's potential.

The importance of the mathematical challenge and its student-dependence in
teaching and learning is reflected in Jaworski's Teaching Triad that synthesizes
three core elements: the management of learning, sensitivity to students, and
mathematical challenge (Jaworski, 1992, 1994). Brousseau ( 1 997), in his Theory of
Didactical Situation, claimed that one of the central responsibilities of a teacher is
devolution of good (challenging) tasks to learners. Both the teaching triad and the
theory of didactical situation stress that teachers ought to provide each and every
student with learning opportunities that fit their abilities and motivate their
learning.

Mathematical challenges may appear in different forms in mathematics
classrooms. These can be proof tasks where solvers must find a proof, definition
tasks in which learners are required to define concepts, or investigation tasks. One
way of helping teachers to use challenging mathematics in their classes is to
provide them with appropriate learning material (e.g., a textbook), making a large
number of challenging tasks available to them (Barbeau & Talor, 2005); but merely
providing teachers with ready-to-use challenging mathematics activities is not
sufficient for their implementation: teachers should be aware of the importance of
mathematical challenges and convinced about them, and they should feel safe
(mathematically and pedagogically) when dealing with this type of mathematics
(Holton et al., in press). Furthermore, teachers should have autonomy in employing
this type of mathematics in their classes (Krainer, 2001; Jaworski & Gellert, 2003).
They should be able to choose mathematical tasks themselves create those tasks,
change them so that they become challenging and stimulating, and naturally they
must be able to solve these problems.

But despite the importance of teacher awareness of the role of mathematical
challenge in teaching and learning, prospective teachers often have limited
experience in challenging mathematics, and sometimes have strong negative
feelings about it (Gellert, 1998, 2000). In other cases, PMTs find challenging
mathematical activities interesting and encouraging, but are not sure whether these
activities are applicable to students. These views are connected to novice teachers'
inclinations to rely on their procedural understanding of mathematics when making
pedagogical decisions about mathematical challenges (Borko et al., 1992) or when
planning or discussing their ideas for teaching (Berenson et al., 1 997). Many PMTs

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

encounter difficulties in coping with challenging mathematics themselves, and
their beliefs about the nature of mathematical tasks contradict the character of the
tasks they are asked to solve (Cooney & Wiegel, 2003).

Changing Approaches to Mathematics Teaching and the Resistance to Change in
Teaching

The beginning of the 21st century is full of ever better and more advanced
technological tools. In a changing world, invention and progress can be anticipated
and developed only by human beings with rich imagination, deep and broad
knowledge, and solid proficiency. In a changing world, learning and teaching
environments, informational resources, interpersonal communications, and the
roles of teachers and students in the classroom adapt constantly to the latest
modifications. Mathematics education is a typical example of a subject that
experiences ongoing, multifaceted change, manifest in the shift toward the
dynamic and investigative nature of mathematical tasks, toward multiple uses of
technological tools for teaching and learning, and toward the dialogic learning
environment (Lagrange, Artigue, Laborde, & Trouche, 2003).

Inquiry and experimentation are basic characteristics of the development of
mathematics, science, and technology. Inquiry (experimentation) tasks in
mathematics classrooms are usually challenging, cognitively demanding, and
enable highly motivated work by students (e.g., Yerushalmy, Chazan, & Gordon,
1990). Borba and Villarreal (2005, pp. 75-76) stressed that the "experimentation
approach gains more power with the use of technological tools" by providing
learners with the opportunity to propose and test conjectures using multiple
examples, obtain quick feedback, use multiple representations, and become
involved in the modelling process.

Educational technologies such as computers and graphic calculators can be
viewed as cultural tools that reorganize cognitive processes and transform social
practices in the classroom (Borba & Villarreal, 2005; Goos, 2005). These tools can
provide a vehicle for incorporating new teaching roles ranging from the
authoritative "master" to the collaborative "partner" (Goos, Galbraith, Renshaw, &
Geiger, 2003) and influence the mathematics curriculum (Wong, 2003).

As approaches to teaching and learning mathematics change, the nature of
mathematical discourse and socio-mathematical norms changes as well (e.g., Cobb
& Bauersfeld, 1995; Wood, 1998; Wells, 1999). The shift toward the dialogic
nature of learning is grounded mainly in Vygotsky's (1978) theory of meaning
making as the result of the learners' communicative experiences. This approach to
learning and teaching is based on theories that go beyond cognitive views of
learning (e.g., Brousseau, 1997; Cobb & Bauersfeld, 1995; Jaworski, 1994; Lave &
Wenger, 1991; Steinbring, 1998; Wells, 1999). In this context, teaching can be
considered as a spiral process that facilitates the students' autonomous learning and
includes planning of learning opportunities for students, presenting challenging
tasks, monitoring the students' handling of the tasks, and reflecting on learning and
teaching.

ROZALEIKIN

As a complex system, teaching is stable and resistant to change (Cogan &
Schmidt, 1999). As a result, changing approaches to school teaching are beset by
many pitfalls and difficulties (e.g., Lampert & Ball, 1999; Tirosh & Graeber,
2003). As is true with any cultural activity, teaching is learned through
participation in activities involving learning and teaching (Stigler & Hiebert,
1998). In part, future teachers learned to teach when they were prospective teachers
with certain perspectives on teaching and learning. When they are challenged by
new teaching approaches, PMTs are often unenthusiastic and reluctant to adopt
new practices and express preferences for the teaching methods used by their own
teachers (Cooney, Shealy, & Arvold, 1998; Hiebert, 1986; Lampert & Ball, 1999).
Education programmes have a special role in supporting educational reform by
developing teachers' knowledge and beliefs (Llinares & Kramer, 2006).

The Need to Intermingle the Different Components of Teachers ' Knowledge in
Teacher Education

Teachers' knowledge and beliefs determine their decision making at all stages of
teaching (Ball & Cohen, 1999; Cooney, et al., 1998; Even & Tirosh, 1995;
Shulman, 1986; Thompson, 1992). The complexity of teachers' knowledge within
the context of the complexity of teaching itself is one of the main problems in the
education of PMTs (Llinares & Krainer, 2006). To demonstrate this complexity, I
use a 3D model of teacher knowledge (Leikin, 2006) that combines three main
perspectives adopted by researchers in discussing teachers' knowledge: kinds of
knowledge (Shulman, 1986), conditions of knowledge (Scheffler, 1965), and
sources of knowledge (Kennedy, 2002).

From the perspective of the kinds of teacher knowledge (Shulman, 1986),
teachers' subject-matter knowledge (SMK) comprises the PMTs' own knowledge
of mathematics and of the philosophy and history of mathematics. Teachers'
pedagogical content knowledge (PCK) includes knowledge of how students cope
with mathematics and knowledge of the appropriate learning setting. Ball and
Cohen (1999) discussed "mathematics knowledge for teaching" that allows
teachers to unpack their SMK in order to develop deep and robust mathematical
knowledge in students. Teachers' curricular content knowledge (CCK) includes
knowledge of different types of curricula and understanding the different
approaches to teaching. Under conditions of changing approaches to school
mathematics, this type of knowledge can endow teachers with flexibility in shifting
among various curricular approaches.

Kennedy (2002) classified teachers' knowledge according to the sources of
knowledge development. Systematic knowledge, she argued, is acquired first
through personal experiences as school students, then through participation in
courses for teachers, reading professional literature, and interacting with
colleagues. Prescriptive knowledge is acquired through institutional policies and is
manifest in tests, accountability systems, and texts of a diverse nature. In contrast,
craft knowledge is developed largely through experience. This type of knowledge
relates to teachers as members in a community of practitioners, and is based mainly

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

on teachers' interactions with their students and on teachers' reflections on these
interactions.

The distinction between knowledge and beliefs as conditions of teacher knowledge
started with Scheffler (1965) and has been presented in the works of mathematics
education researchers (e.g., Thompson, 1992; Cooney et al., 1998). Knowledge has
operational power whereas beliefs are of propositional nature solely. An individual
has proofs for the facts that belong to his/her knowledge, whereas beliefs are
accepted without proof. The additional distinction between formal and intuitive
knowledge is consistent with the views of Atkinson and Claxton (2000), who
discussed teachers as intuitive practitioners, and differentiated between teachers'
intuitive knowledge as determining actions that cannot be premeditated and their
formal knowledge, which has to do mostly with planned teacher actions.

Professional development programmes must be consistent with the complex
structure of teacher knowledge. The complexity of professional development
programmes lies in searching for a reasonable balance between mathematics and
pedagogy and in connecting between them (Peressini, Borko, Romagnano, Knuth,
& Wills, 2004). The balance between systematic and craft modes of development
is also important because if teachers develop their own mathematical
understanding in a systematic mode, only practice can persuade them that
implementation of this type of mathematics in the classroom is valid (Leikin &
Levav-Waynberg, 2007). Because PMTs lack experience as teachers, pedagogical
and craft knowledge are among the most challenging issues in the professional
development of prospective mathematics teachers.

The Difficulty of Becoming a Member of a Community of Practice

From the social perspective, teachers are considered to be members of communities
of practice characterized by common norms, routines, sensibilities, artefacts, and a
vocabulary that are the result of the situated nature of the teachers' practice (Lave,
1996; Lave & Wenger, 1991). This practice is embedded in a cultural enterprise
that is also a complex system of the beliefs about society, educational policies,
auricular requirements, assessments, and the school environment (e.g., Stigler &
Hiebert, 1999). Teachers' understanding of mathematics and pedagogy within the
community of practice is bounded by socially constructed webs of beliefs that
determine the teachers' perception of what needs to be done (Roth, 1991; Brown,
Collins, & Duguid, 1989).

The community of mathematics teachers is usually regarded as one of learners
who continually reflect on their work and make sense of their history, practice, and
other experiences (Lave & Wenger, 1991). In other words, teacher knowledge
develops socially within communities of practice, and in turn determines these
practices. The situation of PMTs is very different from that of practising teachers.
When PMTs begin their studies, they are (in the best case) members of a team.
Krainer (2003) maintains that

ROZA LEI KIN

"Teams" (and project groups) are mostly selected by the management, have
pre-determined goals and therefore rather tight and formal connections within
the team. [In contrast,] "communities" are regarded as self selecting, their
members negotiating goals and tasks. People participate because they
personally identify with the topic, (p. 95)

PMTs may find that they are simultaneously members of different teams in the
different courses they attend. One of the purposes of educational programmes is to
develop the norms, routines, sensibilities, artefacts, and vocabulary that will help
PMTs join their future professional communities. But PMTs rarely emerge from
their mathematics teacher preparation programme as members of these
communities. Moreover, they realize that many teachers work more or less
individually, some of them collaborate because the department or school
challenges them to do so; in general, it is rather rare, that teachers really take part
or form self-selected communities of practice. Furthermore, when starting their
teaching in a school in which a group of mathematics teachers abide by different
norms than the ones taught and learned in the programme, they often return to their
point of departure (e.g., Peressini et al., 2004). PMTs should be prepared for
integration into the collectives they join - especially into those that adopt ideas
contrary to those stressed in their teacher education programme. As newcomers
they must take an active part in advancing those groups towards communities,
prompting innovations, making communities creative and adapting to changes in
society and culture.

In sum, preparatory programmes for PMTs must answer various complex and
sometimes contradictory questions. They should be aimed at developing new
generations of teachers ready to teach new generations of students in a changing
world. They must develop the PMTs' mathematical understanding to enable them
to approach challenging mathematical tasks successfully, design tasks creatively,
and be flexible in the implementation of the tasks in their classes. Teacher
programmes should prepare PMTs to teach in ways that are different from those in
which they learned as students. They need to prepare PMTs to be effective,
confident, and creative users of new educational approaches, rich in technological
tools (Goos, 2005). PMTs should become acquainted with approaches useful for
studying their own teaching practice as well as that of others, and in analysing the
effects of teaching on learning. They must aim at generating enthusiasm, intuitions,
and beliefs about the introduction of challenging mathematics in school in the form
of reform-oriented pedagogy.

DIVERSITY OF SOLUTIONS SUGGESTED BY RESEARCH ON PMT EDUCATION

MTEs offer a wide range of courses and programmes for the professional
development of PMTs, aimed at solving the problems described above.
Corresponding research on knowledge, beliefs, and the education of PMTs is
characterized by the diversity of focal points, types of knowledge involved, level of

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

mathematics addressed, and the research tools used in the investigations. In this
subchapter, I present a review of 30 studies exploring the education of PMTs'. The
choice of the studies was based on the two main issues: (1) the studies focused on
the education of prospective teachers as members of teams, (2) they analysed
development of teachers' knowledge, skills or beliefs rather then examined their
knowledge as a in its present condition. The selected papers were published
between 1998 and 2007 in Educational Studies in Mathematics (ESM), the Journal
of Mathematical Behavior (JMB), and the Journal of Mathematics Teacher
Education (JMTE). Table 1 summarizes the following characteristics of the 30
papers:

• Type of knowledge (skills, beliefs): The distinction was mainly between
research focused on SMK and PCK.

• Level of PMT participants: This column addresses teachers in elementary
school, secondary school, and other populations involved in the studies. At
times, the relation between these characteristics is apparent. For example, in
the first category, studies focusing on SMK were conducted with secondary
school PMTs, whereas studies focusing on PCK involved primary school
PMTs.

• Number of participants: most of the studies were performed with one group
of PMTs who participated in a specific course; several were case studies
reporting on a small number of participants; and some involved a large
number of PMTs who completed a research questionnaire. Note that studies
based on a small number of participants (N=l, 2, 3) were included in the
review because the selected PMTs were representatives of teams.

• The setting in which the study was performed: for example, mathematics
course (MC), didactic course (DC), or teaching practicum (TP). This column
also includes information about the "mathematics of change" in a computer-
based environment.

• Research tools: for example, individual interviews, examples from video- and
audio-recordings, artefacts of the students' activities, individual journals, and
written questionnaires.

• Focal issues of the research: these characteristics are detailed further in this
subchapter with respect to the findings of the studies under consideration.

Note, that some of the authors did not report all these characteristics, which
accounts for empty cells in the table.

The studies are divided into several categories with respect to the roles PMTs
play in the research interventions and the balance between systematic and craft
modes of development implemented in the courses. The categories are not
mutually exclusive, and some studies can belong to more than one category.
Moreover, other categorizations are possible (e.g., Cooney & Wiegel, 2003;
Jaworski & Gellert, 2003), but the focus of the present analysis is on the balance
between mathematics and pedagogy on one hand, and between systematic and craft
modes of learning on the other.

Table 1

Article

Journal

Type
of knowledge

PMT

level

No.

PMTs

Setting

Research tools

Focus
of the study

Development through Personal Experiences as&etraera ?- *" v " *•: •* &'£"* , * -

Applying context-based approaches to course design (CD)

1 . Cavey & Berenson (2005)

JMB

SMK
PCK

secondary

LPS

Case study
l-lnt

CD, growth in understanding right
triangle trigonometry

2. Furinghetti (in press)

ESM

SMK
PCK

secondary

designing
learning
sequences

Mm
Observation

CD, how history affects the construction
of teaching sequences in algebra

3. Heaton & Mickelson (2002)

JMTE

SMK
PCK

primary

MC
statistics

I-Jni

Observation

CD, statistical knowledge, views on

teaching statistics, project-based

learning

4. Lavy & Bershadsky (2003)

JMB

SMK

secondary

(2 lessons)

1-Int
Observation
(protocols)

Types of participant-generated
problems, mathematical difficulties

5. Nicol(2002)

ESM

SMK
PCK

primary

Connect
SMK and

workplace

G-int, I-Int

PMT journals

Researcher field

notes

CD, style of teaching,

changing PMTs* belief that they must

reproduce the style of mathematics

teaching seen in their school days

6. Philippou & Christou ( 1 998)

ESM

SMK
PCK

primary

427

history

Questionnaire
l-lnt

CD, identifying and changing PMT
attitudes and beliefs about math

7. Taplin& Chan (2001)

JMTE

SMK
PCK

primary

problem-
based
learning

Group discussion
Journals

CD, student altitudes toward problem-
based learning and critical pedagogical
incidents

r-
m

2
z

Article

Journal

Type

knowie

dge

PMT
level

No.

PMTs

Setting

Research
tools

Focus
of the study

8. Wubbels, Korthagen, & Broekman
(1997)

ESM

SMK
PCK

secondary

realistic

math

Longitudinal study

comparing two

courses

Quest, I-lnt, video

CD, student and teacher views of

mathematics and mathematics

education, more inquiry oriented

approach

9. Zbiek & Conner (2006)

ESM

SMK
PCK

primary

modelling

1-Int, video, audio,
artifacts

CD, how mathematical understandings

can develop while learners engage in

modelling tasks

Socio-mathematical norms and psychological processes

10. Blanton (2002)

JMTE

SMK
PCK

secondary

geometry
discourse

Observation, video

CD, notions about mathematical
discourse

11. McNeal & Simon (2000)

JMB

SMK
PCK

primary

constructi-
vistTE

Observation, video

Mathematical and pedagogical

development, processes of negotiation

of norms and practices

12. Szydlik, Szydlik, & Benson (2003)

JMTE

PCK

primary

177

MC
norms

Survey
1-lnt

Beliefs about the nature of mathematical
behaviour

13. Tsamir (2005)

JMTE

PCK

SMK

second

DC-
intuitive
rules

Lessons, video,
audio

CD, awareness of the role of intuitive
rules, learning math

14. Tsamir (2007)

ESM

PCK
SMK

second

DC-
intuitive
rules

Lessons, video,
audio

CD, awareness of the role of intuitive
rules, learning math

15. Ponte, Oliveira, & Varandas
(2002).

JMTE

Professional
knowledge
and identity

Attitude test of

use of

information

technology

160

DC-

Intemet and

DGE

Observation,
reflective
discussions

CD, awareness of the role of intuitive
rules, learning math

O
•n

•a
SO

•a

>
n

-J
to

Article

Journal

Type
of knowledge

PMT
level

No.

PMTs

Setting

Research tools

Focus
of the study

Focusing on the Teaching Process ^

Using multimedia cases (MCA)

16. Doenr & Thomson (2004)

JMTE

PCK

Secondary

PMTs
T-educators

28
4

DC
MCA

Questionnaire, I-Int

observation, field

notes

CD, use of cases, teacher educators'
decisions

17. Masingila & Doerr (2002)

JMTE

PCK

after
TP

Class observation,
student notes, final
paper, questionnaire
instructor journals,
researcher field
notes

MCA

for using students' thinking in guiding
classroom experience

18. McGraw, Lynch, Koc, Budak, &
Brown (2007)

JMTE

PCK

Secondaiy

PMTs
Practising
Teachers
Mathematic
ians
Teacher
educators

8
5

4
4

LS
MCA

Observation, videos
transcripts of

dialogs,

semi-structured

interview

Online and face-to-face discussions,

classroom implementation of tasks,

task characteristics and appropriateness,

developing content, and pedagogical

content knowledge

19. Morris (2006)

JMTE

PCK

primary

Individual
work

sessions
videotaped

lessons

Written analysis of

lesson, student

learning,

source of problem

Ability to collect evidence about

students* learning, ability to analyse and

revise instruction

20. Santagata, Zannoni, & Stigler

(2007)

JMTE

PCK

secondary

144

(videotaped

lessons)

Pre/post-assessmen t

What do PMTs learn from the analysis

of videotaped lessons?

How to measure PMTs' analysis ability

and its improvement

Article

Journal

Type
of knowledge

PMT
level

No.

PMTs

Setting

Research tools

Focus
of the study

Teaching individual students (TIP)

21. Ambrose (2004)

JMTE

SMK
PCK

primary

MC - TIP

Surveys,

I-int: pre/post,

written work, field

notes

Changes in beliefs and skills

22. Bowers &Doerr (2001)

JMTE

SMK
PCK

secondary

MC-TIP

Math worlds

software

Participant works,
written reflections
on teaching, daily
journals (AHA!
Insights)

Thinking: as learners (about math with

computers); as teachers about students'

thinking

23. Crespo (2000)

JMTE

PCK

primary

DC -Letter

exchange

with 4th

grade

students

Journals: reflection
on activities,
case reports

(Changes in) learning about students'

thinking,

interpretive practices

24. Lee (2005)

JMTE

PCK

second

DC -TIP

teaching

math with

technology

Videos, written

works (PMTs' and

students')

Teacher's role in facilitating students'

math problem solving with

technological tool

>
2

•a

•a
m
o

-J

r
m

Article

Journal

Type
of knowledge

PMT

level

No.

PMTs

Setting

Research tools

Focus
of the study

Teaching practicum (TP)

25. Blanton, Berenson, & Norwood
(2001)

JMTE

PCK
TE

middle
school

Case study,

observation,

episode-based

interviews,

journals

Supervision in teacher education

26. Goos(2005)

JMTE

PCK

skills

secondary

4 of 18

Survey,

whole-class

interviews,

4 case studies

Working with technology

27. Nicol(1999)

ESM

PCK

primary

14 of 34

MC-TP

Course video,

PMT journals,

TE journals

Learning to teach

(what appears to be problematic)

28. Nicol & Crespo(2006)

ESM

PCK

use of
textbooks

primary

4 of 33

MC-TP

Mnt, pre, med, post,

course work,

class observation

Learning to teach
(use of textbooks and teaching)

29. Rowland, Huckstep, & Thwaites
(2005)

JMTE

SMK
PCK

primary

24 videotaped
lessons

Contribution of knowledge to teaching

30. Walshaw (2004)

JMTE

PCK,
skills

primary

Questionnaire about

recent leaching

practice experience,

discussion

Instances or leaching knowledge in

production, as interpreted by prospective

teachers

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

Development through Personal Experiences as Learners

These studies examined the education of PMTs through personal learning
experiences and reflection on those experiences. These studies assumed that PMTs
should be involved in authentic mathematical activities in order to develop SMK
and PCK, to advance the understanding of constructivist socio-mathematical
norms, of the uses of technology in mathematics education, and of awareness of
psychological issues in the teaching and learning of mathematics. Studies in this
category analysed PMT development through participation in mathematical,
didactic, or psychological courses that integrated different approaches to teaching
and learning mathematics. Courses in this category can be subdivided into two
groups: applying context-based approaches to course design and focusing socio-
mathematical norms and psychological processes.

Applying context-based approaches to course design. Based on the assumption that
PMTs need a context that would allow them to look in a different way at the topics
they will be teaching, the studies in this group suggest a context-based course
design. Most of the studies were carried out within the framework of didactics of
mathematics courses. Among the contexts that were proposed for the course
design, we find the history of mathematics (Furinghetti, in press; Philippou &
Christou, 1998), project-based learning within a statistical context (Heaton &
Mickelson, 2002), realistic mathematics (Wubbels, Korthagen, & Broekman,
1997), mathematical modelling (Zbiek & Conner, 2006), right triangle
trigonometry (Cavey & Berenson, 2005), mathematics in a workplace (Nicol,
2002), and problem-based learning (Taplin & Chan, 2001).

These studies demonstrated that a context-based design of courses for PMTs
was effective when using content previously unknown to the PMTs. For example,
in Zbiek and Conner's study the objectives of the course included both learning
mathematical modelling and learning to develop and implement application
problems and mathematical modelling tasks in future classrooms. They found that
the course opened opportunities for PMTs to grow as "knowers and doers" of
curricular mathematics. Furinghetti analysed how history affected the construction
of teaching sequences in algebra based on activities carried out at the "Laboratory
of Mathematics Education". The aim of the course was to equip PMTs with
understanding of the cognitive roots of the concepts and processes that their future
students were going to encounter in algebra. The study showed that the integration
of history in school mathematics inspired variability in strategies of teaching, and
that the fact that students had not had specific preparation in the history of
mathematics opened diverse opportunities for the development of prospective
teachers' SMK and PCK.

Overall, these studies demonstrated that as a result of systematic implementation
of the suggested context-based approaches in the courses for teams of prospective
teachers the changes that took place in the PMTs' knowledge and beliefs occurred

ROZA LEIKIN

both in the field of mathematics and pedagogy. At the same time, these studies did
not ask whether these learning experiences were powerful enough to equip PMTs
with norms that will help PMTs join their future professional communities.

Socio-mathemalical norms and psychological processes. The studies in this group
acknowledged the importance of involving PMTs in authentic activities focusing
particular socio-mathematical norms. The motivation of these studies was to design
undergraduate experiences organized around reform-minded ways of teaching in
order to close the gap between reform-oriented mathematics and the PMTs
previous mathematical experiences and conceptions about mathematics and the
teaching of mathematics. Two main modes of course design can be observed here.
The first one - a mathematical mode - in which PMTs' learning through
challenging mathematical experiences leads to the development of both SMK and
PCK (e.g., Blanton, 2002; McNeal & Simon, 2000). The second one - a didactic
mode - in which MTEs emphasize changes in approaches to teaching mathematics,
leading to the development of both PCK and SMK through PMTs' experiences
with innovative pedagogy (Szydlik, Szydlik, & Benson, 2002).

The results of these studies show that changes in socio-mathematical norms in
PMTs' courses influence their conceptions of mathematics teaching. Blanton
(2002) showed that the undergraduate mathematics classroom offers a powerful
framework for PMTs to practice, articulate, and collectively reflect on reform-
minded ways of teaching. The study demonstrated that participants construct an
image of discourse as an active collective process by which students build
mathematical understanding and develop their ability to participate in such
discourse. McNeal and Simon (2000) noted that, in the beginning, most
prospective teachers were uncomfortable with the mathematics of the course both
as learners and as future teachers. They argued that the constitution of a classroom
micro-culture supports knowledge development and demonstrated how through
participation in the course, students develop a new relationship with mathematics.
Examining the processes by which PMTs negotiated norms and practices, the
researchers identified and elaborated categories of interaction central to the
ongoing negotiation of new norms and practises, and illustrated how each of these
categories of interaction contributed to the process of negotiation. Szydlik, Szydlik,
and Benson, (2002) found that participants' beliefs became more supportive of
autonomous student behaviours. Participating PMTs attributed their changes in
beliefs to classroom norms that included mathematical explorations, expanding
problem-solving methods, and the requirement for explanation and argumentation.
All these studies stressed once again that cognitive development requires a social
context in which mathematical activities support such development (McNeal &
Simon, 2000).

In contrast with the above studies, Tsamir's (2005, 2007) studies examined
courses from a psychological perspective. Tsamir addressed the accumulating
knowledge of secondary school PMTs in a course from the psychological point of
view of mathematics education, explicitly including the intuitive rules theory.

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

Tsamir demonstrated that emphasis on the psychological processes involved in
mathematics learning and teaching allowed advancing PMTs' knowledge of both
SMK and PCK types.

The mutual relationships between mathematics and pedagogy (including the
didactic and psychological aspects) appear clearly in the studies of this category.
Regardless of whether PMTs participate in mathematical, didactic, or psychology
courses, both SMK and PCK are developed at once. It may suggested that by
attending to the two types of knowledge explicitly, MTEs can develop both PMTs'
knowledge and their awareness of the importance of mathematical challenges and
pedagogical approaches in teaching and learning mathematics.

As members of teams of PMTs' the participants of the observed studies
advanced in their views on socio-mathematical norms, changed their attitudes
towards "different mathematics" or "new pedagogy" and sometimes were shown to
develop their mathematical expertise. Moreover, the learning teams when involved
in challenging learning experiences transformed into communities of learners.
Through the participation in "unusual" mathematical activities prospective teachers
developed shared values, norms, routines, appreciation of the role of mathematical
discussion for the development of mathematical understanding (e.g., Blanton,
2002; McNeal & Simon, 2000). Still, the question of PMTs' readiness to join their
future communities of practice was not addressed. Additionally, Santagata,
Zannoni, and Stigler (2007) found PMTs often found innovative teaching
approaches to be too abstract and unrealistic (e.g., MTEs often hear PMTs saying:
"fVe learned the other way, and why do we need this?" or "These experiences are
good for us as fixture teachers but not for our future students ").

Thus, the content of the courses should be well connected to the classroom
context in which PMTs are going to apply that knowledge. It is a common belief
that in order to bridge the gap between the PMTs' previous experience and the
desired outcome of their education, programmes for them must include field
experiences where prospective teachers are exposed to the complexity of the
classroom and to the reality of having to implement alternative approaches.

Focusing on the Teaching Process

Prospective mathematics teachers must be offered more authentic teaching-related
experiences to prepare them for the complexity and challenges of the school
context (e.g., Darling-Hammond, 1997). The studies in this category explored
courses in which PMTs were involved in analysing teaching processes. Courses
for PMTs vary in the way in which they enable PMTs to learn form experience,
whether they are based on others' teaching experiences (exposure to examples of
teaching) or on one's own: (a) using multimedia cases, (b) teaching individual
students, or (c) making teaching practice an integral part of each educational
programme for PMTs. The use of video cases may be considered a transition from
pure learning of mathematics and pedagogy to learning from the teaching
experiences of other teachers. These studies maintain that observation and
systematic analysis of video cases and videotaped lessons are effective professional

ROZA LEI KIN

development tools. Another group of studies deals with the teaching of individual
students and reflection on those experiences within a team and can be considered
as an intermediate stage between systematic and craft modes of development.
Although PMTs are not in a classroom, they interact with students and learn from
these interactions. The last category is that of teaching practicum. These studies
examine the PMTs' involvement in school teaching as individuals with sequencing
discussion of the teaching experiences in a team. The authors analyse the process
of learning to teach, supervision in the course of practicum, and the contribution of
knowledge to teaching.

Using multimedia cases. Research on video cases continues earlier research that
examined the use of text-based cases in teacher education (e.g., Barnett, 1991;
Shulman, 1992; Stein, Smith, Henningsen, & Silver, 2000). The rationale for the
implementation of video cases includes several considerations. First, it is difficult
to find sufficient high-quality classrooms for placements; a careful choice of video
cases can expose PMTs to good teaching. Second, video cases or complete
videotaped lessons serve as a basis for group discussion, the development of shared
norms, and reflective thinking on the students' mathematical thinking (see also
Seago, this volume). Third, videotaped classroom episodes or whole-class
procedures can develop PMTs' critical evaluation of classroom practice. Fourth,
videos can be played repeatedly to enable a depth of reflection and analysis that are
often impossible to achieve in live observations. Overall, the various uses of videos
allow teacher education programmes to face the challenge of developing PMTs'
conceptual understanding of how students understand subject matter and how it
should be introduced to them (Hiebert, Gallimore, & Stigler, 2002).

Studies on the implementation of multimedia cases demonstrated the following
outcomes: (I) PMTs were able to learn from video cases about students' learning,
to analyse the effects of instruction, and to revise the initial instruction (McGraw et
al., 2007); (2) providing ways to measure PMTs' ability to analyse video cases and
the improvement of this ability along the course (Santagata, Zannoni, & Stigler,
2007); (3) characterizing ways in which video-cases are used by MTEs and the
relationship between the PMTs' background and experiences and their uses of
video cases (Doerr & Thomson, 2004); (4) demonstrating the potential of the case
studies for developing PMTs' ability to analyse critically classroom episodes
(Masingila & Doerr, 2002).

These studies showed that video cases have a positive effect on the development
of PMTs' knowledge and skills and identified the complexity of diverse processes
involved in this development. They stressed once again the complexity of noticing
(in the sense of Mason, 2002) and the diversity of the focuses of attention: the
mathematics of the tasks, the interaction between teacher and students, and the
issues related to whole-class discussions (McGraw et al., 2007; Morris, 2006). The
studies showed that PMTs analyse dilemmas and tensions revealed in teaching
recorded in video-cases based on their own perspectives on teaching (Masingila &
Doerr, 2002).

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

Overall, video-case studies may be seen as a preparatory stage in PMTs'
experiences of analysing teaching practice. I suggest considering video-case
courses as exemplifying an experimental mode of professional development. By
analysing other teachers' craft experiences in laboratory conditions PMTs develop
tools and skills for analysing their own teaching practice. The analysis of video-
cases performed in systematic mode, under the guidance of MTEs, leads future
teachers to important inferences about the nature of teacher-student interactions
and their role in the students' knowledge development. These experiences evolved
PMTs' understanding of the complexity of teaching. As members of teams,
through analysis of the video-cases, PMTs developed sensibilities and a vocabulary
that should prepare them to future teaching practice, and help them to enter there
future communities (e.g., Morris, 2006; Doerr & Thomson, 2004).

It should be noted that video-case studies analysed primarily the impact of the
use of video-cases on PMTs' knowledge of pedagogy. In these studies, little
attention has been paid to PMTs' learning of mathematics. Additionally, the
question of whether in their personal teaching interactions with students PMTs will
be able to use the knowledge and skills advanced through the analysis of video-
cases remains open until PMTs experience the teaching settings in person.

Teaching individual students. As an intermediate stage between video-cases and
teaching practicum, mathematics educators incorporate the teaching of individual
students, complemented by reflective analysis of their experiences within teams. A
diversity of focal issues appears in this group of studies. For example, Bowers and
Doerr (2001) and Lee (2005) analysed secondary PMTs' thinking about the
"mathematics of change" in a computer-based environment, working individually
with young children. Crespo (2000) explored PMTs' learning about student
thinking by analysing their interpretations of the students' works by PMTs engaged
in interactive mathematics letter exchanges with fourth-grade students.

A variety of findings associated with learning through teaching individual
students have been reported. PMTs were surprised to find mathematics teaching to
be more difficult than they had thought, and inferred that providing children with
time to think when solving mathematical problems was an irreducible component
of teaching (Ambrose, 2004). PMTs' foci of attention changed as their roles
changed: as students they were curious about reconceptualising the mathematical
theorem they learned, whereas as teachers they built on their students' explanations
and on the role of the technological tools (Bowers & Doerr, 2001). PMTs tended to
use their problem-solving approaches in their pedagogical decisions: they asked
questions that would guide students to their own solution strategies; recognized
their struggle in facilitating students' problem solving, and focused on improving
their interactions with students. PMTs used technological representations to
promote students' mathematical thinking, and used technological tools in ways
consistent with the nature of their interactions with students (Lee, 2005). PMTs'
interpretations of students' works developed in the course of their experiences: in
the beginning, PMTs attended to the correctness of students' answers and later they

ROZA LEIKIN

focused on meaning; they started from quick and conclusive evaluation and shifted
to a more complex, thoughtful, and tentative approach.

Overall, the studies that analysed PMTs' teaching experiences with individual
students demonstrated advance in PMTs' understanding of the teaching process.
Instructing individual students helped break the well-known conviction loop: to
implement new pedagogical approaches, teachers must be convinced of the
suitability of those approaches in their work with students and, at the same time, to
be convinced of the suitability of those approaches they have to implement them in
school. Experiencing teaching with individual students allowed PMTs to feel more
confident and to gain those convincing experiences. By discussing these
experiences with their team-mates, PMTs realized they all had common
difficulties, surprises, unexpected events and satisfaction. Did PMTs, only when
teaching, begin to understand that teaching was complex and required different
levels of attention? Only in craft mode PMTs started feeling what it meant to be
flexible in teaching and sensitive to the students. The role of teams in these courses
was to enhance PMT's reflective skills, to provide them with mutual support, and
sharpen their critical reasoning.

Teaching practicum. Teaching practicum is one of the professional development
settings that enable PMTs to make the connection between learning and teaching,
but requires negotiation between the school and the college or university culture.
Practicum is a course in which the knowledge of content and pedagogy learned in
systematic mode is implemented, and craft knowledge is developed almost for the
first time in the PMTs' professional career. The teachers find themselves teaching
individually school students and then discussing those experiences with supervisors
or team-mates. I suggest that courses of this type are representative of another
mode of professional development: implementation mode.

Several studies performed in the last decade analysed the effects and
characteristics of diverse components of teaching practicum on the professional
development of PMTs. These studies explored the role of university supervision
(Blanton, Berenson, & Norwood, 2001); pedagogical practices and beliefs about
integrating technology into the teaching of mathematics (Goos, 2005); ways in
which PMTs employ their knowledge of mathematics and pedagogy in their
teaching (Rowland et al., 2005); issues that PMTs find problematic in teaching
mathematics and changes in the types of questions PMTs ask students (Nicol,
1999); and the use of textbooks in learning to teach mathematics (Nicol & Crespo,
2006). Nicol (1999) showed that PMTs - as a result of experiences gained in
practicum - began to consider students' thinking and to create spaces for inquiry
through the types of questions they posed. PMTs also began to see and hear
possibilities for mathematical exploration that evolved as their relationship with
mathematics and students changed. Nicol and Crespo (2006) showed that PMTs'
attempts to modify textbook lessons posed pedagogical, curricular, and
mathematical questions that were not easily answered by reference to textbooks or

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

teacher's guides. Findings indicated that practicum could challenge prospective
teachers to be creative and flexible users of curriculum materials.

In this group, we also find studies that constructed theoretical models and tools
(e.g., Goos, 2005; Rowland et al., 2005). Goos (2005) theorized PMTs' learning
using the notions of Zone of Proximal Development (ZPD), Zone of Free
Movement (ZFM, possible teaching actions), and Zone of Promoted Action (ZPA,
the efforts of a teacher educator that are needed to promote particular teaching
skills or approaches). Using these concepts and the mutual relationships between
them, Goos demonstrated a variety of relationships between a range of personal
and contextual factors that influence the formation of the PMTs' identity as
teachers.

The study by Rowland et al. (2005) proposed a set of four units as a framework
for lesson observation and mathematics teaching development. The four units
were: foundation, transformation, connection, and contingency. Foundation refers
to teachers' awareness of purpose, the theoretical underpinning of learning and
pedagogy, and the use of various tools. Transformation refers to knowledge-in-
action as revealed in manner in which the teacher's own meanings are transformed
to enable students to learn, including the use of analogies and examples. The
teachers perform connections between different meanings and descriptions of
particular concepts or between alternative ways of representing concepts and
carrying out procedures. Contingency is revealed by the ability of the teacher to
respond appropriately to contributions by students during a teaching episode.

Overall, the studies on teaching practicum involved various issues and
participants in the process of learning-to-teach. The studies analysed primarily
PMT's pedagogical skills and beliefs, including their understanding of students
(and their errors), teaching with technological tools, and the use of textbooks. Note
that these studies paid little attention to the development of SMK in the process of
teaching or to the extent to which PMTs implemented in practice the material that
had been studied in systematic mode. The role of the teams in the teaching
practicum courses was especially important for the development of PMTs'
reflective and analytical skills (e.g., Nicol, 1999). The prospective teachers when
discussing their teaching experiences were exposed to the variety of views on
teaching profession, could reify their own position, learn from own and others'
"mistakes". Teaching practicum got PMTs closer to the communities of practice
that they would join in near future.

THE DIVERSITY OF SOLUTIONS AND QUESTIONS THAT REMAIN OPEN

MTEs face inherent dilemmas and challenges when preparing teachers for work in
classrooms. The studies described above demonstrated the diversity of solutions
that MTEs use in order to solve various issues in teacher education and to support
prospective mathematics teachers' conceptual changes. In a majority of the studies
under consideration, these solutions took the form of various professional
development tools that MTEs integrated in the courses designed for PMTs. Their
effectiveness was shown by analysing changes in PMTs' knowledge, beliefs, and

ROZA LEIK1N

attitudes. Other studies designed theoretical models and tools that may be effective
in analysing and describing changes in teachers' knowledge and beliefs. In this
subchapter, I return to the problems highlighted in the first subchapter and outline
further research questions associated with the education of PMTs.

Attending to the Centrality of the Mathematical Challenge

Most of the studies that examined PMTs' development through their personal
learning experiences included mathematical challenge among those experiences
(e.g., McNeal & Simon, 2000; Taplin & Chan, 2001; Zbiek & Conner, 2006).
Little attention has been paid, however, to the concept of mathematical challenge
itself, although this meta-mathematical awareness is complex and crucial for the
ability to design and analyse a lesson effectively (e.g., Holton et al., in press). I
suggest that the notion of "mathematical challenge" as a meta-mathematical and
psychological concept may serve as a springboard for the development of PMTs'
knowledge and beliefs. The following questions are important for advancing their
beliefs in the importance of the mathematical challenge:

What are the PMTs' conceptions of the mathematical challenge and its role in
teaching and learning mathematics? How can programmes for teams of PMTs
foster these conceptions so that PMTs would be eager to implement them in their
future practice? How can these programmes promote PMT's expertise in solving
challenging mathematical tasks and their capability of choosing and designing
those tasks for teaching?

Attending to Changing Approaches in Mathematics Teaching and Learning

PMTs' mathematical knowledge and beliefs about mathematics and about teaching
mathematics are influenced significantly by their experiences in learning
mathematics long before they decided to become teachers (Cooney et al., 1998).
PMTs bring these experiences to their teacher education programmes in the form
of conceptions, and are expected to make changes in their views of mathematics
and pedagogy. I suggest that the ideas of conceptual change theory found in
science education (Posner, Strike, Hewson, & Gertzog, 1982) may be useful in
addressing this issue. Conceptual change acknowledges the importance of prior
knowledge to learning and considers both the enrichment of existing cognitive
structures and their substantial reorganization (Schnotz, Vosniadou, & Carretero,
1999). Such reorganization is conceptualised as being motivated by dissatisfaction
with the initial conception and by the intelligibility, plausibility, and fruitfulness of
the new conception (Posner et al., 1982). When teaching PMTs about learning
alternative approaches to teaching mathematics, MTEs must reconceptualise their
initial views on teaching and learning.

Most of the studies reviewed in this chapter showed the effectiveness of
integrating alternative approaches to mathematics teaching and learning within
courses or programmes for PMTs. They argued that that programmes focused on
alternative approaches to teaching and learning mathematics must require from

TEAMS OF PROSPECTIVE MATHEMATICS TEACHERS

PMTs to experiment with these approaches with their students (Hiebert, Morris, &
Glass, 2003), and to analyse and discuss these experiments (e.g., Bowers & Doerr,
2001; Crespo, 2000; Lee, 2005; Goos, 2005; Nicol, 1999). But in order to be able
to experiment the alternative approaches, PMTs must encounter conceptual change
in the field of pedagogy or of mathematics. Thus, beyond what has been achieved
already in the reviewed studies, the following questions can be raised to further
analyse and strengthen PMTs' learning in the various university and college
programmes:

What are the ways in which PMTs can achieve dissatisfaction with their initial
conceptions of school mathematics and mathematics teaching? How can courses
for teams of PMTs advance their perception of the new approaches as intelligible,
plausible, and fruitful for teaching?

Attending to the Complexity of PMTs ' Knowledge

Most of the studies described course design and examined its effectiveness. The
courses differed in the balance between systematic modes of development (through
learning) and craft modes (through teaching), between knowledge and beliefs,
between mathematics and pedagogy. From this perspective, and together with the
analysis of the research provided in the second subchapter, I suggest that there are
four main modes of professional development that vary with respect to explicit
versus implicit goals, the balance between mathematics and pedagogy, the role of
challenging content, and the mechanisms of teachers' knowledge development.
These are the mathematical, pedagogical (didactic or psychological), experimental,
and implementation modes.

Combinations of different modes can be achieved by PMTs' participation in the
courses belonging to different modes within a programme, or through the
integration of different modes in one particular course for teachers. For example,
such integration is present in studies that combine experimental and mathematical
modes (Ambrose, 2004; Bowers & Doerr, 2001), mathematical and
implementation modes (Nicol, 1999; Nicol & Crespo, 2006), or pedagogical and
implementation modes (Masingila & Doerr, 2002). Balancing between craft and
systematic modes is especially important: whereas systematic knowledge provides
a stable base for the development of craft knowledge, only craft knowledge
contains the necessary convictions and beliefs about the applicability of what has
been learned systematically. The balance between systematic and craft modes can
help solve the "conviction loop" (see above).

Answering the following questions would further advance the design of the
programmes for the professional development of PMTs:

What combinations of systematic and craft modes of development are most
effective in the courses for PMTs? What combinations of mathematics and
pedagogy are the most effective in the preparation of PMTs? How can the different
modes be integrated so that they support each other?

ROZA LEIKIN

Becoming a Member of a Community of Practice

All the studies reviewed in this chapter explicitly or implicitly acknowledged the
importance of integrating newcomers in the school system. Llinares and Kramer
(2006) and Peressini et al. (2004) stressed that the process of recontextualization of
what has been learned in teacher education programmes for prospective teachers
into what will be taught in the classroom is extremely important. There is not
enough evidence that the professional development of PMTs prepares them for
such integration and recontextualization. And there is not enough evidence that
PMTs who changed their views about approaches to mathematics teaching and
learning will implement these approaches in their future classes.

When they first begin to teach school, teachers usually learn from more
experienced teachers. At the same time, experienced teachers can learn from the
newcomers how to change mathematics teaching and learning so that it fits better
the new reality, new technologies, and the latest cultural artefacts and advances. It
is unclear, however, to what extent future teachers are ready to be adaptive agents
of these approaches. When beginning their teaching careers, new teachers grasp the
conflict between systematic knowledge (constructed in the teacher education
programme) and the prescriptive knowledge they develop within the school
system. They usually find it hard to cope with the complexity of the system, and
with the gap between what has been learned in teacher education and the reality of
the school. Longitudinal studies that could answer the following question may
further advance teacher education programmes:

What are the springboards and pitfalls in the transitions from systematic to
experimental learning, from experimental learning to teaching practicum, and from
teaching practicum to the real classroom? How, when educated in teams, PMTs
may be prepared to become members of communities of practices?

Gaining a better understanding of the factors that promote and hamper the
professional development of PMTs will support MTEs in their complex task of
preparing their students for teaching careers at all grade levels.

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Mathematics, 63, 89-1 12.

Roza Leikin
Faculty of Education
University of Haifa
Israel

SUSAN D. NICKERSON

4. TEAMS OF PRACTISING TEACHERS

Developing Teacher Professionals

The focus of this chapter is on teams of practising teachers brought together in
formally arranged situations organized by management, such as, subject
coordinators, school-based or district-based administrators. The chapter begins
with a description of the goals for developing teacher professionals. I then
illustrate the current practice of professional development of the past ten years
with examples from several countries. Common aspects are evident in the structure
of the professional development programmes. A case study brings the issues of the
complexity of studying practising teachers into focus. Finally, I use these examples
to explicate the message about teams and relevant environment, framed in terms of
inter-dependence and the co-constructed context.

GOALS FOR DEVELOPING TEACHER PROFESSIONALS

Current reform initiatives mainly initiated by the National Council of Teachers of
Mathematics (1989) call for changes in the "core dimensions of instruction"
(Spillane, 1999; S pi I lane & Zeuli, 1999). Mathematics teachers are expected to
establish in their classrooms "communities of learners" where students explore
mathematics in depth and teachers facilitate students' mathematical learning.
Students are expected to construct mathematics for themselves and develop a
means of determining the appropriateness of solutions and procedures based on
arguments used to justify the solutions and procedures. This vision of mathematics
learners suggests students make meaning of mathematics and comprises a
corresponding emphasis on achievement for all students (Stein, Silver, & Smith,
1998; Tirosh & Graeber, 2003). This theme of more learner-centred and
conceptually-focused instructional practice is international in scope with
mathematics progressively seen as a critical competency for greater numbers of
more diverse students (Adler, 2000; Adler, Ball, Krainer, Lin, & Novotna, 2005).

A common goal of teacher development programmes, therefore, is to develop
support for student thinking by helping teachers develop interactive and dialogic
contexts for learning and promoting students' thinking with appropriate questions
and statements (Sowder, 2007). Teachers' learning to teach for student
understanding must integrate knowledge of: mathematics content (including
concepts, processes, and methods of inquiry); student thinking (understanding the
ways in which students thinking could develop); and instructional practice (nature
and effects of their teaching) (Carpenter, Blanton, Cobb, Franke, Kaput, &

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 89-109.
© 2008 Sense Publishers. All rights reserved.

SUSAN D. N1CKERS0N

McClain, 2004; Jaworski & Wood, 1999). Teachers, often having only been
participants in a classroom with a traditional or an instrumental approach to
mathematics teaching, are expected to teach in ways they themselves have not
experienced. Therefore, many professional development programmes have as a
goal assisting teachers in paradigmatic shifts in epistemology and instructional
practice. The goals of supporting changes in teachers' practice, philosophy, and
beliefs is fundamentally based in the notion that changed practice, raised
awareness, and changes in beliefs about mathematics, mathematics teaching, and
learning can result in a more effective environment for student learning (Jaworski
& Wood, 1999).

This chapter focuses on professional development experiences for teams of
practising mathematics teachers; in particular, teams are mostly selected by
management, with pre-determined goals, which therefore create rather tight and
formal connections within the team (Krainer, 2003). Although teams may develop
into collaborative communities, they are not inherently so. Teams of practising
mathematics teachers may consist of school-based groups or teachers drawn from
across a school district or region. Subject-coordinators, school-based or district-
based administrators are examples of management that bring a team together
within a formal structure. Teacher educators' and management's choice of what to
offer or require is shaped by a perception of teachers' needs, by the larger context
in which schooling occurs, and resources (human, social, and capital) committed to
such endeavours (Borasi & Fonzi, 2003; Nickerson & Brown, 2008). Individual
teacher participant's decision to participate is affected by factors such as his or her
perception of meaningfulness, feasibility, work demands, management and
collegial support (Kwakman, 2003). Although teams of mathematics teachers are
mostly selected by management for work structured towards predetermined goals,
in practice, the goals and structure of the work are equally affected by teachers'
and managements' participation and the design of successful professional
development can and should evolve and change (Loucks-Horsley, Hewson, Love,
& Stiles, 1998). As such, the professional development programmes in which
teams of teachers are engaged is the joint construction of the teacher educators who
design it, administrators and management who decide what to offer or require, and
the teachers who choose the programmes and the manner in which they agree to
participate (Borasi & Fonzi, 2003). In this same sense, context is not deterministic
but interactively constructed among the participants (Jones, 1997).

Therefore, the mathematics teacher professional development programmes as
constituted are reflective of the specific political and cultural contexts in which
they are embedded. Cooney and Krainer (1996) describe how the nature of the
programmes for practising mathematics teachers is constructed from macro
problems and micro problems. Macro problems emanate from society in general
and are related to economics, politics, culture, and language. Micro problems are
directly related to problems within mathematics teacher education, such as
curricula or teacher training. The design of professional development programmes
is reflective of an attempt to address both macro and micro problems.

TEAMS OF PRACTISING TEACHERS

For example, national initiatives affect policy and privilege some foci in the
distribution of resources. Mathematics teachers' professional development in many
Western countries is motivated by having not compared well to other countries in
international comparisons of students' mathematics achievement in the 1990s.
Consequently, the governments, for example, New Zealand, Australia, some
European countries and the United States responded to such rankings with an
impetus to improve student achievement (see e.g., Bobis et al., 2005; Borasi, Fonzi,
Smith, & Rose, 1999; Keitel & Kilpatrick, 1999). Some countries have a focus on
repairing inadequate prospective teachers' preparation, which includes the
mathematical knowledge of teachers and, in the case of South Africa, extends to
reconstruction of identity (Adler & Davis, 2006). In other countries, mathematics
education and consequently professional development programmes for teachers are
shaped by the goal of eradicating economic and technological disparity (Atweh &
Clarkson, 2001; Tirosh & Graeber, 2003). Thus, teams of teachers are, by
definition, engaged in professional development structured as top-down
implementations. As such, political, social, and economic concerns contribute to
the structure of mathematics professional development for teams of teachers.

The nature of mathematics teacher professional development is also shaped by
the mathematics education community's beliefs about mathematics, mathematics
teaching, and learning. For example, Ball (1997) discusses mathematics education
reform in the United States as based on concern about students' achievement in
mathematics and current economic, political, and social pressures for greater
numbers of students who can use mathematics competently. Yet, the reform is also
shaped by our ideas of what constitutes mathematics learning and knowledge.
Constructivist learning theories permeate new directions for teacher professional
development experiences (Ball, 1997). In many teacher education programmes
around the world, teachers are seen as constructors of knowledge with a need for
opportunities for reflection in order to learn (Cooney & Krainer, 1996). Teacher
educators of practising teachers acknowledge the challenge of paradigmatic
changes in instructional practice, the need to make connections between
professional development and work on site, and the benefits to teachers of
substantial support from colleagues (see e.g., Knight, 2002; Putnam & Borko,
2000; Tirosh & Graeber, 2003).

What follows is a survey of studies of professional development programmes
and approaches that illustrate current themes of mathematics teacher professional
development for teams of teachers in many countries. Approaches to professional
development are frequently characterized according to the content of the
workshops or focus of the work with teachers (see e.g., Borasi & Fonzi, 2003;
Kilpatrick, Swafford, & Findell, 2001; Sowder, 2007). Most studies of teacher
development adopt an individual teacher as the unit of analysis, describing changes
in an individual teacher's knowledge of mathematics, beliefs, or instructional
practice (e.g., Cohen, 1990; Sowder & Schappelle, 1995). From another
perspective, professional development is a social matter, enhancing collective
capability (Knight, 2002). This perspective on teacher development situates
teachers' work in context and views professional development as the building of

SUSAN D. NICKERSON

communities of collaborative, reflective practice, wherein teachers are joined with
colleagues to create effective mathematics environment for their students' learning
(McClain & Cobb, 2004; Stein, Silver, & Smith, 1998).

Researchers have identified some of the critical aspects of effective professional
development. Notable among these is the need to foster collaboration to encourage
the formation of communities of collegial learners (e.g., Borasi & Fonzi, 2003;
Wilson & Berne, 1999), to involve administrators and other stakeholders as they
are critical to arranging resources (e.g., Gamoran, Anderson, & Ashmann, 2003;
Krainer, 2001), and the need to carefully consider alignment with the
organizational context in which these teachers work (e.g., McClain & Cobb, 2004;
Sowder, 2007; Stein & Brown, 1997; Walshaw & Anthony, 2006). In what
follows, I describe several professional development programmes that illustrate the
importance of organizational factors and joint activities when teams of practising
teachers are engaged in professional development (Krainer, 2001). This survey is
intended to highlight these themes while simultaneously revealing the breadth of
possible paths.

PROFESSIONAL DEVELOPMENT WITH TEAMS OF PRACTISING TEACHERS
WITHIN THE WORK OF TEACHING

Although the structure of the mathematics professional development described here
varies with regard to length and focus and in the manner in which the goals are
addressed, the central goal of these programmes is to support practising teachers in
reorganizing their instructional practice to become more learner-centred and
conceptually focused. Teachers learning to teach in a manner that supports student
understanding means that professional development often focuses on engaging
teachers in doing mathematics, understanding student thinking and the ways in
which it develops, and scaffolding attempts at changed instructional practice.

Teacher education programmes have focused on and continue to focus on the
central aspect of teachers' knowledge of mathematics (Jaworski & Wood, 1999;
Sowder, 2007). This knowledge of mathematics is framed more broadly than
understanding concepts. The mathematical knowledge needed by teachers
encompasses the ability to use knowledge of mathematics to foster effective
learning for students (Adler & Davis, 2006; Ball & Bass, 2000). One important line
of research of the last decade concerns the nature of the mathematical knowledge
needed for teaching, which in turn influences the design of teacher education as
mathematics teacher educators try to provide opportunities to learn these
specialized ways of learning and knowing mathematics (Adler & Davis, 2006).
Thus, the programmes with a mathematical focus encompass concepts, process,
methods of inquiry, beliefs about mathematics and mathematics learning. In some
countries, professional development has a focus on initiating an alternative
conception with regards to mathematics teaching and learning (see e.g., Farah-
Sarkis, 1999; Mohammad, 2004; Murray, Olivier, & Human, 1999).

As an example, in the context of South Africa, the practising teachers and the
administrator placed limits on the time available for professional development.

TEAMS OF PRACTISING TEACHERS

Murray et al. (1999) describe a two-day workshop attended by primary (K-3, 5-8
year olds) mathematics teachers and head of subject upper elementary teachers in
South Africa. Within this narrow window of time, teacher educators had a goal of
changing teachers' perceptions of mathematics "[...] and equipping teachers for
radically different classroom practice" (Murray et al., 1999, p. 33). Given the
teachers' impoverished preparation consisting of traditional mathematics
experiences and low self-perception of mathematical ability, the teacher educators
tried to address, in their workshop, perceptions of how mathematics is learned and
used, as well as the teachers' perceptions of their own mathematical ability. By
posing problems that were challenging to teachers and then supporting reflection
on their experiences as learners, the designers hoped to justify a problem-centred
approach to teaching and to share information to support teachers in establishing a
problem-centred classroom. Murray et al. (1999) noted the importance of engaging
teachers in mathematical activities that suggested different solution strategies and
were aligned to the syllabus. The format encouraged reflection and connections to
students' views in similar activities, and the development of a vision of the
teacher's role in fostering a problem-centred approach to learning mathematics.

The teachers' evaluations of the workshop suggested that the activities were
successful in providing a vision of a "starting place" in their own classrooms.
Teachers who perceived themselves as mathematically weak reported being deeply
moved by their experience with mathematical sense-making. While the real test of
the workshop's success was in changed instructional practice, the authors believed
that the fundamental groundwork was laid. They acknowledged that the challenge
of changed practice is ultimately highly dependent on factors such as supervisor
and peer support. The teams of practising teachers were brought together with
persons who were heads of mathematics in their schools to develop a vision of how
mathematics is learned with understanding and the role of the teacher in supporting
this understanding. After participation, the teachers requested more opportunities
for professional development of longer duration.

In stark contrast to the time available with practising South African teachers,
teacher educators in Israel describe an extensive programme for practising middle
and high school mathematics teachers and mathematics department chairpersons.
Each round of the Kidumatica programme consisted of full-day weekly meetings
for three years, and had a goal of raising the level of teachers' content knowledge
and of promoting collaboration among teachers teaching at different grade levels.
Fried and Amit (2005) describe a "spiral" activity employed to provide
opportunities for teachers to see a problem situation developed for different grade
levels. A single problem situation was illustrated and modified for each of seventh
through eleventh grades. The teacher educators addressed a perceived problem in
instructional practice wherein teachers tended to present problems as one-
dimensional entities that embody a single technique or concept. The team met for a
portion of the time into large across-grade groups to encourage reflective
professional conversations. These conversations were not just about a particular
task, but rather concerned the links between the mathematics for students of
different grades or achievement levels.

SUSAN D. NICKERSON

One of the goals of arranging across-grade groups was to knit together a broad
mathematics teaching community within an individual school and a region. Fried
and Amit (2005) stressed the importance of the multiple perspectives brought by a
broad grade-level range of teachers both to develop a deeper, connected
understanding of the mathematics and also to develop respectful relationships with
colleagues across the grade spectrum. The professional development enhanced
collective capability of the team in understanding the range of mathematics
connections and challenged the notion that if one understands the concepts at a
higher level, then one has the mathematical knowledge needed for effectively
teaching it to lower middle-grade students.

A number of professional development initiatives are characterized by the
central goal of making students' thinking the focus of and the impetus for teachers'
reflection on their own instructional practice (e.g., Fennema et al., 1996; Loucks-
Horsley et al., 1998; Sowder, 2007; Zaslavsky, Chapman, & Leikin, 2003).
Teaching mathematics for understanding requires knowledge of students and how
their mathematical thinking develops. Examining student thinking focuses
teachers' attention on the consequences of instructional practices and provides
opportunities to understand the discrepancies between what students understand
and what teachers wish them to learn (Loucks-Horsley et al., 1998). Teachers can
then build upon the concepts and skills expressed in student thinking, using it to
guide instructional decisions.

Whitenack, Knipping, Novinger, Coutts, and Standifer (2000) report on a team
of practising primary teachers brought together to learn about students' thinking.
There were three phases to the programme; in the first phase, the practising
primary teachers were part of a larger district-wide group invited to hear two
presentations by pre-eminent mathematics educators who addressed issues of
teaching and learning mathematics with understanding and creating a learning
environment for all students to engage in meaningful mathematics. The teacher
educators sent out a district-wide invitation to all K-2 teachers requesting they
submit an application to participate in the programme. The professional developers
worked with a K-7 district mathematics coordinator, who had a rich history with
some of the teacher participants, to plan sessions (Whitenack, personal
communique).

During the second phase, 27 practising teachers participated in a one-week
summer institute. Participants viewed videotapes of students being interviewed
while engaged in mathematical problem solving then discussed the videos in terms
of students' number development. A mini-case study assignment was designed as a
culminating activity for the summer institute. In contrast to case-based professional
development formats wherein teachers analyse and discuss a presented case,
teachers in the summer institute were asked to work with a partner to develop mini-
case studies of students' thinking. The videos included examples of six students (5-
9 years old) solving various problems. For example, teachers could select to
investigate the strategies used by one child or identify instances in which several
students using the same strategy. This provided an opportunity for teachers to
develop hypotheses about the strategies students used and the kind(s) of

TEAMS OF PRACTISING TEACHERS

mathematical reasoning such solutions required and conjectures about what it
means for students to know and do mathematics.

During the third phase, the project team met with teachers for eight 3-hour
sessions during the first four months of the school year. These meetings afforded
opportunities for teachers to share new ideas they were exploring, and to reflect on
instructional materials that might support students' understanding of place value,
multiplication, division, and geometrical concepts. The teachers designed and
conducted interviews with students from their own classes and implemented
lessons that addressed concepts that had surfaced during interview sessions. The
mathematics teacher educators worked with the district mathematics coordinator as
a means to support teachers' learning about students' thinking. Participation in the
project facilitated peer collaboration focused around an analysis of students'
thinking, and scaffolded teachers' trying on new lenses for looking at the student
work by assisting teachers embedded in the work of teaching.

Another professional development approach involving teacher-teams was also
based on the belief that increasing teacher's awareness of students' thinking
contributes to improvement in teaching, however in this case with middle and
secondary level students. Tirosh, Stavy, and Tsamir (2001) developed a research-
based seminar to introduce a theory that would assist practising middle school and
high school teachers in understanding and predicting students' responses to
mathematical and scientific tasks. During the course of the seminar, teachers were
first introduced to the Intuitive Rules Theory, used to explain how students react in
similar ways to mathematical and scientific tasks. Teachers learned about how
incorrect answers involving comparison and subdivision tasks can be explained by
this theory, the educational implications of this theory, and then they engaged in
discussions of teaching by analogy and conflict. Specific examples of research in
the context of the intuitive rules are discussed. Finally, each teacher was asked to
select a topic and define research questions either to conduct a micro-study related
to validating a known or identifying a new intuitive rule, or the development and
assessment of teaching interventions to counter unproductive intuitive rules.
Members of the group met weekly to collectively discuss each other's proposals
and research questions. Several teachers collectively analysed data and presented
the results to the others. Although the team of practising teachers met with experts
outside of the teaching community with an initial structured agenda, in the final
phase they selected an investigation related to the work of their teaching. These
practising teachers collectively developed a new lens for collaborative inquiry into
student reasoning.

Thus far, the programmes discussed illustrate professional development that is
grounded in and related to the work of teachers' own teaching and involving
collaboration with outside experts. The programmes are reflective of the growing
realization of the importance of involvement of heads of department or other
stakeholders. They illustrate different means of support toward reorganization of
instructional practice, often by asking teachers to "try out" aspects of practice that
they are encouraged to "take back" to the classroom. The following programmes I
describe are more integral to the institutions in which the teachers work. In each of

SUSAN D. N1CKERS0N

these cases, a level of professional development and scaffolded attempts at
changed instructional practice occur at the level of the school and within the site.
Two programmes for Australian teachers focused on the use of research-based
frameworks for young students' number learning in the early years of schooling, an
assessment interview to profile a child's knowledge, and whole-school approaches
to professional development.

The two initiatives, Count Me In Too (CMIT) in New South Wales and the
Victorian Early Numeracy Research Project (ENRP), had two goals: to help
teachers understand students' mathematical development and to improve students'
achievement in mathematics. A key aspect of CMIT was the Learning Framework
in Number (LFIN) developed by Wright (1994) and based on Les Steffe's
psychological model of the development of students' counting-based strategies
(Bobis et al., 2005). Teachers were assisted in using an assessment to profile a
student's knowledge across the spectrum of key components of LFIN; this profile
was then be used to guide instruction. Typically, the programme involved a district
mathematics consultant and a team of three to five teachers from each school with
the district mathematics consultant assisting in planning. The project expanded
from a pilot in 1996 in 13 schools to almost 1700 schools by 2003. One initial
obstacle to implementing CMIT was the misalignment between the programme's
content and the national syllabus. In 2002, a new syllabus was released which was
closely aligned with the CMIT project. During this time, the focus in the
professional development on number extended to include a measurement strand
and a space (geometry) strand (Bobis et al., 2005).

With a similar focus on the use of research-based frameworks, the Victorian
Early Numeracy Research Project (ENRP), which ran for three years from 1 999-
2002, introduced teachers to a framework of growth points in young students'
mathematical learning developed by the project leaders (Bobis et al., 2005). This
included five or six growth points in each strand of Number, Measurement, and
Space. Similarly, it involved a task-based assessment interview, and a multi-level
professional development programme aimed at developing a common "lens"
through which teachers could view students' reasoning in multiple settings. The
professional development programme involved engagement at national, regional,
and school levels.

The ENRP involved approximately 250 teachers from 35 project schools. The
support for teachers formally occurred on three levels, state, regional and school,
(Bobis et al., 2005). All 250 teachers from the state of Victoria met with the
research team each year for five full days spread across the year. The focus of these
meetings was on understanding the research framework, the interview, as well as
appropriate classroom strategies, content, and activities to meet the needs of their
students. At the regional level, teachers gathered on four or five occasions each
year usually for two hours after school. These meetings brought together three to
five school "professional learning" teams and were facilitated by a member of the
university research team. Teams were made up of all Prep-2 teachers (teachers of
students 5-8 years old) in each school, an early numeracy coordinator, the principal
and the early year's literacy coordinator in some schools. The meetings focused on

TEAMS OF PRACTISING TEACHERS

sharing particular activities or approaches, mathematical content, and an
articulation of the tasks to be completed before the group met again. Finally, at the
school or classroom level, the cluster coordinator spent time on teaching,
observing, and planning. The coordinator at each school conducted regular
meetings of the "professional learning team" to facilitate communication, maintain
continuity and focus.

Researchers examined the nature of the work of the professional learning teams
by analysing coordinators' "significant event" folio entries in which they reflected
on current mathematics education issues at their school. Other data sources were
interviews of the coordinators and surveys administered to principals of the
participating schools. The professional learning teams varied in size, operation, and
meeting frequency. Some teams had the same team members and coordinator
across the three years of the project, others had a transient team and some had a
different coordinator each year (State of Victoria, 2003). Early in the project,
professional learning teams were enthusiastic, yet somewhat overwhelmed by the
requirements of project participation, while the student interviews provided insight
into students' thinking, the information also proved daunting for teachers. The
teams became aware of and were initially uncomfortable with the notion that there
were many different approaches to teaching. Eventually, teams acknowledged
common goals for students while accepting professional differences in teaching. In
the beginning of each subsequent year, the enculturation of new team members was
seen as a major challenge. The further the project progressed, the greater became
the discrepancy between the "oldtimers" and the "newcomers". The coordinators
worked within limited release time to address this discrepancy but the strategies
and success in mediating this varied across sites. However, both principals and
coordinators noted increased dialogue around mathematics with more willingness
to share ideas, opinions, and resources. The researchers concluded that the teams
served an important role in supporting teachers in improving the teaching of
mathematics (State of Victoria, 2003).

These two programmes were motivated by national government interest in a
remedy for poor achievement in mathematics in international comparisons. The
initiatives drew on research-based learning frameworks that describe a trajectory or
pathway of students' early mathematical learning. The frameworks and assessment
tools provided teachers with a common lens for discussions across groups. Both
programmes were whole-school approaches to professional development with
outside experts and site-based coordinators supporting teachers' practice.

Many professional development programmes of the past decade engage teachers
in concrete examination of instructional practice more broadly considered than
student thinking alone. Chissick (2002) described a three-year professional
development programme for secondary mathematics teachers in Israel. The goal
was to change the instructional practices of teachers toward a use of innovative
instructional practices, including more prominent use of technology and promotion
of teamwork. Teams from 13 schools were assigned a "facilitator" for one day a
week for three years, and the school's head of mathematical department worked
with the facilitator to lead weekly training workshops on innovative instructional

SUSAN D. NICKERSON

practices and to assist teachers in experimenting with new instructional practices
and technology. The facilitators and department heads received support for their
leadership role in monthly meetings.

Ghissick (2002) investigated effective implementation of reform mathematics
teaching practices, effective use of technology and increased teamwork in
mathematics teaching. The teachers, head of departments, and school heads were
asked to complete structured and semi-structured questionnaires. Additional data
collected included weekly reports from project facilitators and field notes of
observations of meetings. Chissick reported significant change in teamwork culture
and some changes in classroom practice (specifically the use of more open-ended
tasks and more student-centred teaching). This study also explored teachers' views
of themselves as learners. Case studies at two of the schools provided more in-
depth data analysis and a closer look at a few teachers that included classroom
observations, interviews regarding teacher beliefs and attitudes toward change,
mathematics and mathematics teaching, personal history and self esteem. Thus, this
research included a focus on an individual's role in implementation.

Another trend of the past ten years has been an effort to take successful models
of practising teacher professional development and use them with a team of
practising teachers. Here I focus on successful professional development for
teachers in East Asia adopted for use in the United States. Teacher development in
Japan, called kenshu, describes peer collaboration, review and critique of actual
lessons; this is often referred to as "Lesson Study" (see Yoshida, Volume 1). The
first thing teachers generally do is establish a lesson study goal. Study lessons are
collaboratively planned by about four to six teachers, implemented by one teacher
(usually the teaching is videotaped and observation notes are taken) and afterward
the lesson is discussed. Administrators and principals participate in this discussion
because they are considered "peers". The participants discuss their observations
and decisions about how to improve the lesson. An invited observer provides an
"outside prospective". An outside observer is most often an instructional
superintendent, an individual appointed by prefectures to regularly visit schools
and advise teachers in a region. The outside advisor can also be a university expert
or teacher on leave hired to provide professional development (Fernandez &
Yoshida, 2004). In Japan, practising teachers organize the most common type of
professional development with the formation of a study promotion committee
drafting a yearly study plan, which is the negotiated with the teachers at different
grade levels. There are also interschool programmes organized by district-wide
subject area associations of teachers (Shimahara, 2002).

Lesson study as practiced in Japan can be obligatory or voluntary. For example,
one type of obligatory programme is an internship for beginning teachers.
Internships are legally required for teachers and provide a one-year probation
wherein beginning teachers have reduced responsibilities and a mentor who is
selected from among experienced teachers. As part of the internship, interns
observe mentor teachers teaching and implement study lessons before their more
experienced colleagues. The beginning teachers learn and practice the different
roles of the teacher in teaching a lesson (Shimizu, 1999). The internship is

TEAMS OF PRACTISING TEACHERS

designed by the prefectural education centre or local board (Shimahara, 2002;
Yoshida, 2002). Though there are obligatory top-down initiatives, lesson study in
Japan is embedded in the culture of teaching (Shimahara, 2002). Three premises
are used as organizing principles for Japanese professional development. The first
is that teaching is a collaborative, peer-driven process that can be improved
through this process. The second is that peer planning is critical to teaching. The
third is that teachers' active participation is a critical element of professional
development and teaching. Although these premises constitute the normative
framework of Japanese professional development, school-based teacher
professional development varies a great deal depending upon the leadership of
teachers in the local context and the level of teaching (Shimahara, 2002). In
general, lesson study is utilized more at the elementary school level and is less
common at the secondary level.

On the basis of their growing understanding of how lessons are conducted and
prepared in Japanese classrooms, Hiebert and Stigler (2000) suggested that
educators in the US consider a form of lesson study to build professional
knowledge; thus variations of lesson study are rapidly proliferating across the
country (Chokski & Fernandez, 2004; Fernandez, 2005). As lesson study comes to
be viewed as effective approach to professional development, school and district
administrators select this structure as a means to provide a professional
development experience connected to the classroom work of teachers. Because
collaboration is not part of the culture of teaching in the US, management and
teacher educators need to bring teachers together to engage them in lesson study.
Lessons study requires management to assist with scheduling, allocating funding
and arranging for substitute teachers and these endeavours are often initially
obligatory for teams of teachers.

Podhorsky and Fisher (2007) described a lesson study implementation in an
elementary school in a low socio-economic area, offered in response to No Child
Left Behind (NCLB) legislation, a high stakes accountability programme. Lesson
study was selected because it provides a collaborative environment for teachers to
focus on curriculum and student learning to facilitate an increase in student
achievement. Teacher participants (30 teachers of grades 1-5) met weekly for
roughly a year to plan, teach, and critique lessons. The participants were also part
of a university class, but the approach became one that was implemented school-
wide. The school administrator at the time was very involved (Fisher, personal
communique, June, 2007).

The teachers were observed as they engaged in lesson study and were
interviewed individually and in focus groups. Teachers and school site
administrators were surveyed using Likert-scale questionnaires assessing
perceptions of this model of professional development. From the surveys and
interviews we learn about the participants' perspectives on the strengths and
challenges of the lesson study process within their context. The strengths of lessons
study as identified by the teachers included: an emphasis on meaningful lessons; an
impetus for implementing short and long term goals; and a focus on student
assessment. Teachers also cited the benefits of the community aspect of

SUSAN D. NICKERSON

participation in lesson study, increased reflection on teaching practices, and
excellent preparation for attaining national teaching certification. The challenges of
implementing lesson study in this urban school were the planning time required for
the research lesson and, in particular, the significant amount of time required
outside of the school setting. Lesson study requires common preparation periods
with grade-level teams, common curriculum, and external guidance in conducting
lesson study; resources not readily available to teams of practising teachers in other
schools throughout the United States. However, this situation is slowly changing.
According to a survey conducted in 2004, a majority of US lesson study groups
met at least once a week, most during the school day (Chokshi & Fernandez,
2004). As lesson study becomes part of the culture of teaching in the US, it can
evolve into a more teacher-led process of professional development.

In another approach to professional development, teams of practising teachers
are often brought together to implement challenging curriculum. Balfanz, Maclver,
and Byrnes (2006) reported on a study of the first 4-years of a mathematics reform
project, the Talent Development Middle School Model Mathematics (TD)
programme, which was initiated in three, high-poverty urban US middle schools in
the context of whole school reform. This programme included professional
development that was directly linked to implementation of three reform
mathematics curricula in grades 5 through 8 (10-13 year olds). The mathematics
reform (curricula, corresponding professional development and whole school
initiative) was instigated to meet the national milestones of the federal No Child
Left Behind (NCLB) legislation. The professional development approach was that
of providing "model lessons" using the curricula.

The professional development was led by peer teachers and experienced users of
the curricula and took place over three days of summer training that was followed
by monthly 3-hour workshops on Saturdays. The monthly sessions were focused
on lessons that would be taught in the participants' classes the following month.
The facilitator guided the teachers as participants through the upcoming lesson and
modelled important aspects, including the mechanics of the activities, questioning
and then providing an opportunity for the participant teachers to ask questions and
discuss with each other past experiences of teaching. The teachers also had
substantial implementation support in the form of curriculum coaches that spent 1-
2 days a week at a school helping teachers in their classrooms. The curriculum
coach co-taught, modelled, assisted with lesson planning, provided feedback and
worked with the teachers to make modifications to the curriculum to address his or
her students' specific needs. By the completion of the fourth year of the initiative,
two teacher leaders from each school were ready to help with on-site
implementation. Part of the teacher-leader training involved shadowing the
curriculum coaches in their work with teachers.

Even with all of the support structured into this programme, high levels of
implementation were difficult to achieve. Across the four years of the study, about
two-thirds of the teachers achieved the minimum recommended hours of
professional development each year. The initial goal of having teachers' complete
instruction in the use of 6 to 8 units of each grade level's instructional materials

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was not achieved. Two-thirds to three-fourths of the classrooms in these high-
poverty middle schools obtained at least a medium-level of implementation, but
with significant variation across sites. Of significance to this discussion, the
Balfanz et al. (2006) noted important factors of institutional context that may have
affected the implementation: school leadership; scheduling; staffing and resources.
As an example, during the four years of the reported intervention, only one school
of the three had the same principal and one school had three principals, with
varying degrees of commitment to the programme. Staffing patterns changed as the
administration either successfully or unsuccessfully provided the resources, and
assigned or reassigned teachers. Teacher turnover was one of the biggest
challenges; by the fourth year of the study only 31% to 59% of the homerooms
across the three schools had mathematics taught by a teacher who had participated
in the reform effort all four years.

Summary

The studies described illustrate themes of mathematics professional development
for teams of practising teachers in several countries. The programmes are
structured toward predetermined goals of improving student achievement by
constructing effective mathematics environments for student learning.
Consequently, the focus is often on developing understanding of connections
among concepts and topics in mathematics, increasing awareness of students'
mathematical thinking and the pathways of development, and supporting teachers'
changing instructional practice. Whether the approaches to professional
development involve engaging teachers in doing mathematics, or examining
student thinking, or introducing lesson study as a mechanism for improving
teaching, or modelling the teaching of reform curriculum, some common themes
have emerged. Here we see, teams of practising teachers engaged in programmes
structured to foster collaboration, including not just peers but heads of departments
and other significant site leaders. The programmes illustrate a growing awareness
of the critical role of alignment with organizational context (see Cobb & Smith,
this volume). The following case study is an example of these common themes and
importantly, illustrates that the goals and structure of the work are equally affected
by teachers' and managements' participation.

CO-CONSTRUCTED CONTEXT: A CASE OF MATHEMATICS SPECIALISTS

The story that emerges from the studies on teams of practising teachers across
multiple school sites is one of differential enactment for seemingly similar
programmatic activities, highlighting the co-constructed nature of the professional
development experience. I have chosen a case study for elaboration. The case study
describes a programme designed to assist teachers in becoming mathematics
specialists. The initiative emerged in a context of addressing poor student
achievement at a local level. The focus was on engaging a team of teachers in
doing mathematics and supporting reorganization of their instructional practices.
Several school districts and universities in the United States have partnered to

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develop professional development programmes for mathematics specialists,
individuals with specialized preparation in mathematics (see e.g., Nickerson &
Moriarty, 2005; The Journal of Mathematics and Science, 2005). Mathematics
specialists can have work assignments such as a lead teacher or coach. In some
specialist work assignments, teachers may teach only mathematics or they may
teach mathematics and one other subject, such as science. In this way, teachers can
focus on being knowledgeable or expert in the teaching and learning of one, or at
most two, subjects.

Nickerson and Moriarty (2005) described an initiative in which 32 upper-
elementary teachers in a large urban school district in the United States were hired
as additional staff at eight low-achieving schools. The teachers, as the only
teachers of mathematics in these schools, travelled from classroom to classroom
with carts of materials "visiting" other teachers' classes to teach three mathematics
classes, each lasting 90 minutes. During their first year of teaching as a
mathematics specialist, they participated in a professional development programme
that had a focus on building teachers' knowledge of mathematics and mathematics
for teaching with connections to practice. The principals arranged for shared
professional development time at each school site. The 60-90 minutes each day
was intended to facilitate teacher's sustained growth in knowledge and practice.
The teachers met for two weeks one summer and then about three hours a week for
one year in coursework designed to help teachers reconceptualize the mathematics
they were teaching and to deepen understanding of mathematics pedagogy. The
coursework also entailed teachers' learning about how students' mathematical
thinking develops. Coursework and on-site activities provided opportunities for
reflecting on practice by engaging in collaborative reflective teaching cycles and an
analysis of student work, their own and from research. In addition, the mathematics
specialists had on-site coaching support from the teacher educators about once a
week.

Furthermore, the coursework, site-based support, and daily, shared professional
development time was intended to facilitate teachers' sustained, generative growth
in content knowledge and practice. Like other programmes described in this
chapter, this team of practising teachers was brought together by administrators
with pre- determined goals. The professional development was in support of a
larger reform initiative, and was planned with the school district mathematics
instructional team. The designers of the initiative had the expectation that the
provision of these resources would promote the formation of teachers' professional
communities of collaborative, reflective practice. Nickerson and Moriarty (2005)
used the construct of teachers' professional community as defined by Secada and
Adajian (1997). Teachers' professional communities are described along four
dimensions: (I) shared sense of purpose, (2) co-ordinated effort to improve
students' mathematical learning, (3) collaborative professional learning, and (4)
collective control over decisions affecting the mathematics programme. The
researchers undertook an analysis of teachers' activities and experiences as situated
within institutions as opposed to a structural analysis (Cobb & McClain, 2001).

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Based on an analysis of a number of data sources including weekly field-notes
of teacher educators and researchers regarding visits to schools, interviews with
teachers and mathematics administrators, teachers' written coursework and
reflections, teachers' mathematics exams, a survey regarding curriculum
implementation, and an interview with the school district leadership team.
Nickerson and Moriarty (2005) investigated the relationship of professional
development, encompassing both formal and informal sources of support, to
teachers' knowledge and practice. Nickerson and Moriarty (2005) reported that
professional communities formed at some sites and not others. Five aspects of
teachers' professional lives emerged as significant in the formation and strength of
teachers' professional community: (1) the relationship the mathematics teachers
had with the school administration and other classroom teachers, (2) the respect for
and access to the knowledge of other mathematics teachers, (3) the presence or
absence of a teacher leader at the site, (4) a shared base of mathematical and
pedagogical knowledge, and (5) shared high teachers' expectation for students. The
strength and nature of teachers' professional community is significant in local
interpretation of reform and the development of collective professional values and
goals (see e.g., Talbert & Perry, 1994).

One important aspect of teachers' professional lives was the relationship that a
team had with a principal and other teachers at the site. The teachers applied for
positions at particular schools. The school principals made the hiring decisions. At
a few schools, the teachers were at the site and were encouraged to apply while
other principals hired all the staff from outside. School-based administrators were
able to align resources for the teaching of mathematics - choosing teams of
participants, arranging shared professional development time at each site, and
facilitating the teams' ability to exercise collective control over decisions
concerning the mathematics programme. The other teachers further contributed to
or hindered the formation of a shared vision. At some sites, administrators arranged
for minimal shared professional development time that was sometimes overtaken
by other responsibilities. There was evidence of administrators who did not
understand the goals of the initiative and inhibited the power of teachers to make
decisions regarding the mathematics programme.

A second important aspect related to teachers developing respect for and access
to the knowledge of other mathematics teachers because it involves teachers seeing
themselves as members of a community with contributions to make to the
collective capabilities (Nickerson & Moriarty, 2005). This sense of self as a
valuable member of a team contributed to collaborative professional learning at
some sites. However, one team of teachers expressed a view that they were
implementing school district mandates and spent little time together unless
mandated by an outside presence. A third important aspect was the presence or
absence of a teacher leader who appeared to play a fundamental role in supporting
collaborative professional learning and shaping a shared sense of purpose. A fourth
important aspect was teachers' mathematical knowledge that appeared to affect
their collective control over decisions regarding the mathematics programme and
their ability to effectively discuss instruction and student learning. Teacher teams

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differed in how they spent the professional development time together. Some
teachers used the time to help each other with mathematics and to share the
successes and failures of their classes. Other teams used the time to look ahead,
prioritize topics, and examine mathematics across grades. Finally, and
significantly, shared high expectations for students were successfully jointly
constructed by some teams and administrators and not others.

The goals and structure of the work were equally affected by teachers' and
managements' participation. Teacher educators and administrators designed an
initiative shaped by administrator's selection of team participants and alignment of
resources. The programme was jointly constructed by teachers' sense of self as a
mathematics teacher, teachers' respect for others' contributions, and their ability to
effectively discuss instruction and student learning. Teacher educators and school
district administrators designed the initiative, local site administrators decided what
to require, and the teachers who chose the mathematics specialist programmes
shaped the initiative by their varying and evolving participation. The case study
suggests that the fostering of collaboration, the involvement of administrators, and
the alignment of organizational context do not individually account for differential
enactment and point to the complexity of studies on teams of practising teachers.

STUDYING TEACHER DEVELOPMENT IN CONTEXT

The focus in this chapter has been on studies of the approaches to professional
development for teams of teachers, with a particular focus on the past ten years.
Issues of political, social, and economic significance contribute to the structure of
mathematics professional development for teams by providing resources for reform
curricula, new technologies, and orienting administrators' focus on mathematics.
This survey of studies of the past decade highlights the nature of professional
development approaches for teams of teachers. The studies illustrate an aim of
fostering collaboration among colleagues around central issues such as developing
a common lens for viewing and discussing students' early learning, connections
among mathematical content areas, and improving instructional practice. The
studies of the past decade reveal increasing involvement of management,
principals, heads of department and other stakeholders, as part of crafting the
vision of effective mathematics teaching. Furthermore, these studies speak to the
need to align programmes with the organizational context in which teachers work.

The research that has emerged from the studies point to the complexity and
highlight the interdependence of institutional context, relevant management, and
teachers themselves as learners. Teams brought together to foster formation of
communities of collaborative learners are affected by and affect the different ways
that teachers participate in the team. Likewise, as Krainer (2001, p. 282) suggested
the management's vision and alignment of resources critically affects and is
affected by institutional context:

Principals and other important stakeholders such as regional subject
coordinators or superintendents with different roles and functions in school

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system, have their own ideas and beliefs about the nature of learning,
teaching, mathematical knowledge, and reform. [...] it is essential to pay
more attention to their role in the professional development of teachers, both
practically and theoretically.

The research in social context suggests the importance of consistent intentions
and motives among teacher educators, administrators, and teachers (Cobb &
McClain, 2001). Yet, many of these initiatives describe multiple layers of
administrators, wherein the administrators who decided what to require were
distinct from the site administrators who are proximate to teachers. As a district
superintendent describes (Nickerson & Brown, 2008, p. 20):

We've got. I don't know. We've got 1, 2, 3 partnerships over 26 teachers
right now. We've challenged principals next year that they all will have at
least one partnering teacher when school starts in September. Principals are
uncomfortable with that because they have to figure out how to make that
happen. We challenge them to make that happen. We think, you know, trust
us. In the end you will be glad you did.

Although management selected or required the programmes required, several
researchers described the varying degrees of commitment by the administrators
that work most closely with teachers (see e.g., Balfanz et al., 2006; Nickerson &
Moriarty, 2005; Walshaw & Anthony, 2006).

In closing, we must acknowledge that the nature of the professional
development programmes for teams of practising mathematics teachers is reflective
of different contexts. There are differences in the issues for developed western
countries with substantial resources directed to large-scale recruitment of teams of
teachers and developing countries with limited time and resources. Whatever the
nature of these professional development programmes, as Perrin-Glorian, DeBlois,
and Robert (this volume) conclude, it is important to study teaching within the
larger societal and political context and to recognize that when practising teachers
must change because of externa) reforms, it can be very difficult to integrate new
instructional practices with institutional and social expectations. This suggests that
mathematics educators and researchers can learn much by taking up the challenge
of understanding the complex interrelationships among teachers, peers, teacher
educators, administrators, and institutional contexts.

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5. FACE-TO-FACE LEARNING COMMUNITIES OF
PROSPECTIVE MATHEMATICS TEACHERS

Studies on Their Professional Growth

Focusing on participants in mathematics teacher education, this chapter focuses
on face-to-face learning communities of prospective teachers. Important elements
of a learning community synthesized from the literature are addressed in the start-
ing section and used as a frame for discussion in the final section. Two cases from
Taiwan and Portugal, which comprise prospective teachers for secondary and
elementary schools, are presented. Examples of learning communities both initi-
ated by participants and created by institutional arrangements are given, with con-
sideration for their outer contexts. The process and product of learning from one
another within a community from the two cases are synthesized with respect to the
inner issues of face-to-face learning communities.

INTRODUCTION

The responsibility of mathematics teachers is to foster students' learning. However,
teachers and prospective teachers are also learners. In particular, prospective teach-
ers have to learn how to carry out their job and later, as practising teachers, need to
learn to deal with new mathematics topics, technologies, needs from students, and
demands from curriculum and society. Both teachers and prospective teachers learn
- in practice - for practice, and - from practice. For prospective teachers, this in-
cludes not only teaching practice in school contexts, but also learning practice dur-
ing university courses and in informal situations. They learn as they carry out the
activities set up in the courses of their teacher education programme and during
field work as they design instructional units and educational materials and tasks,
observe classroom situations, interview students and work together with their col-
leagues and supervisors.

In this chapter, learning is viewed as both an individual and a social process.
People learn as they interact with the physical and social world and as they reflect
on what they do. Therefore, learning originates from the activity of an individual
carried out in a given social context (Lerman, 2001). Prospective teachers learn
from their activity and their reflection on their activity, and such learning takes
place in a variety of places, as they interact with others, notably their university

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 111-129.
© 2008 Sense Publishers. All rights reserved.

FOU-LAI LIN AND JOAO PEDRO DA PONTE

teachers, colleagues, school mentors, school students, and other members of the
community.

A {earning community is a special context in which practising teachers and pro-
spective teachers may learn. The most important feature of a learning community is
that its members learn from one another. There may be differences in age, experi-
ence, status, and professional roles, but the unifying element is that all assume that
they are learners; they are keen to learn together and, most importantly, to learn
from the others.

These learning communities may take a variety of forms. They may develop
from a group that developed habits of working together or a group that was particu-
larly constituted for the purpose of learning or carrying out some project. Such a
group may be formed spontaneously by the initiative of its participants, or may be
created by some institutional arrangement. Therefore, the group of mathematics
teachers in a school, formed on a purely administrative basis, may become a learn-
ing community if the teachers begin valuing the fact that the participants can learn
a great deal from each other. Similarly, a class for prospective teachers, that by
itself is a highly contrived setting, can constitute a learning community if the in-
structor and the prospective teachers develop relationships of learning from each
other (Jaworski, 2004).

People may work together in a variety of ways, either competitively or collabo-
ratively. For example, they may constitute teams, that "are mostly selected by the
management, have pre-determined goals and therefore rather tight and formal con-
nections within the team" (Krainer, 2003, p. 95); In contrast, communities "are
regarded as self-selecting, their members negotiating goals and tasks. People par-
ticipate because they personally identify with the topic" (p. 95). A group that be-
gins as a team may develop a working culture that transforms it into a community.
Learning is generally regarded as a key element in becoming a member of a com-
munity but it may also be regarded as an important feature of the activity of the
whole community. Therefore, a learning community is a community where learn-
ing is valued as an important outcome of the groups' activity.

The notion of learning community in many respects is close to that of discourse
community. For example, for Putnam and Borko (1997), a discourse community is
a group of people that learned to think, talk, and act in a similar way - in our case,
as mathematics teachers. The authors suggest that prospective teachers can model
joint cognitive activities, doing "careful analysis of the cognitive roles played by
each participant" (p. 1276). Another related notion is that of community of practice
(Lave & Wenger, 1991), where newcomers learn from old-timers by participating
in the tasks that relate to the practice of the community and, with time, newcomers
move from peripheral to full participation. Still another related notion is that of
inquiry community (Jaworski, 2005), in which a group of professionals question
their existing practices and explore alternative practices. In this chapter, we use the
notion of learning community because our focus is on prospective teachers' face-
to-face learning in formal and informal teacher education activities.

Diversity and heterogeneity among the participants of a learning community of-
ten make it difficult to find a common language, to adjust purposes and ways of

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

working. However, such diversity may be very beneficial to the work of the com-
munity. As Ponte et al. (in press) highlight, different experiences, viewpoints, and
expertise may make the group more powerful in identifying and generating solu-
tions to deal with relevant issues, thus leading to quite significant learning from the
participants. For example, Clark and Borko (2004) describe how a community was
generated in a professional development institute for middle-school mathematics
teachers that focused on algebra content knowledge. In their view, the participants
at the beginning formed a rather diverse group but the tasks proposed on the first
day of the institute fostered their active participation in activities such as "explain-
ing and clarifying ideas, building off of others' ideas, admitting weaknesses, giving
praise to others, and laughing" (p. 229), that the authors regard as indicators of the
establishment of a community.

Working with colleagues, aiming at professional learning, is also an essential
part of the model proposed by Hiebert, Morris, and Glass (2003, pp. 21 1-212) for
generating and accumulating knowledge for teaching and for teacher education
(and we may see prospective teachers and practising teachers both involved in this
process):

Becoming a professional teacher, in our view, means drawing from, and con-
tributing to, a shared knowledge base for teaching. It means shifting the focus
from improving as a teacher to improving teaching. This requires moving
outside the individual classroom, surmounting the insularity of the usual
school environment, and working with colleagues with the intent of improv-
ing the professional standard for daily practice. This also requires redirecting
attention from the teacher to the methods of teaching. It is not the personality
or style of the teacher that is being examined but rather the elements of class-
room practice.

Jaworski and Gellert (2003) indicate that, to teach effectively, prospective
teachers need to integrate different kinds and layers of knowledge. Such knowl-
edge develops through the work of the university and school, of prospective teach-
ers and their university tutors, mentors and other teachers. In their view, all the
participants involved, including prospective teachers, may contribute to this devel-
opment: "all participants are learners, their roles developing in relation to their
critical evaluation of them" (p. 270).

Ponte et al. (in press) indicate that there arc four key issues in learning commu-
nities. The first is the purpose of the group and its relation to the personal purpose
of its members. In fact, a very important condition for one person to learn is his or
her wanting to learn and, similarly, a very important condition for one group to
learn is its desire to learn. Therefore, we need to consider the purposes that led to
the creation of the group, to what brought people to the group and how they iden-
tify with the purposes of the group. And, of course, we need to consider the fact
that the group's purposes change with time. The second issue concerns the knowl-
edge that develops from the activity of the community based on shared practices.
We need to pay attention to what participants are really learning and how they

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FOU-LA1 LIN ANDJOAO PEDRO DA PONTE

learn it. Is it superficial or significant knowledge? The third issue is how learning
happens in the group. Do group activities involve intensive moments of studying,
discussing, and reflecting, or are the participants mostly carrying out routine activi-
ties? Is learning developed through negotiating meanings and sharing reflections or
just by memorizing and imitating the others? Finally, the fourth issue concerns the
roles and relationships of the participants. In a learning community, the mutual
involvement and commitment of members to the progress of the group are essential
features (Ponte et al., in press):

The learning community is stifled if some members do not feel confident
enough to expose their concerns, do not ask for help, and refrain from par-
ticipation in the group or if, on the contrary, other members participate "too
much", occupying all space, helping others too much or in an improper way,
etc. A proper style of leadership is a critical element to the working of any
group [...]. There are always participants that play a more prominent role in
one stage or another of any group, but the group itself may establish a collec-
tive leadership, assuming the most important decisions after a thorough dis-
cussion of the issues, and a distributed leadership for practical activities, as-
signing specific group members the conduction of such or such activity.

The development of e-learning and other distance education settings generated a
great deal of interest in learning communities for practising and prospective teach-
ers (Llinares & Vails, 2007; Ponte, Oliveira, Varandas, Oliveira, & Fonseca, 2007;
Borba & Gadanidis, this volume). However, much less attention has been paid to
face-to-face learning communities of prospective teachers; this is the focus in this
chapter (see also Llinares & Oliveiro, this volume). We begin by presenting two
case studies that illustrate learning communities for prospective teachers as learn-
ers. The first, from Taiwan shows how prospective secondary and elementary
teachers form learning communities to study for exams, and the second one, from
Portugal, illustrates the different learning communities to which a prospective
teacher may belong during their teacher education programme.

A CASE FROM TAIWAN

In this section, the social and political contexts of prospective teacher education in
Taiwan are described. Based on the specific conditions under which prospective
teachers are certified, certain learning communities are formed.

The Supply of Prospective Teachers in Taiwan Exceeds the Demand

According to the statistical data of 2006 annual reports on teacher education (Dai,
Kuo, Yang, Lin, & Wei, 2006), 72 universities and colleges with teacher education
programmes, in general, a two years study, graduated a total of 17,000 certified
elementary and secondary school teachers. Among them, about 4,000 were selected
by schools as regular teachers. From the 13,000 of prospective teachers who were
not selected, about 2,500 are teaching in schools as substitute teachers and more

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

than 10,000 do not have any teaching position in schools. These teachers are
named "stray teachers" by the media for the fact that they are certified as teachers
but are not hired by any school. This has become a social issue.

The social status of teachers in Taiwan is relatively moderate; teachers are gen-
erally deemed as middle class. Their initial salary is about 30% more than that in
almost all other jobs; those who specialize in humanities may even exceed other
humanity-related jobs by 80%. Teaching is generally a lifelong career. With a sta-
ble pay, few change for another career as the Taiwanese society has an unstable
economy. A reliable job with a good retirement system is attractive to young peo-
ple. A teacher qualified to retire will have a monthly pension that is 85% to 95% of
his or her original pay. Therefore, it is likely that a teacher with 30 years seniority
would receive retirement pension for another 30 years. These incentives attract
many distinguished youngsters to teacher education programmes. Besides, low
birth-rate and concomitant low population growth rate in Taiwan bring about a
decrease every year in the number of people in each age group, from the maximum
of about 420,000 births in 1976, to a minimum of 205,000 births in 2006. Due to
this, the number of school-age children lessens each year, and so does the demand
for teachers. While the demand for teachers declines, the incentives for people to
become teachers remain strong. Under such circumstances, when the number of
certified prospective teachers that graduate is not efficiently limited, the supply of
teachers naturally exceeds the demand.

Examination for Teacher Certification

Owing to the new social issue of "stray teachers", the Ministry of Education set a
limit on enrolling students for teacher education programmes. At the same time,
the Ministry of Education started to evaluate these programmes. The ones that
score low would have their enrolment decreased at first. Two years later, they
would be re-evaluated. The ones that fail to pass the evaluation would have to close
their education programmes. In addition, the Ministry of Education instigated an
examination for teacher certification. Before 2005, prospective teachers became
certified teachers once they completed the teacher education programme which
included one year of teaching practice in school. Since 2005, however, the period
of teaching practice for prospective teachers was cut in half, and the examination
for teacher certification was added and administered by the Ministry of Education.
Those who pass the exam receive the teacher's certificate.

The examination for teacher certification comprises of four subjects: Mandarin-
Chinese; general education theory and system; youth (child) development and
counselling; and secondary (elementary) curricula and pedagogy. Words inside and
outside of parentheses are the subject names for elementary school teachers and
secondary school teachers respectively. In 2006, there were 7,857 persons regis-
tered for taking the exam; 4,595 of them passed it, which means a passing rate of
58% (Dai et al., 2006).

115

FOU-LAI LIN AND JOAO PEDRO DA PONTE

Examination for Teacher Selection

Following the examination for teacher certification, those who get the teacher certi-
fication attend the examination for teacher selection held in each of the 23 counties
and cities. The examinations for senior high school teacher selection are hosted by
each school. There are two phases to the selection exam. The first phase tests the
prospective teachers' content knowledge (e.g., mathematics teachers will have an
exam on mathematics). Those who pass the first phase can proceed to the second
phase, which is a teaching demonstration and interviews. Generally speaking, on
average, each elementary level prospective teacher takes more than three examina-
tions for teacher selection. The number of those that obtain a teaching position is
between 1% and 2% of the total. Secondary school teachers are a little better off.
The ratio of those that obtained a new teaching position is around 5% to 6% in
2006.

Du Shu Hui (BltW), Learning Communities for Exams

In the following, we regard the prospective teacher education programmes at Na-
tional Taiwan Normal University. In mathematics education, the university pre-
pares for the examinations for teacher certification and teacher selection as well as
for the entrance examination for graduate schools. Personal communication with
colleagues and prospective teachers about how the graduates prepare for the ex-
aminations mentioned above, indicated that active students usually form Du Shu

Hui (IKftft), or learning communities. Prospective teachers participating in Du
Shu Hui are more likely than others to pass these examinations. Du Shu Hui are
usually initiated by one or several students. The motive of initiating a Du Shu Hui
is to prepare for certain examinations. Many of these Du Shu Hui last nearly a
whole year. Under the present social and political contexts of Taiwan, these pro-
spective teachers, in order to further their education or to become teachers, form
their Du Shu Hui as a learning strategy. The interactions and the norms for interact-
ing, the knowledge sharing, the operation of collaborative learning, and the out-
come and feedback of the learning communities are interesting issues in terms of
social dimension.

Du Shu Hui is a Chinese noun (9ltt0) meaning study group. If translated liter-
ally, Du means to read, Shu means books and Hui means meeting. According to the
definition of learning community given by Krainer (2003), Du Shu Hui is a learn-
ing community in that:

- Du Shu Hui are self-selecting, their members negotiate goals and tasks.

- Prospective teachers participate because they personally identify with the
topic.

The following is an overview of some Du Shu Hui we picked out as examples of
how these work as learning communities.

Method of collecting data. In order to understand prospective teachers' Du Shu
Hui, a structured interview was conducted in November, 2007. The structure of the

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

interviews consists of three parts: the incentives and goals for setting up learning
communities, their organizational operation, and the outcomes and reflection of
these communities.

Subjects from departments of mathematics (education) in three universities, two
for elementary school teachers and one for secondary school teachers were selected
and interviewed. One of the mathematics educators from the departments of these
subjects conducted the interviews. Each interview lasted about 30 to 40 minutes.

Four Examples ofDu Shu Hui. After being certificated, most of prospective secon-
dary mathematics teachers take examinations either for teacher selection or for
further study in graduate institutes, majoring in mathematics or mathematics educa-
tion. In the examination for secondary mathematics teacher selection, solving
pedagogical mathematics problems, designing instruction units and 15-minute
teaching demonstration are the main activities. From the data we collected, we
chose three Du Shu Hui, each of which respectively focused on one of the three
activities during their community meetings. A brief description of the Du Shu Hui
are shown in Table 1. This table also shows one Du Shu Hui which focused on
studying advanced mathematics for entrance examination of graduate institute.

Some specific features of each Du Shu Hui are described as following. During
each community meeting, the five members of the Du Shu Hui teaching demon-
stration took turns to demonstrate 15 minutes teaching. The other four members
could spend as much time as they wanted on commenting others' teaching demon-
stration. They are critical friends (Jaworski, 1999) to each other. During the inter-
view with Shi-An, he reflected that "I was influenced by other members. They
helped me rectify my teaching approach. This is rather important. I wouldn't have
passed the exam if not for the rectification." and "everyone has blind spots. The
members helped me see mine."

In Du Shu Hui on designing instruction units, the reason there were only two
members was that each one would like to design more instruction units to under-
stand more fully the teaching content. One of them has to travel more than 100
kilometres to meet the other at a cafe each week. Both of them were selected as
senior high school mathematics teachers in the summer of 2007. Though they are
teaching in different schools, they remain close friends. One member reflected in
the interview that: "Now, whenever I have problems concerning teaching I talk it
over with Wen-Rong. We discuss about issues including mathematical contents,
teaching method, and developing test items."

In the Du Shu Hui for solving pedagogical mathematics problems, four mem-
bers are still in their fifth practicum year. They meet one whole day a week. They
are solving pedagogical mathematics problems appeared in previous examinations
for teacher selection. Because one of them practice teaching at the best gifted class
in Taiwan, tough problems very often are assigned to those gifted students to do;
afterwards they discuss students' answers and learn from those students. The
community implicitly consists of not only the four prospective teachers but also the
class of more than 40 mathematically gifted senior high school students. Those
gifted students are also learning by solving problems. The members we inter-

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FOU-LAI LIN AND JOAO PEDRO DA PONTE

viewed expressed that "One of us has very good understanding of mathematical
concepts. His clear linking of geometry and algebra made a great impact on me."
and "One of us has excellent conceptual generic examples; they are inspiring."

Table 1. Examples ofDu Shu Hui, prospective secondary teachers

^s^Activity

Du Shu Hui for exams

Teaching dem-
onstration

Designing in-
struction units

Advanced
mathematics

Solving peda-
gogical mathe-
matics problems

Goal

To pass teacher
selection

To join
graduate study

To pass
teacher selection

Members

Duration

September 2006
-June 2007

December 2006
-March 2007

November 2006
-March 2007

March 2007~on
going

Frequency

1 weekly meet-
ing (3-4 hours
each)

Saturday mor-
ning biweekly
(3-4 hours
each)

2 meetings
weekly (3-4
hours each)

A whole day per
week

Mode

Took turns to
practice teach-
ing (15
min/person) +
commented on
others' demon-
stration

Discussed over

their pre-
written instruc-
tion units

- divided the
textbook content
by schedule

- took turns to
lecture

- collected hard
questions and
solved by a par-
ticular member

- focused on
doing test ques-
tions before the
exam

- do past exam
questions (2-3
exam papers) dur-
ing each meeting

- (20 min. demon-
stration+others
comment) * 4

- discuss the tough
questions assigned
to the gifted stu-
dents

Outcome

All passed

3 joined graduate

study; 1 became

a teacher

will take exams in
2008

In the Du Shu Hui for advanced mathematics, the four members started to take
turns to give lectures. But gradually most of tough problems in the textbook were
collected to be solved by one of the members. He became a preceptor for the com-
munity. He himself expressed that: "It is only by explaining the texts to others in
detail does one know where his/her weaknesses lie. Other people's criticisms and

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

opinions help one to find misunderstandings of texts. This is the best part during
the discussions."

Three of them passed the entrance exam for mathematics graduate institute, the
other one passed the exam for teacher selection. Two members were interviewed
and expressed that: "By attending this learning community, the members have
more incentives to study because they oversee each others' progress - one would
be urged to study when seeing others do so." The advantage of keeping the com-
munity small is: "They could hear their peers share any mathematical thoughts.
They were impressed by some of the brilliant thoughts."

Examples of Learning Communities of Prospective Elementary Teachers

After interviewing prospective mathematics teachers from two education colleges
that educate elementary school teachers, it is found that learning communities in
these schools are not as common as in NTNU. It is worth investigating whether
these students are less keen on taking the exams due to low passing rates (1% to
2%) or the university traditions (10 years ago prospective teachers from these col-
leges were assigned teaching positions after they graduated). However, 30 of the
graduates (about 5% of the total graduates) from National Taipei University of
Education, one of the two universities interviewed, passed the examination for
elementary teacher selection of year 2007. There were about 300 vacant positions
in 2007. NTUE had a celebration for the graduates' "good performance". Some of
the students of NTUE said that they joined a school club named "Math Camp". The
objective of Math Camp is that during summer and winter vacations the partici-
pants go to elementary schools in remote areas to provide social service for the
students there. The organization and operation of Math Camp basically follow
Krainer's definition of learning communities (2003). The following are brief re-
ports of the interviews.

In Table 2 is a brief description of two learning communities of prospective ele-
mentary teachers: Math Camp and Du Shu Hui for enhancing understanding
mathematics. We will further describe some specific features of those two commu-
nities as follows.

In the Math Camp community, prospective elementary teachers are practising
school organizational operation. Members are put into groups of five to ten and
organized in structure similar to a school administrative system. Particularly, each
member is designing mathematics activities. During community meetings, they
discuss over each member's design of activities. Those activities intend to imple-
ment to an elementary school for the children there during summer and winter vo-
cation. The schools chosen for running a math camp are in remote districts. A
group of them reflected that

By trying to design various mathematical learning activities, we acquire
knowledge that is not in the elementary mathematics teacher education pro-
gram. Since the camp is held in different places every year, we learn about
regional differences. For example, we have seen the cultural diversities in the

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FOU-LAI LIN AND JOAO PEDRO DA PONTE

students of Hakka village schools and coastal Min-Nan schools. These diver-
gences contribute to the differences in students' response in learning. This
helps us know that it is necessary and important to teach students in accor-
dance with their aptitude.

Table 2. Examples ofDu Shu Hui, prospective elementary teachers

^sActivity

Du Shu Hui

Math camp

Enhancing mathematics under-
standing

Goal

To learn mathematics teaching, ob-
tain collaboration experiences, and
enhance member-to-member rela-
tionships

To enhance undergraduate
mathematics understanding

Members

5- 10 per group
(to recruit new members yearly)

11
( 1 sophomores + 1 junior)

Duration

All the year round

September 2007-
(during school terms)

Frequency

Fixed time, once a week

3 hour-meeting, once a week

Mode

- hold mathematics camps for ele-
mentary school children during
summer and winter vacation .

- activity design, activity execution,
and novice member training

- the junior takes position as the
instructor; the sophomores dis-
cuss over his or her instruction

Outcome

- members relationship enhanced

- knowledge in the non-elementary
mathematics curriculum acquired

- cultural diversities learned

- the junior: the only one to pass
the first quiz

- concentration on studying raised

The Du Shu Hui for enhancing undergraduate mathematics understanding, ten
sophomores and one junior mathematics education students have participated. The
community meeting have developed in the way that the junior student takes posi-
tion as the instructor which others discuss over what he or she speaks. It seems that
an instructorship is often developing whenever a community is focused on ad-
vanced mathematics. The junior student responded to the interviewer that

I have to go through the texts and questions in advance and then teach the
others during each meeting. While I teach 1 examine whether I was on the
right track. I have gained more than others during the process (teaching be-
comes learning). Among the members 1 was the only one to pass the first
quiz.

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

Furthermore, he expected that "Hope that the sophomores would go over the
questions before each meeting, otherwise they would not have much progress."

A CASE FROM PORTUGAL

Teacher Education in Portugal

The political and social situation of Portugal bears some resemblances but also
important differences to that of Taiwan. Like Taiwan, in Portugal the population is
aging and declining and every year the need for teachers decreases, both at the
primary and secondary school levels. Recently, there was a national effort to uni-
versalize education and thus new institutions were created to supply the necessary
teachers; there was a shortage in many school subjects, including mathematics.
Teaching was then became an attractive career, as it not only provided a reasonable
salary but also had unique features of flexibility, reduced weekly schedule and ex-
tra holidays in Christmas, Easter and Summer. All this changed dramatically in the
last few years. The shortage is over and now there is a large surplus, with many
unemployed teachers that can do not find a place in schools. The weekly schedule
was extended and teachers are now required to participate in school activities even
when students are absent. The government has announced an external mathematics
exam to ascertain if the teachers to be recruited have the necessary mathematical
competence. The Government also decided to adopt the Bologna framework, 1 and
revised the structure of teacher education. Programmes to prepare school teachers
take five years (for secondary school candidates) or four and a half years (for pri-
mary) to complete and teachers now get a professional masters' degree. Profes-
sional practice (or practicum) in schools was reshaped to consist of about three-
fourths of a year from one full year) but prospective teachers now have to write an
extended report to conclude their study.

These changes are taking place at this moment in time; thus it is too early to
know what the effects will be. So far, the main noticeable effect is a decrease in the
number of teacher candidates, especially at the secondary school level - due to the
large number of unemployed teachers and also the long period of time that it took
the government to certify the new study plans that were submitted by all teacher
education institutions.

The Teacher Education Programme of the Faculty of Sciences, University of
Lisbon

The
ences

prospective mathematics teacher education programme of the Faculty of Sci-
s of the University of Lisbon prepares for teaching at grades 7 to 12. This five-

The Bologna framework is a movement of general reform in higher education in Europe, also includ-
ing some non-European countries, aiming to promote student mobility, programme comparability and
renewal of teaching and learning processes.

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FOU-LA1 LIN AND JOAO PEDRO DA PONTE

year programme, as it happens in other countries, has a three-stage model: (i) Dur-
ing the first three years, prospective teachers follow scientific-oriented courses
(covering the standard branches of pure and applied mathematics, with emphasis in
advanced algebra and infinitesimal analysis); (ii) In the fourth year they take edu-
cational courses, some addressing general educational issues (pedagogy, psychol-
ogy, sociology, history, and philosophy of education) and some others dealing with
mathematics education issues (mathematics curriculum, instructional materials,
classroom work, assessment, and teaching number, algebra, geometry, statistics
and probability); and (iii) The most important part of the fifth year is a supervised
practicum in a school. During the first three years, with only a few exceptions, pro-
spective teachers take the same courses as pure mathematics majors. During the
fourth year, the programme seeks to provide prospective teachers with theoretical
frameworks to analyse educational issues with special attention to current prob-
lems, and to provide the essential elements to plan and carry out the daily activity
of a mathematics teacher. It also puts prospective teachers in contact with educa-
tional practice through two fieldwork courses (one in each semester). The fifth
year, the programme includes a year long practicum, during which prospective
teachers are responsible for teaching in one class and become progressively in-
volved in all aspects of the professional activity of a mathematics teacher. The pro-
gramme still includes other elements that complete the educational, scientific, cul-
tural, and ethic preparation of prospective mathematics teachers. Elements for the
following examples were drawn from different studies carried out by the second
author, as part of a research programme in teacher education.

Face-to-Face Learning Communities in Regular Courses

Within this programme, prospective teachers have several opportunities to consti-
tute face-to-face learning communities. Each course, during the first four years,
either in mathematics or in educational subjects, has made such possibility implicit.
Often, prospective teachers create informal groups to study, to work on assign-
ments, or to carry out more extended tasks. In the fourth year, most education
courses explicitly encourage these groups and some of them constitute rather stable
communities of prospective teachers that tackle in turn the tasks related to different
courses. One of the most interesting opportunities for prospective teachers to con-
stitute learning communities is provided in the fourth year by the fieldwork course
Pedagogical Actions of Observation and Analysis. 2 In each semester, the course
runs for 3 hours a week, providing an opportunity for prospective teachers to re-
flect about educational phenomena based on school observations. The aim is that
they begin to regard these phenomena from the point of view of the teacher and to
develop their capacity of observation and reflection about educational situations.
Contrary to all other disciplines of the programme, this one does not have a fixed

In Portuguese, AccSes Pedagogicas de Observac3o e Analise (APOA).

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curriculum. Its activity is mostly based on observing and reflecting about observa-
tions and is jointly undertaken by prospective teachers and instructors. Given the
nature of the work, the classes have between 12 and 16 prospective teachers. Group
work - involving usually three or four prospective teachers - is the most common
working pattern used all through the year. Visits to the school are first prepared,
then carried out, and later discussed in classes at the university. Prospective teach-
ers present in class the results of extended observations on issues of their choice.
The most usual form of classroom interaction is informal discussion with active
participation by prospective teachers. The role of the instructor is to propose tasks
and to lead discussions. Each semester ends with presentation and discussion of
projects (in oral and written form).

The group of 12 to 16 prospective teachers, together with the instructor, consti-
tutes a learning community. Several factors contribute to that. The fact that there is
no prescribed curriculum enables that the planning of work be carried out in a
flexible way with the contribution of all participants. Since the activity extends for
a full academic year, there is plenty of time for the participants to get to know each
other and to adjust to the working requirements of this course. Most of the prospec-
tive teachers experienced working in small groups from the time when they were in
high school. They now come back to this kind of activity for which most of them
adjust rather quickly. The balance between the moments of working as a whole
group and in small groups of 3-4 elements, all with their own more specific divi-
sion of labour, has proved adequate for carrying out tasks such as observing, re-
cording data, analysing observations, and reflecting.

A study by Ponte and Brunheira (2001) indicates how some prospective teachers
regard this activity:

While I circulated in the corridors, among pupils, teachers, and staff, I had the

opportunity to look at things differently and see things that I had never noticed

before. (Beatriz)

This visit [to the school] [...] now made me enter a world that I already knew,

but with other eyes, in another role, a little [as I will do] in the future as a

teacher. I no longer felt like a pupil although I [still] do not feel like a teacher.

(Ana)

The observation allowed us to look at the classroom in a completely different

way, a "teacher's" look. It was there that we began paying more attention to the

type of class, to the physical conditions, to the teacher's methodology, to the

pupils' reactions. (Eduardo)

Although I left secondary school 5 years ago I can already see that it went

through great changes [...]. (Dora)

The practicum in the fifth year constitutes another important event in this pro-
gramme. The prospective teacher, together with one, two, or three other colleagues
is assigned to a school. He or she is responsible for teaching two classes, and at the
same time participates in seminars and other activities with his/her supervisors
from the school and the university. In each school, a micro learning community is

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FOU-LAI LIN AND JOAO PEDRO DA PONTE

formed by this small group of prospective teachers and the school supervisor. The
university supervisor is not present on a daily basis but tends to become more of an
external consultant. The practicum plays an essential role in developing the profes-
sional competencies of the teacher candidate and in supporting the construction of
his or her professional identity, and promoting a reflective and active professional
attitude.

The different practicum groups, together with their supervisors, meet regularly
about once a month. This large group (its size may vary from 30 to 50 or even 70
participants) plans the programme of work for the year, discusses issues that
emerge from the activity of the practicum groups, and invites outside experts
(sometimes a secondary school mathematics teacher) to carry out seminars on spe-
cific topics. The most important activity is the meeting organized for the end of the
year, with a format similar to that of a mathematics teachers' professional meeting,
in which prospective teachers present to each other some aspects of their work,
through posters, oral communications, and workshops.

The practicum group constitutes another kind of learning community, especially
when there are collaborative relationships between the prospective teachers and the
supervisors that provide the appropriate challenge and support. For many prospec-
tive teachers, the practicum is the most important element in their teacher educa-
tion programme; this is understandable, since it provides a confrontation with the
reality of practice, requiring the mobilization, revision, and integration of previous
knowledge developed in separate experiences and leading to the development of
new practical knowledge necessary to conduct the professional activity. One pro-
spective teacher, Nelia (the study is described in Ponte et al., 2007), indicates that
one of the aspects that contributed to the success of her practicum was the fact that
she already worked with her two colleagues in many university courses. She also
indicates that at her school there were two other mathematics practicum groups and
there was a good collaboration among all of them.

An Informal Face-to-Face Learning Community Setting

Nelia reports that, as a prospective teacher in the last two years of her studies, she
participated in a research project in mathematics education. This project, conducted
by a university mathematics teacher educator, involved practising and prospective
teachers as well as a large group of doctoral students. Overall, the project had
about 40 members organized in different sub teams that provided intensive activity
of planning research studies, collecting data, presenting seminars, writing and dis-
cussing papers; thus this was another important learning community in Nelia's pro-
fessional journey:

In this project we have several themes, we analyse several things. Besides
maintaining a contact with on-going research [...] it is an opportunity to talk
to people with rather different backgrounds. Therefore, we have a meeting
once a month and in those meetings we may share the experiences concern-
ing classroom practice as well as concerning research. Therefore we may al-

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

ways talk to somebody, share our questions with somebody, know what the
others are doing, what questions they also have, and this keeps my inquiry at-
titude active, that leads me searching for more things [for my practice].

It is interesting to note how this project had a profound impact on this prospec-
tive teacher. The formal activities of the project were important, as were the infor-
mal contacts with other members and especially the work that she carried out in
collaboration with her close colleagues.

In summary, this mathematics teacher education programme provides several
opportunities for prospective teachers to participate in different face-to-face learn-
ing communities. These communities vary in size, from very small to rather large
groups, vary in the intensity of their activity, and also vary in the extent to which
they support prospective teachers' learning. The most successful learning commu-
nities seem to be those that combine some sort of formal aims and structure with a
significant flexibility in carrying out the activities and involve different levels of
working together from small groups for undertaking specific tasks to large groups
for sharing and discussing more general issues.

CONCLUSION AND DISCUSSION

In order to make sense of learning communities for prospective mathematics teach-
ers, the outer contexts and inner issues need to be considered. In the following, we
focus on these two aspects separately.

The Outer Contexts of Face-to-Face Learning Communities

A learning community often is rooted meaningfully in its outer context. On the one
hand, different functions of communities may be established with respect to differ-
ent relations of supply and demand of beginning mathematics teachers within a
society. Popularly established Du Shu Hui with the goal of preparing for examina-
tions in the case from Taiwan resulted from the outer context of a supply of teach-
ers far greater than available teaching positions. However, whenever the supply is
less than the demand, establishment of a face-to- face learning community often is
created by institutional arrangement akin to a team and then transformed into a
community. The case from Portugal provided an example of learning communities
that were organized along such approach. An educational system in which an over-
supply of teachers exists, prospective teacher education programmes are naturally
extended to include the need for students to perform well on examinations for
teacher certification and teacher selection. Various types of learning activities may
take place from each component in such a programme: participating within a learn-
ing community is one such activity. Examples of learning communities taking
place in regular courses, research projects and teaching practices are shown to be
major activities as in the case from Portugal. Thus, the two cases, Taiwan and Por-
tugal, provide a diversified set of examples of learning communities with regard to

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FOU-LAI LIN AND JOAo PEDRO DA PONTE

the outer contexts of the different relations of the societal need for teachers, and the
components in a broader sense of prospective teacher education programme.

The inner Issues of Face-to-Face Learning Communities

Four inner issues for learning communities were identified by Ponte et al. (in
press) at the beginning of this chapter. However, a more fundamental principle is
that, in both the cases from Taiwan and Portugal, apart from each of distinct goals
for learning, were similar learning communities in the sense that the prospective
teacher participants were learners that learned from one another. The following
provides a synthesis from the examples of the learning communities given by the
two cases mentioned according to the four inner issues.

The first issue is the purpose of the group and its relation to the personal pur-
pose of its members. The cases of Taiwan and Portugal both illustrate that the aim
of the learning communities is for the professional learning of prospective teachers
by generating and accumulating knowledge about teaching and teacher education,
as is indicated by the model proposed by Hiebert et al. (2003). The members of the
learning communities in these cases did participate in framing their tasks. This is
clear in the case of Taiwan. In the Portuguese case, prospective teachers from the
field-based course Pedagogical Actions of Observation and Analysis learned
through the process of observing and reflecting on educational situations, as well
during their practicum activities and in the research project. In all cases the tasks
were negotiated by participants with teacher educators or project leaders.
The second issue concerns the knowledge that develops from the activity of the
learning community based on shared practices. Learning activities within a learn-
ing community can be sequentially separated into three phases: entry, interacting
and reflecting. Professional knowledge of teaching can be distinguished into prac-
tice knowledge and thought knowledge. Practice knowledge is context-dependent
and has to be gained from real teaching practice (Goffree & Oonk, 2001). Thought
knowledge is revealed when one is thinking about teaching without facing stu-
dents. At the entry phase of learning communities provided by the Portuguese
field work course, participants focused on practice knowledge. In Taiwan, at the
entry phase of Du Shu Hui for examinations, participants designed instruction units
and demonstrated teaching which focused on thought knowledge. To what extent
that different entry knowledge might influence the knowledge that is generated and
accumulates during interacting and reflecting phases is worth further investigation.
For example, at the entry phase, participants in the Math Camp designed various
mathematical learning activities for implementing in a school; thus they focused on
thought knowledge. Participants reflected on the fact that they saw the cultural di-
versities in students from different cultural backgrounds. These variations contrib-
uted to the differences in students' responses in situations of school learning. The
knowledge participants of Math Camp generated was "local knowledge of teach-
ing" that could not be generated outside the practice (see e.g., Krainer, 2003, p.
98). The prospective teachers indicated from these learning communities that they

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FACE-TO-FACE COMMUNITIES OF PROSPECTIVE TEACHERS

acquired knowledge that was not included in the elementary mathematics teacher
education programme.

The third issue concerns how learning happens in the group. This crucial issue
is somehow related to the fourth issue that concerns the roles and relationships of
the participants. The roles and relationships of participants in a community might
change simply by making adjustments in their operation. Such changes are neces-
sary in order to meet the members' interests and learning needs. Let us regard, for
example, the Du Shu Hui for Advanced Mathematics and the Du Shu Hui for En-
hancing Mathematics Undergraduate Understanding: Each of these two learning
communities had one member that had a greater mathematical competence than the
others. Both began with equal competence among the participants such that they
criticized each other and helped each other solve problems, but evolved to a learn-
ing activity which was dominated by a student superior in mathematical compe-
tence taking over as the leader. This is most evident in the Du Shu Hui for Enhanc-
ing Mathematics Undergraduate Understanding. Among the members, the junior
student played the role as an instructor. What he learned during the process of in-
struction is that he became aware of what he did not understand, which he then
discussed with other members. This is how learning happened in a community with
a definite leadership.

In the case of Portugal, the learning communities were study groups of school
courses and a research project. The prospective teachers in these communities were
learning through an open and flexible discourse among them. For example, the
study groups in the field-based course had no fixed curriculum, enabling a flexible
planning of work with the contribution of all participants. And as the prospective
teacher who participated in a research project described there was plenty of oppor-
tunity to talk to others, know what they were doing, learn about their questions, and
thus cultivate an inquiry stance.

Learning may happen through reflecting on critical friends' comments in the
interacting phase of a community's meetings. In the Du Shu Hui for examinations,
prospective teachers started to learn together collaboratively. During the inter-
views, some prospective teachers expressed that the closer to the days of examina-
tion for teacher selection, the participants became aware that they were indeed
competitors for the sparse number of teaching positions. However, they further
expressed that this competitive relationship among participants did not change the
operation within the learning community, particularly, the role of being a critical
friend to one another on teaching demonstrations and in the analysis of instruction
units. This regulation of operation showed that the relationship of critical friends
between participants remained in the Du Shu Hui despite the competition for lim-
ited teaching positions.

ACKNOWLEDGEMENTS

The first author thanks Yuh-Chyn Leu, Yuan-Shun Lee and Wen-Xiu Xu for con-
ducting interviews and Yu-Ping Chang and Jia-Rou Hsieh for preparing the manu-
scripts of cases from Taiwan.

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Fou-Lai Lin

Department of Mathematics

National Taiwan Normal University

Taiwan

Joao Pedro da Ponte

Departamento de Educacao da Faculdade de Ciencias

University of Lisbon

Portugal

129

SECTION 3

COMMUNITIES AND NETWORKS

OF MATHEMATICS TEACHERS

AS LEARNERS

STEPHEN LERMAN AND STEFAN ZEHETMEIER

6. FACE-TO-FACE COMMUNITIES AND NETWORKS
OF PRACTISING MATHEMATICS TEACHERS

Studies on Their Professional Growth

In this chapter we examine the research on and by mathematics teachers working
together on their practice. We examine both groups of teachers within a school,
that we are calling face-to-face communities, as well as wider networks across
schools, in regions or even national initiatives. We give examples of research and
we raise a range of issues that call for consideration when embarking on the use of
such networks and communities. Of particular concern is the sustainability of such
work beyond the engagement of teachers in research or professional development
activities. By "sustainability" we mean the lasting continuation of achieved
benefits and effects of a project or initiative even after its termination (DEZA,
2005). We draw on the literature and our own experiences to suggest critical
factors that may lead to sustainability. It may be the case, we suggest, that teachers
need to engage in some type of research mode with issues that face them on a
regular basis. The difficulties of sustainability might suggest more systematic
support at state level for learning communities engaging in such activity in every
school and across school networks.

INTRODUCTION

Researching practice in teacher groups within a school, when this takes place, may
take a number of forms and serve different purposes and goals. It is perhaps not
unusual to find mathematics departments in high schools wanting to analyse their
student achievement data to identify areas for development. There are likely to be
experiments with new resources that may be examined systematically. In general,
though, we must say that teachers in most countries at all levels are under
increasing pressure of expectations and demands, with performance management
criteria, targets for students' achievements and other constraints. Time for
systematic research by teachers is difficult to find, even when the will is there. It is
therefore particularly important to identify what has and is being done in this area
in order to inform others of the feasibility of such work and to provide some
evidence of teachers' experience in researching their practice as members of a
community.

In this chapter, we will examine the literature on communities of practising
mathematics teachers to identify the kinds of organisational structures that have
been established, the issues that have been identified for development, the aims of

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 133-153.
© 2008 Sense Publishers. All rights reserved.

STEPHEN LERMAN AND STEFAN ZEHETMEIER

the initiatives, methodologies, and outcomes. Sustainability is a key issue in such
initiatives, particularly where the initial motivation is for higher study or driven by
a research project rather than the more intrinsic goal of improvement of teaching
and learning per se. We will also discuss the sustainability of professional
development projects and identify communities and networks as key factors
fostering the long-term impact of such initiatives.

WITHIN-SCHOOL COMMUNITIES

Perhaps the most extensive framework for the study of teaching and learning is the
Lesson Study (see also Yoshida, volume 1 ; Nickerson, this volume) which has been
implemented in Japan for many years as a method for teachers' professional
development and taken up more recently in the US (Stigler & Hiebert, 1999; Lewis
& Tsuschida, 1998; Fernandez & Yoshida, 2004; Puchner & Taylor, 2006). There
are also cases in which Japanese teachers provided support to US teachers to
engage in Lesson Study (e.g., Fernandez, Cannon, & Chokshi, 2003). Briefly,
lesson study involves a group of teachers in a school analysing in great depth how
to improve specific lessons within particular topics, such that over a period of time
the way concepts are taught changes. Initial development of a lesson will typically
be followed by one teacher teaching the revised lesson, observed by the other
teachers. They meet subsequently to review what happened, make further changes,
and possibly teach the re-revised lesson again.

Other studies in the US (see also e.g., Peterson, 2005) have investigated the
impact of Lesson Study on prospective teachers and indicated that this can support
growth in understanding of what to teach which can, in turn, lead to growth in
understanding of how to teach (Cavey & Berenson, 2005). Puchner and Taylor
(2006) found that the Lesson Study work can have an impact on teacher efficacy
since teachers have the potential to discover through their lesson planning that their
work does have an impact on their students' engagement and learning activities in
class. In a major review of the potential of lesson study to contribute to
instructional achievement Lewis, Perry, and Murata (2006, p. 3) suggest that we
have few examples of full Japanese lesson study cycles on which to base research
and development:

Yoshida's (1999) dissertation case of mathematics lesson study in a Japanese
elementary school (which formed the basis for Stigler's and Hiebert's chapter
on lesson study and is now available in Fernandez & Yoshida, 2004); and a
case of science lesson study in a Japanese elementary school [...].

Lewis, Perry, and Murata (2006) call for more detailed exemplars, which can
help us try to understand the mechanisms and cycles of design-based research. The
bibliography of review indicates, however, that whilst these deep studies are not
yet available, a great deal of work is taking place using Lesson Study, in
mathematics and other curriculum subjects.

There is a long tradition of action research in education amongst practising
teachers (see Krainer, 2006a; Benke, HoSpesova, & Ticha, this volume) with the

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aim of improving one's own teaching, drawing inspiration from Lewin (1948) and
SchSn (1983). The goal of action research is to bring about change in one's own
setting, be that classroom, department or indeed whole school (Stenhouse, 1975;
Elliott, 1978). Generalisability is not a major concern, site-specificity is recognised
as what matters. As such, the action research movement, which "can be traced back
to Stenhouse's (1975, p. 142) reconceptualisation of curriculum development from
an objectives to a process model" (Adler, 1997, p. 88) saw itself as, first, making
educational research relevant to the classroom and, second, as giving teachers a
voice in research, indeed the voice in research, as teachers were claiming power
over what was researched and by whom, and the right to make the decisions over
what the research was intended to change. It was, for some teachers, a response to
the university researcher who came into their classroom, recorded some data, and
then went away to publish in journals which no teacher read, in the furtherance of
their career. There are strong links between action research and reflective practice
(Leitch & Day, 2000), whereby in the act of teaching one reflects and acts, usually
unconsciously, called reflection-in-action, and later perhaps one reflects on action
(Sch5n, 1983), this latter leading to a recognition of the need to change something,
coming therefore into line with action research. One might describe action research
as reflection-o/j-reflection-0/7-action. It is intentionally set against an instrumental
or technical view of action research. A search of the leading relevant journal
Educational Action Research reveals very few studies in mathematics education
although the approach is known within the community (e.g., Crawford & Adler,
1996; Lerman, 1994). The two articles in the 2006 issue of the Journal of
Mathematics Teacher Education by Goodell (2006) and Stephens (2006) draw on
reflective practice and action research in their studies of prospective teachers'
learning and their own practice, indicating that teacher educators can find action
research a fruitful research orientation to support prospective teachers' learning.
Goodell refers to the approach she took to her own practice as a teacher educator as
self-study but this also has very similar features to action research, this latter term
being more present in the mathematics education research literature (Schuck, 2002;
Dinkelman, 2003). Stephens (2006) demonstrates the role of prospective teachers'
reflections on their students' learning for developing awareness of their own
mathematics.

To return to the way in which teachers' action research can be more than
instrumental or technical, the revolutionary potential of action research was taken
up by Carr and Kemmis (1986). They used Habermas's (1970) three-fold
classification of social action: technical; practical; or emancipatory and they
emphasised the need to take an emancipatory approach, which called for collective
action to change education. However, action research is today often motivated by
higher study goals or it may be instigated by engagement in research projects,
usually in collaboration with university researchers. The sustainability of action
research beyond the course of study or after the researchers leave, or if the funding
runs out is clearly an issue, one we take up later in this chapter.

We must note here also the external drivers for face-to-face researching
communities. We have indicated that the regulatory systems of education in many

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STEPHEN LERMAN AND STEFAN ZEHETMEIER

countries have restricted teachers' freedom to engage in research in their
departments on their pedagogy, curriculum, assessment or other aspects.
Conversely, that same regulatory system in many countries (e.g., UK, the
Netherlands) is requiring schools to produce development plans for school
improvement and in most cases preparing those plans and acting on them devolves
to departments. This can be an opportunity for subject-specific quality
development and quality assurance that will have to be taken on board and acted
upon by senior management in the schools. We report below on some research in
this area. It may well be that a community may develop from the work called for
by the regulatory system.

In general, we might note that cooperation between mathematics teachers for the
improvement of their pedagogy and their students' learning is possibly less
common than in other school subjects because of the tendency to teacher autonomy
(Lortie, 1975), particularly in mathematics. However, Visscher and Witziers (2004)
examined the performance of school students in mathematics differentiated by how
cooperatively mathematics departments in those schools operated, using a six-point
scale for assessing the form and degree of cooperation. "A positive relationship
was found between departmental policy, on the one hand, and student achievement
on the other" (p. 796).

EXAMPLES

To this point, we have set out some theoretical considerations in relation to
teachers working together in face-to-face communities and networks to change
mathematics teaching and learning. In this section we will present some examples
of within-school research and across-school projects on mathematics teaching and
learning. It is our intention to select examples that highlight key features of
successful groups, "success" being defined locally, that is, by the participants
themselves. It seems to us that it is in the spirit of the theme of this chapter that we
do not judge particular initiatives by objectively developed measures, if such
measures could be developed at all, but by the subjective evaluations of the
participants, both university staff and teachers. We will "collect" together the key
features from each of the examples and, following a review of the literature and of
other initiatives, we will present them all, emphasising what, for us, are the most
critical factors that enable the success of any programme to be sustained beyond
the end of the specific project, namely the establishment of communities or
networks.

In the first three examples, all of face-to-face communities, we have chosen
ones that among them cover the main possibilities in terms of initiation of such
networks. The first example was the initiative of a group of teachers in a school,
the second is a mutual collaboration between university researchers and school
teachers, and the third is the initiative of university researchers. Our fourth
example sets the scene for the study and analysis of across-school networks which
follows.

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Example 1

Arbaugh (2003) describes her work with a study group. A study group is defined as
"a group of educators who come together on a regular basis to support each other
as they work collaboratively to both develop professionally and to change their
practice" (p. 141). A mathematics department chair in a US high school contacted
the university researcher, Fran Arbaugh, for support in the department's
development of a way of teaching geometry that was more student-centred and
inquiry-based than their current practice. Over a period of six months the group
met ten times, drawing on what is called the Mathematics Task Framework (Stein,
Smith, Henningsen, & Silver, 2000) which "focuses on the levels of cognitive
demand required by mathematical tasks and the various phases tasks pass through
in their instructional use" (p. 142). The seven teachers who participated throughout
reported that they valued the development of community and relationships that
supported their learning. They particularly benefited from sharing experiences
about aspects of their teaching, such as managing whole-class discussion, and
appreciated the way the group worked, whereby no one person dominated, as is
usually the case in other forms of education for practising teachers. They talked of
the opportunity to think things through, to question and to experiment. They all
spoke about how the study group enabled them to bring the theories, as used in the
research community and practice together, facilitated by the research articles
provided by Arbaugh. There is clear evidence of curriculum reform and change and
a growing sense of their professionalism. One teacher commented (Arbaugh, 2003,
p. 153):

I'd not spent a lot of reflection time before. When I did, I mainly thought of
how the kids could learn better. Now I look at things I could do to help them
learn better. I look more at how I can create an opportunity for them to learn.

Key features of the success of this activity, according to Arbaugh, were the
financial support that enabled the teachers to be free from their teaching to hold the
meetings and the fact that the teachers were all from the same school. Arbaugh
highlighted the tension for herself between being the expert needed to introduce
new content and the autonomy and empowerment that teacher groups needed to
really benefit from working within a community. Arbaugh notes that the group
continued alone for a further year to work on algebra. We find it interesting that
there is no mention of action research in the article, even though the research seems
to have followed that approach. Features of the activity not mentioned specifically
by Arbaugh but important for this review are the focus on content-specific material
and the opportunity for addressing pedagogical content knowledge.

Example 2

We have referred to Japanese Lesson Study above. A modified version, called
Action Education, has been developed and used in China (Huang & Bao, 2006).
These authors, in designing their approach, argued that the established and well-

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known benefits of Lesson Study can be, and need to be, enhanced by the
participation of experts who can provide input on new content and on pedagogic
issues, whilst the control of what is changed and improved in the mathematics
lesson remains in the hands of the teachers. In their model, the process begins with
a teacher teaching a lesson, with a focus on an issue that requires reflection and
examination, in front of a group of other teachers and experts (Huang & Bao, 2006,
p. 280):

A fundamental feature of "Action Education" is that the unfolding of the
program is mediated within the community by the whole process of
developing an exemplary lesson (Keli), including the lesson planning, lesson
delivery and post-lesson reflection, and lesson- re-deli very [...] a
collaborative group (the Keli group) that consists of teachers and researchers
is established through discussion between researchers and a group of
interested teachers.

Huang and Bao locate their approach within the literature on action research. It
is a requirement of the whole learning process that the teacher group writes a
narrative article that summarises their experiences, the changes, and their findings.
The authors give an example of work on teaching Pythagoras' Theorem in which
the group was formed from teachers in the school and a group of researchers from
the local university. The teachers were aware that using an inductive approach to
the proof tends to fail due to the problems of measurement, whilst expecting
students to find a proof for themselves is unreasonable. They designed a series of
lessons around the following diagram and the associated proof of Pythagoras'
Theorem:

Figure I. Diagram for the proof of Pythagoras' Theorem.

Using a series of worksheets building from numerical examples to the general,
students' learning was scaffolded to a proof of the theorem.

The findings of the study revealed a number of interesting and important
changes. The first to be commented upon in the article is the change in what Huang
and Bao (2006, p. 290) call the teaching paradigm:

Through a comparison of the time distribution between the previous lesson
and the revised lessons in terms of teacher talk, teacher-student interaction,
student exploration and student practice, it was found that the time for teacher
talk, and student practice went down from 51.2% to 26.7%, and from 28.2%

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to 3.2%, respectively, while the times for teacher-student interaction, and
student exploration went up from 16.8% to 23.5% and from 3.8% to 46.6%,
respectively.

They argue also that students learnt not only the proof but also why it constitutes
a proof. They go on to illustrate the teachers' reflections and hence their learning
through Keli. In their conclusions, Huang and Bao (2006, p. 295) point to some of
the problems faced by this approach to development for practising teachers:

[...] where do the qualified experts come from? How to organize a practice
community including teachers and researchers? How to re-schedule the
teaching program for doing Keli without disturbing the normal teaching
program? How to simplify the Keli model for use by teachers, so that the
phases of the model can be easily understood and implemented effectively by
teachers?

They also emphasise the importance of the role of the expert:

It also shows that the expert's roles and follow-up action are important for
participating teachers' professional learning and changes in classroom
practice (p. 295).

Key features, beyond those mentioned by the authors, are the focus on both
content and pedagogical knowledge and skills and an open, learner-centred
implementation component.

Example 3

Drawing on a situated theory of learning, Kazemi and Franke (2004, p. 205)
describe the work of a group of teachers across a year examining their students'
work in an attempt to develop their understanding of their students' mathematical
thinking. They work with a notion of transformation of participation in studying
the change in the teachers over that time.

The transformation of participation view takes neither the environment nor
the individual as the unit of analysis. Instead, it holds activity as the primary
unit of analysis and accounts for individual development by examining how
individuals engage in interpersonal and cultural-historical activities.

The intervention was the initiative of the university researchers and was based
on pilot work using the Cognitively Guided Instruction (CGI) approach (Carpenter,
Fennema, Franke, Levi, & Empson, 1999). The researcher, called facilitator in the
paper, worked with the teachers to choose a common problem, modified by each
teacher to suit her or his students. The group then met to discuss the strategies their
students had used, and what those strategies revealed about their students' thinking.
The researcher might add some findings from research, and assist the teachers in
seeing commonalities and differences between their own findings and the research.

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Data were collected of the group meetings and in the classrooms. Kazemi and
Franke (2004, p. 213) highlight:

Two major shifts in teachers' workgroup participation emerged from our
analyses. The first shift in teachers' participation centered around attending to
the details of children's thinking. This shift was related to teachers' attempts
to elicit their students' thinking and to their subsequent surprise and delight
in noticing sophisticated reasoning in their students' work. The second shift
in teachers' participation consisted of developing possible instructional
trajectories in mathematics that emerged because of the group's attention to
the details of student thinking.

As Key Features for teachers' learning, Kazemi and Franke emphasise two
factors in particular as mediators of teachers' learning: the role of the facilitator in
guiding their reflections and appropriate use of the students' work as material upon
which to reflect. Interestingly also, from our point of view, is the indication by the
authors that the initiative was sustained beyond the year of interaction with the
facilitator. From our point of view we would add: the focus on both content and
pedagogical knowledge and skills; an open, learner-centred implementation
component; and the prolonged duration of the activity.

Example 4

Background. The Austrian Ministry of Education put in place the project 1MST
("Innovations in Mathematics, Science, and Technology Teaching"; see Krainer,
2007; Pegg & Krainer, this volume). The project represents a nation-wide support
system in the areas of mathematics, science and technology as well as related
subjects. This project is working on the levels of professional, school, and system
development, integrates systematically the principles of evaluation, gender
sensitivity, and gender mainstreaming, and comprises several central programmes.
One of these programmes is the setting up and supporting of Regional and
Thematic Networks of teachers and schools. While the concept of Regional
Networks is targeted at local and regional level, the Thematic Networks aim for
cooperation between teachers across the whole nation. These Regional and
Thematic Networks were successively established in 2004 and have gained
importance in recent years, as they enable the economical use of available human
and material resources. Beyond this, the establishment of such networks is
expected to facilitate the setting of regional subject-related or cross-subject goals
which direct the support in the teaching of mathematics, science and technology.
Each Regional or Thematic Network is coordinated by a steering group, consisting
of teachers, university staff, school authorities, and further relevant persons
(Krainer, 2005). More than 8000 teachers have already participated in network
meetings.

The implementation of Regional and Thematic Networks aims at three goals: (a)
enhancement of attractiveness and quality of teaching and school development; (b)
professional development of teachers; (c) increased numbers of participating

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teachers and schools by enhancing regional and national communication structures.
The design of the networks is based on several principles: (a) utilisation of existing
human, institutional and material resources; (b) accountability of participating
persons and organisations; (c) target-oriented thinking and acting together with
systematic evaluation (balance of action and reflection); and (d) autonomous
thinking and acting of persons or organisations in close interplay with the shared
goals and principles (balance of autonomy and networking). In cooperation with
school authorities and universities, teachers are implementing innovative projects
aimed at these goals and principles (Rauch & Kreis, 2007).

In the focus: The thematic geometry network Recent technological development
and innovation processes show impact on the teaching and learning of mathematics
as well as on teacher education (see e.g., Llinares & Oliveira, this volume; Borba
& Gadanidis, this volume). For example, the increasing number of accessible
internet resources and miscellaneous packages of CAD (computer aided design)
software are changing the context of teaching and learning geometry. This change
tends to result in several consequences. On the one hand, in Austrian secondary
schools descriptive geometry is not part of mathematics and represents an
independent subject. The development of new technologies leads to fundamental
changes in the subject's curriculum, which in turn have to be implemented by the
teachers. On the other hand, the implementation of these new contents and
techniques requires highly skilled teachers. Even though many secondary
mathematics teachers are highly engaged and innovative, they often lack
knowledge and practice regarding these new issues, simply because in Austria
hardly any descriptive geometry courses for teachers' professional development are
offered.

To address these issues and to support the concerned teachers, the ADG (the
professional association of geometry in Austria) decided to implement action by
establishing the Thematic Geometry Network, a face-to- face network of practicing
mathematics (in particular geometry) teachers and teacher educators interested in
subject-didactics (Gems, 2007). This is seen as a step towards becoming an
Austrian Educational Competence Centre (AECC). So far in Austria six such
centres are established, among these is one for mathematics education. The
geometry network was created in November 2005 and aims at improving
communication between descriptive geometry teachers, quick and direct exchange
of information concerning recent subject-related development, and the organisation
of a database containing professional development programmes and expert pools.
The spirit and guiding idea of the Thematic Geometry Network is characterised by
shared intentions and goals, mutual trust and respect, voluntary participation, and a
common exchange principle.

On a concrete level, the network designs, initiates, implements, and evaluates
various projects. All activities are coordinated by the network's steering group,
consisting of teachers representing participating school types and levels. This
group acts as both an inward central administrative network node (e.g., for

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planning meetings or distributing learning materials), and an outward contact point
(e.g., providing information or handling public relations). The steering group
together with different subgroups on national, regional, and local level frame the
structure of the network (Mailer & Gems, 2006). The Thematic Geometry Network
initiates and organises annual nation wide meetings of geometry teachers: Within
the scope of these "subject didactics days" the teachers and working groups share
information and ideas concerning geometry teaching and learning, curriculum,
prospective teacher training, and professional development.

Various working groups within the network deal with issues concerning
geometry. One particular group prepares the content, curriculum, and
implementation of a modified initial teacher training model. Another working
group develops and provides support and assistance regarding 3D-CAD-software
for students and teachers by designing and offering online tutorials and examples.
A third group coordinates CAD modelling contests for students of secondary
schools on regional and national levels. Yet another working group provides and
maintains a touring geometry exhibition that contains objects, working and
experimental stations, as well as concomitant information and material. In the
future, the Thematic Geometry Network intends, on the one hand, to deepen and
consolidate the actual projects and, on the other hand, to expand its activities by
focussing on issues of competences, standards, assessment, and evaluation. In
particular, by becoming a competence centre (AECC) the network could serve as a
support structure for practicing teachers (Gems, 2007).

There are two Key Features for the success of this network initiative. The first is
the existence of teacher leaders that are willing to overcome the lack of an
adequate support system for teachers' professional development. The second is the
window of opportunity when IMST offered the establishment of Thematic
Networks. In short: It is a combination of internal human resources and external
support.

KEY FACTORS IN FOSTERING SUSTAINABILITY OF PROFESSIONAL
DEVELOPMENT

Rationale for Examining Sustainability

Most reform projects are initiated to enhance the quality of teaching and learning in
the specific setting of schools and across regions and nations. Ingvarson, Meiers,
and Beavis (2005, p. 2) state that

Professional development for teachers is now recognised as a vital
component of policies to enhance the quality of teaching and learning in our
schools. Consequently, there is increased interest in research that identifies
features of effective professional learning.

In particular, the formation of face-to-face communities and networks, as
described in the examples above, is seen as one of the most promising ways to
reach the goal of enhanced teacher quality. In the next section, the following

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questions are addressed: 1) "What kinds of effects on teacher quality are possible?"
and 2) "How can this impact be sustained?"

The expected effects of such projects by both the facilitators and the participants
are not only related to the professional development of individual teachers to
improve teacher quality, but also to the enhancement of the quality of whole
schools, regions and nations. Expected outcomes are not only focused on short-
term effects that occur during or at the end of the project, but also on long-term
effects that emerge (even some years) after the project's termination (Peter, 1996).
The desideratum of all such projects and community building activities for
providing teachers support and qualification is to enhance the learning of students.
As Mundry (2005) states, "We recognize professional development as a tool
focused on improving student outcomes" (p. 2); "Funders, providers, and
practitioners tend to agree that the ultimate goal of professional development is
improved outcomes for learners" (Kerka, 2003, p. 1). This strategy, to achieve
change at the level of students (improved outcomes) by fostering change at the
teachers' level (professional development and community building), is based on
the assumption of a causal relationship between students' and teachers' classroom
performance, "high quality professional development will produce superior
teaching in classrooms, which will, in turn, translate into higher levels of student
achievement" (Supovits, 2001, p. 81). Even though a variety of external factors
influence students' outcomes (e.g., the socio-economic background of students'
parents), the above mentioned hypothesis was verified in several studies such as
Fennema and Loef (1992). Hattie (2003) states, "It is what teachers know, do, and
care about which is very powerful in this learning equation" (p. 2).

Most evaluations and impact analyses of professional development are
formative or summative in nature; they are conducted during or at the end of a
project and exclusively provide results regarding short-term effects. Apart from
these findings which are highly relevant for critical reflection of the terminated
project and necessary for the conception of similar projects in the future, an
analysis of sustainable effects is crucial and the central goal of professional
development; "too many resources are invested in professional development to
ignore its impact over time" (Loucks-Horsley, Stiles, & Hewson, 1996, p. 5). But
this kind of sustainability analysis is often missing because of a lack of material,
financial and personal resources. "Reformers and reform advocates, policymakers
and funders often pay little attention to the problem and requirements of sustaining
a reform, when they move their attention to new implementation sites or end active
involvement with the project" (McLaughlin & Mitra, 2001, p. 303). Despite its
central importance, research on this issue is generally lacking (Rogers, 2003) and,
"Few studies have actually examined the sustainability of reforms over long
periods of time" (Datnow, 2006, p. 133). Hargreaves (2002, p. 120) summarises
the situation as follows:

As a result, many writers and reformers have begun to worry and write about
not just how to effect snapshots of change at any particular point, but how to
sustain them, keep them going, make them last. The sustainability of

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educational change has, in this sense, become one of the key priorities in the
field.

Levels of Impact

When analysing possible effects of professional development, the question of
possible levels of impact arises. Which levels of impact are possible and/or most
important? How can impact be classified? Ball (1995) points out that teacher
educators and facilitators should take a "stance of inquiry and experimentation" (p.
29) themselves regarding these questions of impact. Recent literature provides
some information as to the answers to these two questions; the following levels of
impact are identified (Lipowsky, 2004):

• Teachers' knowledge: This level includes different taxonomies of
teachers' knowledge (e.g., subject knowledge and general and subject-
specific pedagogical knowledge; e.g., Shulman, 1987), or attention-based
knowledge (Ainley & Luntley, 2005), including knowledge about learning
and teaching processes, assessment and evaluation methods and classroom
management (Ingvarson et al., 2005).

• Teachers' beliefs: This level includes a variety of different aspects of
beliefs about mathematics, and its teaching and learning (Leder,
Pehkonen, & Torner, 2002), as well as the perceived professional growth,
the satisfaction of the participating teachers (Lipowsky, 2004), perceived
teacher efficacy (Ingvarson et al., 2005) and the teachers' opinions and
values (Bromme, 1997). Shifter and Simon (1992) highlight that change
of teachers' beliefs is indeed common and desired, but is not necessarily
an accomplished goal.

• Teachers' practice: At this level, the focus is on classroom activities and
structures, teaching and learning strategies, methods or contents
(Ingvarson et al., 2005).

• Students' outcomes: This level is related to professional development's
central task, the improved learning and consequential results for students
(Kerka, 2003; Mundry, 2005; Weiss & Klein, 2006).

Classification and Analysis of Impact

Classification and analysis of impact is based on two major types of effects: short-
term effects that emerge during or at the end of a project; and, long-term effects
that occur after the project's termination. Effects that are both short-term and long-
term are considered by some to be sustainable. However, as Fullan (2006) points
out, short-term effects are "necessary to build trust with the public or shareholders
for longer-term investments" (p. 120). Although short-term effects are important
and it may be that it is only possible to accomplish short-term impact, this does not
provide for sustainable impact and the result would be to "win the battle, [but] lose
the war" (p. 120), because sustainability, in this case, means the lasting

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continuation of achieved benefits and effects of a project or initiative beyond the
termination of a professional development project or effort (DEZA, 2005).
Hargreaves and Fink (2003) state, "Sustainable improvement requires investment
in building long term capacity for improvement, such as the development of
teachers' skills, which will stay with them forever, long after the project money has
gone" (p. 3). Moreover, analysis of sustainable impact should not be limited to
effects that were planned at the beginning of the project; it is important to examine
the unintended effects and unanticipated consequences that were not known at the
beginning of the project (Rogers, 2003; Stockmann, 1992).

Factors Fostering Sustainability

To give an overview and to summarise the literature concerning factors
contributing to and fostering the sustainability of change, the following four
elements of professional development projects are used to classify these factors:
participating teachers, participating facilitators, the project or initiative itself, and
the context that embedded the first three elements (Borko, 2004). Rice (1992)
states that "these factors relate to the nature of teachers as people, schools as
organisations and change processes themselves including many variables which
facilitate or constrain change" (p. 470). Finally, we argue that the core factors
fostering sustainability are community building and networking.

Participating teachers. A professional development project should meet the
teachers' needs and interests (Clarke, 1991; Peter, 1996) and it should be coherent
with teachers' other learning activities (Garet, Porter, Desimone, Birman, & Yoon,
2001), fit into the context in which they operate, and provide direct links to
teachers' curriculum (Mundry, 2005). The teachers should be involved in the
conception and implementation of the project; this allows teachers to develop an
affective relationship towards the project by developing teachers' ownership in the
proposed change (Clarke, 1991; Peter, 1996); it prepares and supports them to
serve in leadership roles (Loucks-Horsley et al., 1996); and it focuses on the
teachers' possibility to influence their own development process (empowerment)
(Harvey & Green, 2000). These features act to facilitate attendance and trust,
which in turn affects the teachers' future decisions concerning their development
process. This ends up in a spiral process, which continually enhances the level of
teachers' empowerment.

An "inquiry stance" is another factor that fosters impact on sustainability
(Farmer, Gerretson, & Lassak, 2003, p. 343; Jaworski, this volume). Teachers
understand their role as learners in their own teaching process and try to
understand, reflect, and improve their practice. This stance requires professional
and personal maturity as well as the possibility to critically reflect one's own
decisions and activities (Farmer et al., 2003). This notion was also used by
Cochran-Smith and Lytle (1999, p. 289) when describing the attitude of teachers
who participate in communities towards the relationship of theory and practice:

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"Teachers and student teachers who take an inquiry stance work within inquiry
communities to generate local knowledge, envision and theorise their practice, and
interpret and interrogate the theory and research of others".

Participating facilitators. Similar to the teachers, are also the participating
facilitators of the professional development programme. They also should take a
"stance of inquiry" (Ball, 1995, p. 29) towards their activities, reflect on their
practice and evaluate its impact (Farmer et a!., 2003). In addition, another
important factor is the facilitators' knowledge, understanding, and whether they
have a well-defined image of effective learning and teaching (Loucks-Horsley et
al., 1996).

Project or initiative. Each innovation, intervention, or project (e.g., the formation
of communities and networks) is unique regarding its design. Therefore, impact
analysis needs to include these differences in organisational and structural
characteristics. Rogers (2003) studied the process of diffusion of innovations and
pointed out that the impact of an innovation depends on several characteristics
(Relative Advantage, Compatibility, Complexity, Trialability, and Observability).
Fullan (2001) also described similar characteristics (Need, Clarity, Complexity,
Quality and Practicality) that influence the acceptance and impact of innovative
processes.

- Relative Advantage: This includes the individually perceived advantage of the
innovation (which is not necessarily the same as the objective one). An
innovation with greater relative advantage will be adopted more rapidly
(Rogers, 2003).

- Compatibility and Need denote the degree to which the innovation is perceived
by the adopters as consistent with their needs, values and experiences (Fullan,
2001; Rogers, 2003).

- Complexity and Clarity indicate the adopter's perception of how difficult the
innovation is to be understood or used (Rogers, 2003) which relates to
concomitant difficulties and changes (Fullan, 2001). Thus, more complex
innovations are adopted rather slowly, compared to less complicated ones.
Clarity (Fullan, 2001) supports reducing complexity and perceiving advantages.

- Trialability means the possibility of potential adopters to experiment and test the
innovation on a limited basis. Divisible innovations that can be tested in small
steps therefore, represent less uncertainty and will be adopted as a whole more
rapidly (Rogers, 2003).

- Quality and Practicality makes an impact on the change processes. High quality
innovations that are easily applicable in practice are more rapidly accepted
(Fullan, 2001).

- Observability: The more the results of the innovation are visible to other persons
(e.g., parents, principals) and organisations, the more likely the innovation is
accepted and adopted (Rogers, 2003).

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For example, if face-to- face communities and networks are expected to make an
impact, they should provide a high level of relative advantage, should meet the
teachers' needs and their goals, and structures should be clearly and easily
understandable, they should be practical and usable, of high quality and their
results should be visible to others.

Context. The particular importance of the school context that embeds participating
teachers, participating facilitators, and the project or initiative itself, as a factor
fostering the sustainability of innovations and change processes is well
documented (e.g., Fullan, 1990; McNamara, Jaworski, Rowland, Hodgen, &
Prestage, 2002; Noddings, 1992; Owston, 2007). Teachers promoting change (e.g.,
by participating in face-to-face communities and networks) need administrative
support and resources (McLaughlin & Mitra, 2001). Support from outside the
school by parents or the district policy is also an important factor (McLaughlin &
Mitra, 2001; Owston, 2007). Moreover, school-based support can be provided by
students and colleagues (Ingvarson et al., 2005; Owston, 2007), and in particular
by the principal (Clarke, 1991; Fullan, 2006; Krainer, 2006b). As Owston (2007)
states, "Support from the school principal is another essential factor that
contributes to sustainability" (p. 70). Moreover, to foster sustainability not only at
the individual (teacher's) level but also at the organisational (school's) level,
Fullan (2006) proposes a new type of leadership that "needs to go beyond the
successes of increasing student achievement and move toward leading
organizations to sustainability" (p. 1 13), and calls these leaders "system thinkers in
action". In particular, these school leaders should "widen their sphere of
engagement by interacting with other schools" (p. 113) and should engage in
"capacity- building through networks" (p. 115).

Community building and networking as key factors fostering sustainability. The
review of the literature on sustainability of structures for developing teaching and
learning indicates that there is a great deal of research on factors that can lead to
such initiatives continuing beyond the initiation stage. There is plenty of evidence
to demonstrate that the community as a whole and the participants in particular
benefit and learn from any initiatives to develop the teaching and learning of
mathematics. That such projects may be initiated by higher level study, the
research interests of university faculty or the immediate needs of teachers in a
school or network of schools is the nature of the professional educational
environment in most countries.

Research findings, illustrated in large part by the examples given above, indicate
that there are several characteristics of successful initiatives; we are particularly
interested here in features that might foster the sustainability of a professional
development programme or project: it should focus on content knowledge (Garet et
al., 2001; Ingvarson et al., 2005), use content-specific material (Maldonado, 2002),
and provide teachers with opportunities to develop both content and pedagogical
content knowledge and skills (Loucks-Horsley et al., 1996; Mundry, 2005).

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Moreover, an effective professional development initiative includes opportunities
for active and inquiry-based learning (Garet et al., 2001; Ingvarson et al., 200S;
Maldonado, 2002), authentic and readily adaptable student-centered mathematics
learning activities, and an open, learner-centered implementation component
(Farmer et al., 2003). Further factors fostering the effectiveness and sustainability
of the programme are: prolonged duration of the activity (Garet et al., 2001;
Maldonado, 2002), ongoing and follow-up support opportunities (Ingvarson et al.,
2005; Maldonado, 2002; Mundry, 2005), and continuous evaluation, assessment,
and feedback (Ingvarson et al., 2005; Loucks-Horsley et al., 1996; Maldonado,
2002).

In particular, research indicates that communicative and cooperative activities
represent key factors fostering sustainable impact of professional development
programmes. This general result is supported by several authors and studies, even
if the categories used to describe these activities are different: Clarke (1991), Peter
( 1 996), and Mundry (2005) point to cooperation and joint practice of teachers,
Loucks-Horsley et al. (1996) and Maldonado (2002) highlight the importance of
learning communities, McLaughlin and Mitra (2001) identify supportive
communites of practice, Arbaugh (2003) refers to study groups, and Ingvarson et
al. (2005) stress professional communities as factors contributing to the
sustainability of effects. In particular, providing rich opportunities for collaborative
reflection and discussion (e.g., of teachers' practice, students' work, or other
artifacts) presents a core feature of effective change processes (Clarke, 1991;
Farmer et al., 2003; HoSpesova & Ticha, 2006; Ingvarson et al., 2005; Park-Rogers
et al., 2007). In this regard, research findings, exemplified by some of the authors
in our examples, point to the issue of power relations between the external
facilitator(s) and the teachers in the school. There is no one prescription for the best
form of such relations, but it is clear that teachers must feel themselves empowered
and autonomous and that the expert(s) play a role that supports their work rather
than dominates it. Once again, this issue recalls the motivation of the action
research movement but with the emphasis on collaboration and cooperation, not
one of excluding the academic. Both, internals and externals need to be regarded as
experts.

Reducing the long list of key features, on the basis of the examples we have
given above as illustrative of the field, we suggest that there is evidence to show
that large scale, centralised projects across schools, such as the Thematic Geometry
Network, are sustainable. Within-school developments such as lesson study are
also sustainable provided that the developments are supported by school resources
(time being the most important) and that they are seen by the teachers as what they
choose to do for the improvement of their mathematics teaching and their students'
learning, because they see the benefits and the impact.

We believe that the job of a teacher should incorporate a lifelong process of
learning about and developing one's teaching and two examples of sustainable
structures, lesson study and Thematic Geometry Network, are indications of what
works. Not all centralised interventions are benign of course, but where the

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participants (teachers in the main but also students and school managers), see the
benefits of on-going research and development, sustainability is possible.

CONCLUSION

Reviewing the work we have described and analysed in this chapter, we end here
with some final, concluding observations:

- There are few examples of either face-to-face or cross-school networks research
when compared to the rest of the body of research on mathematics teacher
education.

- Where there are examples, the effects seem to be extremely positive on
teachers' perspectives and understanding and, where there is evidence, on
students' achievements.

- Providing the opportunity for time out of the classroom for meetings may be a
key to success but is costly.

- Engagement of academics as experts appears to be crucial, at least because
when teachers do research as a group without an academic it may not be written
up and disseminated. In addition, teachers' groups often call for support in
aspects of research activity, such as access to literature, research design and data
analysis.

- Community building and networking represent the core factors fostering
sustainable impact of professional development programmes.

- In order to go deeper with these observations, we unquestionably need more
research on how fruitful initiatives can be sustained.

- Whether dissemination of the outcomes and experiences of one network can or
will be taken up by another network or by individual teachers remains an
unresolved question. If it is the case that teachers' change and growth, perhaps
what we should call teacher learning, only takes place when teachers engage in
these activities themselves and not through reading the work of others, we may
need to radically re-think and advocate that systematic national initiatives foster
learning communities in every school as part of its work.

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Stephen Lerman
Department of Education
London South Bank University
United Kingdom

Stefan Zehetmeier

Institutfur Unterrichts- und Schulentwicklung

University ofKlagenfurt

Austria

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7. VIRTUAL COMMUNITIES AND NETWORKS OF
PROSPECTIVE MATHEMATICS TEACHERS

Technologies, Interactions and New Forms of Discourse

The chapter discusses the use of communication technologies and the development
of new forms of discourse in the context of prospective mathematics teachers'
learning. Two key ideas are identified from current studies and used to frame the
processes that are generated when prospective teachers use new communication
tools: the notion of community and the features of knowledge building processes
and discourses, within a sociocultural framework. Three examples are discussed in
details: creating and sustaining virtual communities and networks, constructing
meaning through online interactions, writing and reading blogs and videopapers.
Finally, the chapter draws together key factors that should be considered when
computer-supported communication tools are introduced in mathematics teacher
education and that seem to shape the characteristics of online interactions,
construction of knowledge and creation and support of communities of practice.

INTRODUCTION

The use of information and communication technologies in Higher Education and
in initial teacher education programmes has increased over the last few years. A
range of new computer-based communication tools are now available for teacher
educators to adapt and transform into pedagogical tools aimed at developing new
approaches to teacher education (Blanton, Moorman, & Trathen, 1998; Mousley,
Lambdin, & Koc, 2003). Communication tools are tools that typically handle the
capturing, storing, and presentation of communication, usually written but
increasingly including also audio and video. They can also handle mediated
interactions between a pair or group of users. 1 Communication tools can be either
synchronous or asynchronous, and include bulletin boards, e-mail, chats, virtual
video-based cases, computer-mediated conferences and forums. These tools can be
embedded in interactive learning environments that support the creation of virtual
communities and networks. These new communication tools not only facilitate
access to information, but also have the potential to change the personal and social
relations amongst individuals and the way we understand the process of becoming
a mathematics teacher (e.g., knowledge and identity, Borba & Villareal, 2005).

1 http://en.wikipedia.org/wiki/Social_software

K. Krainerand T. Wood (eds.), Participants in Mathematics Teacher Education, 155-179.
© 2008 Sense Publishers. All rights reserved.

SALVADOR LLINARES AND FEDERICA OLIVERO

Current studies explore a number of ways in which communication tools can be
used in the context of teacher education. Nowadays, new technologies can be used
to support interaction among prospective teachers. For example, computer-
mediated communication may provide support for online discussions (Byman,
Jarvela, & Hakkinen, 2005) during problem solving activities or for the creation of
virtual communities. New communication tools such as forum and bulletin boards
allow extension of classroom boundaries and provide opportunities to develop
skills that might enable prospective teachers to learn from practice and develop
knowledge-building practices (Derry, Gance, Gance, & Schaleger, 2000). Bulletin
boards and online discussion also enable content-related communication including
course materials, resources and activities and are used by prospective teachers as
source of support for their development. Questions are generated about the social
and cognitive effects of interactions in this online social space. In particular, it is
interesting to look at how different forms of participation may operate towards
mediating meanings in conversations within prospective teachers' learning. In
these social interaction spaces, reciprocal understanding and the process of
becoming a member of a community support the possibilities of taking and sharing
different perspectives that require an understanding of others' points of view when
they join in the same activity. By using new communication tools, new forms of
discourse are emerging, as for example argumentative discourse, videopapers and
blogs, which are used both as tools to represent and communicate knowledge,
practice and research in a new form but also as tools for prospective teachers'
reflection, self-reflection and assessment and for sharing good practice.

The question of how these new forms of discourse and new forms of
participation operate to mediate meaning construction in conversations and to
create and sustain virtual communities is central to our understanding of the
contribution of interactive learning environments and new communication tools in
teacher education. In this sense, learning in collaborative settings is based on the
assumption that learners engage in specific discourse activities and that learning
stems from the relationship between the nature of the participation and the content
of these discourses (learning is seen as becoming a member of a community that
shares knowledge, values and skills). This situation assumes that computer-
mediated communication is a tool used to mediate prospective teacher learning and
reflective thinking and that it can mediate and transform teachers' experiences
(Blanton et al., 1998).

The introduction of new communication tools in mathematics teacher education
is generating new research questions and is calling for new analytical procedures.
What emerges from the literature is on the one hand the scarcity of research in this
area, and, on the other hand, the attempts to use theoretical constructs from
sociocultural perspectives of learning to explain the processes taking place when
these tools are implemented. Common features to current studies are the
description of the activities and assignments tackled by prospective teachers, the
enumeration of the messages exchanged and the subsequent development of
analytic categories drawing on discourse analysis (Schrire, 2006; Strijbos, Martens,
Prins, & Jochems, 2006). Research also suggests that prospective teachers should

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learn how to contextualise these tools as learning means rather than using them in
isolated initiatives. When these communication tools are used as mediators in
prospective teachers' learning then some issues are generated about the nature of
this kind of learning. Mathematics educators are attempting to provide claims
about both the individual and collective construction of meaning in communities
and on the relationship between discourse and knowing. Another important point
for discussion is what role these tools play in relation to the specific nature of
mathematical knowledge and knowledge of mathematics education.

The chapter is structured taking into account these general features emerging
from current studies on the use of tools and the creation of virtual communities and
networks in the context of teacher education. In the second subchapter, we describe
how the sociocultural perspective on mathematics teachers' learning is being used
in current research studies. We identify two key ideas that can help us understand
the generated processes: the notion of community and the features of knowledge
building processes and discourses. These ideas underline different aspects of
learning that are considered in those studies: learning as identity (becoming),
learning as practice (doing) and learning as meaning (experience). Three examples
of experiences with three different tools are introduced and discussed separately.
Although we use a common theoretical framework to frame the three examples,
each subchapter will highlight the specificities that each tool brings to the
framework, as emerging from the literature. The third subchapter focuses on how
virtual communities are created and sustained (learning as identity, becoming). The
next subchapter deals with the question of how online interactions support the
construction of meanings when prospective mathematics teachers are involved in
solving specific learning tasks (learning as meaning, doing); and the fifth
subchapter focuses on how the use of new forms of communication and discourse
(blogs and videopapers) supports prospective mathematics teacher learning
(learning as meaning, how experience becomes knowledge through discourse).
Finally, in the last subchapter we discuss some emerging issues and suggest ideas
for further research in mathematics teacher education.

USING SOCIOCULTURAL PERSPECTIVES TO INTERPRET PROSPECTIVE
TEACHERS' LEARNING

Sociocultural theories of learning and development offer useful conceptual tools
for studying prospective teachers' learning when new communication technologies
are introduced in mathematics teacher education. This perspective views learning
both as a process of meaning construction and as a process of participation in
mathematics teaching practices (Greeno, 1998; Lerman, 2001; Llinares & Krainer,
2006). Sociocultural theories underline the social processes underpinning learning
and consider that learning is mediated by participation in social processes of
knowledge construction scaffolded by social artefacts or tools. These tools can be
both technical tools and conceptual tools and are considered mediators of learning
interactions in educational settings. Two notions appear to be essential when
analysing the learning and development of prospective mathematics teachers: the

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notion of "communities" and the notion of "knowledge building" practices in
interactions.

Communities of Learning

Communities of learning are formed by people who engage in a process of
collective learning in a shared domain of human endeavour. The emergence and
sustainability of some kind of community among prospective teachers seems to be
an important mechanism in the process of becoming a mathematics teacher and in
the transition from a university context to a professional context. The idea of
community of practice introduced by Wenger (1998) in relation to learning in
apprenticeship situations might be a useful analytical tool, but its translation to a
context in which teaching is a deliberate process, as is the case in teacher
education, is not an easy task (Graven & Lerman, 2003) and has fostered the
necessary differentiation between communities, teams and networks (Krainer,
2003).

One of the main features of the notion of communities of practice is that they
are groups of people who share a concern for something they do and leam how to
do it better as they interact on a regular basis. One relevant aspect in this
characterisation is the notion of sharing a goal, as for example acquiring
knowledge, skills and dispositions that are necessary to teach mathematics. An
institutional context assumes the intentionality of learning and the existence of an
expert or facilitator, but from a general perspective the definition of a community
of practice according to Wenger allows for, but does not assume, intentionality or
the existence of a facilitator. Although this constitutes a theoretical difference
between how mathematics educators may use this notion and how this notion is
used in other contexts, the construct of "community" provides new avenues that
may help understand better prospective teacher learning and offers suggestions for
teacher educators about how to design opportunities for learning. In the context of
teacher education, learning is the reason why the prospective teachers come
together, therefore teacher education programmes should define a shared domain
of interest, as for example learning to analyse mathematics teaching in terms of
student learning. In addition, this goal can also be considered as a framework for
teacher preparation programmes that aim at helping prospective teachers learn how
to teach from studying teaching (Hiebert, Morris, Berk, & Jansen, 2007), how to
interpret classroom practices (Morris, 2006; Sherin, 2001), or how to conceptualise
a contemporary view of mathematics teaching (Lin, 2005). So, at the very least, the
intentionality should be explicit when mathematics teacher educators use this
approach to think about the process of becoming a mathematics teacher and to
design opportunities for learning.

Three characteristics need to be fulfilled so that communities of practice emerge
and are sustained and collaborative learning processes are knowledge productive
(Wenger, McDermott, & Snyder, 2002): (i) a focus on shared interests and domain,
(ii) the involvement in joint activities, discussions and sharing of information, (iii)

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the development of a shared repertoire of resources (experiences, stories, tools,
ways to address recurrent problems).

The first characteristic of a community of practice is the existence of a shared
domain of interest that generates the idea of membership as a commitment to the
domain. In the case of prospective mathematics teachers, we should see these
conditions as part of the process of becoming a mathematics teacher, and so, of the
process of generating ways of seeing the activity of mathematics teaching with a
teacher's eye. Sometimes, this process is supported by reflections on the actions
and experience of others through video-cases that encourage prospective teachers
to participate in and reflect on discourse centred on mathematical ideas (Lin, 200S;
Seago, this volume). In this process of learning to analyse teaching in terms of
student learning, prospective teachers can generate a shared competence that
characterises them as teachers. Hiebert et al. (2007, p. 47) conjectured the features
of this domain in terms of four skills and knowledge "rooted in the daily activity of
teaching, that when deployed deliberatively and systematically, constitute a
process of creating and testing hypotheses about cause-effect relationships between
teaching and learning during classroom lessons".

Recently, information and communication technologies have been used to
develop this characteristic of a community of practice. In the process of induction
of primary mathematics teachers in professional communities, the constitution of
an online mathematics community may provide both opportunities for sharing and
communicating and access to quality resources (Dalgarno & Colgan, 2007; Goos
& Benninson, 2008).

The second characteristic of a community of practice is the way in which the
members pursuing their interest in their domain engage in joint activities and
discussions as a way of sharing information and building relationships that enable
them to learn from each other. Analysis of mutual engagement amongst
prospective teachers when they are solving specific tasks has pointed out that what
is really important is identifying how, through mutual engagement, the prospective
teachers define tasks and develop meanings for the different elements of
mathematics teaching. The key characteristic "interact and learn together" is being
introduced in the design of web-based learning environments (Wade,
Niederhauser, Cannon, & Long, 2001), since it is considered that interaction and
cognitive engagement during online discussion are critical for constructing new
knowledge (McGraw, Lynch, Koc, Budak, & Brown, 2007; Zhu, 2006). From this
perspective, communication tools can begin to mediate prospective teachers'
thoughts, actions and interactions.

Finally, the third characteristic of a community of practice is the development of
a shared repertoire of resources, experiences, representations, tools and ways of
addressing professional problems linked to mathematics teaching. Developing
different ways to analyse teaching and to notice and interpret classroom
interactions, or to interpret students' mathematical thinking is a process in which
prior experience and beliefs are entwined. However, by using an instructional
scaffolding process, it is possible that prospective teachers develop new ways of
conceptualizing mathematics teaching. For example, Lin (2005) argues that the

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prospective teachers in his research, when constructing pedagogical
representations, were able to articulate students' difficulties with a specific topic
from multiple perspectives.

New Forms of Discourse and Knowledge Building

According to the sociocultural perspective about teacher learning (Wells, 2002),
"knowledge building" has to do with ways in which prospective teachers are
engaged in meaning making with others in an attempt to extend and transform their
collective understanding. In this sense, knowledge building involves constructing,
using and progressively improving different representational artefacts with a
concern for systematicity, coherence and consistency (Garcia, Sanchez, Escudero,
& Llinares, 2006; Llinares, 2002; Sanchez, Garcia, & Escudero, 2006). From this
perspective, knowing is the intentional activity of prospective teachers who make
use of and produce representations in a collaborative attempt to understand and
transform their world (Wells, 2002). According to Wells (2002), the experience
needs to be extended and reinterpreted through collaborative knowing, using the
informational resources and representational tools of the wider culture, in our case
mathematics education.

Knowledge construction in collaborative settings is based on the assumption
that learners engage in specific discourse activities and that the nature of the
participation and content of this discourse is related to the knowledge thereby
constructed (Sfard, 2001; Wells, 2002). We use and adapt here the notion of
Discourses as developed by Gee (1996, p. viii), according to whom Discourses are
"ways of behaving, interacting, valuing, thinking, believing, speaking, and often
reading and writing that are accepted as instantiations of particular roles (or types
of people) by specific groups of people".

Prospective teachers might create points of focus around which the negotiation
of meaning and reciprocal understanding become organised by generating
processes such as noticing, representing, naming, describing, interpreting, using
and so on, what Wenger (1998) calls reification. The process of reification shapes
the prospective teachers' experience of creating "objects" about mathematics
education that they use to notice and interpret mathematics teaching and learning.

Some research has shown that the construction of knowledge and the
development of the skills needed in order to generate a more complex view of
teaching is a process in which the interrelationship and integration of ideas about
teaching and learning are progressively included in the analysis and reflection by
prospective teachers (Garcia et al., 2006; Sanchez et al., 2006; van Es & Sherin,
2002). These ideas are viewed as "tools to think about" and handle mathematics
teaching and learning situations. The progressive use of theoretical ideas as
conceptual tools in activities of analysing and interpreting teaching and learning
situations, and the progressive modification in the type of participation in the
spaces set up for social interaction are manifestations of the knowledge
construction process (Derry et al., 2000). Here we are paying special attention to
the activity of knowing through making and using representational artefacts (e.g.,

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the theoretical information provided in online discussions) as a means of guiding
joint action and of enhancing collective understanding (as can be seen in the
creation of a videopaper). The new interaction and communication tools (such as
virtual debates, bulletin board discussions, videopapers, blogs) contribute to
generating an ongoing discourse amongst prospective teachers that enables
viewing "the said" as a knowledge artefact that contributes to the collaborative
knowledge building of the participants in the activity. These new forms of
discourse generated by the currently available communication tools use "writing"
as an instrument for collaborative reflection and as a tool for inquiry. The new
"type of text" generated by these communication tools can be used as an
"improvable object" that favours the generation of a progressive discourse that acts
as the focus of collaborative knowledge building (Wells, 2002).

CREATING AND SUSTAINING COMMUNITIES OF LEARNING

The use of communication technologies in teacher education has seen an attempt
by researchers to develop a view of teacher learning as a social and cultural
phenomenon. Within this framework, some teacher educators are now studying the
role played by virtual learning communities through the use of electronic
discussion boards. In particular, recent studies look at how these new
communication tools support professional reflection, how communities are
established and supported through online and face-to-face interactions, and what
type of support an online community formed by prospective teachers and
practising teachers can provide to prospective teachers (Dalgarno & Colgan, 2007;
Goos & Benninson, 2008; Schuck, 2003).

Schuck (2003) argues the role played by computer-mediated conferencing (e.g.,
electronic conferencing boards 2 ) in challenging prospective mathematics teachers'
beliefs and poses the question of how teacher educators should express an opinion
or suggest a course of action. In Schuck's study, a group of prospective primary
teachers were encouraged to post questions to a forum, either about the use of
technology or about the content of the course they were studying, with the aim of
developing their understanding of mathematics and of mathematics teaching. The
analysis of how the prospective teachers used the forum and of the content of the
messages posted in the discussion board showed that the use of the forum to
achieve this objective was irregular and that the role played by the students varied.
Although there were prospective teachers who did not participate in the forum, for
those who used the forum the discussions were useful to encourage reflection, to
share teaching experiences without having to be on campus, and they also
encouraged the process of justifying and explaining points of view. Because of
these reasons, Schuck argues that accessibility to a forum is an important factor in
developing a community of learners. This author raises the issue of whether
participation should be compulsory due to the benefits identified, but in the end

Electronic conference boards are web-based conferencing tools.

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proposes that the reasons not to participate should also be respected. In this last
case, the prospective teachers should value the participation in the forum as an
alternative way of learning. Another issue to take into account is the level of
structure imposed on the use of and participation in bulletin boards. Questions that
emerge from the study are: what conditions may restrict the use of bulletin boards
and the free exchange of ideas, what conditions may determine the emergence of a
community and what should be the role of mathematics teacher educators in
supporting or suggesting new avenues.

Also working with prospective primary teachers, Dalgarno and Colgan (2007)
examine what is the support provided by an online mathematics community to
prospective teachers. In their study, a group of prospective primary mathematics
teachers sought opportunities to continue their professional development through a
forum that they could access once they were out in schools after graduation. The
needs identified by the prospective teachers were: discussion with experts, access
to suggestions and help about mathematics content and to mathematical resources
selected by the experts for use in the classroom (repository of exemplars and
resources), having a place where they could share lesson plans and activities that
they themselves had created (repository of novice teachers' ideas about teaching
and learning). An online community called Connect-ME was created and was
constituted by prospective teachers and beginning teachers. Connect-ME offered a
means to meet the expressed needs through mixed-method delivery mechanisms.
According to Dalgarno and Colgan (2007, p. 12), the online community Connect-
ME provided novice teachers "with a safe, communicative community for sharing
resources and ideas and an environment where they can proactively seek the help
they need".

This also highlights the significance of emotional and personal connections.
Dalgarno and Colgan suggest three essential elements that help sustain this type of
connections: (i) the initial community members should have a personal link to, and
a loyalty and respect for, the project facilitator; (ii) the facilitator should continue
to communicate with all members of the online community even after they
graduate; and (iii) the online forum should be created and developed at the
"grassroots level", but its growth ought to be the results of previous personal
connections.

Goos and Benninson (2004, 2008) also study the interface between secondary
school mathematics prospective teachers and beginning teachers and how such
type of community is established and maintained through online and face-to-face
interaction. These researchers look at an online community of practice established
via Yahoo! Groups 3 with the aim to: encourage sessions and professional
discussion outside class times and during the practicum periods; provide
continuing support, by remaining accessible to its members after graduation. One
characteristic of this website was that the authors imposed minimal structure on
communication. Goos and Benninson are in fact interested to know how and why

1 http://uk.groups.yahoo.com/

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

prospective teachers and beginning teachers might choose to use this form of
communication.

One participation structure linked to the bulletin board that Goos and Benninson
considered interesting was the fact that prospective teachers used the bulletin board
to organise and negotiate the agenda of a debriefing session to take place after they
returned to university. Prospective teachers from different cohorts, beginning
teachers and teacher educators attended this debriefing session, discussing
pedagogical challenges, identifying sources of assistance, and comparing the
effectiveness of different teaching approaches. Afterwards, the bulletin board was
used to provide a summary of the debriefing session for those who had been unable
to participate. This internship debriefing session was organised the following year
too, which is what Goos and Benninson interpret as the beginning of a professional
routine and a part of the shared history of the community.

Goos and Benninson (2004, 2008) also suggest a possible factor that might have
influenced the creation and sustainability of this community, for example, their
own role in shaping the interactions between the participants by offering models of
online professional exchanges through forwarding messages from other e-mail
discussion lists used by mathematics teachers, encouraging prospective teachers to
share teaching resources and their mathematics teaching experiences.

The studies mentioned in this subchapter start shading light on the processes of
creating and sustaining communities of practice through online environments. Four
factors that emerge are: (i) the provision of accessible and flexible online forums,
discussions and bulletin boards which can be appropriated and adapted to satisfy
the teachers' needs; (ii) the participation of both prospective and practising
teachers to the same community, together with "experts", which may enable the
construction of professional knowledge and practices, together with the creation of
a shared repertoire of mathematics resources; (iii) the co-existence of online and
face-to-face interactions, which also enables the creation of emotional and
personal connections that foster continuous participation in the exchanges and
discussions and the development of a shared history; and (iv) the provision of
models of professional exchanges and interactions to get the teachers started and
provide an initial structure for the discussion.

ONLINE INTERACTION AND KNOWLEDGE BUILDING

The question of how forms of participation operate to mediate meanings in
conversations is central to our understanding about the role of interactive learning
environments in teacher education. Research on knowledge construction in
collaborative settings is based on the assumption that learners engage in specific
discourse activities and that the nature of the participation and the content of these
discourses are related to knowledge construction (Llinares, 2002; McGraw et al.,
2007; Santagata, Zannoni, & Stigler, 2007). From the point of view of
sociocultural perspectives on learning, it is assumed that prospective teachers
construct arguments in interaction with their partners in order to build knowledge

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SALVADOR LLINARES AND FEDERICA OLIVERO

about mathematics teaching, as well as to develop the skills needed to learn from
practice.

Llinares and his colleagues designed several learning environments considering
this theoretical perspective and using "design experiments" as a methodological
approach (Callejo, Vails, & Llinares, 2007; Cobb, Confrey, DiSessa, Lehrer, &
Schauble, 2003; Llinares, 2004). Figure 1 displays the web-structure of one of
these learning environments integrating video-clips of mathematics teaching,
asynchronous computer mediated discussion, theoretical information related to the
given task and links to written essays about mathematics teaching (Vails, Llinares,
& Callejo, 2006). This particular design gives prospective teachers the opportunity
to engage in the process of meaning making with other colleagues, in an attempt to
extend and transform their collective understanding in relation to some aspects of a
jointly undertaken activity. Video-clips or teaching vignettes are used to situate the
individual cases in the context of classroom practices. In addition, this web-
learning environment provides prospective teachers with theoretical information
and questions aimed to generate online discussions and to promote an inquiry
orientation towards the observation of mathematics teaching (which is one of the
objectives of teacher education).

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The adopted approach creates a setting in which prospective teachers come
together in a virtual collaborative action and interaction setting. The website allows
them to watch videos and to download reports in text format (transcription of the
lesson in the video-clip, the activities used by the teacher during the lesson, and
documents with theoretical information about the characteristics of mathematics
teaching), at any time and from anywhere by logging in the learning environment.

In the study described in Llinares and Vails (2007), the prospective primary
teachers had the chance to observe aspects of a mathematics lesson from different
perspectives, starting from their own initial conceptions (based on their experience)
and moving onto positions in which they use conceptual elements introduced in the
training programme (theoretical information). The progressive use of the
conceptual tools in the online discussions about the analysis and the interpretation
of the teaching and learning situations and the progressive change in the way
students participated in the spaces set up for social interaction, are manifestations
of the development of the skills needed to learn from practice and of the
knowledge construction process. What was emphasised in the design of the
learning environments in Llinares and Vails' study was the activity of knowing
through making and using representational artefacts (the provided theoretical
information) as means of guiding prospective teachers' joint action and enhancing
collective understanding through online discussions. The relationship between
online interaction and the construction of meaning is rooted in the assumption that
the semiotic process through which the ideas are formulated and communicated
towards the achievement of a goal during an activity is part of the construction of
meanings.

This organisation of the learning environments attempts to reflect the
progressive and evolutionary nature of the process of construction of the
knowledge needed for teaching (Goffree & Oonk, 2001) and of the required skills
to learn from practice, in which conceptual tools are progressively integrated in the
activities of analysis and reflection (Garcia et al., 2006). In this web-learning
environment, when prospective primary teachers contributed to the online
discussion they had to interpret the preceding contributions and to formulate a new
contribution extending, questioning or qualifying what had already been said. Such
progressively generated discourse mediated knowing. For example, in the
exchanges below the prospective primary teachers refined and amplified previous
contributions about how the idea of equity should be understood in the classroom
context, when they analysed a video-clip in which a primary teacher managed the
interactions among primary students posing problems from a commercial brochure.

I do not entirely aeree (LUISA - 12:05:00 09/01/2006)

I partly agree with the comments made by my colleagues Estefania and
Angela, but although all the students participate and interact, the teacher did
not carry out her task properly. The role of the teacher in the classroom is to
get everyone to share while working on the problem (which I think she does,
like my colleagues). But the teacher, in my opinion, does not pay attention to

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SALVADOR LLINARES AND FEDERICA OLIVERO

what the children say, nor does she show any real interest in their
suggestions. During the class, two other children offer possible data to help
in solving the problem. One boy says that there are 14 chocolates in the box,
and another says there are 50. The teacher ignores these two children's
suggestions, and proceeds to solve the problem using only the supposed
existence of 8 chocolates. In my opinion she should have paid more attention
to these two children and should also have solved the problem using their
data. Not doing the exercise in this manner means that she does not respect
the principle of fairness, because she has not listened to the contributions
made by two of the children.

To LU1SA (ANGELA - 14:28:06 09/01/2006)

In one way you are right. The teacher does not accept all the children's
proposals for the problem, she only accepts that of the child who suggests 8
chocolates. At one point in the video the teacher says "There could be 8
chocolates, or there could be more [...]" and "8 chocolates is a number which
might be the one, or it could be different [...]", so 1 think that by saying that,
she is trying to show that she has paid attention to the other opinions, but as
she can't put all of them on the blackboard she decided to concentrate on the
number proposed by one of the children, simply in order to set up the
problem and solve it. I think that if the video were to continue and the teacher
were to approach the problem again she would pay attention to a different
child, and that if she required more data, she would use those suggested by
several different children.

The interactions amongst the prospective primary teachers, motivated by other
students' contributions, were increasingly focused - in this case on the notion of
equity as a valued-added dimension in teaching that can promote understanding -
indicating that the structure of this type of environments, including the contexts
and the activities, seemed to encourage the prospective primary teachers to engage
in meaning making with others in an attempt to extend and transform their
collective understanding of mathematics teaching.

The study confirms that interaction occurs among prospective primary teachers
when it is generated by a given task, which in this case was the analysis of
mathematics instruction and of its effects on children's mathematical competence.
The tasks and online discussions were intentionally integrated into the course in
order to lead to a high degree of interaction. Llinares and Vails (2007) argue that
the focus on shared interest helped the prospective teachers to engage in the joint
activities of identifying and analysing different aspects of mathematics teaching
thus enabling them to construct a shared understanding of the situation.

The activities that were designed within these online learning environments
required the prospective primary teachers to identify key aspects in a mathematics
lesson and interpret them, something that seemed to encourage interaction. Here
the role of the theoretical information was to help prospective teachers to begin to

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

"notice" (Pea, 2006). From Wells' (2002) theoretical perspective, the process of
knowledge-building is based on the assumption that students are engaged in
specific discourse activities related to knowledge acquisition through discussions
in which they focus on issues directly related to their future teaching. The
interactions in the online discussions showed how the prospective teachers' initial
personal interpretations could progressively be modified to construct common
knowledge, when they perceived the ideas about mathematics teaching as
functional in relation to the task that they had to undertake.

These findings suggest that the different types of task and conditions of online
discussion in the learning environments in which prospective primary teachers
participated (the discussion questions and the video-clips) seem to exert an
influence on the nature of the interaction (Schrire, 2006). The results of Llinares
and Vails' analysis imply that the degree of prospective teacher involvement in
interactive processes is related to the type of task intended to justify their
participation in online discussions. Different structural factors that seemed to
contribute to the construction of meaning were clear established goals with
thematic prompts, and time to write.

The existence of clear goals with thematic prompts seems to support the
hypothesis that prospective teachers should use online discussions as a tool in their
learning environments. That is to say, prospective primary teachers should
consider participation in online discussions to be useful and beneficial for carrying
out assigned tasks. The findings suggest that prospective primary teachers could
identify in the task a focus on a shared interest - the goal - that justified their
engagement in joint activities, and considered the online discussions as social
spaces in which it was possible to develop a shared repertoire of experiences, tools
and ways of addressing the analysis of videotaped case studies (Wenger, 1998). In
this context, the personal interpretations were questioned and clarified in the online
discussion, and assumptions and inferences were challenged in an attempt to
construct a communal answer supported by common knowledge. This process was
given further importance by the fact that the progressive discourse was conducted
in writing.

The questions for discussion that were set in the learning environment led the
prospective primary teachers to respond to each others' messages, agreeing or
disagreeing about different points of view. This enhanced their ability to see things
from another's viewpoint and they began to develop a more complex view of
teaching, as could be inferred from messages that became more focused on the
specific topics over time (Byman et al., 2005) showing that writing was used as a
tool for collaborative reflection where the text of the messages acted as an
"improvable object" to the focus of collaborative knowledge building (Wells,
2002).

In another context, McGraw et al. (2007) uses discussion prompts to stimulate
critical analysis in a multimedia case and facilitate online discussions. In the
project described by McGraw at al., online forums were used to discuss a
multimedia case amongst prospective mathematics teachers, practising
mathematics teachers, mathematicians and mathematics teacher educators. The

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analysis of the messages posted in the online forums and of the transcripts of the
face-to-face discussions identified episodes of dialogic interaction in which
individuals explicitly responded to the ideas and opinions of the previous writers.
McGraw and her colleagues suggest that integration in discussion groups of
members with different perspectives and level of experience enabled the
generation of multiple episodes of dialogic interaction in each discussion group.
Moreover, the interplay between theoretical and practical knowledge as a
manifestation of knowledge building was evidenced by movements in the
discussions between case specific observations and more general observations or
use of theoretical knowledge. In relation to this last aspect, the variations in level
of noticing in the different members of the group - prospective teachers, practising
teachers, mathematicians, teacher educators - seemed to influence the development
of know I edge.

Summing up, two characteristics emerge as relevant as concerns online
interactions and knowledge building: (i) providing structured guidance through
tasks and discussion questions with thematic prompts seems to enable the
participants in online discussions to reflect on and integrate multiple aspects of
teaching; (ii) prospective teachers might benefit from engaging in discussions with
more knowledgeable persons, grounded in a case of classroom practice, that can be
accessed through an online environment from anywhere at anytime. The
interaction amongst the multiple perspectives that may emerge, and be made
explicit in the written messages appearing in the online discussions, from people
with different levels of knowledge while "seeing" the same multimedia case, is
vital to meaning making and knowledge construction.

NEW FORMS OF DISCOURSE THAT SUPPORT PROSPECTIVE TEACHER
LEARNING: BLOGS AND VIDEOPAPERS

The relationship between the forms of discourse and knowledge building needed to
become a mathematics teacher and the emergence of communities when new tools
such as blogs and videopapers are used in mathematics teacher education have
recently become an object of research (Beardsiey, Cogan-Drew, & Olivero, 2007;
Makri & Kynigos, 2007; Nemirovsky, DiMattia, Ribeiro, & Lara-Meloy, 2005;
Olivero, John, & Sutherland, 2004; see also Borba & Gadanidis, this volume).

Blogs are (personal or organisational) web pages organised by dated entries
whose items are links, commentaries, papers, personal thoughts and ongoing
discussions. Blogs are considered as learning spaces with digital, sharable, and
reusable entities that can be used for learning and are available to prospective
teachers anytime and anywhere (learning objects). Recently, research interest has
emerged about the potential and possible roles of blogs in the professional
development of mathematics teachers and about the necessary relation between the
use of these new tools of communication and the intentional ity of their use in
mathematics teacher education (Makri & Kynigos, 2007). Makri and Kynigos
focus their research on the ways in which prospective teachers write about both
their subject and its pedagogy and study the discourse that is developed in

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

teachers' blogs. In Makri and Kynigos' course the prospective teachers were given
writing tasks addressing their epistemological and pedagogical beliefs and their
subject related knowledge (e.g., the pedagogical value of using software in
mathematics). The prospective teachers were encouraged to publish their answers
on a blog and comment on the work of their peers by sharing opinions and
engaging in discussion.

The prospective teachers in this study used the explanatory and expository
genres and their writings showed a structured cognitive presence since they
combined factual knowledge and conceptual and theoretical knowledge which
emerged collaboratively. The researchers suggest that the use of the blog
introduced changes in the social orchestration of the course at an affective,
interactive and cohesive level. Although research of this type is still recent, Makri
and Kynigos identify different profiles of prospective teachers in relation to the
emerging forms of social interaction indicating the degree of appropriation of the
blog by the prospective teachers. The three profiles identified, blog enthusiasts,
blog frequent visitors and blog sceptics, indicate that it is necessary to study in
depth the changing social practices and roles and the new role of the instructor, as
it has also been pointed out by studies on creating and sustaining communities of
practice (Schuck, 2003). The sociocultural perspectives of learning assume that the
social context affects the nature of learning activities, so when the social context is
modified by introducing new forms of social interaction on the web it is assumed
that this will influence the capacity to engage prospective teachers in collaborative
activity, reflection, knowledge sharing and debate.

Besides tools that facilitate social interaction through writing, such as blogs,
other multimodal tools that integrate different forms of discourse in the same
environment, such as videopapers, are beginning to be investigated in the context
of teacher education. Videopapers (for an example, see Figure 2) are multimedia
documents that integrate and synchronise different forms of representation
including text, video and images, in one single non-linear cohesive document
(Nemirovsky et al., 2005; Olivero et al., 2004).

Combining the video with the text in a videopaper creates a fluid document that
is more explicit than the text or video alone, while remaining contained and
controlled by the author. Since their initial development in 1998 as an alternative
genre for the production, use, and dissemination of educational research, research
has investigated their potential and use in a variety of contexts ranging from
teacher education to professional development to research collaborative practices
(e.g., Barnes & Sutherland, 2007; Beardsley et al., 2007; Galvis & Nemirovsky,
2003; Nemirovsky, Lara-Meloy, Earnest, & Ribeiro, 2001; Smith & Krumsvik,
2007).

4 Videopapers are created with the free software VideoPaper Builder 3 (Nemirovsky et al., 2005),
downloadable from http://vpb.concord.org

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Considering videopapers as "products", Olivero et al. (2004) discuss the role
they might have in teacher education and how they might support the
representation and development of communities of practice where new ideas can
be expressed and experienced. Reading about new and innovative approaches to
teaching and learning can influences prospective teachers' beliefs and principles
and impact on the process of "becoming a teacher", but reading does not provide
any re-assuring image where the action is modelled vicariously. Moreover,
prospective teachers might also lack the confidence to experiment if they have few
realistic models to work from. Videopapers can provide prototypical instances of
(innovative) practices combined with a range of supporting warrants. The study 5
reported in Olivero et al. (2004) uses videopapers as a way to represent and
communicate innovative uses of ICT in mathematics teaching, based on research
findings, to both prospective and practising mathematics teachers. After reading
one of these videopapers, a teacher typically comments:

What is going on in classrooms is being communicated and it does make the
project look real, real pictures in real classrooms. Seeing it, helps you to
make sense out of it and gives you real models that you know are more than
just tips for teachers.

Such videopapers were seen as "more than tips for teachers" as they afforded
the opportunity to develop a different kind of knowledge for teaching - knowledge
not of "what to do next", but rather "knowledge of how to interpret and reflect on

' More information about the project can be found in Sutherland et al. (2004).

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

classroom practices" (Sherin, 2004, p. 17), because of the interplay between
theoretical and practical knowledge they incorporated. Similarly to Vails et al.
(2006), the teachers appreciated the theoretical knowledge embedded in the
videopaper as a tool that could help them "notice" relevant practices, meanings,
knowledge in the video:

I have shown this to a number of teachers in Bristol - maths teachers - 1 was
working with, they wanted something stimulating on proofs. So I did it but
they wanted the bit on the research reading and the thinking cut out because
they thought teachers would find it irrelevant and too time consuming. That's
a problem because the thinking and the discussing and reading were so
important. They are missing out [...] well sort of missing out parts of the
story.

Another teacher continues explaining how he thinks this kind of videopapers
can be useful to teachers, who, thanks to the videopaper, would benefit from the
results of the research project, in which the teachers collaborated with the
researchers, even without being part of it:

So they [teachers] would benefit I would hope from coming into my
classroom, well we can't afford that because we can't get supply cover so they
can watch my classroom instead [through the video in the videopaper]. They
would benefit from talking to Tim [researcher], well they can't do that but
they can understand what he is trying to say [through the text in the
videopaper]. And they'd benefit from having my lesson plan which we
worked on but they can adapt it for themselves. So they have everything -
even the research evidence backing up the principles and practice. And also
they can see the outcome, which is very important. If you do this they would
do this, and this is the reason why we're doing it, this is me doing it and this
is what they produce at the end.

This quote suggests how in this case the videopaper is a tool that may afford the
creation of and provide a space for the representation of communities of practice
that bring together researchers, practising teachers and prospective teachers.

Another project (Armstrong et al., 2005; Barnes & Sutherland, 2007) looks at
the creation of videopapers to represent a collaborative research process in which
both researchers and teachers interpreted classroom episodes within a series of
mathematics and science lessons. Similarly to what McGraw et al. (2007) found in
relation to the discussion of multimedia case studies, having people with different
knowledge and expertise looking at the same data provides the possibility for
different understandings of a lesson to be put forward and productively interact to
create knowledge. The subsequently created videopaper embodies this process,
enabling the multiple perspectives to coexist and to be grounded in the reality of
the classroom (Galvis & Nemirovsky, 2003). While these projects see the process
of creating a videopaper as an individual process, Zahn et al. (2006, p. 738)

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SALVADOR LLINARES AND FEDERICA OLIVERO

describe a pilot project on collaborative learning through advanced video
technologies and analyse the use of the DIVER system 6 by prospective teachers:

DIVER is based on the notion of a user diving into videos, for example,
creating new points of view onto a source video and commenting on these by
writing short text passages. Diving on video performs an important action for
establishing common ground that is characterised as "guided noticing".

The "diving" process can be shared and collaboratively developed.

Although videopapers are mainly seen as objects or "products", research has
shown that what is also important in terms of the development of prospective
teachers' knowledge are the phases of actively creating a videopaper. A number of
studies look at the use of videopapers as a reflective learning tool for prospective
teachers in different subjects (including mathematics) and its advantages and
disadvantages over more conventional use of videos, observation tasks and
assignments (Beardsley et al., 2007; Daniil, 2007; Lazarus & Olivero, 2007),
adopting very similar methodologies. In these projects, groups of prospective
teachers were asked to choose one lesson from those they taught in their teaching
practice and create a videopaper showing self-reflection on their practice, as an
alternative to the traditional assignment that consisted of writing an essay with the
same aim. All studies demonstrate the profound insight that is possible when
teachers use a medium, like videopapers, that allows them to represent and share
the vitality of their classrooms by means of capturing, preserving, and representing
events in ways that connect with their world, where different forms of knowledge
are continually being juxtaposed, as opposed to traditional text-based documents
(Olivero et al., 2004). Videopapers offer an easy to use tool for teachers to create
commentaries around teaching episodes, including reflection on their own practice
and reference to the underpinning theoretical ideas. This provides an essential
medium for teachers to help them improve their understanding and interpretation
of their practice (Carraher, Schliemann, & Brizuela, 2000; Derry et al., 2000; Pea,
2006), leading to knowledge construction and meaning making.

Research also shows that the process of creating a videopaper is crucial in
supporting teacher reflection and is qualitatively different from simply watching a
video from a whole lesson to reflect on one's practice (Daniil, 2007; Lazarus &
Olivero, 2007). These studies shows that it was the editing process, for example,
the process of having to select relevant clips to discuss, that fostered prospective
teachers' reflection, by initiating a more analytical process that might not
necessarily happen if they had just watched a whole lesson without needing to
produce a written text. Besides, the multimodal character of videopapers suggests
that the process of meaning making and knowledge construction, through
reflection, comes from the two modes (video and text) together.

http://www.diver.slanford.edu

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

Besides the process of creating a videopaper, also the process of reading a
videopaper has been an object of research. Video has been used in teacher
education and professional development in different forms since its introduction in
the 1960s (Sherin, 2004). However, watching a video without text is different from
reading a videopaper. Reading a videopaper can be described as a dialogue
between the reader and the author through the text, the video clips and the Play
buttons that refer the reader to particular clips while reading the text. Smith and
Krumsvik (2007) present the analysis of the reading processes of and discussion
around a videopaper integrating key educational theories that prospective teachers
are normally exposed to. Video illustrations of these theories, taken from the
authors' own teaching practice, are incorporated in the videopaper. This
videopaper was read by prospective teachers, teacher educators, and researchers in
conferences. Smith and Krumsvik argue that this way of using videopapers
contributes to bringing together communities that would not normally exchange
ideas around teaching; they also found that the prospective teachers appreciated the
fact that the practice field was brought to the university and the fact that they could
"see" the reality of the profession rather than just "hear" about it.

Overall we can say that videopapers mediate reflection on practice and
therefore enculturation in the mathematics teaching community for both the creator
and the reader. The mediation occurs through the multimodal character of
videopapers, which enables the processes of becoming, doing and experiencing.
Videopapers crystallise the reflection of the teacher creating them, through video
and text, and stimulate reflection in the teachers reading them, through the
dialogue and interaction with the video and the text created by the author. Multiple
perspectives are elicited, so contributing to the development of professional vision
and knowledge of mathematics teachers.

The tools discussed in this subchapter differ from the online learning
environments previously described in that they are artefacts that are constructed by
the prospective teachers themselves, with the purpose of representing knowledge
and eventually communicating and sharing it, as opposed to given artefacts within
which students may interact. Therefore, what emerges as important is the process
of writing a blog or creating a videopaper, first as private tools embodying
personal reflections and representing what is "seen" and "noticed" and then as
public tools communicating and sharing these reflections with others. The
collaborative aspect of these tools resides in this double face (creating and
communicating) and in the interactive process of reading a blog or videopaper,
through which multiple perspectives are elicited around a shared context,
represented for example by the video clip in a videopaper. Because these forms of
discourse are relatively new, they need to be gradually appropriated by both writers
and readers before they can really become an integral part of teaching and learning
processes.

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SALVADOR LLINARES AND FEDERICA OLIVERO

EMERGING PERSPECTIVES AND FUTURE VIEWPOINTS

In the previous subchapters, we have identified key factors that should be
considered when computer-supported communication tools are introduced in
mathematics teacher education and that seem to shape the characteristics of online
interactions, construction of knowledge and creation and support of communities
of practice.

The first key factor is the level of structure imposed on the context of
communication in the community and in the designed learning environment. The
form in which the activities in online interaction spaces are structured and the
establishment of clear goals, seem to mediate the process of knowledge building in
interactive learning environments and the creation and sustainability of
communities of practice. In some cases, allowing prospective teachers to build a
space that meets their needs helps the emergence of the sense of belonging to a
community and supports its sustainability over time (Goos & Benninson, 2004).
Here, the emergent design of the community contributed to its sustainability by
allowing the prospective teachers to define their own professional goals and
values. In this case "communities" emerge since prospective teachers have more
freedom to share their interests.

However, in other contexts, it was the pre-determined structure of the activity in
which prospective teachers were engaged that contributed to the development of
reflective dialogue, as for example in the learning environments integrating online
discussions and the analysis of segments of mathematics teaching. A structural
factor that seems to influence the involvement of prospective teachers in this type
of learning environment was that the prospective teachers had a clear
understanding of the goal of the activity (Llinares & Vails, 2007). In this
intervention in a mathematics methods course, the object of reflection that the
prospective teachers had to focus on was made salient since the initial phase and
the conditions were public. In this case, the thematic prompts given to the
prospective primary teachers seemed to contribute to the development of a shared
understanding of the situation. Another characteristic that determined the level of
participation of the prospective teachers was the nature of the questions posed and
the theoretical information included in the learning environment (thematic
prompts). The nature of the questions posed in the learning environment
contributed to keep the messages focused and to generate different interactions that
favoured the reification of meaning about mathematics teaching as a knowledge
building process. But a common aspect of all the interactions generated in different
social interaction spaces is that the activity towards the analysis of mathematics
teaching or the identification of one-self as belonging to a community of practice
were mediated by tools, like written text and technological tools (bulletin boards,
online discussions, writing in a blog, creating a videopaper). Prospective teachers
indicated that having to write and think about the different issues posed also
contributed to generating a reciprocal understanding and a sense of belonging to a
community. One thing that research has begun to point out is that the type of
participation and the reification processes in these new learning environments

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VIRTUAL COMMUNITIES OF PROSPECTIVE TEACHERS

seem to be dependent on the structure of the activity in which prospective teachers
are engaged (Zhu, 2006). But the focus on a shared interest helps prospective
teachers to engage in the joint activities enabling them to share the understanding
of a given situation (Llinares & Vails, 2007).

The second key factor to our understanding of prospective teacher learning is
the role played by new communication tools and by the nature of the generated
discourse. Communication through forums, bulletin boards, blogs or videopapers
mediates the process of becoming (learning as identity), doing (learning as
practice) and experience (learning as meaning) of the prospective mathematics
teachers. The sociocultural perspective on teacher learning underlines the role
played by the artefacts built by prospective teachers as communication
instruments. The prospective teachers' written contributions in an online
discussion, the process of creating a videopaper or a blog as a way of
communicating their understanding of facts and situations mediate the process of
knowledge building and, in some cases, the sense of belonging to a community.
The important thing is that individuals might come to understand a topic better,
share resources or have opportunities to define their own professional goals and
values, when they have to write in order to communicate to others - writing for
others using the new ways of communication that new technologies provide.
Writing, as a manifestation of the use of the new available communication
technologies, is understood from this perspective as a tool for collaborative
reflection and, at the same time, problem solving. Therefore, "writing" about a
topic is considered to be a powerful way of knowing. Posting a message in an
online discussion, exchanging ideas and experiences in bulletin boards, creating a
videopaper in a reflective context, or creating blogs as ways of sharing knowledge
and resources through a progressive discourse that takes place in "writing",
constitute the spaces where prospective teachers' personal interpretations are
questioned and clarified in an attempt to construct "common knowledge". The
"text" created within these news ways of communicating functions as an
improvable object, that provides the focus for progressive discourse and
simultaneously embodies the progress made; this might allow prospective teachers
to become acquainted with and understand the topic they are writing about and can
be seen as a dialogic process of knowledge (Andriessen, Erkens, van de Laank,
Peters, & Coirier, 2003; Wells, 2002). In these new contexts, discourse enables the
development and articulation of shared values.

In summary, the introduction of information and communication technologies in
teacher education also involves the development of sociocultural perspectives on
learning that provide new avenues through which teacher educators may attempt to
understand the process of becoming a mathematics teacher. Despite the scarcity of
research on the topic, the studies discussed in this chapter have shed light on key
factors related to the introduction and use of these new tools. Questions have
emerged too and these call for further research.

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ACKNOWLEDGEMENTS

The contribution of S. Llinares was supported by Ministerio de Educacion y
Ciencia, Direction General de Investigation, Spain, under grant no. SEJ2004-
05479.

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Salvador Llinares

Departamento de Innovation y Formation Diddctica,

University of Alicante

Spain

Federica Olivero
Graduate School of Education,
University of Bristol
United Kingdom

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MARCELO C. BORBA AND GEORGE GADANIDIS

8. VIRTUAL COMMUNITIES AND NETWORKS OF
PRACTISING MATHEMATICS TEACHERS

The Role of Technology in Collaboration

The collaboration of practising teachers in a virtual environment introduces the
technology tools themselves both as mediators and as participants - ay co-actors -
in the collaborative process. Although there is a growing literature on the
collaboration of practising teachers, the role of virtual technology tools is typically
not addressed In this chapter, we turn our attention to two cases, one in Brazil and
one in Canada, as we explore how tools mediate and interact in the way teachers
collaborate and construct knowledge. A challenge in this exploration is that
technological tools change dramatically over short periods of time. Some aspects
of teachers ' online learning that are brought to light by the two cases from Brazil
and Canada are: (I) virtual collaboration can happen in very different ways and
using very different tools and methods; (2) online technology tools can transform
abstract mathematics objects like polygons into tangible objects of communal
attention and action; (3) collaborative knowledge construction tools like wikis help
re-shape the collaborative process and transform roles played by teachers and
instructors; and (4) multimodal communication through drawing tools, rich text,
and video changes the "face " of mathematics. The virtual, non-human objects that
are part of collaborative collectives of humans-with-media are not tools that we
simply use for predetermined purposes. Humans-media interactions, which are
quickly evolving with changes in the online world, are organic, reorganizing and
restructuring our understanding of what it means for practising mathematics
teachers to collaborate in a virtual environment.

INTRODUCTION

Since the mid 1990s, as the WWW became available in the virtual world, there has
been resurgence and a redefinition of the idea of distance education. This modality
of education is of course much older and used regular mail and television as the
main means of communication between students and teacher. The main
characteristic of this kind of education is that, while ideas were transmitted,
students and teachers did not share the same space, and until the 1990s, the
interaction was limited to teacher-to-students dialogue and did not include student-
to-student dialogue. The Internet introduced the possibility of online collaboration
among teachers in "distance education" settings, using synchronous (occurring at
the same time) and asynchronous interaction with different modes of

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 181-206.
© 2008 Sense Publishers. All rights reserved

MARCELO C. BORBA AND GEORGE GADANIDIS

communication such as chat rooms, forums, wikis, videoconferences and
multimodal ones in which text, pictures, video and voice are combined in different
ways.

Distance education with the strong use of the Internet has been renamed online
education. This modality of education has been used in undergraduate courses, as
reported by Engelbrecht and Harding (2005), mainly with the independent learning
model, which stresses the download of didactical material by learners. In such a
model, the emphasis is on posting "instructional material on the web". Expressions
such as "self-learning" and others that negate the role of the teacher are associated
with this model of online education.

Interestingly, parallel to the growth of online learning in mathematics teacher
education and in teacher education in general, there has been a growing interest in
the collaboration of teachers. Krainer (2003), for example, notes that "Increasingly,
papers in teacher education refer to some kind of 'communities' among teachers"
(p. 94). There is also growing evidence that collaboration among teachers is a key
ingredient for their professional development (e.g., Krainer, 2001; Peter-Koop,
Santos- Wagner, Breen, & Begg, 2003). In trying to understand teacher professional
development, many distinctions have been made among terms such as cooperation,
collaboration, collegiality, teams, networks, and communities to address issues of
power, conflict, conflict resolution and reflection (Begg, 2003; Krainer, 2003;
Santos- Wagner, 2003; Lave & Wenger, 1991). Because of the recent availability of
online collaborative tools, it is not surprising that the role of technology has not
been addressed in most of the work on teacher collaboration.

However, some exceptions can be found in the last few years. Literature that
addresses issues of teacher collaboration in online and face-to-face settings
identifies a variety of methods for creating a collaborative focus: using multimedia
cases (McGraw, Lynch, Koc, Budak, & Brown, 2007; Llinares & Olivero, this
volume), identifying pedagogical issues of common interest (Arbaugh, 2003;
Groth, 2007), using student work as a focus of reflection and discussion (Kazemi &
Franke, 2004), and mathematical content (Lachance & Confrey, 2003; Davis &
Simmt, 2006). A gap in the literature on the collaboration of practising
mathematics teachers, and the focus of our chapter, is the role of virtual
environments and tools both as factors mediating teacher collaboration and as co-
actors in the collaborative process.

Our work on online teacher collaboration is based on a perspective that
knowledge is constructed in interactions with others, what has been labelled as a
Vygostikyan, sociocultural approach. By "others" we also refer to digital tools that
permeate our new media culture. Borba and Villarreal (2005) see humans-with-
media as actors in the production of knowledge and they note that humans-with-
media form a collective where new media also serve to disrupt and reorganize
human thinking. They base their view in authors such as Levy (1997), who
suggests that technology itself is an actor in the collaborative process. Levy sees
technology not simply as a tool used for human intentions, but rather as an integral
component of the cognitive ecology that forms when humans collaborate in a
technology immersive environment.

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In our research (Borba & Penteado, 2001; Gracias, 2003; Borba, 2005; Borba &
Villarreal, 2005; Gadanidis & Namukasa, 2005; Santos, 2006; Gadanidis,
Namukasa, & Moghaddam, in review) we have used online education models that
value the interaction among teacher educators and prospective or practising
teachers by using synchronous and asynchronous interactions. Another
characteristic of models we used in our courses and of the research that we
developed involves exploration of what is new in computer technology. The
medium used is considered in a deep way as a co-actor, that is an active, modifying
agent that transforms the collaborative process, in the same way that writing is seen
as being modified by the medium used, be it paper and pencil, or computer word
processing. Other authors in this area, such as McGraw et al. (2007) and Rey,
Penalva, and Llinares (2007) do not seem to take into consideration that the effect
of media is important; they assume writing in online environments to be "neutral",
as far as media is concerned, in their analysis of the way mathematicians,
mathematics teachers, prospective educators and practising teachers interact. It is
too early to try to draw strong conclusions regarding the role of the Internet in
mathematics teacher education as there is little research on media as a co-actor in
online education. Therefore, in this chapter, we would like to provide examples to
inspire those concerned with teacher education to think about this issue which, as
recently as 2005, was not a major topic in the ICMI study that took place in Brazil
(http://stwww.weizmann.ac.il/G-math/ICMI/log_in.html), although studies such as
Pateman, Dougherty, and Zilliox (2003), and Hoines and Fuglestad (2004) began
to appear at conferences such as the Psychology of Mathematics Education (PME).
We will first give examples that come from both Brazil 1 and Canada separately and
then a final one that involved collaboration between both research teams. In these
examples, we will show that, although we have experienced different kinds of
online courses and used different types of interfaces, there is a common underlying
goal of providing tools for interaction and supporting a collaborative culture among
participants and among teachers and researchers. We will also show how a given
tool shapes the nature of interaction, stressing, therefore, the role of media in the
way teachers collaborate and the way knowledge is produced among participants.

EXPERIENCES IN USING ONLINE MATHEMATICS EDUCATION IN BRAZIL

Brazilian mathematics education has organized itself, among other means, in
research groups. This type of organization has helped us to focus our efforts in
conducting research. For instance, the first author of this paper participates in
GPIMEM, a fifteen-year-old research group, registered in our national research
group database. Since 1993, we have studied the role of different software in
mathematics education, and how pedagogical approaches that involve students in
choosing problems to be solved are in resonance with the use of information

Some of the examples presented in this chapter from the Brazilian side have been presented since
2004 at PME conferences. See for example Borba (2005); Borba and Zulatto (2006); Borba (2007).

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technology, including the Internet. Since the late 1990s, we have been involved
with online education, specifically. First, we started studying the scarce literature
available then in a search for one model of online courses to guide our work. Since
2000, we offered online courses as a means of experiencing distance education, as
we continued to study the increasing number of studies and descriptions of online
distance education. Courses such as these are of paramount importance in Brazil
due to the size of the country and the concentration of knowledge production in the
southeastern region, where the states of Sao Paulo and Rio de Janeiro are located.
Internet- based courses are one way of connecting research centres such as Sao
Paulo State University (UNESP) with people in remote locations, where the closest
university may be more than several hundred kilometres away. This kind of
practice and research illustrate how online education can be a path for social
equity.

To present the way GPIMEM views online education, we will briefly depict the
way we conceptualise technology in mathematics education as a result of our
experience with software we used in the classroom. We have developed the
theoretical notion of humans-with-media (Borba & Vilarreal, 2005) as a means of
stressing the idea that knowledge is constructed by collectives which involve
humans and different technologies of intelligence (Levy, 1993), such as orality,
paper-and-pencil, and information and communication technology (ICT). Different
humans, or different technologies, result in different kinds of knowledge
production. Knowledge production involves humans and some medium. This
notion has provided important insights as we examined how different interfaces,
such as graphing calculators, function software, or dynamic geometry software,
affected knowledge production (Borba, 2004). We, as a group, illustrated how
different media shape the way mathematics is produced. For example, students-
with-graphing-calculators are more likely to raise conjectures and discuss
mathematical ideas related to them (see e.g., Borba & Villarreal, 2005) than
students working without such mathematical modelling and representation tools;
also, collectives of humans-with-geometry-software-paper-and-pencil have
integrated simulation and demonstration, as shown by Santos (2006). We were
curious to know whether the Internet would also be equally as active in the way
mathematics is done as we have argued that geometry and function software have
been.

Based on research developed by GPIMEM we discuss the following issues: (1)
our view about the role of technology in knowing; (2) our model of online courses
based on interactions; (3) the importance of synchronous relationships; and (4) the
active role of different internet interfaces on the construction of mathematical
knowledge by teachers. We elaborate on each of these below and discuss them in
the context of teacher collaboration.

Chat as the Main Interface for Online Courses

In order to learn how to teach online and to develop research, some members of
our group have been researching how collectives formed of humans-with- Internet

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construct knowledge. For this purpose, among others, we offered several online
courses for mathematics practising teachers. These courses were based on our
previous experience with technology, which led us to believe that real-time
interaction with the teacher is paramount when one works with technology. This is
why, in most of our courses, we emphasized that at least one part of the course be
synchronous, either through chat or videoconference. We have offered online
courses such as Trends in Mathematics Education. These courses have fostered the
development of communities that discuss issues related to the topics presented:
teaching and learning of functions and geometry using software,
ethnomathematics, modelling, adult education in mathematics, critical mathematics
education, and so on. Eight versions of the "Trends course" have been offered, all
of them with updated literature discussions along with adopting different virtual
environments and evolving changes in the model adopted.

Each course brings together approximately 20 practising teachers online at
regularly scheduled times over a period of about three months. The first author of
this chapter and a collaborator taught all the eight courses offered until 2007.
Almost every week, chat sessions of about three-hours long were scheduled. The
teachers who took the course each time were, for the most part high school
teachers, but university level teachers, teacher educators and others, such as
curriculum developers, also took part in a group that could thus be labelled as
heterogeneous. It was common in these chat sessions to have simultaneous
dialogues, since different teachers would pursue different aspects of a given
problem, or would pose a different problem, or talk about something that happened
recently in their classroom. Courses were, therefore, designed in such a way that
interaction was the key word. In 2003 and 2004, we used a free online
environment, Teleduc 2 , which requires a Linux server, but can be accessed by
computers that use different platforms. Chat rooms became the principal means of
synchronous interaction in the course. Preparation for a session would be done
through asynchronous interactions, mainly e-mail and regular mail. For example,
prior to a session on ethnomathematics, participants were mailed a book by
D'Ambrosio (2001), the major proponent of ethnomathematics. All participants
were expected to have read the book before the session, and two of them would be
responsible for raising questions to generate discussion. After the class, a third
teacher would generate a summary for the class that would be published in the
virtual environment of the course and be made accessible to all participants.

However, when the objective of the class involved doing mathematics a
different kind of preparation was required. Problems involving the use of function,
for example, were sent beforehand to the teachers, and they attempted to solve
them before the class; during the chat session, different solutions were discussed.
When we decided to include problem solving sessions, as part of our online course,
we also started to investigate how mathematics produced via chat might be shaped

TelEduc is a free plataform for online courses developed by Nied and the "Institute de Computacao da
Unicamp", chaired by Dr. Heloisa Vieira da Rocha. It is available for download at:
http://hera.nied.unicamp.br/teleduc.

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MARCELO C BORBA AND GEORGE GADANIDIS

by this particular medium, and this became an important research question for us as
mentioned before.

The problems that we posed to the teachers taking the online courses were
designed to be solved with the use of plotters such as Winplot, 3 or a geometry
software, Geometricks (Sadolin, 2000). Prior to a chat meeting, teachers could send
their solutions to the virtual environment used in a given course, which might then
be posted, for example, in a tool called portfolio. However, it was not possible to
simultaneously share a figure with the other course participants. Such a situation
generated discomfort for some participants. In the 2003 class, prior to a scheduled
chat meeting with all 20 teachers participating in one of the courses, a problem was
posed to them regarding Euclidean geometry. Different solutions and questions
were raised by all participants, but one teacher's reflections caught our special
attention; during the discussion, Eliane, 4 said: "I confess that, for the first time, I
felt the need for a face-to- face meeting right away [...] it lacks eye-to-eye contact."
She then followed up, explaining that discussing geometry made her want to see
people and to share a common blackboard. In this case, there was no follow-up
discussion to clarify what she meant. From her comment, we started to think about
an initial answer to our question regarding the transformation of mathematics in
online courses: the clash between Euclidean geometry, a symbol of space in our
culture, and virtual space, a symbol of the beginning of the 21 st Century, may
permeate things such as doing mathematics. Some of the teachers did feel a need to
share a screen during the synchronous interactions. Just seeing a solution in a
portfolio, and commenting on it in another part of the virtual environment was not
enough. As we will see later in this chapter, there are interfaces in the virtual world
that can overcome part of this discomfort. For our research, it meant that the
possibilities of the tool, of the virtual environment, shaped the way teachers were
producing mathematics collectively.

A more detailed example will show a different facet of the way chat may
transform the way mathematics is displayed, and in our view, this signifies changes
in mathematics.

In the 2004 class, we posed the following problem 3 to the teachers who
participated in the course, which is based on a true story that happened in a face-to-
face classroom, taught by the first author of this chapter:

Biology students at UNESP, S3o Paulo State University, take an introductory
course in pre-calculus/calculus. The teacher of this course asks the students to
explore, using a graphing calculator, what happens when the values 'a', 'b'
and 'c' in y = ax 2 + bx + c are changed. Students have to report on their
findings. One of them [Renata] stated: "When b is greater than zero, the

http://www.gregosetroianos.mat.br/softwinplot.asp

Eliane Matesco Cristovao, High School teacher, from the 2003 class.

Translation of this problem and of the excerpt from Portuguese into English was done by the author
and Anne Kepple.

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VIRTUAL COMMUNITIES OF PRACTISING TEACHERS

increasing part of the parabola will cross the y-axis [...]. When b is less than
zero, the decreasing part of the parabola will cross the y-axis." What do you
think of this statement? Justify your response.

The mathematics involved in the conjecture, and its accuracy according to
academic mathematics, is developed in detail in Borba and Villarreal (2005). But it
is interesting to see how the teachers, participants of this online course, dealt with
it. Some aspects of it were eliminated since they were seen as irrelevant to the
understanding of the dialogue, or because they were part of a conversation that was
not associated with the solution of the problem.

Carlos, a high school teacher, started the debate at 19:49:07 (these numbers
indicate the hour, minutes and seconds when the message reached the on-line
course), reporting on what one of his students, in a face-to-face class, had said:
"When a is negative, or b is positive, the parabola goes more to the right, but when
a is negative and b is also negative, the parabola goes more to the left." He
challenged the group to see if the student's sentence could lead them to solve the
problem. Since the debate was not gaining momentum, the professor of the course,
the first author of this chapter, tried to bring the group back to what Carlos had
said:

(19:53:15) Marcelo Borba: The solution that Carlos' student presented
regarding 'a' and 'b'. Does anyone have an algebraic explanation for it?

(19:54:53) Tais: It has something to do with the x coordinate of the vertex of
the parabola.

(19:55:30) Carlos: after a few attempts (constructing many graphs changing
the value of 'a', 'b', and 'c') the students concluded that what was proposed
by Renata [the Biology student who first stated the conjecture in one of
Borba's face-to face-classes] is really true.

The issues at stake are distinct. Carlos tried to do what the professor proposed to
the group, but Tais raised a new issue, the vertex idea. As can be observed on the
excerpt below, the two issues also have intersections:

(19:57:07) Tais: Xv = -b / 2a [...] if 'a' and 'b' have different signs, Xv is
positive.

(19:59:16) Norma: I constructed many graphs and I checked that it is correct,

afterwards I analyzed the coordinates of the parabola vertex Xv = -b / 2a, and

developed an analysis of the 'b' sign as a function of 'a' being positive or

negative, then I verified the sign of the vertex crossing. [...] with the

concavity upwards or downwards, and checked if it was increasing or

decreasing. [...] did I make myself clear?

I Norma presented her ideas, which according to the interpretation developed, are

| similar to the one made by Tais, and can be labelled the vertex solution. After

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MARCELO C. BORBA AND GEORGE GADANIDIS

further discussion about this, the professor presents another solution based on the
derivative of y, y' = 2ax + b:

(20:07:03) Marcelo Borba: Sandra, [...] I just saw it a little differently. I saw
it [...] I calculated y'(0) = b. [...] and therefore when 'b' is positive the
parabola will be increasing and analogously [...].

Since a few people said they did not understand this comment, the educator
went back to explain his solution.

(20:10:59) Marcelo Borba: [...] as I calculate the value of y', y 1 > 0, then the
function is increasing, and therefore I consider y'(0), which is equivalent to
the point at which y crosses the y-axis, and y'(0) = b, and therefore 'b' decides
the whole thing!!!! Got it?

(20:29:24) Badin: The parabola always intercepts the y-axis at the point
where the x coordinate is zero. In order for this point to belong to the
increasing "half of the parabola (a > 0), it should be left of the x v , this means
x v should be less than zero. Therefore, -b / 2a < is equivalent to -b <
(remember, a > 0). But -b < is equivalent to b > 0. In other words, if b > 0,
the point where the graph crosses the y-axis is in the increasing part of the
parabola. The demonstration for a < is analogous.

At this point, some of the teachers had been discussing the problem and both
solutions - the vertex and the derivative - for 40 minutes. The large spaces shown
by the clock between the different citations from course participants indicate the
size and number of sections that were eliminated in this paper, as there were about
four messages per minute. For 10 more minutes, additional refinement and shared
understanding of the solutions were presented. More examples of people's writing
about their understanding in the chat are available in the naturally recorded data.
Educational issues regarding the use of Winplot to explore the problem and
generate conjectures were discussed. But what is new about the Internet in this
case?

Before going further, the reader should be aware that some sentences were
omitted to make it easier to follow the interaction, and that the translation from
Portuguese into English suppressed most of the informality and typos that normally
occur in this kind of environment. There were other actors involved in the
discussion and refinement of the solutions of the problem, but for the purpose of
clarity, only a few are included here. When we compare the solution presented by
the teachers - the vertex one - to the original situation that took place in a normal
classroom situation in 1997, there are similarities and differences. Students used
graphing calculators to generate many conjectures for the problem relating
coefficients of parabolas of the type y = ax 2 + bx + c to different graphs. Similarly,
the teachers used Winplot (or other software, in some examples) to investigate the
problem just described, and later the problem related to Renata's conjecture. In the
face-to-face classroom, the professor (author) led the discussion, and eventually

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VIRTUAL COMMUNITIES OF PRACTISING TEACHERS

presented the vertex solution (as he did not know the answer either, at first). The
students never wrote the explanation for the conjecture. In an on-line learning
environment based on chat, writing is natural, and everyone involved had to
express themselves in writing (see also Llinares & Olivero, this volume). Although
we know that some aspects of writing in a chat situation are different compared to
writing with paper and pencil, there is a fair amount of research showing the
benefits of writing for learning (see e.g., Sterret, 1990). However, the data
presented here is insufficient, and the design of the study is inappropriate, to
support arguments about "benefits". Still it can be argued that chats transform the
mathematics that is produced by the participants of the course. The chat tool,
together with human beings, generate a kind of written mathematics that is
different from that developed in the face-to-face classroom, where gestures and
looks form part of the communication as well. We have been building evidence,
with other examples, of how collectives of humans-with-Internet-software generate
different kinds of knowledge, which does not mean that the mathematical results
were different. But if the process is considered, most of us at GPIMEM believe that
we may be on the way to discovering a qualitatively different medium that, like the
"click and drag" tool of the dynamic geometry, offers a new way of doing
mathematics that has the potential to change the mathematics produced, because
writing in non-mathematical language becomes a part of doing mathematics. At
this point, it is too early to confirm this, but we believe that this "working
hypothesis" (Lincoln & Guba, 1985) regarding the transformation of mathematics
by the Internet is one that we have been pursuing in research developed within
virtual environments like the ones described, but also in different ones, such as the
ones we present below.

In virtual collaboration, the participants that influence the nature and the focus
of the collaborative process are not only the human participants, but also the
technological tools and the collaboration affordances and constraints that they
introduce. In Borba and Penteado (2001), and Gracias (2003), we show that
"multialogues" - understood as simultaneous dialogues - are a characteristic of
interaction in chat rooms. Unlike in a regular class, when more than one person
talks at the same time only when there is group work, in chat, theoretically all can
"talk" at the same time. In a videoconference environment, new kinds of
collaboration emerge. Interaction and collaboration, two key words of our model of
online courses we have, can also be supported drawing on the literature on teacher
education, where authors such as Hargreaves (1998), Larrain and Hernandez
(2003), Fiorentini (2004), and Llinares and Krainer (2006) claim that collaboration
and sharing are powerful actions that generate new knowledge. If we bring these
ideas to online courses, we have a strong argument for generating courses that
emphasize interaction not only with the leader of the course, but among
participants. In a cyclic model, we researched different types of courses offered,
chose one that emphasizes collaboration, and investigated how different platform
interfaces for such courses shape the knowledge that is produced, or in other
words, how different collectives of humans-with-media produce different kinds of

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MARCELO C. BORBA AND GEORGE GADANIDIS

mathematics and collaboration. As we will see next, different interfaces mean
different possibilities of online education.

Video Conference as the Main Interface for Online Courses

A course, entitled "Geometry with Geometricks", was developed in response to a
demand from mathematics teachers from a network of schools sponsored by the
Bradesco Foundation spread throughout all the Brazilian states. The teachers from
these 40 schools, which include some in the Amazon rainforest, have access to
different kinds of activities, such as courses that are administered at a pedagogical
centre located in the greater Sao Paulo area. Following the improvement of Internet
connections in Brazil, the administrators of the school network realized that online
courses could become a good option, since sending teachers from different parts of
Brazil (which is larger in area than the continental US) to a single location to take
courses was neither cost nor pedagogical ly effective. Lerman and Zehetmeier (this
volume) point to cost as an important factor for programmes that involve practising
teachers when goals include encouraging teachers to reflect on their practice
outside the classroom. The cost factor is related to the size of the country, and the
pedagogical consideration is related to the fact that teachers would usually
participate in the courses for a short period of time, with little or no chance of
implementing new ideas while still taking the course. Our model, based on online
interaction and applications in their face-to-face classes in middle and high school,
gained respect gradually within this institution.

The pedagogical headquarters of this network of schools approached us, asking
for a course about how to teach geometry using Geometricks, dynamic geometry
software originally published in Danish and translated into Portuguese. It has most
of the basic commands of other software such as Cabri II and Geometer Sketchpad,
and it was designed for plane geometry. As we know from previous research on the
interaction of information technology and mathematics education, just having a
piece of software available, and a well-equipped laboratory with 50
microcomputers, as is the case of these schools, is not enough to guarantee their
successful use, even if the teachers are paid above average compared to their
colleagues from other schools.

In our research group, we designed a course using an exploratory problem
solving approach similar to that discussed by Schoenfeld (2005); it was divided
into four themes within geometry (basic activities, similarity, symmetry and
analytic geometry). Problems usually had more than one way of being solved, and
they could be incorporated at different grade levels of the curriculum according to
the degree of requirements for a solution, and according to the preference of the
teacher. Both intuitive and formal solutions were recognized as being important,
and the articulation of trial-and-error and geometrical arguments were encouraged.
We "met" online for two hours on eight Saturday mornings over a period of
approximately three months. Besides this synchronous activity, there was a fair
amount of e-mail exchanged during the week for clarification regarding the
problems proposed and technical issues regarding the software; in addition, also

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pedagogical issues regarding the use of computer software in the classroom were
raised, for example: should we introduce a concept in the regular classroom and
then take the students to the laboratory, or the other way around? Pedagogical
themes were also discussed during the online meetings, in particular in one session
in which, instead of the students working on problems during the week, they had to
read a short book about the use of computers in mathematics education (Borba &
Penteado, 2001). We encouraged teachers to solve problems together in face-to-
face or online fashion.

The Bradesco Foundation had already purchased an online platform that allowed
participants to have access to chat, forum, e-mail, and video conference that
allowed the download of activities, as well. In our course, participants could
download problems, and they could also post their solutions if they wanted to, or
they could send them privately to one of the leaders of the course (the authors of
this paper). The platform allowed the screen of any of the participants to be shared
with everyone else. For example, we could start showing a screen of Geometricks
on our computer, and everyone else could see the dragging that we were
performing on a given geometrical construction. A special feature, which is
important for the purpose of this paper, was the capability to "pass the pen" to
another participant who could then add to what we had done on a Geometricks file.
In this case technological possibilities transformed the way collaboration could
happen. Different teachers, who were taking the online course, could lead a
problem solving activity. As it will be described, there were times when one
teacher would have the pen, and another would be commenting or giving
instructions on how to proceed with a given problem.

This example can illustrate how the convergence of different ideas generates the
collective construction of knowledge about geometry (content knowledge) about
use of a given geometry software in the classroom (pedagogical content
knowledge), and about use of the geometry software itself (technological
knowledge). This problem involved symmetry. A Geometricks file had already
been given to them with the figure MNOPQ, presented below (Figure 1). Teachers
were asked to find the symmetric figure, in relation to axis "q". Teachers were
reminded in the text that the symmetric figure had to remain symmetric even after
being dragged.

Figure I. Picture of the file given to teachers.

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MARCELO C. BORBA AND GEORGE GADANID1S

As mentioned earlier, teachers were given each problem before the synchronous
sessions and could interact with one of the teachers of the course by e-mail or other
means. This allowed us to sometimes choose issues to start the debate and, at the
same time, we could also limit undesired exposure of some errors. For this
problem, the vast majority of solutions used a "count dot approach" in which they
counted how many dots a given vertex was distant from "q". The result is
visualized in Figure 2. We invited a volunteer to make a construction. We passed
the pen to one participant who offered to do so, who in turn was helped by another
who was acting like a sports narrator. After the construction was done, we asked
questions such as: is MVWZQ symmetric to MNOPQ? The "argument" was quite
intense, and the participants were divided. The issue about dragging emerged, and
we came to the conclusion that if the dragging of a vertex is considered to be
essential, that solution would not generate a symmetric figure (see Figure 3).

. . >»/ \p.

- M ■»

• • V< >2

•>

p'.

Figure 2. The first
solution of the teachers.

Figure 3. Dragging and
passes the test of dragging.

Different solutions were developed by different actors who took over the pen
and/or whose voices emerged, including the one presented in Figure 4 - where
circumferences are used playing the role of a compass in dynamic geometry - that
would still be symmetric after dragging since, as we pull a point away from the
axis, for example, the symmetric point will do the same, since it is part of the
circumference. This issue is relevant, and for authors such as Laborde (1998) and
Arzarello, Micheletti, Olivero, and Robutti (1998), a straight quadrilateral with
four equal sides and four angles is only considered a square in a dynamic geometry
software if it passes the "test of dragging", which means, in this case, that it is still
a square after different kinds of manipulation are performed. For these authors, if
the resulting figure is no longer a square, we did not have a construction of a
square but just a drawing. We agree with these authors that this is an important
issue, even though in our course, we also emphasized the relevant role of "drawing
solutions" depending on the grade level, the complexity of the problem, or as a

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VIRTUAL COMMUNITIES OF PRACTISING TEACHERS

path for more complex solutions. Other examples such as these were developed
during this course. Teachers were overwhelmingly positive about the idea of
collective problem solving during the synchronous videoconference sessions,
especially when they compared the experience with others in which just we, the
leaders, would present our solution or present one of their solutions in a more
expository manner of teaching. Our own assessment, as leaders, was that this
virtual collaboration created a bonding among teachers that was not experienced
when we did not use this technical feature. The issue about drawing versus
construction is not new in the literature; what we believe to be original in the
episode reported is the fact that it came from an online course, and that it resulted
from collaboration among teachers. In such collaboration, they learned from each
other, which is considered important by authors such as Lin and Ponte (this
volume).

: : :, ^xs

/'./.\. . .y. . ./..-.-■

Figure 4. The second solution.

Hargreaves (2001) proposed that teaching is a paradoxical profession. On the
one hand, everyone expects from teachers, even in developing Third World
countries, efforts to help the construction of learning communities in the new
"knowledge society". Even though most teachers were not educated in a time when
microcomputers were extensively available, they are required to teach using
computers and software in the classroom. On the other hand, there is discourse in
governments and segments of society that claims we should have a "shrinking" of
the state due to the needs of this knowledge society, and teachers become one of
their first victims with funding cuts for sectors such as education. Teachers should
enhance the knowledge society, even though they are some of their first victims.
Whether teachers are aware of this dilemma or not, they feel the need, and there is
institutional pressure, to constantly be working on their professional development.
Integrating software into the face-to-face classroom is still a challenge as teachers
enter a risk zone (Penteado, 2001) in which the students sometimes know better
how to manipulate computers and often come up with original solutions to a given

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MARCELOC. BORBA AND GEORGE GADANIDIS

problem, or even create new problems which are not easy to handle. Soon the
demand to incorporate the Internet in the classroom should rise as it seems like we
are moving towards a "blended" learning approach, in which virtual and face-to-
face interaction will happen "in" the classroom.

It makes sense, therefore, that teachers experience online courses, in particular if
one considers the cost factor, as in the case described above. Reimann (2005), in
his plenary at PME 29, emphasized the need to build artefacts on the Internet that
could foster collaboration. We agree and believe that we are helping to construct
collaborative practices within online continuing education courses offered to
mathematics teachers. This type of collaboration was possible due to a design in
the platform that allowed us to do so. Of course, having the possibility is just one
step, and a pedagogical approach that enhanced participation should always be
pursued. But we want to emphasize that we should also be creating demands for
technical developments in online platforms, and this is an area in which our
research group, GPIMEM, has started working in the last few years.

We would like to say that once design features are incorporated either into
software, such as dragging, or into online platforms, like the "pass the pen" option,
participants become co-actors in the process of creating and recreating knowledge.
Different media, different people, imply different perspectives in the knowledge
constructed. That is why we believe that using the construct of humans-with-media
is useful to analyse educational practices that use technology, since they foster the
search for specific uses of an available interface. "Pass the pen" was a unique
characteristic of this platform that made possible the co-construction of
mathematical knowledge in an online course.

Engelbrecht and Harding (2005) have presented a study that shows that many
courses offered are based on independent learning with very little interaction
among participants. Those who take courses like these are expected to download
material and learn by themselves, and they are then assessed through some kind of
standardized test. Many teachers can be enrolled in courses like these, thus
generating profit for the organizers, and they dismiss the role of interaction in
teachers' professional development. At the other end of the spectrum are courses
that use the Internet as a means of generating quick feedback among participants.
For example, forum is used for asynchronous relations so that participants express
their ideas and doubts and post solutions to problems. Synchronous relationships
that employ chats, videoconference, and other tools, make it possible to share ideas
in real time, even if people do not share the same space. Courses such as these are
based on the idea that media should not be domesticated, which means that we, as
mathematics educators, should try to design curricula that fully explore the
possibilities of new media or interfaces. In the examples presented, we emphasized
how mathematics gains different characteristics, and teachers collaborate in
different ways, depending on the interface used. At GPIMEM, we are in the
process of learning from these experiences to shed light on face-to-face
mathematics education, as we are more knowledgeable about the role of different
media in the production of knowledge. As the reader will see, there are
commonalities and particularities as we learn some from the Canadian experience.

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THE CASE OF ONLINE PRACTISING MATHEMATICS TEACHER EDUCATION IN
ONTARIO, CANADA: ISSUES AFFECTING TEACHER COLLABORATION

Our experience, in the Canadian case, based on online education courses offered to
practising teachers at the Faculty of Education, University of Western Ontario, in
Canada, points to the following issues: (1) online courses versus face-to-face
courses; (2) asynchronous versus synchronous online course communication; (3)
text-based versus multimodal communication; and (4) read-only versus read/write
communication. We elaborate on each of these below and discuss them in the
context of teacher collaboration.

Context

Our Faculty of Education has three teacher education programmes: a teacher
education programme, a continuing teacher education programme, and a graduate
programme (masters and PhD). Our continuing teacher education programme
offers over 150 online courses to Ontario teachers, with approximately 5,000
teachers taking our courses annually. To support the development of community
and collaboration, the number of participants in each of the courses we offer is
capped at 25 teachers. These courses are part of a provincially mandated and
certified regimen of additional qualifications which lead to teacher professional
development and in some cases to salary increases. Six of these courses are
specifically for mathematics teachers. In addition, we offer three mathematics-for-
teachers courses which are not part of the provincial regimen of courses. Our
graduate programme offers a fully certified online Master of Education degree in
addition to its traditional face-to-face masters and PhD degree programmes (where
some courses are also offered either fully or partially online). Our prospective
teacher education programme offers some components of its mathematics and
language arts programme online, where we replaced large group lectures with
online content and discussion (Gadanidis & Rich, 2003).

Face-to-Face Versus Online Classrooms

In the case of courses for practising teachers, our experience indicates that most of
them prefer online as opposed to face-to-face courses. Over ten years ago, our
continuing teacher education programme was fully face-to-face, and we offered our
courses in the evenings, at weekends and during the summer break. We offered
courses on campus as well as in remote areas, with approximately 2,700 teachers
taking our courses annually. However, once we started offering courses online,
teachers opted for these rather than the face-to-race courses. Currently, our
continuing teacher education programme is approximately 95 percent online. It is
important to note that we do still offer the face-to-face courses, however teachers
choose not to take them. It is also important to note that there are other providers
offering the same courses, both face-to-face and online, and they have experienced
a similar trend. The main reason for this shift from face-to-face to online, based on

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surveys we conduct on a regular basis, is that teachers lead busy lives, in and out of
school, and having to attend classes at set times is a scheduling burden.

Asynchronous Versus Synchronous Classrooms

Teachers also tell us that the asynchronous nature of our online program makes it
appealing. Rather than having to schedule their lives around an arbitrarily
scheduled class, they can choose to participate during a time in the day or night
that is most convenient for them. The indication we have from teacher feedback is
that if our online courses used a synchronous mode, many teachers would either
opt to take online courses from another provider that used an asynchronous mode
or they would take a face-to-face course. It is important to note that we have a
significant number of Ontario teachers that teach overseas in countries such as the
Middle East and Singapore and it would not be possible for them to participate in
synchronous discussions because of time zone differences. The fact that most of
our practising teachers choose our online rather than our face-to-face courses, and
indicate that they prefer asynchronous rather than synchronous online discussion
environments, does not necessarily mean that one mode is educationally better than
another, or that it is a simple either-or choice between modes (as hybrid
environments can and do exist). However, it is interesting to consider some of the
differences between these modes and how the differences may impact on teacher
online collaboration.

Face-to-face and synchronous online discussions are temporal experiences.
They occur in real time and they have a linear quality. In the Brazilian case we
noted that online synchronous chat is different to face-to-face synchronous
communication because the former allows for more than one discussion theme to
be conducted simultaneously, with chat postings woven into a complex tapestry of
ideas being discussed simultaneously. This would be analogous to the (impossible)
face-to-face situation where teachers work in small groups but somehow everyone
can hear what everyone else is saying without the dialogues overlapping. The
tapestry of multi-theme postings in an online synchronous chat may cause some
confusion or disorientation. However, as we noted in the Brazilian case, this multi-
linear, multi-tasking environment can provide a novel, complex and rich
experience of collaboration.

In the Ontario case, a chat tool does exist in the e-learning platform we employ.
However, the chat tool is rarely used. The asynchronous online discussion used in
our courses introduces some interesting affordances for collaboration. Unlike
typical synchronous discussions (especially when the synchronous communication
is oral rather than textual), where it is possible for a small number of people to
dominate the discussion, asynchronous discussion makes it possible for everyone
to contribute to every discussion theme. In fact, this is an expectation in our on|ine
courses. A significant part of the assessment (typically about 30%) focuses on the
discussion component, and teacher participation is assessed based on its frequency,
regularity and quality. Another difference is that in an asynchronous environment,

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teachers can take more time to think about the ideas of others and to craft their own
responses before posting in the online discussion.

Textual Versus Multimodal Communication

Until the end of 2004, our online courses used a platform where communication
was primarily textual. However, in 2004, we made a decision to offer some of our
mathematics course for elementary teachers in an online setting. This created a
challenge because of the lack of an ability to communicate visual aspects of
mathematics, like diagrams. Consequently, late in 2004, we developed a new
online platform (Gadanidis, 2007) that offered multimodal communication by
allowing users to embed the following within discussion postings: rich text (using a
text editor that is similar to those found in a word processor); diagrams (using a
built-in draw tool); video or audio (using a built-in tool that allowed the capture of
video or audio from a web cam); as well as multimedia content, such as JPEG
images and Flash interactive content. The immediate difference this made to online
discussion was the visual appeal. Opening a discussion thread the reader was faced
with postings where text was bolded, coloured and formatted, and accompanied
with colourful mathematical drawings and images. Figure 5 shows a drawing
created by an elementary teacher to illustrate her conception of what parallel lines
may be transformed onto a sphere, a flat piece of paper and a rolled piece of paper.

paplt

Figure 5. An elementary teacher uses the Draw Tool to show
three representations of "parallel" lines.

Figure 6 shows the diagram an elementary teacher created to explain to another
teacher how slope is calculated. Does this type of communication make a
difference? Kress (2003) and Kress and van Leeuwen (1996, p. Ill) suggest that in
a digital environment "meaning is made in many different ways, always, in the
many different modes and media which are co-present in a communicational
ensemble". In terms of mathematics meaning making, being able to communicate
via images and user-created diagrams adds layers of meaning and elaboration that
enhance how ideas are expressed and understood. In addition, the ability to add
video in the discussion postings (using a web cam) makes the online discussion

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feel personal. For example, prior to a face-to-face graduate seminar in Brazil, video
postings were used to introduce the Canadian instructors and the Brazilian students
to one another and to introduce ideas to be discussed in the seminar. Also, the
ability to add video allows for embodied communication of mathematics, through
gestures and the use of physical materials. Figure 7 shows how elementary school
students in Brazil used the video capture tool to illustrate how L patterns can be
used to physically represent odd numbers and their sums as square patterns. With
the steady growth of bandwidth, the mode of online interaction and the content
generated are increasingly multimodal. It makes sense that online discussion and
collaboration among mathematics teachers would use the multimodal
communication tools available.

ender o que eu quero

Ldrissa

Jessica

Figure 6. An elementary teacher 's
explanation of slope.

Figure 7. Students in Brazil posting
videos of physical patterns.

It should be noted that because the "Mathematics-for-Teachers" courses
developed in the Ontario case were aimed at elementary teachers, the drawing tool
was sufficient for most needs to communicate visual aspects of mathematical ideas.
If this platform was used for discussing more complex mathematics, this could be
accommodated by embedding Flash applets in discussion postings that allow
teachers to explore more complex ideas using graph plotters, probability
simulations, etc.

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Read-Only Versus Read/Write Communication

The Web is today in the process of transforming itself from a read-only to a
read/write environment. In terms of online education, this transformation is
perhaps best reflected through the use of wilds. A wiki is a collaborative website
that can be edited by anyone who has access to it. When in late 2004 we created a
platform that allowed for the multimodal communication discussed above, we also
created the ability to use wiki discussion postings. When someone creates a
discussion posting, they have the option of making it a wiki posting. This means
that everyone else in the discussion can edit that posting.

The wiki feature has been used in a graduate course called the "Analysis of
Teaching" to create collaborative tasks that focus on mathematics pedagogy. For
example, we have used the transcript of a typical grade 8 US mathematics lesson as
depicted in the 1995 TIMSS Video Study (see excerpt below), by posting this
transcript in a wiki posting and then asking teachers to collaboratively edit the
transcript in ways that would improve the quality of the lesson. This is an
interesting and rather complex collaborative writing experience in that as teachers
start editing the transcript, other teachers have to adapt their own ideas to fit with
the edits made up to that point. This does not mean that they cannot edit the edits
of other students, but rather that they have to keep the changing flow of the lesson
in mind as they make edits: they cannot simply make an edit to one section without
taking into account the edits that precede and follow. That is, students need to
understand the edits of other students - seeing the lesson through the pedagogical
lenses of others - and then negotiate a synthesis of their edits with existing edits.
Thus the final edited lesson is the result of sophisticated collaborative editing
process.

Teacher: Here we have vertical angles and supplementary angles. [...]. Angle

A is vertical to which angle?

Students: 70.

Teacher: Therefore, angle A must be [...].

Students: 70.

Teacher: 70, right. Go from there. Now you have supplementary angles.

Don't you? [...]. Now, what angle is supplementary to angle B, I mean, I'm

sorry, Angle A?

Students: B.

Teacher: [...] and so is [...]?

Students: C.

Teacher. Supplementary angles add up to what number?

Student: 180.

The TIMSS Video Study (Stiegler, 1999) activity is one of the first collaborative
writing activities we use in the "Analysis-of-Teaching" course, to get teachers
comfortable with writing collaboratively and editing the ideas of others. Because
they are editing a transcript that is not their own, it is easier for them to make edits
without feeling that they might hurt someone's feelings. Later on in the course,

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students post transcripts from their own teaching, and discuss these in wiki
postings. Our experience at the graduate level is that teachers are very hesitant to
edit the ideas of others or to have their own ideas edited. One other thing that we
do to help shift their perspective is to discuss how the peer review process works
for scholarly writing. For example, when we submit a paper for possible
publication in a scholarly journal, we receive feedback from the reviewers about
edits that we could make to improve the paper. Sometimes, these edits are written
on a separate piece of paper, sometimes they are written in the margins, and
sometimes there are suggested edits in the text of the paper. This is typically a
tremendous learning process, both for the reviewer and for the author. We thus try
to emphasise that peer review and peer editing is a natural and rewarding scholarly
experience. Nonetheless, some students never fully engage in the process, and
instead make superficial edits or react negatively to edits of their own work. It
takes more than a couple of collaborative writing activities to shift some students'
view of graduate work as personal and private and of the instructor as the only
person who has the right to give feedback and make editing suggestions.

A LOOK TO THE FUTURE

Borba and Villarreal (2005) have argued how different software interfaces interact
with our cognition and reorganize our thinking. Our thinking about online
education has been disrupted and reorganized as we have used and thought with
emerging online affordances like multimodal and read/write communication. For
example, using a wiki in our online teaching is a very different experience than
teaching in a physical classroom. It is also very different from using simple text-
based discussion platforms with read-only discussion postings. Using a wiki does
not only disrupt and reorganize our thinking about how we structure classroom
interaction, it also becomes a lens that changes how we see other aspects of our
online teaching, such as course content, evaluation practices, our role as
instructors, and generally what constitutes knowledge and how it is or should be
constructed collaboratively in an online environment. The emerging multimodal
and collaborative tools of online teaching and learning environments are not simply
tools that we use for predetermined purposes. Rather, they can be seen as co-actors
in the cognitive ecology of online environments, existing in a complex, organic
relationship between humans and technology (Gadanidis & Borba, 2008).

Collaborating without Geographical Boundaries

From the examples presented in the Brazilian cases and in the Canadian cases, the
reader may be led to believe that online education is a solution for many of the
problems of teaching education, and that it brings no problems. In the Canadian
case, for example, where teachers have the option of choosing an online or a face-
to- face version of a course, they overwhelmingly opt for the online version. In the
Brazilian case there has been very positive assessment of the online courses made
not only by the teachers who participated in the courses, but also by administrators

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of the private schools who paid for the courses attended by their teachers, as
reported by Borba (2007). On the other hand, in other courses, in which there was a
fair number of teachers dropping out, members of the research group based in
Brazil attempted to know the reasons why teachers left the course, but they could
not, since there were not enough respondents. In addition, in the Brazilian case, a
few problems have already been identified, such as the lack of a common
geometric figure to share synchronously in some environments, but at the same
time we have also shown how different interface such as videoconferencing can
solve such a limitation. But in other papers we have pointed out limitations such as
technical difficulties to deal with a given platform or even inability to type fast
enough in order to participate in a session (Borba, Malheiros, & Zulatto, 2007).
Other authors such as Ponte and Santos (2005) and Ponte, Oliveira, Varandas,
Oliveira, and Fonseca (2007) have shown how different teachers live this
experience in different ways. They report that although some teachers are positive
about the online experience, there are others who are far from that, for reasons
which include the fact that once you "say/write" something it is recorded
electronically and accessible for everyone to read. So we can say that the modality
of online courses for teachers is still open for debate and for research. Moreover,
we need to investigate whether styles of learning are connected to the medium
used. It may be the case that some teachers prefer online and others like face-to-
face-courses.

One aspect of teachers' online learning that is brought to light by the two cases
from Brazil and Canada is that collaboration can happen in very different ways and
using very different tools and methods. For example, if you read only the Brazilian
case, you might appreciate the value of synchronous communication as a tool for
online collaboration. On the other hand, if you read only the Canadian case, you
might appreciate the value of asynchronous communication as a tool for online
collaboration. Another aspect that is brought to light by both cases is that the online
world is changing at a rapid pace. In both cases, the technological tools used
changed dramatically over short periods of time. The reason for this is the rapid
development of new online technologies coupled with the rapid growth of Internet
access and bandwidth. The text-based, read-only online world of a few years ago is
rapidly evolving into a multimodal, read/write social networking environment
(Sprague, Maddux, Ferdig, & Albion, 2007). This is bound to have an impact on
the virtual collaboration of practising mathematics teachers. The question is how
widespread will this impact be? In a review of online education, Sprague et al.
(2007, p. 158) suggest "that so-called 'early-adopters' of technology may have
made up the majority of faculty and students who have so far been involved in the
online education phenomenon". It will be interesting to look back five or ten years
from now and see whether the use of the new collaborative affordances of the
WWW is also limited mostly to the "early-adopters" or whether their use pervades
online mathematics teacher education. In the same direction, it should be
interesting to confirm whether, in fact, the online world has been "an actor" in
transforming mathematics. This has been proposed by Borba and Villarreal (2005)

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in the case of the introduction of given software, and in the way we are proposing
for online mathematics teacher education.

Given the ongoing development of the WWW as a social/collaborative
environment, we believe that a promising path for research is the one that
investigates the synergy between the virtual tools used and the collaborative nature
of the online education/professional development of practising mathematics
teachers. Moreover, the virtual tools that are part of collaborative collectives of
humans-with-media are not tools that we simply use for predetermined purposes:
they are active "participants" in the collaborative process. Human-media
interactions, which are quickly evolving with changes in the online world, are
organic, reorganizing and restructuring our understanding of what it means for
practising mathematics teachers to collaborate in a virtual environment. For
example, we have seen in the Brazilian case how online technology tools can
transform abstract mathematics objects like polygons into tangible objects of
communal attention and action. What changes mathematically and what changes in
terms of the collaborative process when mathematical objects become communal
objects? We have also seen in the Canadian case how multimodal communication
through drawing tools, rich text, and video changes the "face" of mathematics.
How does mathematics change when it is expressed in multimodal forms? How
does a media-rich environment affect the collaborative process?

In typical online teacher education settings, the mode of communication is
textual. Texts on chats and forums are changing the way mathematics is expressed.
Bringing everyday language to the forefront of mathematics communication is
already transforming mathematics that is constructed in online mathematics
education. In the cases presented, communication also involved diagrams, pictures,
video and interactive content. These modes of communication are increasingly
changing the mathematics that is constructed in such online environments. We
believe that these are examples of how non-human authors become co-actors in the
production of mathematical knowledge.

In the Brazilian case we can see the "passing the pen" tool transforming
collaboration as teachers can collectively and synchronously solve a problem in a
natural way. In the Canadian case, we can see teachers learning to have communal
production, which can be also seen as being shaped by the possibilities that the
wiki-like environment they used for asynchronous communication. We believe that
the examples presented in this chapter illustrate how tools become co-actors in the
way teachers collaborate and construct knowledge. Design of online tools is a topic
we should be interested as these non-human objects can become parts of
collectives of humans-with-media that produce knowledge. To study the role of
online tools on the role of collaboration in teacher education seems to be a
promising path for research, as the examples presented are not sufficient for us to
assure that online tools shape mathematics and collaboration. At this point, it
seems that we have to see virtual tools as co-actors in the collaboration of
practising teachers as a working hypothesis to be confirmed, rejected or re-
elaborated in the future.

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VIRTUAL COMMUNITIES OF PRACTISING TEACHERS

Also, since the social nature of the WWW appears to be permeating our society
(with the emergence of social networking sites such as MySpace and YouTube), it
might be worth exploring less formal or perhaps emergent collaborative
relationships among mathematics teachers. It might also be worth exploring the
effect of the pervasive nature of WWW on equity of access, as both the Brazilian
and the Canadian cases involve both urban as well as geographically remote areas.

ACKNOWLEDGEMENTS

Although they are not responsible for the content of this chapter, the authors would
like to thank Ricardo Scucuglia, Marcus Maltempi, Silvana Santos, Sandra
Barbosa, Ana Paula Malheiros, Regina Franchi and Rubia Zulatto, members of
GPIMEM, UNESP, for their comments on earlier versions of this chapter. We
would also like to thank CNPq, FAPESP and SSHRC, funding agencies of the
Brazilian and Canadian government for partially funding research presented in this
chapter. Finally we would like to thank the editors of this book for the review of
this chapter.

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Marcelo C. Borba

Department of Mathematics

Graduate Program in Mathematics Education

Sao Paulo State University

Brazil

George Gadanidis
Faculty of Education
University of Western Ontario
Canada

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SECTION 4

SCHOOLS, REGIONS AND NATIONS AS
MATHEMATICS LEARNERS

ELHAM KAZEMI

9. SCHOOL DEVELOPMENT AS A MEANS OF

IMPROVING MATHEMATICS TEACHING AND

LEARNING

Towards Multidirectional Analyses of Learning across Contexts

This chapter focuses on supporting the teaching and learning of mathematics
through school development. School development entails organizational learning
and the use of social, structural and material resources to support teaching and
learning: how are schools or mathematics departments organized to support the
teaching and learning of mathematics? I begin by reviewing why and how
researchers have conceptualized the school as a productive setting for supporting
and improving the teaching and learning of mathematics. I draw on literature from
organizational learning and professional communities to show how recent studies
theorize improving instruction and student learning. Much of the literature
emphasizes mechanisms for such improvement to be highly dependent on how
schools employ resources and how they are organized to support teachers'
collective learning together outside of the classroom. The chapter ends by
proposing new directions for research on professional inquiry and its relation to
the improvement of teaching and learning mathematics and school development.

INTRODUCTION

In this chapter, I take up the issue of how schools cultivate organizational supports
for teaching that aims to successfully engage all kinds of learners in complex
mathematical learning. In particular, I hold a view of mathematical learning that
respects students' intellectual integrity and considers schools to be places for
vibrant intellectual engagement in which students and teachers develop
sophisticated understanding of mathematics, identities as mathematical thinkers,
and are engaged in ongoing inquiry. My interest in school development is to
understand what allows the school community to continually strive to improve
mathematics instruction in order to strengthen student learning. Much of the
literature on school development naturally focuses on supporting teacher learning
and building teacher capacity (see e.g., Borko, Wolf, Simone, & Uchiyama, 2003;
Lerman & Zehetmeier, this volume; Nickerson, this volume). Thus, a large portion
of this chapter is devoted to research focused on cultivating professional
communities where teachers are the major players. In considering school
development, 1 view membership in a school's professional community to include
teachers, instructional leaders (e.g., coaches), and administrative leaders. A school

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 209-230.
© 2008 Sense Publishers. All rights reserved

ELHAM KAZEMI

development perspective emphasizes the interdependence of professionals in the
school community - how schools carve out time and space and employ social,
intellectual, and material resources to support joint inquiry among teachers,
instructional leaders and administrators (Collinson, in press; Lam pert, Boerst, &
Graziani, under review). I review how researchers have conceptualized and
described this joint inquiry and propose directions for future research.

DEVELOPING A SCHOOL'S ABILITY TO SUPPORT MATHEMATICAL LEARNING

What Is Organizational Learning?

One of the strengths of the mathematics education literature has been our careful
attention to supporting teachers to improve their instruction. My goal in this
chapter is to harness our understandings of supporting individual teachers to think
about school-wide professional learning (see also Knapp, 1997; Krainer, 2006;
Oliveira & Hannula, this volume; Perrin-Glorian, DeBlois, & Robert, this volume).
One clear area of growth for mathematics education research on school
development is to develop a framework for explaining how activities engaged by
teachers and other professionals at the school level achieve organizational purposes
and affect student learning outcomes (Boreham & Morgan, 2004). Collinson and
Cook (2007) define organizational learning as "the deliberate use of individual,
group, and system learning to embed new thinking and practices that continuously
renew and transform the organization in ways that support shared aims" (p. 8).
According to Boreham and Morgan (2004), when organizations learn, "co-workers
transcend the boundaries which separate them from colleagues, establish a
common (and expanded) understanding of the object of their joint activity and
make a collective decision on how to achieve it" (p. 313). In recent work, Lampert
et al. (under review, p. 35) have been examining how organizational assets -
material, intellectual, and social - become resources to support successful teaching
of complex learning; they explain this view in the following way:

Assets are effective in supporting the common practice of ambitious
instruction when they are widely used as tools for solving common
instructional problems. When common use is enabled by the organization of
practice, assets can shape the culture of teaching and learning. For this
reason, our analysis has focused not only on the nature of material, social,
and intellectual assets, but also on how instruction at the school is organized
so that assets can become resources that are routinely drawn upon to support
a challenging approach to teaching and learning across the school.

Lampert et al.'s views echo other researchers who suggest that we need to move
beyond making direct causal links between resources available at a school level
(such as adequate funding, leadership, highly qualified teachers, curricular
materials, release time for teachers) to student achievement. Instead, the argument
is to examine how such resources are used to support instruction, which mediates
student achievement (see also Cohen, Raudenbush, & Ball, 2003).

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Professional Inquiry as a Key Resource in Supporting School Development

The existence of robust professional inquiry at the school level has received a
considerable amount of attention as one of the key resources in supporting school
development. Promoting collective professional inquiry within a school emerged in
part as a response to a large body of organizational and classroom research that
documented the successes and failures of teachers' attempts at implementing
reform practices designed to change classroom instruction. Changing daily routines
inside classrooms, the goals and purposes of mathematics instruction, and
dominant discourse patterns and relationships between teachers and students have
proven difficult. Significantly impacting classroom instruction is challenging
because of what organizational researchers have labelled the loose coupling
between traditional school structures, management, and teachers' everyday
decision-making. While teachers may enjoy considerable autonomy, this loose
coupling in traditional organizational structures creates barriers for instructional
reform (Weick, 1976). More recent work on teachers' responses to reform
initiatives suggests that the congruence, intensity, pervasiveness and voluntariness
of the reforms themselves have an effect on how teachers change their classroom
practices (Cobum, 2004). Decoupling may be one of many responses teachers have
to reform initiatives. Others include rejecting reform visions, assimilating,
accommodating, or developing parallel structures to reform practices.

To counter forces that keep teachers isolated and encourage idiosyncratic
responses to instructional initiatives, researchers have attempted to build models
that are built into the regular workday, arguing they have more promise of being
maintained, sustained and integrated into teachers' practice than an occasional
weekend or summer institute (see also Jaworski, this volume; Lerman &
Zehetmeier, this volume). A focus on schools as a unit of change means that a
collective intellectual culture needs to be cultivated among key professionals
(administrators, instructional leaders, and teachers) who work within the
organization. Basing professional inquiry within the workplace can allow school
professionals to work together to make sense of demanding mathematics teaching
and coordinate their efforts across grade levels. Importantly, collective inquiry
serves to break down the dominant "egg crate modef of schooling - teachers
housed together in a school yet working individually in their own classrooms with
little coordination and collaboration with colleagues in the same building. The
dominant cellular model has promoted isolation and norms of privacy (Lortie,
1 975). That said, it is important to note, as many researchers have pointed out, that
simply bringing teachers and leaders together in regular meetings does not mean
they will be able to critique and learn from their practice. How learning
opportunities are structured and enacted make all the difference in teacher learning.

To the extent that we have conducted cross-national comparisons of teaching,
we know that variation does exist in what counts as "doing" school mathematics.
The case could be made that a view of mathematics as a static body of procedures
to memorize and apply with fidelity is a vision that many mathematics educators in
the US try to move away from in their professional development work with

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teachers. Much of the literature about supporting instructional improvement is
aimed toward helping teachers learn to treat mathematics as a sense-making
enterprise that involves argumentation, reasoning, and justification at its core (see
also Seago, this volume). Cross-national comparisons have helped us see that
expository teaching approaches are not necessarily universal. In elementary
classrooms in Japan, for example, significant time is spent on justifying and
connecting mathematical ideas (Hiebert, Stigler, Jacobs, Giwin, & Gamier, 2005).
It may be fair to say that a procedural approach to teaching and learning
mathematics is nonetheless ubiquitous (Giwin, Hiebert, Jacobs, Hollingsworth, &
Galli more, 2005). Although some variation exists internationally in cultural scripts
for teaching (Clarke, Emanuelsson, Jablonka, & Mok, 2006; Clarke, Keitel, &
Shimizu, 2006), static notions of learners, especially durable notions of ability and
competence may be more pervasive across the globe. Schooling institutions are
typically organized to classify, sort, and promote students with higher
mathematical ability and higher achievement (e.g., Spade, Columba, & Vanfossen,
1997). Schooling practices can embody a view of ability that often takes
intelligence as an inherent and stable trait (Boaler, 2002; Horn, 2007). Reforming
mathematics instruction at a school level is often coupled with trying to upend
static views of students' ability and intelligence in order to subvert tracking,
grouping, and classification systems that keep students in ability bins (e.g., Boaler,
2002; Horn, 2007; Oakes, 1985; Rousseau, 2004; Sfard & Prusak, 2005). This view
sees schools and classrooms as "necessarily cultural and social spaces that can
perpetuate social inequities by positioning multiple forms of learning and knowing
as 'having clout'" (DIME, 2007, p. 407). Promoting school development has the
potential to enable school professionals to interrogate how they view students in
relation to instruction and how their actions and policies privilege certain students
over others.

Visions of Ambitious Teaching, Teacher Learning and the Meaning of Inquiry

Current research on teaching and learning puts forth a vision of teaching
mathematics that has been labelled a number of different ways: teaching for
understanding, reform-oriented teaching, standards-based instruction, problem-
centred instruction, inquiry-oriented teaching. These terms have been used to
convey both a level of intellectual rigor, and the nature of the classroom
community needed to achieve that rigor. Lampert and colleagues (under review)
use the term ambitious teaching to mean, "adjusting teaching to what particular
students are able to do (or not)" (p. 2) in order to engage all kinds of students in
complex problem solving activity, mathematical reasoning, and justification; they
identify the challenges of ambitious teaching to include (p. 35):

- Teachers need to move around flexibly in the multiple dimensions of subject
matter in relation to student performance, adjusting teaching to learning.

- Teachers need to create an environment where students are willing and
motivated to take the risks that intellectual performance entails.

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- Teachers need to take what students can do as an integrated indication of what
they know and what they need to learn rather than breaking subject matter into
meaningless bits of information.

To meet such challenges, it seems reasonable to argue that teachers must have
access to substantive learning opportunities for themselves. The literature on
teacher learning is replete with uses of the term inquiry. Richardson (1994)
distinguishes formal educational research from teachers' own practical inquiry by
describing how practical inquiry aims to help teachers change their instructional
practice or increase understanding by studying their own contexts, practices, or
students. When conducted with colleagues, practical inquiry can develop a local
sense of shared norms and local standards of practice. Cochran-Smith and Lytle
(1999, p. 288) describe inquiry as stance:

Teachers and student teachers who take an inquiry stance work within inquiry
communities generate local knowledge, envision and theorize their practice,
and interpret and interrogate the theory and research of others. Fundamental
to this notion is the idea that the work of inquiry communities is both social
and political; that is, it involves making problematic the current arrangements
of schooling; the ways knowledge is constructed, evaluated, and used; and
teachers' individual and collective roles in bringing about change.

The critical perspective reflective in Cochran-Smith and Lytle's view of inquiry
as stance is reflected in other views of professional inquiry designed to help
teachers not only develop new knowledge and skills but interrogate their views and
question and critique the political nature of schooling as it relates to students
learning mathematics (e.g., Gutierrez, 2007; Horn, 2005; King, 2002). This more
critical stance of inquiry is an attempt to support teachers to consider how schools
and learning opportunities affect access and equity in students' academic identities
and trajectories in successfully learning mathematics and pursuing mathematics as
they continue through schools.

Viewing schools as sites for teacher learning rather than as places where
teachers simply work is well supported by sociocultural theories of learning.
Drawing from the notion that knowledge and meaning are constructed through
practice, Franke, Carpenter, Fennema, Ansell, and Behrend (1998, p. 68) argue for
supporting learning that is self-sustaining and generative; they claim that teachers'
supported efforts to engage in classroom practices guided by student learning can
serve as a basis for their own continued growth and problem solving of classroom
dilemmas:

It is in developing an understanding of their practices in relation to their
students' learning that teachers develop the understanding necessary to
generate new ideas. If a teacher struggles to understand why the students are
successful, how they are solving problems, how their thinking develops, and
how instruction might help students to build on their current conceptions,
connections are made, understanding develops, and the potential for more

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connections becomes possible. Thus, there exists a basis for the teacher to
learn and continue to grow.

Through Franke and colleagues' work, we have learned how teachers can use
their own practices to make sense of student learning. However, we need to
understand further how teachers' classroom practices and broader school-based
professional communities can provide the basis for teachers to continue to develop
their practice. In their work, Franke and colleagues argued that teachers who are
generative develop detailed understanding of their own students' thinking, organize
that knowledge and view it as their own to create, adapt, and change. These
teachers learn from interacting with their students; they are focused in the ways
they listen to, interpret, and make use of their students' thinking. In this chapter on
school development, 1 consider how teachers' participation in multiple and
potentially overlapping communities of practice shapes and re-shapes their
identities and constitutive skills and knowledge as teachers of mathematics.

THEORIZING ABOUT PROFESSIONAL COMMUNITY

The main mechanism for supporting teachers' generative learning for ambitious
teaching has been through building and sustaining professional communities of
teachers. Theorizing teacher learning through participation in professional
community draws on sociocultural theories of learning which take participation as
a key construct (see also Jaworski, this volume). Lave and Wenger (1991) define a
community of practice as "a set of relations among persons, activity and world,
over time and in relation with other tangential and overlapping communities of
practice" (p. 98). A community of practice can be defined through mutual
engagement in a joint enterprise which develops shared repertoires of practice
(Wenger, 1998). Lave (1996) describes learning as "changing participation in
changing 'communities of practice'" (p. 150). Learning is not a process of
acquiring or transmitting knowledge. Rather learning is apparent in the way
participation transforms within a community of practice (Rogoff, 1997). The shifts
in participation do not merely mark a change in a participant's activity or
behaviour; a shift in participation also involves a transformation of roles and the
crafting of a new identity, one that is linked to but not completely determined by
new knowledge and skills (Lave, 1996; Lave & Wenger, 1991; Rogoff, 1994,
1997; Wenger, 1998). Lave (1996, p. 157) states, "crafting identities is a social
process, and becoming more knowledgeably skilled is an aspect of participation in
a social practice. By this reasoning, who you are becoming shapes crucially and
fundamentally what you 'know'." Knowledge and the development of skill are
clearly important in understanding learning. Developing skill and knowledge is in
service of changing participation in a particular community.

Lave and Wenger (1991) describe transforming participation in terms of
movement from legitimate peripheral to full participation. As a legitimate
participant, one is connected and belongs to the community of practice in question,
but as a peripheral participant, one engages less fully in the community. The

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peripheral participant has access and can move towards full participation, thus
developing an identity of full participation. Full participation entails "developing
an identity as a member of a community and becoming knowledgeably skillful"
(Lave, 1991, p. 65). Analysis of learning focuses on the structuring of the
community's work practices and learning resources; learning is detectable in
members' participation in the work of the community.

What Does Professional Community Mean for Mathematics Teaching and
Learning?

As mathematics education researchers have drawn on the community of practice
theory, they have identified key features of professional community, the strength of
which inspires instructional innovation and commitment to students. Several
reviews exist which compare these features and common to all these communities
is a shared sense of purpose and collective and coordinated collaborative activity
with a commitment to students (e.g., Dean & McClain, 2006; Sowder, 2007). Dean
and McClain (2006, pp. 13-14) define these as:

A shared purpose or enterprise such as: ensuring that students come to
understand central mathematical ideas while simultaneously performing more
than adequately on high stakes assessments of mathematics achievement.

A shared repertoire of ways of reasoning with tools and artefacts that is
specific to the community and the shared purpose including normative ways
of reasoning with instructional materials and other resources when planning
for instruction or using tasks and other resources to make students'
mathematical reasoning visible.

Norms of mutual engagement encompassing both general norms of
participation as well as norms specific to mathematics teaching such as the
standards to which the members of the community hold each other
accountable when they justify pedagogical decisions and judgments.

Gutierrez (1996) calls high school mathematics departments with strong
professional communities as "organized for advancement'' because their collective
activity makes a difference in advancing student learning and achievement. In
addition, several researchers emphasize teachers' ability to wield influence and
control over important decisions that affect a school's activities, policies, and
curriculum (Erickson, Brandes, Mitchell, & Mitchell, 2005; King, 2002; Little,
1999; Secada & Adajian, 1997).

Cultivating Professional Inquiry at the School Level

This chapter underscores the idea that school development necessarily involves
learning. Boreham and Morgan (2004) argue that organizations learn because
members of the community are able to coordinate their perspectives and actions

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towards achievement of common goals. They found that organizations learn by
developing relational practices, "the kind of practice[s] by which people connect
with other people in their world, and which direct them to interact in particular
ways" (p. 3 1 5). These relational practices include:

- opening space for creation of shared meaning

- reconstituting power relations

- providing cultural tools to mediate learning.

Those relational practices can be connected to developing social and intellectual
resources (opening space for creation of shared meaning and reconstituting power
relations) and material resources (providing cultural tools to mediate learning). In
what follows, I draw on the professional community and inquiry literature to
identify the following dimensions that make possible these relational practices.

- Activating school leaders

- Navigating fault lines, dealing with micropolitical issues among teachers

- Developing and sustaining a focus

- Engaging parents as intellectual and social resources

Activating school leaders. When the school is the centre of change, theories of
action for supporting professional learning communities (PLCs) among
mathematics teachers necessitate contention with school cultures and institutional
realities. Gamoran and colleagues (2003) contend that PLCs need access to
resources - material, human, and social - if they are to remain viable. Here the role
of school leaders in facilitating the availability and use of such resources can be
critical. School leaders refer not only to school principals or heads but also to
curriculum leaders, mathematics coaches, teacher mentors, or faculty coordinators.
There is variance documented in the literature as to how much direct involvement
in a professional community principals have versus other mathematics leaders
(Burch & Spillane, 2003; King, 2002; Krainer & Peter-Koop, 2003; Wolf, Borko,
Elliott, & Mclver, 2000). In some cases, principals participate in teacher meetings
in order to learn alongside teachers and spend time in classrooms. In others, the
principal is supportive by allocating school level resources in providing space and
time and by making goals of teacher interactions congruent with school and district
level goals (Coburn & Russell, in press). Whether and how leadership strategies
interact with supporting professional development in a PLC at the school level in
mathematics, is a burgeoning area of research. Findings underscore the situated
way in which leadership strategies interact with local conditions.

Halverson (2007) documented the way school leaders make use of certain
structures to enable the building of a professional community. Key to Halverson's
(2007) analysis is that a professional community is a "form of organizational trust"
(p. 94) resulting from the kinds of interactions teachers have to consider in using
alternative instructional strategies to improve student learning. His analysis of the
role of school leaders in supporting a professional community focuses in part on
leaders' use of artefacts, such as role positions, daily schedules, meetings and
meeting agendas. Moreover, he found that school leaders sequence the use of these

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artefacts in different ways to initiate interactions, facilitate development of mutual
obligations, and provide feedback about how those obligations are being met at a
systemic level. One of the key material resources in supporting teachers' ability to
work together is time and space and to do so in the rushed pace of the normal work
week. Halverson found that leaders strategically made use of local contingencies in
order to carve out space and time for teachers to initiate and legitimate time to talk
about instruction - in one case through Breakfast Club meetings and in two others
through grant-writing projects to respond to a new accountability mandate.
Examples abound in the literature of other means that school leaders employ to
initiate conversations, including such things as analysis of student work (e.g.,
Kazemi & Franke, 2004), implementing common units of lessons (e.g., Borasi,
Fonzi, Smith, & Rose, 1999), study groups (e.g., Arbaugh, 2003), and regular
department meetings (e.g., Horn, 2005). Coburn and Russell's (in press) use of
social network analysis also supports the importance of school leaders in allocating
human resources such as coaching expertise in collective professional interaction.
They found that the allocation of coaches affected depth of interactions in PLCs.

Navigating fault lines. Establishing professional communities is not just a matter
of decreeing that one exists. There is nothing neither positive nor harmonious
inherent in the term community, and if we consider the countless times researchers
have characterized the work of teaching and school reform as complex, we should
expect that forming inquiry communities within schools should not be a trivial
matter. Research on PLC and school-based reform has highlighted how managing
tensions and conflicts are critical for the viability of the PLC (Rousseau, 2004).
Grossman, Wineburg, and Wool worth (2001) identified the navigation of fault
lines as one of the central concerns of building a supportive teacher community.
These fault lines are not necessarily interpersonal conflicts among teachers but may
be tensions regarding disciplinary goals and views of teaching and learning
mathematics that exist in the school or department (e.g., Rousseau, 2004). Because
participating in a PLC is about changing school cultures and breaking norms of
privacy while at the same time building norms of critical colleagueship,
researchers have also attended to the ways such participation affects
transformations in teacher identity (e.g., Drake, 2006; Kelchtermans, 2005; Battey
& Franke, in press). Researchers argue that teachers' individual reactions to a
PLC's demands are mediated by social and cultural contexts as well as teachers'
working dynamic identities. "Teachers' identities carry personal histories, emotion,
values, and knowledge and they shape how teachers participate in professional
development and their classrooms" (Battey & Franke, in press, p. 27).

Research has discussed the paradoxes and conflicts that are bound up in
cultivating PLCs. Many researchers have noted that in order for teacher inquiry to
have a school-wide effect, it must move beyond individual teachers. But mandating
participation in teacher inquiry at the school-level can also backfire. Here
leadership must be strategic in inviting a critical mass of participation that can have
a pronounced affect on school culture (Berger, Boles, & Troen, 2005; Krainer,

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2001). The role of key school leaders (whether principals, school facilitators, or
coaches), again becomes critical in creating a press for teacher inquiry and in
developing collective ownership (Berger et al., 2005; Nickerson & Moriarty,
2005).

Developing and sustaining a focus. If schools are successful in carving out time
and space for teachers to meet together, in managing conflicts and tensions, the
next dimension I wish to highlight is how researchers describe the importance of
developing and sustaining a focus on students. Because time and space are precious
resources in schools, collegial interactions need to be focused in order to be
productive. Some studies indicate that material resources such as curriculum units
or lessons, student work, videotaped lessons can be leveraged successfully in order
to achieve this focus (e.g., Borasi et al., 1999; Kazemi & Franke, 2004; Seago, this
volume; Sherin, 2004). Lin's (2002) study is illustrative. The school-based
professional inquiry she described focused on grade-level collaborations. First-
grade teachers met to develop, observe, and reflect on lessons from the Taiwanese
mandated learner-oriented reform curriculum. To provide sufficient focus for their
collective inquiry, the first-grade teachers selected a common lesson to plan and
observe in each other's classrooms. They used these common lessons and
observations to write classroom cases that in effect focused and deepened their
conversations, beyond what observation alone would have accomplished. Three
types of cases emerged from this collaborative inquiry: (1) analysis of students'
varied solutions and strategies; (2) students' interpretations of other students'
thinking; and (3) comparisons of two different instructional approaches to teaching
the same topic. Cases were developed in several phases; each successive phase
refined and elaborated the teaching context and the questions for discussions. The
creation of teaching cases situated in teachers' own classrooms when all teachers
were able to teach the same lesson enabled the teachers to very carefully analyse
the effect of task design and sequencing on students' mathematical thinking;
students' ability to use symbolic and pictorial representations to solve problems;
and for teachers to compare and think deeply about how their instructional choices
(e.g., "Is binding straws important for students' ability to count by tens?")
impacted student performance. This was dependent, of course, on teachers'
willingness to examine their own practice and to raise questions for one another.
This kind of inquiry is dependent on cultivating norms of inquiry, navigating fault
lines, and developing resources for time to meet.

Engaging parents as intellectual and social resources. The role of parents in
school development has received much less attention than the work of teachers,
administrators, and leaders. This makes sense given the amount of attention of
coordinating professionals who work at the school itself. Nonetheless, recent work
on parents has illuminated a number of issues related to parents' engagement,
raising possible considerations for parents' role in supporting school development.
Improving mathematics instruction involves a dramatic transformation from
viewing mathematics as a fixed body of procedures to memorize and apply with

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fidelity to a discipline that is fundamentally about complex problem solving,
justification, and argumentation. Studies investigating parents' views have revealed
that parents can feel disempowered in relation to reform-oriented mathematics
(Martin, 2006; Remillard & Jackson, 2006). At the same time, interventions that
have aimed to work with families around mathematics as a complex problem
solving discipline have reported significant increases in families' feelings of
empowerment (Civil & Bernier, 2006). Reports of building school capacity may
include events such as parent nights designed to mitigate these anxieties by
beginning dialogue with families about goals of ambitious mathematics teaching
and learning (e.g., Kramer & Keller, 2008). We have much more to learn about
how engagement with families can support school development efforts, in what
kinds of social and political contexts, and to what ends. Might leadership, for
example, leverage family views, practices and questions to catapult teacher inquiry
(e.g., Anderson, 2006) or could family practices undercut professional community?

Studying the Practices of School-Based Professional Communities

Cultivating a viable professional community naturally begs questions about how
members of the school community actually interact with one another as they
discuss teaching practice. What might it look like for teachers and leaders to
engage in critical colleagueship (Lord, 1994) oriented to improving mathematics
teaching and learning? I find the work of Little and Horn (2007) to be particularly
instructive in thinking about the practices of a professional community. They are
developing a conceptual scheme to examine what actually happens in collective
professional learning communities. How do participants actually work together?
What do they say and do? How do they interact with one another around artefacts
of practice and how do they talk about classroom instruction? Little (2002) offers
two central questions for examining interactions within teacher communities:

1 . What faces of practice are made visible through talk and with what
degree of transparency?

2. How does interaction open up or close down opportunities to learn?

The face of practice refers to "those parts of practice that come to be described,
demonstrated, or otherwise rendered in public exchanges among teachers" (Little,
2002, p. 934), which may include artefacts such as student work. Transparency of
practice conveys "how fully, completely, and specifically various parts of practice
are made visible or transparent in the interaction" (Little, 2002, p. 934). These two
questions seem central to us in order to understand what views of practice are made
available to teachers through their collective inquiry. As I describe below, I
contend that the understanding of the inner workings of school-based professional
development communities should be then related to teachers' participation in and
out of the group setting. This relational view can help us understand who teachers
are becoming through this process (Battey & Franke, in press; Enyedy, Goldberg,

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& Welsh, 2006), and how they are enacting their developing identities, skills, and
knowledge with their students in the classroom.

Little and Horn (2007; see also Horn, 2005) have documented in detail how the
rendering of classroom interactions in professional conversations shape
opportunities for teacher learning. By contrasting informal conversations in
mathematics department meetings in two different high schools, they compare how
teachers use replays and rehearsals to reason publicly about their instructional
practice and to consider alternative interpretations and re-formulations of
pedagogical dilemmas and problems in ways that propel their teaching forward.
During replays, teachers recount "blow-by-blow accounts of classroom events,
often acting out both the teacher and students' roles" (Horn, 2005, p. 225).
Through rehearsals, teachers act out classroom interactions that might occur in the
future, anticipating what they might say and how students might respond. In
careful analysis of teacher-to-teacher talk, Horn documents how teachers move in
and out of these modes as they reconsider their pedagogical choices and ready
themselves for continued experimentation. However, the presence of replays and
rehearsals during collegial interactions is not sufficient for such experimentation.
How such replays and rehearsals function in framing pedagogical issues, questions,
dilemmas and frustrations is what matters.

Prior work by Kazemi and Franke (2004; see also Franke & Kazemi, 2001;
Franke, Kazemi, Shih, Biagetti, & Battey, 2005) documented detailed images of
the evolution of collective inquiry among elementary teachers analysing their
students' mathematical work in monthly school-based meetings in which each
teacher had posed a similar mathematical problem (focused on number and
operations) to his or her class. The analysis of the teacher talk revealed that the
workgroup conversations evolved as teachers learned to talk about their work.
Salient developments included the following:

- Teachers first had to leam to attend to the details of their students' thinking.
Even though meetings were structured from the very beginning to detail
students' strategies, teachers did not come to the first meeting prepared to do so.
Instead, many teachers assumed that the pieces of paper themselves would tell
the story. It was further evident in the way they posed the problems that many
assumed that conversations with students about their solutions were not
necessary. Maintaining the structure of the workgroup promoted an emphasis on
documenting the details of student thinking. Kazemi and Franke (2004)
intentionally facilitated the discussions so that teachers would return to the idea
of noticing how students were learning to break apart and put together numbers
using their knowledge of the base ten structure of the number system. In order
for teachers to interpret their students' reasoning, they began to use the student
work as a trace rather than a complete record of their students' reasoning.
Without such a mathematical focus, meetings may not encourage teachers to
follow a particular course of experimentation in their classrooms.

- Teachers' close consideration of student reasoning opened up opportunities to
deliberate mathematical and pedagogical questions. Examining student work,
as structured in this professional development, allowed teachers to surface their

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confusions and uncertainties, not just about student reasoning but also about
mathematics and classroom practice. The discussions opened up opportunities
for teachers to notice the mathematical ideas students were using. This led to the
group's engagement with sophisticated computational strategies they were
noticing in their classrooms. The use of student work provided an entry point for
teachers to explore mathematical ideas and have opportunities to make sense of
efficient student-generated algorithms. Pedagogical issues related to helping
children develop more sophisticated strategies also surfaced once the group saw
students in teachers' own classrooms using such strategies.
- Diversity in teachers' experimentation served as a resource for learning.
Teachers differed in how they reported on their engagement with students in
their classrooms. Not all the teachers experimented with these ideas in the same
way. Because the group had multiple ways of relating the professional
development discussions to their classroom practices, the experiences of some
teachers generated new conjectures about what to try in the classroom. The
frustrations teachers shared in the group also underscored the challenges they
faced in helping children articulate and build their ideas. This diversity became
a resource for teachers to compare and question each other's practices.
Understanding how teachers' interact with one another in PLCs and how those
interactions evolve and develop learning opportunities for teachers is vital for
research and development work in fostering collective inquiry at the school level.
In the next subchapter, I argue for relating this evolving research base to the well-
known research on teachers' classroom instruction.

CONNECTING PROFESSIONAL LEARNING COMMUNITIES WITH CLASSROOM
TEACHING AND LEARNING

Earlier in this chapter, I proposed the view that school development deals with the
development of organizational supports for ambitious mathematics teaching and
learning. Much of the literature emphasizes mechanisms for such improvement to
be highly dependent on how schools employ resources and how they are organized
to support teachers' collective learning together outside of the classroom. What I
would like to offer in the remainder of this chapter is a proposal for attending to
one aspect of the relationship between joint inquiry at the school level and
classroom instruction. My discussion is not meant to be comprehensive; it
identifies one shortcoming of how we have typically thought about the impact of
teachers' and leaders' joint inquiry on mathematics instruction and student
learning. Some of our current research base is limited in scope to a unidirectional
view of the impact of cultivating professional inquiry at the school level to
teachers' classroom practice. I propose that what needs to be addressed and
developed are ways to characterize and document the multi-directional influence
between participation in joint inquiry and the individuality of classroom practice.
In this last subchapter, I outline the rationale for this approach and describe some
of the necessary theoretical and empirical work that lies ahead.

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Teachers are simultaneously involved in multiple activity settings, including
their own classroom, school, and district. When they are involved in sustained
inquiry with colleagues and leaders at the school, this constitutes another important
mediating context. I use the construct of activity settings to focus on the boundaries
and relationships between the classroom, school and district (see also Cobb &
Smith, this volume). Activity settings overlap; that is, they do not exist as insular
social contexts but rather as sets of relationships that coexist with others
(EngestrSm, 2001). Activity settings can have temporal, conceptual, and physical
boundaries (Grossman, Smagorinsky, & Valencia, 1999). It is this dynamism
across activity settings and how that shapes teacher learning that I am concerned
with in this chapter. To develop these ideas further and specify a way of talking
about learning, I draw on Cook and Brown's (1999) distinction between knowledge
and knowing. Knowledge, in their view, is something that we "possess". We
"deploy" this knowledge in our actions. In their words, "Knowing refers to the
epistemic work that is done as part of action or practice, like that done in actual
riding of a bicycle or the actual making of a medical diagnosis" (p. 387).
Knowledge, then, can be seen a tool of action because individuals or groups can
use knowledge (whether tacit or explicit) to discipline their interactions with the
world. This distinction seems both relevant and important in thinking about teacher
learning. Much has been written about the kinds of specialized knowledge that
teachers need, among them, knowledge of the discipline, their students, and
instructional strategies (Ball & Bass, 2000; Shulman, 1986). Professional inquiry
clearly needs to develop teachers' knowledge, and we have been rightfully
concerned with figuring out what kinds of knowledge teachers gain through
inquiry.

Cook and Brown (1999) would agree that knowledge is essential for practice but
it is not sufficient for explaining what it takes to be good at what you do: "An
accomplished engineer may possess a great deal of sophisticated knowledge; but
there are plenty of people who possess such knowledge yet do not excel as
engineers" (p. 387). In addition to all the kinds of knowledge that teachers need,
they also have to be able to teach. For me, this means that we have to attend to the
interplay between knowledge and knowing in the professional community and in
teachers' classroom context. In addition, we need to attend to the interplay between
teachers' development of knowledge and ways of knowing in the professional
development and classroom contexts over time. We need to link the knowledge and
ways of knowing that teachers develop together with what happens as teachers try
to use the knowledge and ways of knowing they gain in joint inquiry in the context
of their classroom teaching. Teachers may develop similar ways of examining and
talking about students' mathematical thinking through inquiry with colleagues in
their school but we clearly need to concern ourselves with how they are drawing on
that knowledge when they interact with students, or in Cook and Brown's terms,
how knowledge is deployed in the service of disciplining action (knowing).
Moreover, I argue that researchers should examine what teachers are learning
during and after conversations with colleagues, looking at the coevolution of
participation between classroom practice and professional inquiry. I claim that this

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coevolution between the contexts of professional collaboration and classrooms
should itself be a key unit of analysis as we try to explicate the mechanisms by
which teachers learn in and through professional inquiry. By seeing how teachers'
participation across these contexts co-evolves, we will have better views of the
relationship between joint professional inquiry, learning and instruction, and school
development.

The Implications of a Multidirectional Analysis for Studying and Designing
Collective Inquiry

A multidirectional analysis of professional learning across contexts where teachers
and leaders work together and the classroom leads us to the following implications
for the study and design of these efforts. We should: (1) understand and elicit the
diversity of teachers' experimentation and the effect of depictions of that work in
joint inquiry and (2) examine the situated nature of primary artefacts.

Understanding the diversity of teachers ' experimentation. In order to understand
the relationship between the contexts of joint inquiry and the classroom, and how
teachers' and leaders' participation across these settings co-evolves, we must
understand individual teachers' classroom experimentation, and how this
influences their collaboration with colleagues. How do teachers deploy their
knowledge in the classroom? What ways of knowing do they demonstrate in their
instructional practice? What do teachers bring to the collective as a result of their
experimentation? In addition to documenting the diversity of individual teachers'
classroom experimentation, we also need to document and study what actually
happens when teachers and leaders come together and their collective learning
trajectories as they participate in this context. It is essential that we document the
diversity of teachers' classroom experimentation and study the nature of how this
experimentation relates to their experiences over time with their colleagues and to
their developing identities - what kinds of teachers do they want to become? What
ways of knowing are developed over time? How and what knowledge do teachers
develop of subject matter, students' thinking, and practice as they engage in
collective analysis around common objects of inquiry?

While the argument here is about research on teachers' joint learning, there are
also implications for the design of collective inquiry (see also Jaworski, this
volume; Seago, this volume). I argue not only that teachers' experimentation
should be studied but that leaders of teachers' joint inquiry should incorporate
depictions of teachers' classroom experimentation in collaborative engagement.
Depictions of practice are images or stories that seek to capture the events in the
classroom as they played out, earlier referenced as replays and rehearsals. They are
created intentionally to support the analysis of teaching. Written cases and video-
cases are perhaps the most visible example of depictions available in the literature.
But there are other examples: replays and rehearsals (Horn, 2005) are depictions

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that are created through teachers' talk; teachers' journals can also serve as a
depiction.

If professional educators sought openings to elicit teachers' experimentation in a
principled way, collective inquiry could serve as a place to pursue questions and
dilemmas teachers encounter as they engage in transforming their practice. While it
is easy to advocate that we incorporate depictions of practice and discuss teachers'
classroom experimentation in the context of collective inquiry more extensively, I
recognize that sharing episodes from the classroom can easily and unproductively
spiral into a show-and-tell. Leaders of collective inquiry will need to become more
knowledgeable and skilled about how to use teachers' classroom experiences. For
example, how can the dilemmas teachers face about modifying tasks, managing
pacing, and orchestrating classroom discourse be usefully depicted and used as a
springboard for discussion? How can leaders utilize one teacher's experiences to
support another to develop more focused and reflective attempts to experiment in
the classroom? Many researchers have written extensively about the intentional use
of records of practice (e.g., Sherin, 2004; Lampert & Ball, 1998; Little, 2004),
arguing that we must attend not only to the careful selection of representations but
also how they are negotiated in practice.

Examining the situated nature of primary artefacts. Primary artefacts are objects
that originate (or are produced for use) in instructional practice. In the case of
teaching, primary artefacts include copies of student work, lesson plans,
mathematical tasks, and curriculum materials. They can travel across boundaries,
into contexts where teachers and leaders collaborate, but they are not created solely
for the purpose of collectively analysing teaching. Primary artefacts allow
particular components of teaching to be extracted from the context of instructional
practice, lessening the complexity by narrowing teachers' focus.

Primary artefacts are produced and used in practice, and so ways of knowing
include the use and production of primary artefacts. If we are concerned with
teachers developing new ways of knowing in their classroom practice, then we
should attend to the relationship between ways of knowing in professional
development and in the classroom. And if we are going to use primary artefacts as
a tool in professional development, we must attend to how they are situated in
particular activities, and how this affects their meaning. For example, student work
is a primary artefact commonly used in collective work of teachers. The way
student work is situated in collective inquiry may look very different from its use
in the classroom. Teachers and leaders may sit together to analyse a collection of
pre-selected student work to illustrate the range of solution strategies students used
in their classroom. They may spend extended time debating what students
understand, generating questions they might ask to better understand the students'
thinking, or considering which strategies they would choose to highlight in a whole
class discussion. In contrast, in their classrooms, teachers may only have a few
minutes to survey students' written work in order to make assessments and
instructional decisions. The teacher typically engages in this work alone, in the
midst of a lesson, while students are working on the task. While collective inquiry

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may certainly help teachers gain knowledge they can deploy in this classroom
situation, it may not help them develop the ways of knowing they need to monitor
students in the moment and to interact with them in ways that assess and advance
students' mathematical thinking. Researchers and leaders must attend to the
meaning teacher's make of primary artefacts across contexts as these artefacts are
situated in different activities. We need to better understand how the ways of
knowing involved in these activities differ, and how they influence one another.

CONCLUSION

Writing about school development is necessarily a synthetic enterprise. In this
chapter, I conceptualized school development as a school's efforts to support
ambitious teaching and learning. Specifically, I took a learning perspective on
school development. This perspective highlights how schools support professional
learning. Collinson (in press, p. 7) states:

Vibrant, innovative organizations work at developing their organizational
capacity by establishing an environment in which members, and thereby the
organization, can continuously learn and improve. Developing members,
along with careful recruiting and hiring of fresh talent, ensures innovation
and renewal.

The way we understand and study professional learning in schools has
significant implications for the way we structure and support teachers' collective
learning opportunities, the goals we create for inquiry, and as researchers and
educators, the ways we collaborate with schools to improve mathematics teaching
and learning. To understand teacher collective learning within the context of
school-based professional development, I have argued that we need to develop
conceptual frameworks to take into account both the dynamics of individual
teacher learning and vulnerabilities to developing their instructional practices as
well as the resources and settings that support learning.

One noteworthy aspect of writing this chapter on school development was the
opportunity to review research that draws on both classroom level research and
organizational and policy implementation research. New collaborations forged
between classroom researchers and policy or organizational researchers (e.g., Cobb
& Smith, 2007; Cobb & Smith, this volume; Gamoran et al., 2003; Little & Horn,
2007) can enrich our view of designing for a tighter coupling between teacher
learning and whole school development that would ultimately benefit professional
culture within the school and students' mathematical learning. That said - here are
a number of issues that remain to be incorporated into our studies of and designs
for school development.

- We need to explicate significant differences between working with schools at
the primary and secondary level and how those impact the ways schools support
ambitious teaching and learning. There seem to be different challenges with
respect to teachers' skills and identities as mathematics teachers, tracking or
grouping practices, testing practices and their consequences, curriculum

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organization, and relationships among teachers, school leaders, and parents
(e.g., Lee, Smith, & Croninger, 1997; Spade, Columba, & Vanfossen, 1997).

- As was evident in this chapter, the research literature on school development
attends predominantly to the role of teachers and school leaders. How parents
and families figure into school development and supporting ambitious teaching
remains underdeveloped and undertheorized.

- Our understanding of school development can be strengthened through further
study of how prospective and novice teachers are involved in collective inquiry
as a way to recruit and continue to develop practices of the school community
(see Leikin, this volume).

Our field's ability to address these issues and others over the next decade will
advance our understanding of school development and inform the next generation
of interventions aimed at supporting ambitious teaching.

ACKNOWLEDGMENTS

I would like to thank Allison Hintz for her invaluable assistance identifying
research used in the literature review. Work with Megan Franke, Amanda
Hubbard, and Magdalene Lam pert has been instrumental in developing some of the
theoretical ideas. I am grateful for the helpful comments of Paul Cobb, Gilah
Leder, Heinz Steinbring, and Terry Wood on previous versions of this chapter. I
am especially indebted to Konrad Kramer for his help in formulating the focus of
this chapter.

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USA

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10. DISTRICT DEVELOPMENT AS A MEANS OF

IMPROVING MATHEMATICS TEACHING AND

LEARNING AT SCALE 1

This chapter focuses on research that can inform the improvement of mathematics
teaching and learning at scale. In educational contexts, improvement at scale
refers to the process of taking an instructional innovation that has proved effective
in supporting students ' learning in a small number of classrooms and reproducing
that success in a large number of classrooms. We first argue that such research
should view mathematics teachers' instructional practices as situated in the
institutional settings of the schools and broader administrative jurisdictions in
which they work We then discuss a series of hypotheses about structures that
might support teachers ' ongoing improvement of their classroom practices. These
support structures range from teacher networks whose activities focus on
instructional issues to relations of assistance and accountability between teachers,
school leaders, and leaders of broader administrative jurisdictions. In describing
support structures, we also attend to equity in students' access to high quality
instruction by considering both the tracking or grouping of students in terms of
current achievement and the category systems that teachers and administrators use
for classifying students. In the latter part of the chapter, we outline an analytic
approach for documenting the institutional setting of mathematics teaching that
can feed back to inform instructional improvement efforts at scale.

INTRODUCTION

In educational contexts, improvement at scale refers to the process of taking an
instructional innovation that has proved effective in supporting students' learning
in a small number of classrooms and reproducing that success in a large number of
classrooms. In countries with centralized educational systems, it might be feasible
to propose taking an instructional innovation to scale at the national level.
However, proposals for instructional improvement at the national level are usually
impractical in countries with decentralized education systems because the
infrastructure that would be needed to support coordinated improvement at the
national level does not exist. The case of instructional improvement at scale that
we consider in this chapter is located in a country with a decentralized education

1 The analysis reported in this chapter was supported by the National Science Foundation under grant
No. ESI 0554535. The opinions expressed do not necessarily reflect the views of the Foundation. The
hypotheses that we discuss in this chapter were developed in collaboration with Sarah Green, Erin
Henrick, Chuck Munter, John Murphy, Jana Visnovska, and Qing Zhao. We are grateful to Kara
Jackson for her constructive comments on a previous draft of this chapter.

PAUL COBB AND THOMAS SMITH

system, the US, in which there is a long history of local control of schooling. Each
US state is divided into a number of independent school districts. In rural areas,
districts might serve less than 1 ,000 students whereas a number of urban districts
serve more than 100,000 students. In the context of the US educational system,
when we speak of scale we have in mind the improvement of mathematics teaching
and learning in urban districts as they are the largest jurisdictions in which it is
feasible to design for improvement in the quality of instruction (Supovitz, 2006).
In this chapter, we speak of instructional improvement at the level of the school
and the district with the understanding that the appropriate organizational unit or
administrative jurisdiction beyond the school needs to be adjusted depending on
the structure of the educational system in a particular country.

The central problem that we address in this chapter is how mathematics
education research can generate knowledge that contributes to the ongoing
improvement of mathematics teaching and learning at scale. The daunting nature
of "the problem of scale" is indicated by the well-documented finding that prior
large-scale improvement efforts in mathematics and other subject matter areas
have rarely produced lasting changes in either teachers' instructional practices or
the organization of schools (Elmore, 2004; Gamoran, Anderson, Quiroz, Secada,
Williams, & Ashman, 2003). Schools frequently experience external pressure to
change, a condition that Hesse (1999) has termed policy churn. However, in most
countries, classroom teaching and learning processes have proven to be remarkably
stable amidst the flux. Cuban (1988), a historian of education, likened the situation
to that of an ocean tossed by a storm in which all is calm on the sea floor even as
the tempest whips up waves at the surface.

Researchers who work closely with teachers to support and understand their
learning will probably not be surprised by Elmore's (1996) succinct synopsis of the
results of educational policy research on large-scale reform: the closer that an
instructional innovation gets to the core of what takes place between teachers and
students in classrooms, the less likely it is that it will implemented and sustained
on a large scale. This policy research emphasizes that although research-based
curricula and high-quality teacher professional development are necessary, they are
not sufficient to support the improvement of mathematics instruction at scale.
Instructional improvement at scale also has to be framed as a problem of
organizational learning for schools and larger administrative jurisdictions such as
districts (Blumenfeld, Fishman, Krajcik, Marx, & Soloway, 2000; Coburn, 2003;
McLaughlin & Mitra, 2004; Stein, 2004; Tyack & Tobin, 1995). This in turn
implies that in addition to developing new approaches for supporting students' and
teachers' learning, reformers also need to view themselves as institution-changing
agents who seek to influence the institutional settings in which teachers develop
and refine their instructional practices (Elmore, 1996; Stein, 2004). We capitalize
on this insight in our chapter by emphasizing the importance of coming to view
mathematics teachers' instructional practices as situated within the institutional
setting of the school and larger jurisdictions such as districts. This perspective
implies that supporting teachers' improvement of their instructional practices
requires changing these settings in fundamental ways.

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In the US context, the institutional setting of mathematics teaching, as we
conceptualize it, encompasses district and school policies for instruction in
mathematics. It therefore includes both the adoption of curriculum materials and
guidelines for the use of those materials (e.g., pacing guides that specify a timeline
for completing instructional units) (Ferrini-Mundy & Floden, 2007; Remillard,
2005; Stein & Kim, 2006). The institutional setting also includes the people to
whom teachers are accountable and what they are held accountable for (e.g.,
expectations for the structure of lessons, the nature of students' engagement, and
assessed progress of students' learning) (Cobb & McClain, 2006; Elmore, 2004).
In addition, the institutional setting includes social supports that give teachers
access to new tools and forms of knowledge (e.g., opportunities to participate in
formal professional development activities and in informal professional networks,
assistance from a school-based mathematics coach or a principal who is an
effective instructional leader) (Bryk & Schneider, 2002; Coburn, 2001; Cohen &
Hill, 2000; Horn, 2005; Nelson & Sassi, 2005), as well as incentives for teachers to
take advantage of these social supports.

The findings of a substantial and growing number of studies document that
teachers' instructional practices are partially constituted by the materials and
resources that they use in their classroom practice, the institutional constraints that
they attempt to satisfy, and the formal and informal sources of assistance on which
they draw (Cobb, McClain, Lamberg, & Dean, 2003; Coburn, 2005; Spillane,
2005; Stein & Spillane, 2005). The findings of these studies call into question an
implicit assumption that underpins many reform efforts, that teachers are
autonomous agents in their classrooms who are unaffected by what takes place
outside the classroom door (e.g., Krainer, 2005). In making this assumption,
reformers are, in a very real sense, flying blind with little if any knowledge of how
to adjust to the settings in which they are working as they collaborate with teachers
to support their learning. In contrast, the empirical finding that teachers'
instructional practices are partially constituted by the settings in which they work
orients us to anticipate and plan for the school support structures that need to be
developed to support and sustain teachers' ongoing learning.

INVESTIGATING INSTRUCTIONAL IMPROVEMENT AT SCALE

One of the primary goals of our current research, which is still in its early stages, is
to generate knowledge that can inform the ongoing improvement of mathematics
teaching and learning at scale. To this end, we are collaborating with four large,
urban districts that have formulated and are implementing comprehensive
initiatives for improving the teaching and learning of middle-school mathematics.
We will follow 30 middle-school mathematics teachers and approximately 17
instructional leaders in each of the four districts for four years to understand how
the districts' instructional improvement initiatives are playing out in practice. In
doing so, we will conduct one round of data collection and analysis in each district
each year for four years to document: 1) the institutional setting of mathematics
teaching, including formal and informal leaders' instructional leadership practices,

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PAUL COBB AND THOMAS SMITH

2) the quality of the professional development activities in which the teachers
participate, 3) the teachers' instructional practices and mathematical knowledge for
teaching, and 4) student mathematics achievement. The resulting longitudinal data
on 120 teachers and approximately 68 school and district leaders in 24 schools in
four districts will enable us to test a series of hypotheses that we have developed
about school and district support structures that might enhance the effectiveness of
mathematics professional development. We will outline these hypothesized support
structures later in the next section of this chapter.

In addition to formally testing our initial hypotheses, we will share our analysis
of each annual round of data with the districts to provide them with feedback about
the institutional settings in which mathematics teachers are developing and
revising their instructional practices, and we will collaborate with them to identify
any adjustments that might make the districts' improvement designs for middle-
school mathematics more effective. We will then document the consequences of
these adjustments in subsequent rounds of data collection. In addition, we will
attempt to augment our hypotheses in the course of the repeated cycles of analysis
and design 2 by identifying additional support structures and by specifying the
conditions under which particular support structures are important. In doing so, we
seek to address a pressing issue identified by Stein (2004): the proactive design of
school and district institutional settings for mathematics teachers' ongoing
learning.

In the remainder of this chapter, we focus on two types of conceptual tools that,
we contend, are central to the improvement of mathematics teaching and learning
at scale. The first is a theory of action 3 for designing schools and larger
administrative jurisdictions as learning organizations for instructional improvement
in mathematics. The second is an analytic approach for documenting the
institutional setting of mathematics teaching that can produce analyses that inform
the ongoing improvement effort. 4

DESIGNING FOR INSTRUCTIONAL IMPROVEMENT IN MATHEMATICS

In preparing for our collaboration with the four urban school districts, we
formulated a series of hypotheses about school and district support structures that
we conjecture will be associated with improvement in middle-school mathematics
teachers' instructional practices and student learning. In developing these
hypotheses, we assumed that a school or district has adopted a research- based
instructional programme for middle-school mathematics and that the programme
was aligned with district standards and assessments. In addition, we assume that
mathematics teachers have opportunities to participate in sustained professional

2 In engaging in these repeated cycles of analysis and design, we will, in effect, attempt to conduct a

design experiment at the level of the school and district.

' The term theory of action was coined by Argyris and Schon ( 1 974, 1 978) and is central to most

current perspectives on organizational learning. A theory of action establishes the rationale for an

improvement design and consists of conjectures about both a trajectory of organizational improvement

and the specific means of supporting the envisioned improvement process.

4 These two types of conceptual tools serve to ground the two aspects of the design research cycle,

namely design and analysis (e.g., Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Design-Based

Research Collaborative, 2003).

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IMPROVING AT SCALE

development that is organized around the instructional materials they use with
students. The proposed support structures, which are summarized in Table 1,
therefore fall outside mathematics educators' traditional focus on designing high-
quality curricula and teacher professional development. To the extent that the
hypotheses prove viable, they specify the types of institutional structures that a
school or district organizational design might aim to engender as it attempts to
improve the quality of mathematics teaching across the organization.

As background for non-US readers, we should clarify that large school districts
such as those with which we are collaborating have a central office whose staff are
responsible for selecting curricula and for providing teacher professional
development in various subject matter areas including mathematics. In this chapter,
we use the designation district leaders to refer to members of the central office
staff whose responsibilities focus on instruction. We speak of district mathematics
leaders to refer to central office staff whose responsibilities focus specifically on
the teaching and learning of mathematics.

Table 1.

The proposed support structures

Primary Support
Structure

Facilitating Support
Structure

Hypothesized
Consequence

Teacher Networks

Time for collaboration
Access to expertise

Social support for
development of
ambitious instructional
practices

Shared Instructional
Vision

Brokers

Negotiation of the
meaning of key
boundary objects

Coherent instructional
improvement effort

Accountability Relations
and Relations of
Assistance

Leadership content
knowledge

Effective instructional
leadership practices

De-tracked instructional
Programme

Category system for
classifying students

Equity in students'
learning opportunities

Teacher Networks

We developed our hypotheses about potential support structures by taking as our
starting point forms of classroom instructional practice that are consistent with
current research on mathematics learning and teaching (Kilpatrick, Martin, &
Schifter, 2003). Teachers who have developed high quality instructional practices
of this type attempt to achieve a significant mathematical agenda by building on
students' current mathematical reasoning. To this end, they engage students in
mathematically challenging tasks, maintain the level of challenge as tasks are
enacted in the classroom (Stein & Lane, 1996; Stein, Smith, Henningsen, & Silver,
2000), and support students' efforts to communicate their mathematical thinking in
classroom discussions (Cobb, Boufi, McClain, & Whitenack, 1997; Hiebert et al.,

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1997; Lampert, 200 1). 5 These forms of instructional practice are complex,
demanding, uncertain, and not reducible to predictable routines (Ball & Cohen,
1999; Lampert, 2001; McClain, 2002; Schifter, 1995; Smith, 1996). The findings
of a number of investigations indicate that strong professional networks (see also
Lerman & Zehetmeier, and Borba & Gadanidis, this volume) in which teachers
participate voluntarily can be a crucial resource as they attempt to develop
instructional practices in which they place students' reasoning at the centre of their
instructional decision making (Cobb & McClain, 2001; Franke & Kazemi, 2001b;
Gamoran, Secada, & Marrett, 2000; Kazemi & Franke, 2004; Little, 2002; Stein,
Silver, & Smith, 1998).

There is abundant evidence that the mere presence of collegial support is not by
itself sufficient: both the focus and the depth of teachers' interactions matter. With
regard to focus, it is clearly important that activities and exchanges in teacher
networks centre on issues central to classroom instructional practice (Marks &
Louis, 1997). Furthermore, the findings of Coburn and Russell's (in press) recent
investigation indicate that the depth of interactions around classroom practice
make a difference in terms of the support for teachers' improvement of their
classroom practices. Coburn and Russell clarify that interactions of greater depth
involve discerning the mathematical intent of instructional tasks and identifying
the relative sophistication of student reasoning strategies, whereas interactions of
less depth involve determining how to use instructional materials and mapping the
curriculum to district or state standards.

Teacher networks that focus on issues relevant to classroom instruction
constitute our first hypothesized support structure. In addition, we anticipate that
networks in which interactions of greater depth predominate will be more
supportive social contexts for teachers' development of ambitious instructional
practices than those in which interactions are primarily of limited depth (Franke,
Kazemi, Shih, Biagetti, & Battey, in press; Stein et al., 1998).

Access of Teacher Networks to Key Resources

Mathematics teacher networks do not emerge in an institutional vacuum. Gamoran
et al.'s (2003) analysis reveals that to remain viable, teacher networks and
communities need access to resources. The second and third hypothesized support
structures concern two specific types of resources that facilitate the emergence and
development of teacher networks (see Table 1).

Time for collaboration. The first resource is time built into the school schedule for
mathematics teachers to collaborate. As Gamoran et al. (2003) make clear, time for
collaboration is a necessary but not sufficient condition for the emergence of
teacher networks. Although institutional arrangements such as teachers' schedules
do not directly determine interactions, they can enable and constrain the social

5 The research base for these broad recommendations is presented in a research companion volume to
the National Council of Mathematics' (2000) Principles and Standards for School Mathematics edited
by Kilpatrick, Martin, and Schifter (2003).

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relations that emerge between teachers (and between teachers and instructional
leaders) (Smylie & Evans, 2006; Spillane, Reiser, & Gomez, 2006).

Access to expertise. The second resource for supporting the emergence of teacher
networks of sufficient depth is access to colleagues who are already relatively
accomplished in using the adopted instructional programme to support students'
mathematical learning. In the absence of this resource, it is difficult to envision
how interactions within a teacher network will be of sufficient depth to support
teachers' development of ambitious instructional practices. In this regard, Penuel,
Frank, and Krause (2006) found that improvement in mathematics teachers'
instructional practices was associated with access to mentors, mathematics
coaches 6 , and colleagues who were already expert in the reform initiative. Their
results indicate that accomplished fellow teachers and coaches can share exemplars
of instructional practice that are tangible to their less experienced colleagues, thus
supporting their efforts to improve their instructional practices.

Shared Instructional Vision

In considering additional support structures, we step back to locate teacher
networks first within the institutional context of the school, and then within the
context of the broader administrative jurisdiction. At the school level, it seems
reasonable to speculate that teacher networks will be more likely to emerge and
sustain if the vision of high quality mathematics instruction that they promote is
consistent with the instructional vision of formal or positional school leaders.
Research in the field of educational leadership indicates that this intuition is well
founded. The results of a number of studies reveal that professional development,
collaboration between teachers, and collegiality between teachers and formal
school leaders are rarely effective unless they are tied to a shared vision of high
quality instruction that gives them meaning and purpose (Elmore, Peterson, &
McCarthey, 1996; Newman & Associates, 1996; Rosenholtz, 1985, 1989; Rowan,
1990). In the case of US schools, formal school leaders might include the school
principal, an assistant principal with responsibility for curriculum and instruction, a
mathematics department head, and possibly a school-based mathematics coach.
The notion of a shared instructional vision encompasses agreement on instructional
goals and thus on what it is important for students to know and be able to do
mathematically, 7 and on how students' development of these forms of
mathematical knowledgeability can be effectively supported.

6 Mathematics coaches are teachers who have been released from some or all of their instructional
responsibilities in order to assist the mathematics teachers in a school in improving the quality of their
instruction. Ideally, coaches should be selected on the basis of their competence as mathematics
teachers and should receive professional development that focuses on both mathematics teaching and on
supporting other teachers' learning.

7 A focus on instructional goals takes us onto the slippery terrain of mathematical values (Hiebert,
1999). It is important to note that values are not a matter of mere subjective whim or taste but are
instead subject to justification and debate (Rorty, 1982).

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PAUL COBB AND THOMAS SMITH

Our argument for the importance of a shared instructional vision is not restricted
to the school but also extends to broader administrative jurisdictions. We illustrate
this point by taking the relevant administrative jurisdiction in the US context, the
school district, as an example. As is the case for the relevant jurisdiction in most
countries, there are typically a number of distinct departments or units within the
administration of large districts whose work has direct consequences for the
teaching and learning of mathematics. For example, one unit is typically
responsible for selecting instructional materials in various subject matter areas
including mathematics, and for providing teacher professional development. A
separate unit is typically responsible for hiring and providing professional
development for school leaders. The unit responsible for assessment and evaluation
would also appear critical given the importance of the types of data that are
collected to assess school, teacher, and student learning. In addition, depending on
the district, the unit responsible for special education might also be influential to
the extent that it focuses on how mainstream instruction serves groups of students
identified as potentially at-risk. Spillane et al.'s (2006) findings indicate that staff
in different administrative units whose work contributes to the district's initiative
to improve the quality of mathematics teaching and learning frequently understand
district-wide initiatives differently. In such cases, the policies and practices of the
various units are fragmented and often in conflict with each other. This has
consequences both for the coherence of the district's instructional improvement
effort and for the degree to which the institutional settings of mathematics teaching
support teachers' ongoing improvement of their instructional practices. Our fourth
hypothesized support structure therefore concerns the development of a shared
instructional vision between participants in teacher networks, formal school
leaders, and district leaders. We anticipate that mathematics teachers' improvement
of their instructional practices will be greater in schools and broader jurisdictions
in which a shared instructional vision consistent with current reform
recommendations has been established.

Brokers

The development of a shared instructional vision of high quality mathematics
instruction in a school and a broader jurisdiction such as a district is a non-trivial
accomplishment. This becomes apparent when we note that mathematics teachers,
principals, and district curriculum specialists, and so forth constitute distinct
occupational groups that have different charges, engage in different forms of
practice, and have different professional affiliations (Spillane et al., 2006). The
fifth support structure concerns the presence of brokers who can facilitate the
development of a shared instructional vision by bridging between perspectives and
agendas of different role groups (see Table I). Brokers are people who participate
at least peripherally in the activities of two or more groups, and thus have access to
the perspectives and meanings of each group (Wenger, 1998). For example, a
principal who participates in professional development with mathematics teachers
might be able to act as a broker between school leaders and mathematics teachers
in the district, thereby facilitating the alignment of perspectives on mathematics

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teaching and learning across these two groups (e.g., Wenger, 1998). Extending our
focus beyond the school, we anticipate that brokers who can bridge between school
and district leaders and between units of the district central office will also be
critical in supporting the development of a shared instructional vision across the
district. Brokers who can help bring coherence to the reform effort in a relatively
large jurisdiction such as an urban district by grounding it in a shared instructional
vision constitute our fifth support structure.

Negotiating the Meaning of Key Boundary Objects

The sixth hypothesized support structure also facilitates the development of a
shared instructional vision (see Table 1). Mathematics teachers and instructional
leaders use a range of tools as an integral aspect of their practices. Star and
Griesemer (1989) call tools that are used by members of two or more groups
boundary objects. For example, mathematics teachers and instructional leaders in
most US schools use state mathematics standards and test scores, thereby
constituting them as boundary objects. Tools that are produced within a school or
district might also be constituted as boundary objects. For example, the district
leaders in one of the districts in which we are working are developing detailed
curriculum frameworks for middle-school mathematics teachers to use as well as a
simplified version for school leaders. It is important to note that boundary objects
such as state and district standards, test scores, and curriculum frameworks can be
and are frequently used differently and come to have different meanings as
members of different groups such as teachers and school leaders incorporate them
into their practices (Star & Griesemer, 1989; Wenger, 1998). Boundary objects do
not therefore carry meanings across group boundaries. However, they can serve as
important focal points for the negotiation of meaning and thus the development of
a shared instructional vision. The value of boundary objects in this regard stems
from the fact that they are integral to the practices of different groups and are
therefore directly relevant to the concerns and interests of the members of the
groups. From the point of view of organizational design, this observation points to
the importance of developing venues in which members of different role groups
engage together in activities that relate directly to teaching and instructional
leadership in mathematics.

Our sixth hypothesis is therefore that a shared vision of high quality
mathematics instruction will emerge more readily in schools and districts in which
members of various groups explicitly negotiate the meaning and use of key
boundary objects. In speaking of key boundary objects, we are referring to tools
that are used when developing an agenda for mathematics instruction (e.g.,
curriculum frameworks) and when making mathematics teaching and learning
visible (e.g., formative assessments, student work), as well as tools that are used
while actually teaching.

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Accountability Relations between Teachers, School leaders, and District Leaders

The picture that emerges from the support structures we have discussed thus far is
that of a coherent reform effort grounded in a shared instructional vision, in which
networks characterized by relatively deep interactions support teachers' ongoing
learning. Although the activities of teachers as well as of school and district leaders
are aligned in this picture, we have not specified the relationships between
members of these different role groups. The next two potential support structures
address this issue.

The seventh hypothesized support structure concerns accountability relations
between teachers, school leaders, and district leaders. At the classroom level,
instruction that supports students' understanding of central mathematical ideas
involves what Kazemi and Stipek (2001) term a high press for conceptual thinking.
Kazemi and Stipek clarify that teachers maintain a high conceptual press by 1)
holding students accountable for developing explanations that consist of a
mathematical argument rather than simply a procedural description, 2) attempting
to understand relations among multiple solution strategies, and 3) using errors as
opportunities to reconceptualize a problem, explore contradictions in solutions, and
pursue alternative strategies. Analogously, we hypothesize that the following
accountability relations will contribute to instructional improvement:

- Formal school instructional leaders (e.g., principals, assistant principals,
mathematics coaches) hold mathematics teachers accountable for maintaining
conceptual press for students and, more generally, for developing ambitious
instructional practices.

- District leaders hold school leaders accountable for assisting mathematics
teachers in improving their instructional practices.

We anticipate that the potential of these accountability relations to support
instructional improvement will both depend on and contribute to the development
of a shared instructional vision. In the absence of a shared vision, different school
leaders might well hold teachers accountable to different criteria, some of which
are at odds with the intent of the district's instructional improvement effort
(Coburn & Russell, in press).

Relations of Assistance between Teachers, School Leaders, and District Leaders

Elmore (2000, 2004) argues, correctly in our view, that it is unethical to hold
people accountable for developing particular forms of practice unless their learning
of those practices is adequately supported. We would, for example, question a
teacher who holds students accountable for producing mathematical arguments to
explain their thinking but does little to support the students' development of
mathematical argumentation. In Elmore's terms, the teacher has violated the
principle of mutual accountability, wherein leaders are accountable to support the
learning of those who they hold accountable. The eighth hypothesized support
structure comprises the following relations of support and assistance:

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- Formal school instructional leaders (e.g., principals, assistant principals,
mathematics coaches) are accountable to teachers for assisting them in
understanding the mathematical intent of the curriculum, in maintaining
conceptual press for students and, more generally, in developing ambitious
instructional practices.

- District leaders are accountable to school leaders to provide the material
resources needed to facilitate high quality mathematics instruction, and to
support school leaders' development as instructional leaders.

Leadership Content Knowledge

The ninth hypothesized support structure follows directly from the relations of
accountability and assistance that we have outlined and concerns the leadership
content knowledge of school and district leaders (see Table 1). Leadership content
knowledge encompasses leaders' understanding of the mathematical intent of the
adopted instructional materials, the challenges that teachers face in using these
materials effectively, and the challenges in supporting teachers' reorganization of
their instructional practices (Stein & Nelson, 2003). Ball, Bass, Hill, and
colleagues have demonstrated convincingly that ambitious instructional practices
involve the enactment of a specific type of mathematical knowledge that enables
teachers to address effectively the problems, questions, and decisions that arise in
the course of teaching (Ball & Bass, 2000; Hill & Ball, 2004; Hill, Rowan, & Ball,
2005). Analogously, Stein and Nelson (2003) argue that effective school and
district instructional leadership in mathematics involves the enactment of a subject-
matter-specific type of mathematical knowledge, leadership content knowledge,
that enables instructional leaders to recognize high-quality mathematics instruction
when they see it, support its development, and organize the conditions for
continuous learning among school and district staff. Stein and Nelson go on to
argue that the leadership content knowledge that principals require to be effective
instructional leaders in mathematics includes a relatively deep understanding of
mathematical knowledge for teaching, of what is known about how to teach
mathematics effectively, and of how students learn mathematics, as well as
"knowing something about teachers-as-learners and about effective ways of
teaching teachers" (p. 416). They extend this line of reasoning by proposing that
district leaders who provide professional development for principals should know
everything that principals need to know and should also have knowledge of how
principals learn.

We see considerable merit in Stein and Nelson's arguments about the value of
leadership content knowledge in mathematics. However, the demands on principals
seem overwhelming if they are to develop deep leadership content knowledge in
all core subject matter areas including mathematics. This is particularly the case for
principals of middle and high schools. We therefore suggest that it might be more
productive to conceptualize this type of expertise as being distributed across formal
and informal school leaders rather than residing exclusively with the principal. In
other words, we suggest that the depth of leadership content knowledge that

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principals require is situational and depends in large measure on the expertise of
others in the school. In cases where principals can capitalize on the expertise of a
core group of relatively accomplished mathematics teachers or an effective school-
based mathematics coach, for example, the extent of principals' leadership content
knowledge in mathematics might not need to be particularly extensive. In such
cases, it might suffice for principals to understand the characteristics of high
quality instruction that hold across core subject matter areas provided they also
understand the overall mathematical intent of the instructional programme and
appreciate that using the programme effectively is a non-trivial accomplishment
that requires ongoing support for an extended period of time. We speculate that
this limited knowledge might enable principals to collaborate effectively with
accomplished teachers and possibly school-based coaches. Stein and Nelson (2003,
p. 444) acknowledge the viability of this approach when they observe that

where individual administrators do not have the requisite knowledge for the
task at hand they can count on the knowledge of others, if teams or task
groups are composed with the recognition that such knowledge will be
requisite and someone, or some combination of people and supportive
materials, will need to have it.

The ninth support structure is therefore leadership content knowledge in
mathematics that is distributed across the principal, teachers, and the coach. This
hypothesized support structure implies that it will be important for principal
professional development to attend explicitly to the issue of leveraging teachers'
and coaches' expertise effectively.

Equity in Students ' Access to Ambitious Instructional Practices

The student population is becoming increasingly diverse racially and ethnically in
most industrialized countries and in a number of developing countries. An
established research base indicates that access to ambitious instructional practices
for students who are members of historically under-served populations (e.g.,
students of colour, students from low-income backgrounds, students who are not
native language speakers, students with special needs) is rarely achieved (see
Darling- Hammond, 2007). In addition, a small but growing body of research that
suggests that ambitious instructional practices are not enough to support all
students' mathematical learning unless they also take account of the social and
cultural differences and needs of historically marginalized groups of students (see
Nasir & Cobb, 2007). This work indicates the importance of professional
development for teachers and instructional leaders in mathematics that focuses
squarely on meeting the needs of underserved groups of students. In addition, it
has implications for the establishment of institutional support structures that are
likely to result in access to appropriate instructional practices for historically
marginalized groups of students. The final two support structures that we discuss
concern equity in students' learning opportunities.

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De-tracked instructional programme. Tracking, or the grouping of students
according to current achievement, often prevails in schools that serve students
from marginalized groups. However, current research indicates that "tracking does
not substantially benefit high achievers and tends to put low achievers at a serious
disadvantage" (Darling-Hammond, 2007, p. 324; see also Gamoran, Nystrand,
Berends, & LePore, 1995; Horn, 2007; Oakes, Wells, Jones, & Datnow, 1997).
The tenth support structure is therefore a rigorously de-tracked instructional
programme in mathematics.

Category system for classifying students. The final support structure concerns the
categories of mathematics students that are integral to teachers' and instructional
leaders' practices. Horn's (2007) analysis of the contrasting systems for classifying
students constructed by the mathematics teachers in two US high schools is
relevant in this regard because it indicates that these classification systems were
related to the two groups of teachers' views about whether mathematics should be
tracked (see Table 1). Significantly, Horn's analysis also indicates that the
contrasting classification systems also reflected differing views of mathematics as
a school subject. The teachers in one of the schools differentiated between formal
and informal solution methods, and viewed the latter as illegitimate. They also
took a sequential view of school mathematics and assumed that students had to
first master prior topics if they were to make adequate progress. This conception of
school mathematics was reflected in the teachers' classification of students as more
or less motivated to master mathematical formalisms, and as faster and slower in
doing so. The teachers' classification of students in terms of stable levels of
motivation and ability grounded their perceived need for separate mathematics
courses for different types of students.

In sharp contrast, the mathematics teachers at the second school that Horn
(2007) studied tended to take a non-sequential view of school mathematics and
conceptualized it as a web of ideas rather than an accumulation of formal
procedures. These teachers also rejected the categorization of students as fast or
slow because it emphasized task completion at the expense of considering multiple
strategies. In addition, the teachers in this school viewed it as their responsibility to
support students' engagement both by selecting appropriate tasks and by
influencing students' learning agendas. Thus, these teachers addressed the
challenge of teaching mathematics to all their students in the context of a
rigorously de-tracked mathematics programme by focusing primarily on their
instructional practices rather than on perceived mismatches between students and
the curriculum. In doing so, they constructed categories for classifying students
that characterized them in relation to their current instructional practices rather
than in terms of stable traits. Building on Horn's analysis, the eleventh support
structure is a category system that classifies students in relation to current
instructional practices rather than in terms of seemingly stable traits.

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Reflection

We developed the proposed support structures summarized in Table 1 by mapping
backwards from the classroom and, in particular, from a research-based view of
high quality mathematics instruction. In doing so, we have limited our focus to the
establishment of institutional settings that support school and district staffs
ongoing improvement of their practices. This backward mapping process could be
extended to develop conjectures that are directly related to the traditional concerns
of policy researchers. For example, several of the hypothesized support structures
involve conjectures about the role of mathematics coaches and school leaders.
These conjectures have implications for district hiring and retention policies. In
addition, the hypotheses imply that the allocation of frequently scarce material
resources should be weighted towards what Elmore (2006) terms the bottom of the
system (see also Gamoran et al., 2003). As the notion of distributed leadership is
currently fashionable, 8 it is worth noting that the hypotheses do not treat the
distribution of instructional leadership as a necessary good. In the absence of a
common discourse about mathematics, learning, and teaching, the distribution of
leadership can result in a lack of coordination and alignment (Elmore, 2000). As
Elmore (2006) observes, effective schools and districts do not merely distribute
leadership. They also support people 's development of leadership capabilities, in
part by structuring settings in which they learn and enact leadership. As the
proposed support structures indicate, important outcomes of an initiative to
improve the quality of mathematics learning and teaching include "the system
capacity developed to sustain, extend, and deepen a successful initiative" (Elmore,
2006, p. 219).

DOCUMENTING THE INSTITUTIONAL SETTING OF MATHEMATICS TEACHING

The hypothesized support structures that we have discussed constitute a theory of
action for designing schools and larger administrative jurisdictions such as school
districts as learning organizations for instructional improvement in mathematics.
We now consider a second conceptual tool that is central to the improvement of
mathematics teaching and learning at scale, an analytic approach for documenting
the institutional setting of mathematics teaching. In addition to formally testing our
hypotheses about potential support structures, we will share our analysis of the data
collected each year with the four districts and collaborate with them to identify any
adjustments that might make the districts' improvement designs for middle-school
mathematics more effective. To accomplish this, we require an analytic approach
for documenting the institutional setting of mathematics teaching that can feed
back to inform the districts' ongoing improvement efforts.

Spillane and colleagues (Spillane, 2005; Spillane, Halverson, & Diamond, 2001, 2004) proposed
distributed leadership as an analytic perspective that focuses on how the functions of leadership are
accomplished rather than on the characteristics and actions of individual positional leaders. However, as
so often happens in education, the basic tenets of this analytic approach have been translated into
prescriptions for practitioners' actions. In our view, this is a fundamental category error that, if past
experience is any guide, might well have unfortunate consequences (e.g., Cobb, 1994, 2002).

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The analytic approach that we will take makes a fundamental distinction
between schools and districts viewed as designed organizations and as lived
organizations. A school or district viewed as a designed organization consists of
formally designated roles and divisions of labour together with official policies,
procedures, routines, management systems, and the like. Wenger (1998) uses the
term designed organization to indicate that its various elements were designed to
carry out specific tasks or to perform particular functions. In contrast, a school or
school district viewed as a lived organization comprises the groups within which
work is actually accomplished together with the interconnections between them.
As Brown and Duguid (1991, 2000) clarify, people frequently adjust prescribed
organizational routines and procedures to the exigencies of their circumstances
(see also Kawatoko, 2000; Ueno, 2000; Wenger, 1998). In doing so, they often
develop collaborative relationships that do not correspond to formally appointed
groups, committees, task forces, and teams (e.g., Krainer, 2003). Instead, the
groups within which work is actually organized are sometimes non-canonical and
not officially recognized. These non-canonical groups are important elements of a
school or district viewed as a lived organization.

Given the goals of our research, we find it essential to document the districts in
which we are working as both designed organizations and as lived organizations.
One of our first steps has been to document the districts as designed organizations
by interviewing district leaders about their plans or designs for supporting the
improvement of mathematics teaching and learning. In analysing these interviews,
we have teased out the suppositions and assumptions and have framed them as
testable conjectures. The process of testing these conjectures requires that we
document how the districts' improvement designs are playing out in practice,
thereby documenting the schools and districts in which we are working as lived
organizations.

Methodologically, we will use what Hornby and Symon (1994) and Spillane
(2000) refer to as a snowballing strategy and Talbert and McLaughlin (1999) term
a bottom-up strategy to identify groups within the schools and districts whose
agendas are concerned with the teaching and learning of mathematics. The first
step in this process involves conducting audio-recorded semi-structured interviews
with the participating 30 middle-school mathematics teachers in each district to
identify people within the district who influence how the teachers teach
mathematics in some significant way. The issues that we will address in these
interviews include the professional development activities in which the teachers
have participated, their understanding of the district's policies for mathematics
instruction, the people to whom they are accountable, their informal professional
networks, and the official sources of assistance on which they draw.

The second step in this bottom-up or snowballing process involves interviewing
the formal and informal instructional leaders identified in the teacher interviews as
influencing their classroom practices. The purpose of these interviews is to
understand formal and informal leaders' agendas as they relate to mathematics
instruction and the means by which they attempt to achieve those agendas. We will
then continue this snowballing process by interviewing people identified in the

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second round of interviews as influencing instruction and instructional leadership
in the district. In terms familiar to policy researchers, this bottom-up methodology
focuses squarely on the activity of what Weatherley and Lipsky (1977) term street-
level bureaucrats whose roles in interpreting and responding to district efforts to
improve mathematics instruction are as important as those of district leaders who
designed the improvement initiative. The methodology therefore operational izes
the view that what ultimately matters is how district initiatives are enacted in
schools and classrooms (e.g., McLaughlin, 2006).

In addition to identifying the groups in which the work of instructional
improvement is accomplished and documenting aspects of each group's practices,
our analysis of the schools and districts as lived organizations will also involve
documenting the interconnections between the groups. To do so, we will focus on
three types of interconnections, two of which we introduced when describing
potential support structures. Interconnections of the first type are constituted by the
activities of brokers who are at least peripheral members of two or more groups.
As we noted, brokers can bridge between the perspectives of different groups,
thereby facilitating the alignment of their agendas. As our hypotheses indicate, our
analysis of brokers will be relatively comprehensive and will seek to clarify
whether there are brokers between various groups in the school (e.g., mathematics
teachers and school leaders), between school leaders and district leaders, and
between key units of the district central office. Boundary objects that members of
two or more groups use routinely as integral aspects of their practices constitute
interconnections of the second type. As we have noted, there is the very real
possibility that members of different groups will used boundary objects differently
and imbue them with different meanings (Wenger, 1998). Our analysis will
therefore seek to identify boundary objects and to document whether members of
different groups used them in compatible ways.

The third type of interconnection is constituted by boundary encounters in
which members of two or more groups engage in activities together as a routine
part of their respective practices. Three of the hypothesized support structures
focus explicitly on boundary encounters: the explicit negotiation of the meaning of
boundary objects, relations of accountability, and relations of assistance. In
addition to documenting the frequency of boundary encounters between members
of different groups, our analysis will focus on the nature of their interactions.

A recent finding reported by Cob urn and Russell (in press) indicates the
importance of pushing for this level of detail. They studied the implementation of
elementary mathematics curricula designed to support ambitious instruction in two
school districts. As part of their instructional improvement efforts, both districts
hired and provided professional development for a cadre of school-based
mathematics coaches (see also Nickerson, this volume). Coburn and Russell found
that there were significant differences in the depth of the interactions between the
coaches and the professional development facilitators in the two districts. In the
first district, interactions were relatively deep and focused on issues such as
discerning the mathematical intent of instructional tasks and on identifying and
building on student reasoning strategies. In the second district, interactions were

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typically of limited depth and focused primarily on how to use instructional
materials and on mapping the curriculum to district or state standards. Coburn and
Russell also documented the nature of interactions between coaches and teachers in
the two districts. They found that teacher-coach interactions increased in depth to a
far greater extent in the first district than in the second district. In addition,
interactions between teachers when a coach was not present also increased in depth
in the first but not the second district. In other words, the contrasting routines of
interaction in coach professional development sessions became important features
of interactions in teacher networks in the two districts.

In our view, Coburn and Russell's analysis represents a significant advance in
research on instructional improvement at scale. To this point, policy researchers
have tended to frame social networks as conduits for information about
instructional and instructional leadership practices. However, research in
mathematics education makes it abundantly clear that information about ambitious
instructional practices is, by itself, insufficient to support teachers' development of
this form of practice. Coburn and Russell's analysis focuses more broadly on
interactions across groups as well as within social networks, and highlights the
importance of co-participation in collective activities. In addition, their findings
demonstrate that the depth of co-participation matters. Their analysis therefore
establishes a valuable point of contact between research on policy implementation
and research on mathematics teachers' learning. This latter body of work
documents that teachers' co-participation in activities of sufficient depth with an
accomplished colleague or instructional leader is a critical source of support for
teachers' development of ambitious practices (e.g., Borko, 2004; Fennema et al.,
1996; Franke & Kazemi, 2001a; Goldsmith & Shifter, 1997; Kazemi & Franke,
2004; Wilson & Berne, 1999). We anticipate that Coburn and Russell's (in press)
notion of routines of interaction will prove to be a useful analytic tool as we seek
to understand whether the nature of the boundary encounters in which school and
district staff engage in activities together influences how they subsequently interact
with others in different settings.

PROVIDING FEEDBACK TO INFORM INSTRUCTIONAL IMPROVEMENT

In the approach that we have outlined, the analysis of a school or district as a lived
organization involves identifying the groups in which the work of instructional
improvement is actually accomplished and documenting interconnections between
these groups. An analysis of the lived organization therefore focuses on what
people actually do and the consequences for teachers' instructional practices and
students' mathematical learning. In contrast, an analysis of a school or district as a
designed organization involves documenting the school or district plan or design
for supporting instructional improvement in mathematics. This design specifies
organizational units and positional roles as well as organizational routines, and
involves conjectures about how the enactment of the design will result in the
improvement of teachers' instructional practices and student learning. An analysis
of the designed organization documents both this design and the tools and
activities that will be employed to realize the design by enabling people to improve

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their practices. In giving feedback to the four collaborating districts to inform their
improvement efforts, we will necessarily draw on our analyses of the districts as
both designed and lived organizations.

To develop this feedback, we will identify gaps between the districts' designs
for instructional improvement and the ways in which those designs are actually
playing out in practice by comparing our analyses of each district as a designed
organization and as a lived organization. This approach will enable us to
differentiate cases in which a theory of action proposed by a district is not enacted
in practice from cases in which the enactment of the theory of action does not lead
to the anticipated improvements in the quality of teachers' instructional practices
(Supovitz & Weathers, 2004). As an illustration, one of the districts with which we
are collaborating is investing some of its limited resources in mathematics coaches
with half-time release from teaching for each middle school. The district's theory
of action specifies that the coaches' primary responsibilities are to facilitate teacher
collaboration and to support individual teachers' learning by co-teaching with them
and by observing their instruction and providing constructive feedback. Suppose
that the district's investment in mathematics coaches does not result in a noticeable
improvement in teachers' instructional practices. It could be the case that the
theory of action of the district has not been enacted. For example, the coaches
might be tutoring individual students or preparing instructional materials for the
mathematics teachers in their schools rather than working with teachers in their
classrooms. In attempting to understand why this is occurring, we would initially
focus on coaches' and school leaders' understanding of the coaches' role in
supporting teachers' improvement of their instructional practices. Alternatively, it
could be the case that the coaches are working with teachers in their classrooms,
but their efforts to support instructional improvement are not effective. In this case,
we would initially seek to understand how, specifically, the coaches are attempting
to support teachers' learning and would take account of the process by which the
coaches were selected and the quality of the professional development in which
they participated.

As this illustration indicates, our goal when giving feedback is not merely to
assess whether the district's design is being implemented with fidelity, although
our analysis will necessarily address this issue. We also seek to understand why the
district's theory of action is playing out in a particular way in practice by taking
seriously the perspectives and practices of street-level bureaucrats such as teachers,
coaches, and school leaders. In doing so, we will draw on both an analysis of the
district design as a potential resource for action and an analysis of the district as a
lived organization that foregrounds people's agency as they develop their practices
within the context of others' institutionally situated actions (e.g., Feldman &
Pentland, 2003).

DISCUSSION

In this chapter, we have focused on the question of how mathematics education
research might contribute to the improvement of mathematics teaching and
learning at scale. We addressed this question by first clarifying the value of

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viewing mathematics teachers' instructional practices as situated in the institutional
settings of the schools and districts in which they work. Against this background,
we presented a series of hypotheses about school and district structures that might
support teachers' ongoing improvement of their classroom practices. We then went
on to outline an analytic approach for documenting the institutional settings of
mathematics teaching established in particular schools and districts that can feed
back to inform the instructional improvement effort.

We conclude this chapter by returning to the relation between research in
educational policy and leadership and in mathematics education. To this point,
researchers in these fields have conducted largely independent lines of work on the
improvement of teaching and learning (e.g., EngestrSm, 1998; Franke, Carpenter,
Levi, & Fennema, 2001). Research in educational policy and leadership tends to
focus on the designed structural features of schools and how changes in these
structures can result in changes in classroom instructional practices. In contrast,
research in mathematics education tends to focus on the role of curriculum and
professional development in supporting teachers' improvement of their
instructional practices and their views of themselves as learners. In this chapter, we
have argued that mathematics education research that seeks to contribute to the
improvement of teaching and learning at scale will have to transcend this
dichotomy by drawing on analyses of schools and districts viewed both as
designed organizations and as lived organizations. In the interventionist genre of
research that we favour, organizational design is at the service of large-scale
improvement in the quality of teachers' instructional practices. In research of this
type, the attempt to contribute to improvement efforts in particular schools and
administrative jurisdictions constitutes the context for the generation of useful
knowledge about the relations between the institutional settings in which teachers'
work, the instructional practices they develop in those settings, and their students'
mathematical learning. This genre of research therefore reflects de Corte, Greer,
and Verschaffel's (1996) adage that if you want to understand something try to
change it, and if you want to change something try to understand it.

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Paul Cobb

Vanderbilt University
USA

Thomas Smith
Vanderbilt University
USA

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11. STUDIES ON REGIONAL AND NATIONAL

REFORM INITIATIVES AS A MEANS TO

IMPROVE MATHEMATICS TEACHING

AND LEARNING AT SCALE

The chapter considers four examples of large-scale projects involving national
reform initiatives in mathematics drawn from four continents — Europe, North
America, Australia and Asia. Poor student performance on international and/or
national assessment programmes was, in part, a catalyst for each programme. The
underlying issue driving each of these studies was the perceived importance of
improved student learning outcomes in mathematics and science in these countries.
All the projects focus on initiating purposeful pedagogic change through involving
teachers in rich professional learning experiences. The primary purpose of this
chapter is two-fold First, a brief description is provided of each project in order to
give an insight into different countries ' efforts to improve teaching and learning at
scale. Second an analysis and discussion of common features are undertaken
leading to lessons learned

INTRODUCTION AND BACKGROUND

All over the world, countries face challenges in terms of supporting students to
achieve their potential in the important area of mathematics. As we move into the
21 st century, we find new meaning to calls about the importance of mathematics
knowledge and know-how to economic growth.

Glenn (2000, p. 7) writing for a US readership, identified four important and
enduring reasons for the need for students to achieve to their potential in
mathematics and science. These were:

- The rapid pace of change in both the increasingly interdependent global
economy and in the American workplace demands widespread mathematics-
and science-related knowledge and abilities;

- Our citizens need both mathematics and science for their everyday decision-
making;

- Mathematics and science are inextricably linked to the nation's security
interests; and

- The deeper, intrinsic value of mathematical and scientific knowledge shape and
define our common life, history, and culture. Mathematics and science are
primary sources of lifelong learning and the progress of our civilization.

These or similar sentiments have prompted education authorities around the
world to look at better ways of (i) increasing the mathematics skills of students, (ii)

JOHN PEGG AND KONRAD KRAINER

developing more effective programmes to support teachers meet the demands of
modern classrooms, and (iii) increasing the attraction and retention of qualified
mathematics teachers by improving the education of prospective teachers as well as
addressing the professional needs of practising teachers.

Any student who experiences ongoing failure in school faces a myriad of
difficulties in achieving long-term employment, and useful and fulfilling
occupations. Those who exhibit consistent weaknesses in basic skills in
mathematics are particularly vulnerable. Test data provide a compelling case for
the need to develop programmes and approaches that improve mathematics
outcomes for all students and particularly for those who are performing at or below
the international or national benchmarks. Equally important is the need of
mathematics instruction to address affective aspects of learning. It is a great
challenge to raise all students' interest in and joy of mathematics, to let them
experience the benefit and beauty of this subject and to decrease their fears and
feelings of failing, in particular taking into account gender and diversity issues.
Cognitive and affective factors are closely interconnected.

International data point out that the life chances of students to acquire
mathematics competencies are not uniform across a nation. Some students
approach education (and the learning of mathematics) with advantages that are not
found throughout the population. In particular, those who live in poverty, in rural
locations or belong to at-risk minorities often have less chance of success than
others. Hence, driving many reform initiatives is the principle of equity of
educational opportunity regardless of current life opportunities as well as an
expectation of achievement regardless of the current ability level.

The central question for most nations is where best to direct efforts so that
meaningful and sustainable change can be managed in a cost-effective way. While
the problems each nation faces are complex and unique, much can be gained by
analysing different approaches. The large-scale projects described in this text,
highlight how four countries have, in part, attempted to address education issues.
While their programmes are sophisticated in design and implementation, they have
a central, common theme, namely, they are directed at the professional learning of
prospective and practising teachers.

This focus on teachers' learning is not arbitrary: it is evident that reform
programmes must access students through their teachers. However, how large is
teachers' impact on students' learning? Over the past few years, a consistent theme
has begun to emerge concerning the variance identified in the analysis of student
learning over many large-scale projects. Identified factors that contribute to major
sources of variation in student performance (Hattie, 2003) include student (50%),
home (5-10%), schools (5-10%), principals (mainly accounted for in schools), peer
effects (5-10%), and teachers (30%). This research implies that genuine
improvement in learning can be achieved by improving the beliefs, emotions,
knowledge, and practice of teachers. The picture is consistent; it is what
mathematics teachers believe, feel, know, and do, that is a powerful determinant in
student learning.

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REGIONAL AND NATIONAL REFORM INITIATIVES

In the fol lowing, four large-scale projects involving significant national reform
initiatives from four continents are described. Given the complexity and authors'
deeper knowledge of their own initiatives in Austria and Australia, a slightly more
extensive focus is placed on these two cases. The four countries vary greatly in size
(and distances to be travelled), spread of the population, importance of teacher
shortages, and the educational disadvantage experienced by minorities. The first
two projects can be traced back to the results of student assessments in Austria
(Europe) and the state of Ohio (US, North America). The remaining two initiatives
address in different ways the under-achievement of students in rural and regional
areas in comparison to their metropolitan peers. One is from Australia and the
other is from South Korea (Asia).

Hence, as we consider the different initiatives outlined we identify two
underpinning tenets. First, more than at any time in our history, more citizens than
ever need to achieve their mathematically potential if they are to have active and
fulfilling occupations. Second, if we are to bring about sustainable change then
money and effort needs to be directed to encouraging and supporting quality
teaching.

In order to facilitate comparison across the four projects, a common structure
has been employed in the next section. Each description considers:

- Impulse for the initiative and challenge

- Goals and intervention strategy

- Implementation and communication

- Evaluation and impact

- Challenges and further steps

AUSTRIA: IMST - INNOVATIONS IN MATHEMATICS, SCIENCE AND
TECHNOLOGY TEACHING

Austria participated in all three cohorts (primary, middle and high school) of the
1995 TIMSS achievement study. Whereas the results for the primary and the
middle school were rather promising, the results of the Austrian high school
students (grades 9 to 12 or 13), particularly with regard to the TIMSS advanced
mathematics and physics achievement test, shocked the public (Mullis et al.,
1998). These data were a catalyst for the government and education community to
re-evaluate the status of mathematics and science education and saw the
responsible federal ministry launch the IMST - Innovations in Mathematics and
Science Teaching - research project (1998-1999). The purpose of this nation-wide
programme was to address an identified complex picture of diverse problematic
influences on the status and quality of mathematics and science teaching.

Impulse for the Initiative and Challenge

In addition to the disappointing performance of Austria's high school students on
TIMSS, Austria was among those nations with the highest achievement differences
between boys and girls. Also, students showed poor results with regard to items

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JOHN PEGG AND KONRAD KRAINER

that referred to higher levels of thinking, and less than a third of Austrian students
felt that they were involved in reasoning tasks in most or every mathematics
lesson(s).

In seeking to explain these results and findings, Austrian researchers were
convinced that there were manifold causes. For example, the answers to a written
questionnaire by Austrian teachers, teacher educators and representatives of the
education authorities showed that teachers were predominantly seen as dedicated
and as having a lot of pedagogical and didactic autonomy. On the one hand, there
were many creative initiatives being carried out by individuals, groups or
institutions; on the other hand, many of these initiatives were carried out in
isolation, and a networking structure was missing.

Mathematics education and related research was seen as poorly anchored at
Austrian teacher education institutions. Subject experts dominate university
teacher education, other teacher education institutions show a lack of research in
mathematics education; the collaboration with educational sciences and schools is
- with exception of a few cases - underdeveloped. A competence centre like those
found in many other countries was not existent. Also, the overall structure
(including two institutions for the education of prospective teachers that are mostly
unconnected, a variety of different kinds of schools with corresponding
administrative bodies in the ministry and in the institutions for the education of
practising teachers, etc.) showed a picture of a "fragmentary educational system"
of lone fighters with a high level of (individual) autonomy and action, however,
there was little evidence of reflection and networking (Krainer, 2001).

Goals and Intervention Strategy

The analyses led to the four year project IMST 2 (2000-2004) - now called
Innovations in Mathematics, Science and Technology Teaching: the addition of
"Technology" in the project title was to express the fundamental importance of
technologies for mathematics and science teaching. The project (Krainer, Dorfler,
Jungwirth, Kiihnelt, Rauch, & Stern, 2002) focused solely on the upper secondary
school level and involved the subjects, biology, chemistry, mathematics and
physics. IMST 2 was financed by the responsible federal ministry and the Austrian
Council of Research and Technology Development. In order to take systemic steps
to overcome the "fragmentary educational system", the approach of a "learning
system" (Krainer, 2005a) was taken. It adopted enhanced reflection and
networking as the basic intervention strategy. The theoretical framework builds on
the ideas of action research (Altrichter, Posch, & Somekh, 1993), constructivism
(von Glasersfeld, 1991) and systemic approaches to educational change and
system theory (Fullan, 1993; Willke, 1999).

Besides stressing the dimensions of reflection and networking, "innovation" and
"work with teams" were two additional features. Innovations were not regarded as
singular events that replace an ineffective practice but as continuous processes
leading to a natural further development of practice. Teachers and schools defined
their own starting point for innovations and were individually supported by

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researchers and expert teachers. The IMST 2 intervention built on teachers'
strengths and aimed at making their work visible (e.g., by publishing teachers'
reports on the website). Thus teachers and schools retained ownership of their
innovations. Another important feature of IMST 2 was the emphasis on supporting
teams of teachers from a school.
The two major tasks of IMST 2 were

- The initiation, promotion, dissemination, networking and analysis of
innovations in schools (and to some extent also in teacher education at
university); and

- Recommendations for a support system for the quality development of
mathematics, science and technology teaching.

The second task led to a plan for a sustainable support system (Krainer, 2005b).
Consequently, IMST 2 was followed by the project IMST3 (2004-2007) which
included all secondary schools and later by the project IMST3 Plus (2007-2009)
which broadened the support of schools to the primary level.

In IMST3 and IMST3 Plus about three times more schools were supported and
also the participation in regional networks was fostered. It was intended to build
up a network of practitioners, researchers and administrative staff that to help to
support the schools. Another task was to contribute to the implementation of a
better infrastructure for mathematics and science education. The improved
infrastructure was considered a basis and precondition for implementing a
sustainable network of persons and institutions. Therefore, for example, the IMST
project team designed a plan on how to establish competence centres for
mathematics, science and technology education.

Implementation and Communication

The operative implementation of most parts of IMST has been entrusted to the
Institute of Instructional and School Development at the University of Klagenfiirt.
Though having this university institute as a key player in the whole process, the
whole project was understood as an initiative, and influenced by a wide network of
people and institutions in order to improve mathematics and science teaching and
learning in Austria.

In the years 2000-2004, IMST supported about 50 innovation projects at
Austrian upper secondary schools (and partially at other organisations, e.g.,
teacher education institutions) in each school-year. The participation was
voluntary and gave teachers and schools a choice among four priority programmes
(Basic education; School development; Teaching and learning processes; Practice-
oriented research: Students' independent learning) according to the challenges
sifted out in the above mentioned research project. In general, teachers in all four
priority programmes - and also later in a specific programme on gender sensitivity
and gender mainstreaming - were supported by staff members of IMST. The
priority programmes can be regarded as small professional communities that not
only supported each participant to proceed with his or her own project but also
generated a deeper understanding of the critical reflection of one's own teaching,

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of formulating research questions, of looking for evidence based on viable data,
and on methods that help to gather that data.

Since 2004, the direct support of about 1 50 innovative projects is organized
within an IMST fund. Whereas the fund aims primarily at reaching experienced
teachers who are able to disseminate their experiences and results to other teachers
(at their school, in their district or nation-wide), the formation of regional networks
in each federal state aims at reaching a greater number of teachers. In addition, a
gender network and a project "examination culture" have been established in order
to offer advice and professional development activities for teachers. All these four
measures of IMST3 (fund, regional networks, gender network and examination
culture) were continued in the phase of IMST3 Plus. Step by step these measures
are opened to teachers at the primary level.

Throughout all phases of IMST, the project is accompanied by a website, an
annual conference and a quarterly newsletter.

Evaluation and Impact

Evaluation was an integral part of IMST since its start (Krainer, 2007). The self-
evaluation comprised forms of a process-oriented evaluation (generating steering
knowledge for the project management and the project teams), an outcome-
oriented evaluation (working out the impact of the project at different levels of the
educational system), and a knowledge-oriented evaluation (generating new
theoretical and practical knowledge about the interconnection between the
project's interventions and its impact). In many cases, this self-evaluation included
data gathering and feedback by external experts. In addition, several independent
evaluations were commissioned looking at the impact of the project or parts of it.
For example, three international experts evaluated IMST 2 (2000-2004) and IMST3
(2004-2006) at the end of these periods of the project and wrote corresponding
reports. Also parts of IMST3 (e.g., the priority programmes or the networks) were
externally evaluated.

The self-evaluation of IMST was done at different levels, in particular at the
school, classroom and individual teachers' level, and at the Austrian educational
system level. The teachers used different forms of action research methods. In
some cases, they were supported by their mentoring teams by means of external
evaluation (interviews, questionnaires, analyses of videos). Overall, the reports
indicated significant gains of students' and teachers' affective and cognitive
growth.

Specht (2004) reported that IMST 2 is seen as an important, useful and effective
support for instructional and school development in mathematics and science. He
(p. 51) identified concrete changes in teachers' orientations to actions, in particular
concerning their readiness for innovations in teaching, their increased ability to
reflect and self-evaluate, their higher care in choosing teaching contents and their
more intensive collaboration with colleagues.

With the start of IMST3 in 2004, efforts to investigate the impact of the
programme were increased. In particular, student questionnaires and more

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systematic analyses of teachers' reports were introduced. Answers to the teacher
questionnaire demonstrated a high level of satisfaction with the programme.
Teachers were highly motivated even though the additional workload of the
programme could be substantial. IMST students were generally more interested in
their subjects (in comparison to students in the Austrian PISA sample), and
reported less anxiety. However, this may be due to their teachers, who show a high
level of motivation already before they entered the IMST programmes.

A longitudinal study (Muller, Andreitz, Hanfstingl, & Krainer, 2007) based on
the self-determination theory of Deci and Ryan (2002) was performed in all
classes taking part. The study showed that teachers who experience support from
their colleagues and principal assess their students as being more motivated. Their
students felt more intrinsically motivated than students from less supported
teachers. However, if teachers felt pressure from colleagues and the principal,
teachers' and students' intrinsic motivation decreased. Overall, the study showed
that teachers trying to improve their practice should not be isolated.

An external evaluation of the regional networks of IMST 2 , on the basis of a
questionnaire directed at principals and superintendents (Heffeter, 2006, p. 47),
found that this particular measure promotes the communication and change of
experiences among teachers. The study also suggested that the IMST process has
the potential to break up thinking patterns that have become entrenched in the
Austrian educational system.

The external evaluations of IMST 2 and IMST3 by international experts
(Prenzel, Schratz, & Messner, 2007) regarded the project as a national and
international remarkable and successful development programme. Suggestions
focused to a large extent on an increasing emphasis on generating new scientific
knowledge that coincides with the main strategy of IMST.

Challenges and Further Steps

In the remaining time of IMST3 Plus (2007-2009), four major challenges are to be
taken into consideration. Firstly, the work with primary schools has to be started.
This is connected with intensive collaborations with the new established teacher
education institutions (Padagogische Hochschulen) in Austria. Secondly, the
recent formation of IMST networks at the district level (e.g., the VIAMATH
project) generates new questions of adequate support and evaluation. Here, a good
interplay between the IMST fund and the regional networks seems necessary as
well as collaborations with the regional school boards and the teacher education
institutions. Thirdly, the ongoing discussion on the plan for the time after 2009 has
to be finalized. Fourthly, steps towards improving the opportunities for research in
mathematics education at the primary school level have to be taken. It should be
mentioned that Austria has so far no professor for mathematics education at the
primary school level that means that research in that area is rather underdeveloped.
Recently, IMST promotes the establishment of regional centres for mathematics
and science education where teacher educators and researchers working at the
primary and the secondary level are expected to collaborate.

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UNITED STATES, OHIO: SYSTEMIC REFORM OF SCIENCE AND MATHEMATICS

EDUCATION

Traditionally, education in the US was a matter for local communities to control
and fund. The publication of "A Nation at Risk" (NCEE, 1983) initiated a US-
wide desire to move schools towards universal standards for accountability in
mathematics and science teaching and learning. The standards-movement led to
the fact that today both state and federal governments play a much larger role in
US education. While federal documents contained recommendations for improving
teaching practices and student learning for all, the specific means for achieving
theses goals were left to individual states. As part of this pattern, Ohio established
a model curriculum for science, established outcomes to be measured, and recently
developed content standards. Ohio was one of the first states in the US to receive
federal funds under the State Systemic Initiative of the National Science
Foundation (NSF). Ohio's Systemic Initiative, Project Discovery, was initially
funded from 1991-1996 (more info can be found at
http://www.units.muohio.edu/discovery/). The following description is based
largely on Wagner and Meiring (2004) and Beeth, McCollum, and Tafel
(submitted).

Impulse for the Initiative and Challenge

In addition to the federal concerns about mathematics and science learning, there
was also a specific concern in Ohio: despite the fact that this state ranked high
(sixth) in science and technology based industry in the United States, Ohio's
schools ranked only in the mid-twenties in those subjects. Public media reports on
student learning brought attention to the need for reform of educational practices.

Ohio's Department of Education issued standards that asked for a more problem
solving and inquiry based approach to classroom instruction. The Discovery
Project Summer Institutes offered teachers a place to work on their professional
competence.

Goals and Intervention Strategy

The overall goal of Discovery was to improve mathematical and scientific
knowledge of middle school students, and to achieve equity between students of
different ethnic groups as well as gender equity through inquiry based instruction.
At the time of conception, national standards were not yet developed, thus
Discovery preceded the later move to achieve national coherence in the conception
of mathematics and science teaching.

Unlike other approaches to educational reform, the Discovery project did not
simply standardise a curriculum, but addressed the methods in which science and
mathematics are taught. The project's principal goal was to educate the educators
and produce a cadre of teachers capable of implementing inquiry based instruction

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in the classroom. In particular, Discovery focused on increasing teachers' content
knowledge.

The Discovery project went on to emphasize partnerships between teachers and
parents in mathematics and science education and incorporated methods to address
learning gaps between demographic groups.

Implementation and Communication

Discovery installed six-week professional development courses for in-service
teachers, which taught research based inquiry methods for individual middle-
school teachers. These courses were taught by university researchers and science
educators as well as master teachers throughout Ohio. The inquiry based method
focused on interaction and communication between students, discussion of
alternative methods of problem solving, supporting claims with data and less
emphasis on traditional rote memorization. Participants observed and discussed
each other's teaching of classes targeted at inquiry learning. After the course,
participants met several times during the following year working on a portfolio and
to exchange their experiences.

After the initial funding period, Discovery institutes have been continuing to
offer courses, albeit to a lesser extent. The initiative expanded to include teachers
at all grade levels and to include some site-based and other administrators, as well.

Evaluation and Impact

Evaluation of the programme occurred at three levels. Firstly, the scientific and
mathematical knowledge of students participating in the Discovery project was
compared with that of students not participating in the programme. Students in the
programme fared an average of 7% better on standardized tests. Secondly, gender
and ethnicity based performance gaps were assessed over time; and the programme
was shown to decrease the gap in achievement for these groups of learners.

Finally, passing rates of students for different schools were compared. This
comparison showed improvements in passing grades for schools with higher
percentages of Discovery trained teachers. An outgrowth of Discovery has been the
establishment of multiple professional development initiatives and networks of K-
16 institutions through a variety of state and national funding mechanisms.

Challenges and Further Steps

The activities leading to the most remarkable improvements in the study were
considered too expensive to implement at a state-wide level. To mitigate some of
the programme costs the state focused its efforts on the 9th and 10th grades, and
distributed Discovery training materials to schools capable of implementing the
programme. In 1995, the state of Ohio initiated the SUSTAIN project to maintain
the results already achieved by the federally funded project Discovery.

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SUSTAIN's primary focus is in creating a partnership between state universities
and state schools by providing inquiry based education methods to its education
students. SUSTAIN was also designed to foster collaboration among Ohio's higher
education institutions and with public school districts through regional professional
development centres named Centres of Excellence in Mathematics and Science.

At present, considerable investment is directed towards technology based, K-16,
educational and professional development support structures delivered through
both virtual and clinical learning experiences.

AUSTRALIA: SIMERR NATIONAL CENTRE

In 2004, the National Centre of Science, Information and Communication
Technology, and Mathematics Education for Rural and Regional Australia
(SiMERR National Centre) received an establishment grant from the Australian
Federal Government. This remains one of the largest education grants awarded in
Australia and indicates the importance attached to issues concerning rural and
regional education.

SiMERR was established at the University of New England (UNE) in Armidale,
a rural centre, utilising a collaborative model involving groups of academics in
each state (referred to as state Hubs). SiMERR carries out research and
professional development activities with a focus on improving the learning
outcomes of all Australian students, especially those studying in rural and regional
Australia.

Impulse for Initiative and Challenge

The rationale for the SiMERR National Centre was based on compelling evidence
from many sources (e.g., Programme for International Assessment (PISA),
Thomson, Cresswell, & De Bortoli, 2004; Thomson & De Bortoli, 2007; the
Trends in International Mathematics and Science Study (TIMSS), Zammit,
Routitsky, & Greenwood, 2002; and national basic skills test information,
MCEETYA, 2006) concerning the performance of students in rural and regional
Australia, about a third of the Australian student population.

These data quantified the extent of inequities for rural students in learning
outcomes in science and mathematics education and underscore the most
significant challenge currently facing education in Australia - equity of educational
opportunity for all school students regardless of location (e.g., Lyons, Cooksey,
Panizzon, Parnell, & Pegg, 2006; Roberts, 2005; Vinson, 2002).

Table 1 illustrates one example of data currently available. Here the columns
illustrate PISA summary data for Australia in 2003 and 2006 considered in terms
of location. There are significant differences in achievement between students in
each of these location groups.

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Table I . PISA 2003/2006 Mathematics achievement (mean scores) by location (Thomson,
Cresswell, & De Bortoli, 2004; Thomson & De Bortoli, 2007)

Average Score

2003

2006

Australia Overall

524

520

OECD countries

500

498

Metropolitan Australia

528

526

Provincial Australia

515

508

Remote Australia

493

468

Goals and Intervention Strategy

SiMERR was established to carry out strategic and applied research, and work with
rural communities to achieve improved educational outcomes for students.
The vision of the work of SiMERR is formulated in three can-do-statements:

- Parents can send their children to rural or regional schools knowing they will
experience equal opportunities for a quality education;

- Students can attend rural or regional schools realising their academic potential
in Science, ICT and Mathematics; and

- Teachers can work in rural or regional schools and be professionally connected
and supported.

To achieve this mission, SiMERR programmes identify and address important
educational issues of (i) specific concern to education in rural Australia, and (ii)
national concern in mathematics, science and ICT education across Australia by
working in rural schools.

Implementation and Communication

SiMERR members are involved in approximately 120 projects. While some
involve small numbers of schools (often in remote areas), teachers, and students,
other projects span across regions or state jurisdictions.

Many projects have national relevance, not only for rural areas but also more
broadly for all Australian students. It has become clear that in working to address
the needs of rural students, the findings and solutions that are emerging offer ways
of enhancing student-learning outcomes in metropolitan areas as well. In an
exemplary way, this is sketched below in brief descriptions of five large-scale
projects.

/. National Survey of Issues in Teaching and Learning Science, ICT and
Mathematics in Rural and Regional Australia (Lyons et al., 2006).
This project involved extensive questionnaire surveys of teachers and parents of
students from primary and secondary schools across Australia. Every provincial
and remote school, and a sample of metropolitan schools, in Australia were invited
to participate in the survey. Focus group interviews were conducted with a

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representative sample of teachers, parents and students from rural schools in each
state.

The survey data provided critical information about key themes that are
considered to be limiting student outcomes in mathematics for rural and regional
Australia as well as offering some practical ways of addressing these issues. The
recommendations focus on several key areas including:

• Staffing issues such as attraction and retention of teachers;

• Teacher training and qualifications;

• Professional development needs of teachers;

• Resource material needs of teachers;

• Learning opportunities and experiences of students.

2. Identifying and Analysing Processes of Groups of Teachers Producing
Outstanding Educational Outcomes in Mathematics (Pegg, Lynch, & Panizzon,
2007)

This project explored factors leading to outstanding mathematics outcomes in
junior secondary education for students across the ability spectrum. The focus was
on the characteristics of and processes used by groups of teachers. Mathematics
faculties achieving outstanding student-learning outcomes were identified by
drawing upon extensive quantitative and qualitative data-bases. The study involved
intensive case studies to identify faculty-level factors. Seven common themes are
reported and these are the strong sense of team, staff qualifications and experience,
teaching style, time on task, assessment practices, expectations of students, and
teachers caring for students.

The research highlighted a number of potential important issues for schooling
into the future around the needs:

• To provide opportunities to help teachers develop the knowledge and skills
necessary to exercise effective leadership in the role of faculty leader;

• For early career teachers to work with and learn from experienced mid and later
career teachers;

• To facilitate strong group interaction within faculties;

• For relevant school-based professional development;

• For high subject-knowledge standards for new and current teachers;

• To create a culture in which teaching and learning, rather than behaviour
management, dominates all classrooms; and

• To develop common goals among teachers, students and the local community.

3. QuickSmart intervention programme for middle-school students performing at
or below National Numeracy Benchmarks (Pegg & Graham, 2005, 2007)

This research programme is referred to by the generic title QuickSmart because it
teaches students how to become quick (and accurate) in response speed and smart
in strategy use. This teaching programme sought to improve automaticity,
operational ised by students' fluency and facility with basic mathematics facts for
those students in their middle years of schooling below national benchmarks. The

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programme refers to intensive focused instruction associated with the students
being withdrawn in pairs from class for three periods a week over a 30-week time-
frame.

The results found that improving automaticity in basic skills frees up working
memory processing, enabling students to undertake more advanced tasks that were
not specifically focused on during the intervention programme and these positive
effects are still in play years after the intervention.

4. Maths: Why Not? Unpacking reasons for students ' decisions concerning higher-
level mathematics in the senior secondary years (McPhan, Morony, Pegg,
Cooksey, & Lynch, 2008)

The project considers why many capable students are not choosing to take higher-
level mathematics in the senior years of schooling. This lack of numbers runs
counter to the national need for a highly skilled workforce to remain competitive in
the global knowledge economy. Australia is facing a multi-faceted skills shortage
just when there is a need for more students to leave school with a sound grounding
in higher mathematics.

The results provide an important "toehold" to a number of critical issues
underpinning the learning and teaching of senior mathematics in Australia. More
importantly, it offers a means of connecting the learning and teaching of
mathematics from the perspective of current and projected skills shortages. The
project offers new insights into the problem and a platform for constructive
national action.

5. Collaborative innovations in rural and regional secondary schools: Enhancing
student learning in mathematics and science (Panizzon & Pegg, 2008)

This project created networks of rural teachers to form learning communities in
science and mathematics. Each team of teachers in a particular school identified an
important issue they believe was impeding student learning within their own
school. This issue became the focus of the professional learning.

Teachers were supported at an optimum time with help varying from school to
school depending on the needs of the staff and students. Support was provided by
(i) consultants with expertise in curriculum, assessment, and quality pedagogy
visiting and working with the teachers at key points during the eighteen months of
the project, and (ii) teams of teachers met on a few occasions to share their
experiences with other teachers involved in the project. These meetings were
crucial because they facilitated opportunities for teachers geographically isolated to
meet collectively and communicate their ideas, challenges and successes. The
model of professional learning used was seen to be highly relevant and cost-
effective for schools that were widely separated by distance.

Evaluation and Impact

Evaluation of SiMERR occurs through two separated but related processes. The
first concerns sets of agreed milestones concerning progress on a six-monthly

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basis. These targets were mutually agreed to, and offer a broad context within
which SiMERR attempts to address its mission. The second process is related to
individual projects undertaken by academics associated with SiMERR, including
those internally financed through targeted funds within the Centre or from
successful contracts with funding bodies outside of SiMERR.

Tying down "impact" in such a diverse area is fraught with problems. At the
heart of the work of SiMERR is building a network where teachers, educators,
universities, education authorities, and communities can reflect and initiate actions
on improving the current situation in rural areas for teachers and students.

There are important signs that projects are having an influence. A critical
purpose of these approaches is to have an evidential basis from which informed
policy decisions can be made on how funding and actions might best target the real
learning needs of different groups of students. In terms of the five projects outlined
above there are now:

• Recommendations to advise Federal policy as it relates to addressing inequity
in rural students learning outcomes as a result of the SiMERR National
Survey;

• Published books identifying characteristics of faculty departments achieving
outstanding educational student learning outcomes across the student ability
spectrum;

• Recommendations to guide Federal Government policy on ways to encourage
and facilitate more senior secondary students to undertake high-level
mathematics courses;

• Solid evidence that students (including Indigenous students) who have been
performing at or below national benchmarks in numeracy for many years can
be supported and show considerable improvement in basic mathematical
skills and understanding;

• Evidence of the nature of the successes for rural schools in solving issues
relevant to them in teaching mathematics and how this professional learning
can be encouraged and sustained.

As a result of SiMERR activities there is now: a large number of research
activities that have been awarded to academic groups (SiMERR Hubs) to support
rural schools, teachers and students; a stronger national awareness and a higher
media presence about rural concerns in education; and stronger support for
professional teaching associations to provide more targeted professional support
for teachers in rural locations.

Challenges and Further Steps

SiMERR has sought to influence positively the educational outcomes of rural
students whose educational opportunities do not match those of their metropolitan
counterparts, and to reduce the professional isolation of teachers. This has been
pursued through targeted research programmes to inform education policy,

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teaching practice and pedagogy, professional development programmes, and
teaching and learning interventions for teachers and students.

Engendering and maintaining a climate of collaboration and trust among
universities and their staff, education jurisdictions and their schools, teachers and
communities around the country is critical to the success of the SiMERR operation.
The capacity to engage schools to participate in activities is built on networking
with teachers, education authorities and professional education organizations.
These fruitful connections are important in building trust and rapport between
schools and researchers, and they also facilitate discussion and collegiality. They
are also critical players in attempts to move the findings of research to scale (see
Cobb and Smith, this volume).

The model of collaboration developed by SiMERR is in contrast to the highly
competitive practices of universities in other fields of endeavour within Australia.
It is recognised as important for the long-term that individual universities are
supported to maintain and celebrate their own integrity, identity and successes as
well as those achievements of the collective.

SOUTH KOREA: NURI -NEW UNIVERSITY FOR RURAL INNOVATION

South Korea has achieved extremely strong national growth that has resulted in
rapid economic development since the 1950s. However, Kitawaga (2006, p. 15)
pointed out that although dramatic, the post-war revival in South Korea was more
about the developments in Seoul rather than more broadly across the nation. A
recent consequence of this imbalance has seen the Government launch major
decentralisation reforms with strong regional development policies.

In order to address this issue in 2004, the South Korean government allocated
over one billion US-dollars and embarked on a series of projects referred to as the
New University for Rural Innovation (NURI) initiative. At the Kongju National
University NURI funds were allocated to address rural education issues. In
particular, the focus was on the development of a new programme as part of the
Bachelor of Education programme for prospective secondary teachers.

Impulse for the Initiative and Challenge

In the PISA results of 2003, South Korean fifteen year olds were in the top group
of countries in science, mathematics and problem solving, placing them second
overall (OECD, 2006). However, the same PISA data show that there is a low level
of satisfaction towards schooling, and parents have their children undertake
extensive learning activities out of school time. It is estimated that 73% of students
in primary and secondary education receive private tutoring after school hours with
an additional 2.2% of GDP (Gross Domestic Product) allocated to private tutoring.
Private outlays for education in South Korea are the highest of any OECD country
(OECD, 2006, p. 29).

Also of concern was the achievement gap between students enrolled in rural
schools as compared to those in urban areas. While the average score in

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mathematics for South Korean students was 542 compared to the OECD average of
500, those students who lived in communities of 3,000 or less had an average score
of 447. This score compared unfavourably with the OECD average for small
communities of 477. These data in mathematics were further confirmed (Im,
2007a, p. 99) when an effect size gap between rural and regional students of 0.62
was identified.

Goals and Intervention Strategy

The NURI project includes:

- A programme for developing ICT pedagogical skills, involving effective ways
of using computers in education as well as the establishment and management of
teaching and learning systems for e- learning.

- A programme for developing understanding of rural societies and rural schools
by exploring issues of rural education more explicitly.

- A programme to help prospective teachers adjust to rural schools and rural life
by volunteering for service in rural communities. This involves voluntary work
in educational contexts, practicum in rural schools, inviting rural school students
to university campus, development of a practicum manual for classroom
teaching in rural schools.

- A programme for enhancing pedagogical skills in classroom teaching built
around microteaching and the establishment of two laboratories for analysis of
classroom behaviours.

- A programme for learning foreign language, improving teaching school subjects
in English, visiting rural schools in foreign countries.

- A programme for enhancing teacher knowledge and enhancing success rates for
securing teaching positions.

Implementation and Communication

This programme was developed as the education part of the four-year NURI
project and was based on NURI-TEIC (Teacher Education Innovation Centre)
team's meta-analysis (Im, Lee, & Kwon, 2007) of previous studies with Korean
participants on the educational gap between rural and urban areas in Korea. The
programme designed built on the current programme for prospective teachers with
an additional emphasis on developing teachers' classroom skills, involving
learning and practising ICT, and learning to understand rural societies and schools
in a deeper way.

Findings from the implementation program are discussed regularly with
representatives of the seven Departments at Kongju University as well as
colleagues at other universities who are interested in or who have made
contributions to the initiative. Also the NURI-TEIC team maintains strong
cooperative relationships with over 80 community schools, the Chungnam
Provincial Office of Education, and the Kongju city office.

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Evaluation and Impact

There are numerous ways in which the initiative is being evaluated (Im, 2007b).
Most important is that the number of prospective teachers who are passing tertiary
examinations is increasing over the rate in years prior to the advent of the
programmes. There has also been a commensurate increase in the number of other
certificates of attainment, such as ICT competencies, than in the past.

As part of the evaluation of the impact of the NURI project, the prospective
teachers and secondary students have undertaken surveys. In both cases, the results
of the surveys have identified improvements in perceptions. Prospective teachers
have responded positively to the changes in the course finding it more useful and
relevant than in the past. Secondary students reported that the prospective teachers
were more knowledgeable about them and their communities and they were more
appreciative of the efforts of the prospective teachers than in the past.

Challenges and Further Steps

The NURI project at Kongju National University is set to finish during 2008.
However, plans are in place to apply for a further 5-year (2009-2014) post-NURI
grant to the South Korean government. The strength of this new application is on
the track record evidence accumulated. Clearly, many of the benefits of the current
funding in terms of course structures, resources and data on prospective teachers'
development will still be available. However, it remains to be seen at this stage
whether all the initiatives currently being undertaken can be maintained in the
absence of such funding.

One of the great challenges lies with the evaluation of the initiative. Firstly, it
takes time for the full effect of a programme, such as described, to be felt.
Secondly, outcomes in complex areas such as addressing issues associated with
poor student learning in rural education are subject to many competing and
complex interactions. These often fall outside of the education focus of the
intervention and have much to do with the socio-economic viability of the
particular rural area. Hence, attributing success of a program or otherwise is
difficult. These issues increase the complexity of providing justifications for the
spending allocated and proof that tax-payer money is not wasted.

COMPARATIVE ANALYSIS, IMPLICATIONS, SYNTHESIS

The motivation for the initiatives described in this chapter was a perceived
deficiency in mathematics (and science) skills of particular groups of students,
following large-scale international surveys or state-wide surveys revealing regional
inequalities.

The surveys also revealed some structural problems with the overall education
system. In Austria, it was mostly the system's fragmentary nature and lack of
researchers in science education. In Ohio, they identified a lack of overt standards
of education upon which to make judgements. In Australia, there was no specific

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national research body or a comprehensive national agenda dedicated to improve
the learning outcomes of rural students. For South Korea, it was the difficulty of
attracting and retaining teachers in rural areas.

In all four cases, a government intervention followed. Improving the teaching
and learning of mathematics and science became a matter of national policy and
funds were allocated to begin to address the situation. National or regional centres
were established and initiatives to improve standards or to address inequities
commenced.

Participants and Their Roles

The relevant environments and participants in these initiatives were the federal and
state government bodies (which financed and oversaw these initiatives), education
experts from selected universities and (in the case of Australia) professional
teaching organisations, and targeted schools (their teachers and students, partially
also parents and community groups). Significantly, all initiatives build on forming
regional and district structures incorporating local stakeholders (see also Cobb &
Smith, this volume).

In three countries, a particular university was given the role to set up a centre,
which coordinated the entire initiative. In the Australian and the Austrian case, the
leading university linked, through a tender process, to the involvement of academic
staff from several other universities and other institutions throughout the country.

Initial participation in all four countries involved only particular schools and
particular segments of the student population. In Austria, initially it was upper
secondary schools and later extended to all secondary schools, and finally also to
primary schools, in Ohio students of middle-school years, in Australia samples of
rural and regional schools and teachers were chosen for different initiatives, and in
South Korea secondary schools were involved.

Goals and Intervention Strategy

In all four countries, the overall goal was to improve the teaching and learning in
mathematics and science by improving the teachers' skills, establishing teacher
networks through which teachers could communicate with each other and with
education experts and, in some cases, establishing a nation-wide support system
(Austria) or state-wide standards (Ohio).

In Austria, the major goals were: the initiation, promotion, dissemination,
networking and analysis of innovations in schools (and to some extent also in
teacher education institutes) and recommendations for a support system for the
quality development of mathematics, science and technology teaching. Innovation
was the key word; participation was voluntary. Teachers and schools defined their
own starting points and goals and were then supported by researchers and expert
teachers. The emphasis was on supporting teams of teachers from one school rather
than individual teachers. The teachers and schools retained ownership of their
innovations. Another important aim of the Austrian initiative was networking.

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In Ohio, the overall goal was to improve the mathematical and scientific
knowledge of students, and to achieve equity among students of different ethnic
groups as well as gender equity through inquiry-based instruction. The Discovery
project also aimed to develop state-wide standards for mathematics and science
teaching. This involved developing shared methods of teaching science and
mathematics rather than standardising a curriculum. A particular focus was on
educating the teachers (through Summer Institutes) in how to implement inquiry-
based instruction in the classroom.

In Australia, the goal was to enhance teacher growth in rural and regional areas
and to maximise high levels of teaching competence and student learning outcomes
in the critical subject areas of mathematics, science and ICT. An additional goal
was to set up teacher networks to help address professional isolation. The focus on
science and ICT in addition to mathematics gave teachers a greater chance of
interacting with a critical mass of teachers.

The South Korean initiative aimed to enhance pedagogic skills of prospective
teachers. Focus was primarily on improving their e-learning skills, their
understanding of rural societies and rural schools, and enhancing their
microteaching skills.

The theoretical frameworks used differed among these countries. However, all
these programmes built on the assumption that teachers play a key role in the
intended change; thus they (as well as their students) were seen as active
constructors of their knowledge. In Austria, the theoretical framework built on the
ideas of action research and systemic approaches to educational change and system
theory. In Ohio, it revolved around inquiry-based instruction. In Australia, it
involved implementation of teaching methods and approaches supported by
empirical evidence through ongoing research into teaching mathematics and
helping move these activities to scale. In South Korea, various programmes to
enhance teacher knowledge and microteaching skills, as well as their sociological
understanding of rural schools and communities drawn from empirically based
information from the research literature.

The four countries used various intervention strategies to achieve their goals.
The broader the goal the more strategies were used. Thus in Austria the primary
strategy was to support teams of teachers from a particular school through different
programmes. The broader focus was on promoting innovation, dissemination of
knowledge, networking, carrying out analyses of innovations, and building a
sustainable support system. In Ohio, the primary strategy was to involve teachers
in Summer Institute programmes to improve their competence and content
knowledge. An additional focus was on developing standards. In Australia, the
intervention strategy involved collaboration with communities, educational
authorities, professional associations and industry groups with the aim to develop
solutions to problems faced by teachers, particularly those who are professionally
isolated. In South Korea, the main strategy comprised programmes of pre-service
education to complement and extend the traditional mathematics preparation
programme and programmes to help prospective teachers adjust to rural schools
and rural life. The latter involved practice-teaching periods in rural schools,

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JOHN PEGG AND KONRAD KRAINER

inviting rural school students to university campus, and development of practice-
teaching manuals for classroom teaching in rural schools.

Collaboration, Communication, Partnership as Central Notions

In all four countries, there was a strong emphasis on collaboration and
communication between various social agents during all stages of the initiative.
Long-term collaboration was seen as an essential basis for reform and
sustainability of any reforms. By comparison, working on short-term professional
learning activities was implicitly understood as being ineffective and highly
unlikely to result in sustained improvements and teacher growth.

Collaboration in Austria took the form of a wide network of people and
institutions involved in the initiative. Staff and members of IMST supported large
numbers of teachers over several years. To facilitate communication between
teachers they also worked out inter-disciplinary connected concepts for basic
education at the upper secondary level for four subjects. There was collaboration
among teachers and staff members from the first meeting during which goals and
research questions were established to guide later analyses of the effectiveness of
new teaching methods. Teachers also helped each other as "critical friends". The
various programmes operated as small professional communities that supported
each participant. Furthermore, regional networks were established through which
experienced teachers were able to disseminate their knowledge to other teachers.
Communication was also facilitated by the IMST website which includes all
documents written by staff members and teachers, by an annual conference, and by
a quarterly newsletter.

In Ohio, the collaboration brought together in-service teachers and university
researchers and science educators who constructed professional development
courses and taught in those courses. The teachers participating in the programme
observed each other teaching classes and providing feedback to each other. After
the course they met several times during the following year and exchanged their
experiences. The emphasis on collaboration and communication also extended to
classroom practice where inquiry-based methods encouraged interaction and
communication between students. The collaboration also involved the partnerships
between teachers and parents in science education. Finally, the Discovery project
also led to the establishment of multiple professional development initiatives and
networks of K-16 institutions through a variety of state and national funding
mechanisms.

In Australia, the emphasis on collaboration led to the formation of a National
Centre at the University of New England and the nation-wide network of "hubs"
located at several universities. The hub members communicated with each other
and regularly met to share ideas. Each hub has its own website with hyperlinks to
other websites and to SiMERR website. Hub coordinators, disciplinary groups and
project teams also regularly conducted video meetings. In 2005 and 2007, the Hub
members and relevant stakeholder organizations met at National Summits. Other
meetings of members across Hubs coincided with disciplinary-based conferences

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or workshops. The collaboration and communication between various stakeholders
was also stimulated by a higher media profile of issues of rural and regional
education.

Collaboration and communication is also an important issue in the South Korean
initiative. It involves the relevant branches of the South Korean government,
education experts at Kongju National University, educators with common interests
from other universities, province education authorities, Kongju city office, and
teachers. The purpose of the work with these groups is to inject different
perspectives into the development and implementation of the program as well as
ownership of ideas across a broad base helping ensure greater flexibility and
cooperation at all levels of society.

In all cases, the exchange of experiences played a predominant role among
teachers, among university staff, as well as between teachers and university staff.

Evaluation and Impact

Evaluations were an integral part of each initiative and these evaluations revealed
positive outcomes. In Austria, evaluation processes were a central part of IMST
from its inception. The focus was on process-oriented, outcome-oriented, and
knowledge-oriented evaluation. It involved both self-evaluation and evaluation by
external experts. Independent international experts also assessed the overall
effectiveness of IMST. The results of these evaluations indicated that the intended
goals were achieved in the most part as well as possible improvements for the
continuation of the project. Answers to teachers' questionnaires indicated a high
level of satisfaction with the programme. Recently, more research on students' and
teachers' beliefs and growth is being carried out.

In Ohio, the evaluation of the programme centred on the test results of students
in the Discovery Project as compared with students/classes who were not involved.
Students in the programme performed better. Further, when school cohorts were
compared, schools with a higher percentage of Discovery trained teachers also
performed better than those schools that had fewer trained teachers in the
programme.

In Australia, members of SiMERR have been involved in approximately 1 20
projects throughout Australia across the four discipline areas. The results so far
indicate that projects are beginning to have a major influence on teacher
professional learning resulting in improved student-learning outcomes with
increased effect-size measures. Interestingly, in a number of cases, data are
showing that the policy announcements, programmes and the solutions developed
for rural areas are also having an impact when implemented on student-learning
outcomes in urban areas.

The evaluations of the NURI project in South Korea show an increasing number
of prospective teachers who are passing tertiary examinations. Surveys of both
prospective teachers and students indicate high levels of satisfaction. Teachers find
the courses relevant and secondary students report that the teachers are more
knowledgeable of them and their communities. The midterm report of the NURI

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JOHN PEGG AND KONRAD KRAINER

project received strong endorsement from the Ministry of Education of South
Korea.

All projects delivered positive results that demonstrated growth for participant
teachers and their students. These data appear to have had a three-fold impact, on
policy, institutions, and the way teachers fundamentally work. These examples
make the point that when programmes are well developed and involve
collaboration, communication and learning partnerships, real changes can be
expected. However, crucially they still depend on continuing government support
and in most cases the allocation of funds.

The experiences documented above indicate that the presence of an intensive
evaluation not only allows stakeholders to react to the results of the evaluation
during a project (in the sense of a formative evaluation) but also - partly related
with the former aspect - increases the likelihood that a programme (or parts of it) is
prolonged or enlarged, or implemented in the system in one form or another.

CONCLUSION

What are the lessons learned from comparing these four cases? Although,
challenges arose from similar sources of data (e.g., studies like TIMSS, PISA, or
national tests), each country has its own genuine context and specific strengths and
weaknesses. Also, those people charged with leading changes in these countries
bring different skills, and knowledge and belief sets to each enterprise. Thus
projects invariably evolve to take different forms, emphases and directions.
However, in all programmes, collaboration, communication, and partnerships
were seen to have played a major role, not only between teachers and university
staff members of the programme but also within these groups.

Close collaboration was an important aspect among stakeholder groups formed
by the ministry (policy, funding), the school practice (teachers), and the scientific
community (teacher educators, researchers). An intensive kind of evaluation
yielding relevant data to all parties concerned and the discussion about the results
seems to enrich the quality of the project and contributes to its further support by
helping to shape policy decisions.

Communication occurred in: oral forms (e.g., workshops, seminars, conferences,
and network meetings); written forms (e.g., newsletters, reflective papers by
teachers, other publications); and electronic forms (e.g., materials on a website,
chat-rooms, emails, etc). This open approach is supported by research on
"successful" schools showing that such schools are more likely to have teachers
who have continual substantive interactions (Little, 1982) or that inter-staff
relations are seen as an important dimension of school quality (Pegg, Lynch, &
Panizzon, 2007; Reynolds et al., 2002). Similar results can be found in other
national programmes (e.g., in Germany, Prenzel, & Ostermeier, 2006).

A very important issue concerned the insight of the significance of partnerships.
Teachers were not only seen as "participants" of teacher education but as crucial
"change agents" of the education system, regarded as collaborators and experts.
Consequently, they were expected and encouraged to take an active role in their

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professional growth. Teachers were critical stakeholders who were themselves
learners in the process of bringing about improved learning environments for
students. Teachers needed to be sensitively supported by teacher educators through
research-based advice and evaluation that gave meaningful feedback to teachers
and generated new scientific knowledge.

Specific recommendations are not easy to make since conditions and contexts in
these countries are very different. However, it seems worthwhile that in any major
intervention the following questions are considered: What kind of active role do
national programmes ascribe to their teachers as change agents? How can exchange
of experiences among teachers and researchers be promoted? How can
communication (e.g., using stakeholder networks) and infrastructure (e.g., national
centres) be established and further developed? How can evaluation contribute both
to the improvement of the projects' process and their impacts as well as to the
generation of scientific knowledge (which in turn contributes to the improvement
of interventions)? How can the collaboration between different stakeholders of
nations' educational change, above all, policy makers, teachers, and researchers be
designed in a way that all parties - including the students and parents - feel
empowered by national initiatives? How does the involvement in such (mostly)
larger intervention projects change the role of researchers? What form should
evidence take to provide reliable feedback about the success or otherwise of the
initiative? How can the sustainability of innovations be supported and evaluated
(see also Lerman & Zehetmeier, this volume)? We believe that mathematics
educators need to be active participants in the discussion of these kinds of
questions.

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John Pegg

National Centre for Science, ICT and

Mathematics Education for Rural and Regional Australia

University of New England

Australia

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Konrad Krainer

InstitutfUr Unterrichts- und Schulentwicklung

University ofKlagenfurt

Austria

280

SECTION 5

TEACHERS AND TEACHER EDUCATORS

AS KEY PLAYERS IN THE FURTHER

DEVELOPMENT OF THE

MATHEMATICS TEACHING

PROFESSION

GERTRAUD BENKE, ALENA HOSPESOVA, AND MARIE TICHA

12. THE USE OF ACTION RESEARCH IN
TEACHER EDUCATION

In recent years, action research has found its way in teacher education and
mathematics teacher development in particular. In this article, we introduce action
research in its various conceptions. We go on to present studies on the way action
research is used in prospective and practising teacher development. The article
addresses issues educational researchers have to attend to, when supporting
teachers in engaging in action research projects, and discusses what accounts of
action research should attend to enable future meta-analysis on the impact of
action research. The article includes two examples of a support system of action
research, one from Austria, detailing an organizational structure to enable many
teachers at different schools to engage in action research, one from the Czech
Republic, presenting the results of a project of close collaboration of teachers with
researchers, engaging in joint reflection.

INTRODUCTION

In the last two decades action research has seen a revival in the educational
community at large (see e.g., Adler, Ball, Krainer, Fou-Lai, & Novotna, 2005;
Kramer, 2006). 1 With new research on educational change small and large, the
important role of the participants in enabling, shaping and maintaining change
processes has become more and more recognized (Fullan, 2001; Wagner, 1997).
For example, the Czech psychologist, Helus (2001, p. 37) emphasized: "A
successful effort to change the school is only possible if the teacher becomes its
leading agent". Seeing the practitioners at each system level as the pivotal figures
of change processes, it is not surprising that action research is seen as one lever to
better practices in education. Action research promises to support the change of the
most important change agents, to ground changes locally where change is
necessary, and to bring about personal growth that affords the retention of the
pursued changes.

In this chapter, we will explore the use of action research in professional
development generally and then specifically in mathematics education. To that end,
we will first introduce action research and then discuss the educational context of

' For example, in their presentation of the discussions of a thematic group on teacher education of the
European Research in Mathematics Education conference (ERME ), Krainer and Goffree conclude that
they see an "|l]ncreased importance of action research as the systematic reflection of practitioners into
their own practice" (Krainer & Goffree, 1999, p. 230).

GERTRAUD BENK.E, ALENA HOSPESOVA, AND MARIE TICHA

action research: What are the problems action research is meant to solve, what is
the context of action research? Next, we will elaborate on different conceptions and
core elements of action research, to provide a framework for the subsequent
discussion of action research in mathematics education and in the professional
development of mathematics teachers. The examination will be completed by the
presentation and discussion of two programmes or projects that use action research
for (practising) teacher training in Austria and the Czech Republic. Finally, we will
raise concerns and issues about using action research in teacher development in
mathematics education.

CHARACTERISTICS AND ASPECTS OF ACTION RESEARCH

Action research is most frequently traced back to the work of the sociologist Kurt
Lewin (1948), who constructed a theory of action research in the 1940s, although
elements of the theory can be found earlier the writings of John Dewey (for a
discussion of pre-cursors see Masters, 2000). Lewin incorporated key elements of
today's action theory. He pointed out the importance of the participation of
practitioners in all phases of the research process, and saw action research as a
cyclical process of planning, action, and evaluation giving way to further planning,
action and evaluation. Action research in education in particular is typically traced
back to the teacher-researcher movement of Stenhouse (1975) who envisioned
teachers taking an active role in curriculum development. Yet, action research -
under the guidance of researchers - was already used in the field of curriculum
studies in the late 1940s and early 1950s. Since then, action research in general
(and in various forms, see below) has formed its own specialized community
within educational research, with heterogeneous strands, since local (national)
developments are strongly influenced by nationally prevalent theories (e.g., the
approach of "Handlungsforschung" in Germany). Altrichter (1990) presents an
account of the way action research fits into the framework of academic research in
general. As of today, there is no unanimous definition of action research, Adler
(1997, p. 99) even argues that there is a "war of definition" being waged. Given the
different perspectives and associated claims and value judgments (see below), she
and others (e.g., Jaworski, 1998) make the case that other terms like "practical
inquiry" are more appropriate.

However, action research conceptions share a number of characteristics, even
though they may differ in the importance of the adherence to any of them. Other
aspects are notable in that they are contested or changing. In the following, we will
present such characteristics and aspects.

(1) In any conception, action research is practical in the sense that it is a form of
localized problem solving. Something in the local setting is meant to be
understood and if needed, changed for the better. Closely tied with this general
goal is the participation of the actors of the specific social system. This has a
twofold background that may gain different emphases. On the one hand, there is
the social theoretical stance, that participants have a unique access and

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ACTION RESEARCH IN TEACHER EDUCATION

knowledge about their local system, and are thus in a position to generate a
more in depth-insight into their system (Cochran-Smith & Lytle, 1990, 1999).
On the other hand, there is the theoretical and practical insight gained from
organizational studies (Fullan, 2001; Weick, 1995), that successful change
processes are dependent on the support of the agents of a social system.
Engaging agents in action research not only generates different kinds of
knowledge, but also transfers ownership of the resulting recommendations for
change to these agents and increases the likelihood that positive change will
happen. Thus, action research can be framed as libratory (Braz Dias, 1999;
Gutierrez, 2002). 2

(2) In practice, action research conceptions differ in how much responsibility is
conferred to the practitioners, in our case the teachers. Different terminologies
have been used to differentiate different conceptions of the interaction between
for example, university based researchers and school based teachers. One
extreme conception of action research portrays a more traditional relationship
between researcher and practitioner in which the researchers pose a problem,
"do the research" (in communication with the practitioner) and ask the
practitioners to validate the results and implement suggested changes. Another
form of relationship sees the researchers and the practitioners as true
collaborators who are constantly in a process of a joint construction of meaning
about the situation at hand, its problems and possible solutions. This position
entails that the localized working theories of action of the practitioners are just
as valued as established academic theories. Both practitioner and researcher are
seen as having different ways of seeing the world, with the practitioners'
perspective being validated by their histories of action. The quality of proposals
for change are validated by the emergent practice (or falsified by the failure of
the system to respond as expected). Finally, action research can also be
understood as practitioner research without the necessary and continuous
involvement of any academic researcher. Instead, researchers may be consulted
at all phases of the action research process, but the ownership of the overall
enterprise rests firmly with the practitioners. Of course, one may envision many
forms of cooperation and collaboration in between these extremes.

Accounts of actual action research usually show the teacher to be more
active than in the first conception; teachers are at least thinking or reflecting on
their own practices. But theoretical discussions on the breath of action research
may frame almost any valued aspect of teacher participation in a research
project under the heading of action research. In all these cases, the practitioners
are called on to help to approach local problems with locally suited solutions.

(3) A further important element of any conception of action research is the
notion of reflection (Fendler, 2003), which can also be traced back to Dewey.

For an early discussion of action research striving for a liberating education see Carr and Kemmis
(1986).

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GERTRAUD BENKE, ALENA HOSPESOVA, AND MARIE TICHA

Elliott's definition of action research frames it as teachers' systematic reflection
of professional situations aiming on their further development (Elliott, 1981, p.
1). Despite the importance of this concept, there is little theoretical and
philosophical exploration to be found about what exactly "reflection" is
supposed to mean. Instead "reflection" is generally taken to be a self-evident
expression, which stands in a critical juxtaposition to action. Theoretical
explorations frequently take up Schon's ( 1 983) notion of reflection in action,
reflection on action and reflection for action as well as reflection on reflection
in action. With these notions Schon explores the relationship between
"knowing" and "doing", touching on many of the issues, which were later
discussed by Suchman (1987) and others exploring situated cognition. We
always know more about a situation than we can express. At the same time, we
never know everything. When a situation surprises us, and we are able to attune
our plans or unfolding practice to address surprising or new insights into a
situation, Schon is talking about "reflection in action". Reflection on action and
reflection for action both take place outside the pressures of having to act at the
(extended) moment of reflection. The first (reflection on action) is looking
backwards, thinking about what happened. Reflection for action is looking
forward, carefully considering plans for action (instead of just reacting). Finally,
reflection on reflection in action captures the careful consideration of what
happened in reflection-in-action. Why did something surprise us, what elements
of our (automatic) practice made us not expect something, how did we deal with
it, what made it (un)successful? An example: If one is planning a unit taking
into consideration what may happen, one is reflecting for action. If the unit took
place, and one ponders about reasons for unexpected student answers, one is
reflecting on action. If a usually engaged student becomes disruptive in class,
and one manages to think about why and take immediate action based on the
assumed cause that she was frustrated due to a misunderstanding (rather than
responding with some routine reaction to disruptive students), one is reflecting
in action. If one later on reflects why one would come to the conclusion that she
was frustrated, and whether the chosen strategy was effective, one is reflecting
on reflection in action.

One further important element of Schon's account of a reflective
practitioner was his conception of a reflective practice as a practice in which a
practitioner is engaged in a constant conversation with his or her problem
situation. Reflection on action and reflection for action may lead to promising
plans for action. But in practice, life will always surprise us; thus expert
performance requires being able to adjust, to be flexible, to reflect in action -
and to learn from our adjustments to situations by reflecting on reflections in
action (Doerr & Tinto, 2000). The complexity of practice portrayed by Schon is
a beautiful rendering of the situation teachers face in their daily lives, in which
complex social systems with multiple actors make the outcome (or the process)
of most plans quite unpredictable.

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(4) The content, action research is concerned with, varies greatly. In terms of
subject area, action research projects were originally more prevalent in the
humanities (especially concerned with literacy), but now many action research
projects are also found in mathematics and the sciences. Nevertheless, if one
looks into articles published in the journal Educational Action Research, one
will frequently find no indication of the subject considered in an action research
project in the abstract, and sometimes not in the entire article. 3 We believe that
the omission of the subject area is not accidental. Action research as classroom
teacher research will usually problematise some aspects of classroom practice.
This puts the interaction between student(s) and teacher or just between students
into focus. Thus, action research projects frequently set out with pedagogical
issues of classroom management and organization, or the projects trial some
previously conceived classroom innovation and seek to confirm or disabuse the
beliefs inherent in the conception (e.g., Watling, Catton, Hignett, & Moore,
2000). Ball (2000) points out that it takes time, experience, and self-confidence
to see classroom problems as possible problems of subject didactics and/or
ultimately as problems of a lack of subject knowledge.

(5) The notion of reflection and action is also closely tied in with the conception
of the place of beliefs in a theory of change. In general, action research projects
follow explicitly or implicitly the assumption that - when considering the
relationship between beliefs and practice - practice takes precedence. In other
words, it is assumed that a change in beliefs will not necessarily bring about a
change in (teaching) practices, but changes or difficulties in teaching practices
may change beliefs about a situation (or lead to a quest for understanding and a
new formation of beliefs) (Fullan, 2001). Thus, action research projects do in
general not set out to teach someone available expert knowledge on any
particular problematic issue. If teachers are collaborating with or guided by
researchers, expert knowledge might take the form of already synthesized and
contextual ly sensitive contributions or advice, the breath of possibly available
information is usually not made explicit; and reports of action research projects
frequently do not present a survey of available literature on the issue the action
research project was concerned with. Knowledge imported concerns foremost
methodological issues about the action research itself. How does one plan an
action research project, how does one collect evidence and arrive at an
interpretation? The theoretical section of action research projects frequently
addresses this theoretical and methodological frame of "an" action research
project in terms of its process characteristics.

(6) Conceptions of action research differ also in the significance they assign to
peer support and collaboration. Early reports of action research and reports of
action research in mathematics teacher education in the Journal of Mathematics

We have made a data-base keyword search for mathematics in the last ten volumes of EAR; the results
show only 6 articles from 1997-2006.

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GERTRAUD BENK.E, ALENA HOSPESOVA, AND MARIE T1CHA

Teacher Education (e.g., Halai, 1998) present action research projects of
individual teachers. In recent years, more and more reports are about
collaborative action research, like action research projects of groups of teachers,
and some conceptions consider such a cooperation as a necessary element of an
action research project. No person lives for him- or herself, no practice can be
transformed in a social vacuum. Sustainable change needs a community. While
those conceptions do not refute outright that action research could be done by
individuals, the emphasis on (peer) feedback and discussion with peers as a
structural prerequisite for an honest self-appraisal, for example, for self-
reflection, render solitary projects as less potential and hence less desirable.

(7) A notion not frequently used or reflected upon in articles on action research
in mathematics education, but which we deem as an important conceptual
contribution is Elliott's notion of first and second order action research. With
this distinction, Elliott (1991) makes the point that doing action research is
different than facilitating practitioners in their efforts. Thus, researchers
collaborating with teachers on action research project have a different job, and
need to reflect on different issues than the teachers engaged in action research
(Losito, Pozzo, & Somekh, 1998).

In the last decades, we have seen the call for and implementation of "reforms of
education" in many educational systems around the world. Fullan (2001, p. 37)
argued that many of these reforms failed and that "change will always fail until we
find some way of developing infrastructures and processes that engage teachers in
developing new understandings". Furthermore, "[cjhange as a change in practice
entails changing (1) materials, (2) teaching approaches, (3) beliefs" (Fullan, 2001,
p. 70). "To change beliefs, people need at least some experience of new
behavio[u]ral practices they can discuss and reflect on" (Fullan, 2001, p. 45).
Action research offers a way to address this need. What is it about action research,
which lets some researchers see so much promise to change education practice,
while others set their stakes somewhere else?

A strength of action research is that the "action" it is concerned with is the
(behavioural) situation and the behaviour of the teacher engaged in improving the
situation at the same time. Action research provides a voice for teachers to share an
emic view of their experiences with other teachers and the research community at
large. The sharing of experiences has raised a number of methodological questions.
What is the nature of the story being shared, what makes a story worthwhile being
shared? "When does a self-study become research?" (Bullough & Pinnegar, 2001)
Is this research at all? And if it is to count as research proper, what does this entail
(Altrichter, 1990; Melrose, 2001)?

In general, from the point of view of some researchers, questions raised about
self-studies (Feldman, 2003) - and thus about action research, which can be seen as

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a specific form of self-study 4 - are the same as those being raised about case
studies, with the key question of "What is this a case for?", addressing
general izabi I ity. A related issue is the ability to particularize the general (case) to a
particular context or particular issues. 5 Where one community of researchers may
ask the question: "What is this a case for?", another may ask: "Does this research
present enough details about a situation, to tell us something worthwhile about the
context we are concerned about (e.g., mathematics teaching)?"

All of this, of course, presupposes the stance of the educational researcher or the
research community asking for contributions of teacher research to comply with
the standards of research in general, and thus it asks for a high level of (research)
proficiency of the members of this discourse community. If teachers enter into the
education research community, if they raise their own voice within this
community, they have to adhere. Considering the place of teacher in the
educational system, this asks for a lot of competence of any one individual.

Educators who do not want to charge teachers with following all the
prescriptions for doing "good, valid research" will frequently argue for substituting
"action research" with less loaded expressions like "teacher inquiry" (Feldman &
Minstrel I, 2000). On the one hand, this will lighten expectation about "action
research projects": They can be conceived as "good worthwhile projects" without
falling short of standards of research at universities. Teachers engaged in such
projects do not need to feel lacking. On the other hand, there are reports of many
other teachers who feel self-empowered by doing research themselves on par with
academic educational research. Educators who do not see action research projects
necessarily as doing "research proper" usually value action research projects for
their local problem solving and the professional learning happening within the
course of such action research projects. Thus, they see action research more as a
means to professional development than as a means to produce general and
generalizable knowledge into teaching and learning.

Cochran-Smith and Lytle (1990) hold that teacher research is a different game,
which follows different rules and standards of quality; it is a genre all by itself and
should not be judged by time-honoured standards of educational research.

A different approach, which we turn to now, does not see teacher research as
(foremost) providing new insights into teaching and learning - even though those
contributions are valued - but regards teacher inquiry as a way to foster
professional development. In this perspective, teacher research is not an end in
itself, but the means to accomplish something else, and debates on the validity and
generalizability of teacher research miss the point. The question is not about the
quality of the products of teacher research, but the changes doing teacher research
brings about in practitioners and their practices.

4 Strictly speaking, this is not entirely true but depends on the conception of action research as discussed
above. If teachers are investigating their own practice, they are engaged in self-study. This does not
imply that they want to change their practice. But if self-study is instrumental in a developmental
process, it turns into action research.

5 We are grateful to Konrad Kramer for bringing up this point.

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On a theoretical level, some conceptions of professional development (Bruner,
1996; Climent & Carrillo, 2001; Helus, 2001; Jaworski, 2003; Krainer, 1996)
integrate the important concept of reflection into the theoretical conception of what
is required of a competent teacher - for example, they include the "competence of
reflection" as a further core dimension of teacher competence theories (e.g., of
Bromme, 1994; Harel & Kien, 2004).

Action Research as a Means for Professional Development in Mathematics
Education

Judged from the number of articles on research methods in action research and
teacher inquiry which appeared in the Educational Researcher, action research and
teacher inquiry has encountered a growing interest in the last. As for mathematics,
the Journal of Mathematics Teacher Education devoted an entire issue to action
research and teacher inquiry in 2006 (JMTE 9.3); and action research and teacher
inquiry features prominently in the European Society for Research in Mathematics
Education (ERME), for example, in a special volume of the first CERME-
proceedings (see e.g., Krainer & Goffree, 1 999) and the continuous special interest
group on teacher education at its conferences. Nevertheless, most papers on action
research that encourage professional development of practising teachers are not
presented as such. Instead, the papers focus on the problems and gains of the action
research project itself, with changing practices, beliefs and understanding of the
teachers being only part of the parcel. On further reflection, this mode of
presentation is not surprising: Presenting action research projects as professional
development enterprises objectifies teachers, and portrays them as in need of
change. Putting the teacher as the learner into the centre, creating a story of
learning might promote a story of a previously lacking individual's development.
Given the entire philosophy of action research, such a move would counter the
very approach, which rests on an appreciation of the knowledge in practice, and the
practitioner as a professional. Nevertheless, papers on action research do generally
report on changes of belief and practice, even though reports may vary on how
elaborate those accounts are. While we have not found a systematic survey on the
outcome of action research for professional development, a recent series of surveys
on continuing professional development published by the EPPI-Centre (Evidence
for Policy and Practice Information and Co-ordinating Centre) in London sheds
light onto possible outcomes of action research.

Action research is prominent in studies on collaborative Continuing
Professional Development (CPD). In their review of the literature on the impact of
collaborative continuing professional development for teachers K-9, the CPD
Review Group (Cordingley, Bell, Rundell, & Evans, 2003, p. 32) 6 reported that
26% of all the surveyed studies employed action research. In this report, action
research as a method to foster collaborative continuing professional development,

6 The review is not particular to mathematics, but twelve of the 30 studies with a curriculum focus on
mathematics; of the 1 7 studies selected for an in-depth review, six concern mathematics.

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is only second to Peer Coaching (30.5%) closely followed by Workshops (25%)
and Coaching (25%). 7 The report (Cordingley et a!., 2003, p. 4) finds that "the
collaborative CPD was linked with improvements in both teaching and learning;
many of these improvements were substantial". The authors report benefits to
teachers with respect to self-confidence as teachers, a heightened belief in their
self-efficacy as teachers, an increase in the motivation for collaborative work, and
an increase in the willingness to change their practice. Likewise, students
demonstrated higher motivation, better performance, and more positive attitude to
specific subjects as well as more active participation. The authors also point out
some important features that seem to have been conducive to the attainment of the
positive results (and studies lacking elements were less effective):

> the use of external expertise linked to school-based activity;

> observation [e.g., teachers visiting and observing each others classroom,
or researchers videotaping a lesson as a basis for further - joint -
reflection];

> feedback (usually based on observation);

> an emphasis on peer support rather than leadership by supervisors;

> scope for teacher participants to identify their own CPD focus;

> processes to encourage, extend and structure professional dialogue;

> processes for sustaining the CPD over time to enable teachers to embed
the practices in their own classroom setting"

(Cordingley et al., 2003, p. 5)

The report also highlights the importance of expert input, including subject
input, if "an intervention [is] intended to achieve subject specific changes"
(Cordingley et al., 2003, p. 6). Furthermore, it stresses the importance that teachers
are in a position to work on their own expressed learning needs, which also entails
the adoption of differentiation strategies (such that each teacher can truly work on
his or her individual concerns), that the collaboration is sustained and that teachers
find a place where it is save to admit needs (and report problems and possible
mistakes).

While this review is very positive on collaborative continuing professional
development arriving at recommendations which square with principles of action
research, another review (Gough, Kiwan, Sutcliffe, Simpson, & Houghton, 2003)
finds that "while student attainment and learning styles profit from reflection, self-
directed learning, planned action and similar approaches, there is no clear evidence
on whether these approaches influence or change the learner's identity, reflective
capacity or their attitudes about learning" (Gough et al., 2003, p. 64). Yet, (changes
in) identity and reflective capacity are constructs which are difficult to capture;
generally reflection is still a much valued element of action research and teacher
inquiry projects, more recent case studies on the use of reflection in professional
development programmes are positive (Even, 2005; Scherer & Steinbring, 2006;
Ticha & HoSpesova, 2006).

' Note that any study may make use of more than one type of invention.

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In his review of models of professional development, Foreman-Peck (2005, p. 9)
states: "Practitioner inquiry and research is a strong element in professional
development courses for teachers, [and] an important part of teachers' personal
professional development". The above mentioned report (Gough et al., 2003) also
finds that 56% of the surveyed studies which featured professional development
planning (PDP) reported on activities which occurred as part of some coursework
(e.g., an action research project done as a requirement in a teacher education
programme). The authors raise the question "whether the emphasis on course-
specific outcomes in any way restricts the reflection that takes place as part of such
interventions" (Gough et al., 2003, p. 45).

In the following, we want to discuss and problematize different aspects and
dimensions of action research that feature in the literature on mathematics teacher
education. 8

(1) Choice of a topic and temporal dimension. The choice of the topic of action
research is closely tied in with the question of where the incentive is coming
from to do action research. If action research starts with the teacher, issues
focus on localized, specific questions. Sometimes (e.g., Watson & De Geest,
2005) papers report that researchers defining an agenda were looking for
volunteers. In general, little is said about processes and negotiations leading up
to the collaboration. The same is true for the last case, when action research is
done as part of some course requirement. From our own experience, we know
that defining the problem such that it is "workable", that it becomes clear what
issues one needs to consider, what evidence one should attend to and collect
can be a difficult and time-consuming process. Additionally, questions and the
corresponding action research programme may change in the process. At the
same time, we found few reports on aspects addressing these issues. One
notable exception is Feldman and Minstrell (2000, p. 448) who state that this
first phase may take up to one year. Ball (2000) also observes that simply
finding a topic may take a long time. These are important elements. We need
to know more about reasonable time-spans, the time more and less
experienced teachers need to become comfortable with action research or self-
directed inquiry.

(2) Authorship and theorizing the area of investigation. As discussed above,
theories on action research stress the unique characteristic of knowledge

8 For the review, we systematically screened all volumes of the Journal of Mathematics Teacher
Education (JMTE); we looked at all articles for the last 10 years of Educational Action Research
(EAR), which were returned by a data-base search for "mathematics" (6 articles, 2 book reviews); we
included the major articles which were returned by a keyword search for mathematics and "action
research" in the JSTOR-database (in June 2007); we looked at articles keyworded for 'action research'
in the Proceedings of the Psychology of Mathematics Education (PME), 2007 (none!) and the
appropriate sections of CERME). This was complemented by various - but not systematically collected
- articles from the handbooks of teacher education, discussions on various aspects of teacher inquiry in
the Educational Researcher and additional information, which is grounded in our local research
contexts.

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generated through action research and that doing this type of research requires
a different approach, which gives precedence to insights gained by a grounded
or "bottom-up" perspective of a situation, clearly rejecting preconceived
notions as might be engrained in theories brought in from the outside. Yet, as
educational researchers, we still hope to "bridge the gap" (see also Jaworski,
2006; Scherer & Steinbring, 2006), and that those implicit theories which
guide the perception and interpretation of the action researchers become
explicit and thus enahJB, on the one hand, a communication between insights
generated in the situation, and understandings found through traditional
avenues and, on the other hand, a deeper understanding of possible points of
view of engaged and reflective practitioners.

A theoretical-terminological framework strives to capture the different
modes of cooperation between teachers and researchers, and thus which logic
and aims was given precedence (the logic of a certain practice in a field, or the
logic of research). Thus one may talk about participatory research (with
teachers/practitioners taking part in a research project), about cooperative
research (with teachers/practitioners cooperating, such that each may reach
their individual goals) and collaborative research (in which both try to learn
from each other and both strive to achieve their shared, negotiated goals). All
conceptions have clear consequences for authorship, decision processes and
the status of academic and practitioner theories. Theorizing what research does
to a field this way, also highlights how issues of authorship, aims of the
enterprise and considered knowledge are interwoven.

Most papers written in peer reviewed journals end up being written by the
researchers collaborating with teachers. 9 Thus, even if action research
(depending on the definition) may strive for an emic perspective, reports on
action research are usually presented "from the outside". This does not imply a
second-order action research perspective, since only a minority of articles
focuses on the particular practice of the researcher in enabling and supporting
action research projects. Rather, most projects are still reported from an
"objective", almost outside point of view. There are some exceptions (see e.g.,
the report of a teachers' book project in Ponte, Serrazina, Sousa, & Fonseca,
2003), but generally we found that those people, who report about their own
development, came into the process as a teacher being engaged with their own
practice, and ended up becoming at least part-time educational researchers (as
e.g., in the case of Deborah Ball, John Mason, and Jim Minstrell). Minstrel I (in
Feldman & Minstrell, 2000) states in a by-line, that he became so interested
that he earned a Ph.D. What does this imply about action research as a
professional development incentive? And what (kind of) people will be
attracted to action research? With respect to those questions, teacher inquiry
required by some programme provides an interesting context, since this is one

9 See also Adler, Ball, Krainer, Lin, F.-L., & Novotna (2005, p. 371) who found, that "Most teacher
education research is conducted by teacher educators studying the teachers with whom they are
working".

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avenue in which teachers may not have chosen to pursue by themselves. This
is one group which did not necessarily self-select for the substantial
involvement required when questioning and researching ones own practice.
Yet, so far we lack reports which attend to the more reluctant participants, 10 as
well as reports which describe failures in going through an entire cycle of
doing action research (a notable exception is Nickerson & Moriarty, 2005).
How does one have to design a setting which encourages people not already
perceiving a need to take up action research, and supports them while they
struggle through the processes? What makes them drop out? (Christenson et
al., 2002) When does someone become disenfranchised?

The theoretical grounding is quite heterogeneous. However, apart from a
general grounding in core concepts of teacher education - like Shulman's
(1986, 1987) account of teacher competences - there are two concepts which
seem prominent: SchSn's (1983) account of reflection, and Wenger's (1998)
notion of communities of practice. As noted by (Cordingley, Bell, Thomason,
& Firth, 2005), most action research projects are done in a collaborative
context. If the action research itself is not performed by a community of
inquiry (Garcia, Sanchez, Escudero, & Llinares, 2006; Jaworski, 2006), it
studies the impact of the action researcher on a community (of learners). In
both cases, peripheral members of some learning community are learning to
become full members of a (reflective, self-directed) community of
practitioners. If- in the education context - the communities of practice (Lave
& Wenger, 1991) is conceived as a community of inquiry (Jaworski, 2006),
one can see action research becoming subsumed in a more encompassing
concept which integrates the stances of action research with a social learning
and social community point of view. Elliott (1991) has maintained that action
research is a group effort. However, given the above mentioned "war of
definition", with the resulting uncertainty what someone means when they
speak of action research, as well as the critique ventured again the notions of
"action" and "research" in "action research", we may see a (terminological)
move to "communities of inquiry" which at the same time stresses the
importance of having communities support change, and avoids the problems of
using a contested notion. However, it remains to be seen how such a
conception will deal (or exclude) the individual teacher doing research (or
inquiry) on their own classroom without being embedded in a community.
(3) Mathematical content. In the articles we found reporting on action research in
mathematics teacher education, mathematics and mathematical concepts form
an undercurrent, a background, which is frequently not elaborated on.
Mathematical concepts turn up in discussed classroom transcripts, to explore
teachers' thinking and practice (and to ground teachers' joint reflection, see
e.g., Goodell, 2006) and to discuss missed and taken up opportunities to learn.
Nickerson and Moriarty (2005) discuss the importance of subject matter

10 Ross and colleagues discuss that low self-efficacy of teachers leads them to avoid engaging in action
researcher (Ross, Rolheiser, & Hogaboam-Gray, 1999).

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knowledge for the practice of their teachers. In their report, they talk about
Habor View teachers, who had a comparably high level of subject knowledge
at the beginning of the project, and the Palm teachers, who did not (Nickerson
& Moriarty, 2005, p. 133):

Our analysis suggests that teachers' knowledge of mathematics affected
their collective control over decisions related to the mathematics program.
Habor View teachers felt empowered to alter the curriculum. Palm
teachers did not feel that they could. Increased mathematical knowledge
supported teachers' recognition of the need for assistance.

While this claim seems plausible, one has to consider that teacher training
differs vastly across the world. What are the necessary thresholds to empower
teachers to feel comfortable to make the "right choices"? How much
knowledge is "enough"? And how can we support teachers to become aware
that they (may) have "enough" (in their context!) - despite recognizing that
there is still more to be learnt?
(4) Success and failure stories. In mathematics teacher education, action research
projects are usually presented by a number of case studies which demonstrate
successes (with examples focusing on subject didactical elements). Studies
focus on "what happens" during the action research projects, and the resulting
changes. To enable future systematic meta-analyses, reports should be more
detailed about contextual elements like processes leading up to the action
research projects (subject selection, negotiation of topics, relationship between
researcher-teacher, kind and extent of external support) as well as elements of
the unfolding processes (time-spans). So far, we know little about "drop-outs",
possible particular characteristics about or histories of those teachers who
volunteer to undertake action research projects, and of what happens "after".
Not knowing about the necessary time investment for different groups of
teachers (stratified by experience, attitude and other possible factors) as well
as the related outcome, we cannot gauge the relative impact of different
conceptions of action research or other similar approaches (e.g., self-study) as
professional development. In particular, it is still an open question as to how
much (little) input and direction one needs to provide, in order to reach the
reported positive benefits.

TWO EXAMPLES OF ACTION RESEARCH PROJECTS

The following two examples present first a case from Austria of a support system
for teachers to engage in action research projects of their own choosing. The
second case is taken from the Czech Republic, and presents a project, in which
teachers were supported to collaboratively reflect and further develop their
mathematics teaching. Thus, both cases can be seen as being at very different ends
of a continuum (or continuous space): in the Austrian case, teachers (individuals or
teams of teachers) choose topics of their own interest to work on. Their reports are

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GERTRAUD BENKE, ALENA HOSPESOVA, AND MARIE TICHA

rarely concerned with questions of didactics of mathematics; much more common
are action research projects which trial the use of some teaching method (e.g.,
group work, or self-directed learning environments). In the Czech case,
practitioners were invited to work on several topics (application, grasping of
situation, problem posing, and geometry) and teachers then decided to focus on
part-whole relation and fractions. Thus, it was much more oriented towards a
common topic, while supporting teachers in their work and individual
development. And it was naturally more concerned with didactics of mathematics.

Austria: The I MST Project"

In order to promote teacher development in mathematics and the natural sciences,
the Austrian ministry of education launched the IMST3 project (Innovations in
Mathematics, Science and Technology Teaching, 2004-2009; Krainer, 2007) which
provides as one measure a fund for teacher research. This fund succeeded the
IMST 2 project (2000-2004), which already supported teachers doing action
research (Altricher, Posch, & Somekh, 1993) of their classrooms, schools or
educational aspects concerning an entire region (Krainer et al., 2002). In both
phases, the IMST project invited teachers, and teams of teachers (of the same or
different schools) to submit project proposals for a one year project (with the
option to submit a proposal to continue or extend already running projects). With
the IMST3 project (and a growth in size of participants), the application to the fund
makes use of an already highly structured online submission form, which asks, for
example, how gender issues are attended to, and how people intend to evaluate the
results of their changes or the state of affairs. Project proposal workshops across
the country offer advice for teachers who find the required definition of their
projects difficult. Moreover, since 2006 it is possible to participate as a teacher or
team with the purpose of developing a project (proposal) to be submitted the
following year.

Taking in account the primary area of expertise of the involved teacher
educators, at the beginning the fund set out to support teachers of college-bound
high school students and to a lesser degree 9th to 13th grade students in general. In
2004, the call was extended to lower secondary schools. In 2007 elementary school
teachers were included for the first time. Thus, in 2007, the call for projects
became open to mathematics and science teachers at all grade levels.

The project proposals are evaluated by educators and mentor teachers; projects
may be accepted with recommendations for further explication or changes. Each
year about 150 projects are accepted. The contract with the teachers provides them
with a budget for project expenses (as defined in the project proposal) and a small
monetary compensation. It requires them to participate in two workshops and to
submit a project report at the end of the year. Teachers are invited to a start-up-day,
in which they are introduced to their advising teams, a specific advisor teacher (one

" Note that the article of Lerman and Zehetmeier (this volume) also reports on the IMST project.
However, their presentation does not address the details of the fund presented here.

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for about seven projects) and two other team members. These advisor teachers are
experienced teachers at schools or teacher education institutes. Within this year,
they attend two project workshops of their choosing out of, for example, an
orientation workshop, a writing-workshop, an evaluation workshop, or a gender
workshop. Throughout the year, they are asked to hand in an "action plan"
(including their evaluation), which is discussed with their specific advising teacher.
Likewise, the project report at the end of the year may be commented on and be
revised before it is accepted by the advising teachers, and eventually be published
in the internet. Thus, the fond provides some direction while clearly setting out the
requirement to have teams of teachers work on issues of their own choosing. The
support of the advising teacher focuses mostly on running the project: stating clear
goals, working out a plan to proceed, planning the evaluation, and writing up the
experience. Above and beyond teachers can consult experts to work on additional
questions.

In the four years of running the fund, we encountered many of the same issues
and concerns noted in the literature above:

• Action research requires a lot of time and energy from teachers. Thus,
asking teachers to volunteer undertaking an action research project usually
leads to a self-selection of already engaged and enthusiastic teachers.

• In schools in which principals supported teachers and they were
embedded in a community of like-minded teachers, teachers were able to
effect substantial changes, for example, introducing a new student-centred
feedback culture in the school, or introducing observations of each others
teaching. In other schools, changes were restricted to the respective
classroom.

• In a series of interviews (Benke, Erlacher, & Zehetmeier, 2006), we found
that experienced teachers, who had tended to self-critical ly question their
own teaching, reported a heightened sense of self-confidence due to the
project. They felt more assured that they were on the right track, which
also allowed them to be more assertive in discussions with colleagues.

• We did not observe Fullan's or Elmore's problem, but it is certainly
something to keep in mind: "It is a mistake for principals to go only with
like-minded innovators. As Elmore (1995) puts it: '[Sjmall groups of self-
selected reformers apparently seldom influence their peers.' (p. 20). They
just create an even greater gap between themselves and others that
eventually becomes impossible to bridge" (Fullan, 2001, pp. 99-100 and
148).

• As has been found elsewhere, the projects dealt more with pedagogical
questions than with content-related ones (as e.g., mathematics). In one
case, in which a mathematics teacher educator invited teachers to use
materials he had developed explicating core conceptual elements, many
teachers stopped posing their own questions, they "executed" the
materials without realizing that the materials did not require "standard"
pedagogical or didactical approaches. In other words, starting with
"teacher problems" affords side-stepping mathematical didactical issues.

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• In the lMST-fund projects, many teachers started out with a vision what
they wanted to work on. Yet, in order to afford a systematic, data-driven
exploration for the required evaluation, most of them still needed to
further explicate "the problem" to make it concrete enough to be able to
look for evidence for some claim. Other teachers - who had less
experience with a research point of view, before becoming interested in
submitting a project proposal - needed substantial support to turn a vague
interest in "working on their teaching" into a concrete project idea which
they could state in a project proposal.

• The very contextual and locally grounded nature of the projects led to a
flower garden of initiatives. Teachers enjoyed sharing their experiences,
but working on different sites and starting with different problems made
joint efforts for evaluation across contexts almost impossible. Thus, in
most schools local teams designed their own instruments for assessment
addressing their very specific goals. In general, that led to a multitude of
small, locally significant findings, which addressed different aspects of
enjoying, learning and doing mathematics across different age levels and
school types. Taken together with the national context, which does not
require state wide centralized tests at any time during schooling (which
could for example measure achievement gain scores), it was not possible
to make coherent statements on the overall impact on learning of the fund
without a further massive intervention into those classrooms. Instead
IMST had to content itself with measures of attitude, self-confidence, and
subject-related anxiety. In these measures, IMST classes on average
performed significantly better than the average Austria mathematics class
(of that age group and school type; Andreitz, Hanfstingl, & Miiller, 2007).
However, the participating teachers usually demonstrated high levels of
job motivation and interest already when entering the projects, thus the
good results may be due to the special self-selected group of teachers.

• Each project had to hand in a report at the end of the year. Reports can
have various purposes, which might at times conflict with each other. The
report should present the project to the public, at the same time, writing is
a means to engage in reflection. In general, teachers reported that this is
the most difficult step of the entire project. Teachers (in Austria) are not
used to write, reflective accounts of their classroom practices or projects.
Retrospectively, teachers uniformly valued the experience as a vital piece
of their learning from the project (see also Schuster, 2008).

• Benke (2004) found a marked difference between those project reports (of
the precursor of the IMST project, IMST 2 ) which were written by
individuals and those which were written by teams - the latter included
almost no reflective elements; teams tended to report on the results of
reflection, but did not mirror the process of reflecting in writing.
Moreover, the project reports of teachers reflect the value judgements and
judgements of relevance of the writing teachers. These may at time be
quite at odds with judgements of relevance of researchers or educators.

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• An open issue is the further use of the project reports. The reports are all
published on the webpage for all projects. Teachers and their school enjoy
having the project reports for internal and external communication
purposes. Yet teachers are not prone to look up, what someone else has
done or to learn about another project. IMST is presently working on a
strategy to better disseminate the project reports to interested teachers -
which is incidentally an issue action research, as a strategy for
professional teacher development still needs to take up. Even if problems
are locally grounded, and each community of inquiry needs to find their
own answers to their own problems, the answers of the others make us
richer, and we need to learn from them as well.

Czech Republic: A Comenius Project on Understanding of Mathematics Classroom
Culture

The intervention into professional teacher education reported here was developed
within the scope of a more encompassing international project on "Understanding
of mathematics classroom culture in different countries". Within this project, it was
decided to collect video records of teaching episodes. Although originally not
intended, these episodes became core elements of joint reflection and development
of the participating teachers in the national and international meetings. Instead of
videotaping a "regular classroom teacher", the project team began to carefully
discuss and then plan individual lessons which covered the required curriculum.
Thus, lessons turned into "teaching experiments" for all participants. The topic that
was selected jointly for such an inquiry approach was one of the most difficult
concepts in mathematics education at primary school level - the concept of
fractions. 12 Thus, the Czech team agreed that the experimental teaching should
focus on: (a) the creation of the notion of the part and the whole, and (b) the
continuous enrichment of various modes of representation and interpretation.

The cooperation in the Czech team gradually settled roughly on the following
routines:

• Preparation of the teaching experiments usually began at a joint meeting of
the Czech team of teachers and researchers. The group discussed the topic of
the upcoming experiment, the potentialities of the use of various methods
and techniques, and the mathematical content in greater detail if necessary.
The topic of the experiment usually addressed the needs of the teacher who
would then conduct the experiment in his or her class.

• The teaching experiment was usually realized by one teacher in her class. If
two of the teachers were teaching the same grade, the experiment was carried
out by both of them. After the joint session, the preparation and lesson
planning of the teachers was individual.

12 Many teachers tend to use a schematic approach and so they focus on drills of numerical operations
with fractions, and students usually master these operations relatively quickly. But, if we investigate the
level of students' conceptions of fractions, we often find out that it is very low (Ticha, 2003).

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• The experimental teaching was video recorded by one researcher using a
camcorder. During the project, the researchers made 25 video recordings of
lessons (or parts of it). The intrusion into the course of the lesson was
minimal as the students became accustomed to being video recorded.

• The teacher who had taught the recorded lesson was the first to watch the
recording and to select interesting segments which to discuss with
colleagues. She consulted with the researcher who made the video recording
about her choice of episodes before sharing with her colleagues.

• The selected video episodes were (sometimes repeatedly) watched as a group
but reflected upon individually by all members of the team. Therefore, all the
members of the team were prepared for subsequent joint reflection.

• The selected video episodes formed the basis for joint reflection of the whole
team. These teacher discussions were usually recorded by the researchers,
which enabled them to conduct a follow-up analysis of the joint reflection.

• Several episodes were also jointly reflected by the international team.
Throughout the project, the teachers, who taught at different grades levels,

gradually began to change their perspective on the meaning and essence of
mathematics education; in addition the researchers found their own understanding
of the school practice was also changing. In joint reflections on these short video
episodes, the teachers and researchers discussed, for example, possible approaches
to teaching a certain topic and sources of various beliefs. After some time of close
cooperation, the teachers began to recognize their different teaching styles and
philosophies of school mathematics, didactical approaches, subject matter
knowledge and so on (Bromme, 1 994) as fundamental sources of differences; and
they used these insights when reflecting on their different professional orientation
and undergraduate training. The discussion usually centred on two areas: (i)
students' work, and (ii) teachers' opinions on how to approach a specific topic.
What was surprising to the researchers was that the whole discussion was
penetrated by considerations on the essence and meaning of mathematics education
and also by considerations on the sense of joint reflections. They concluded that
joint reflection should be used in particular for: (a) cultivation of the teachers'
behaviour and actions, (b) formation of more perceptive teachers' approaches to
students' ways of thinking and the ability to utilize them in teaching, and (c)
becoming more conscious of moments valuable from the point of view of students'
cognitive processes. However, teachers still needed some guidance on how to
reflect on pedagogical situations (Scherer, Sobeke, & Steinbring, 2004). The
project team recommends that prospective teachers should be systematically
prepared for reflections on teaching and should be familiar with the relevant
literature. Practising teachers (participating in the project) themselves admitted that
they would appreciate some instruction and guidelines that they could follow in
order to have more productive reflections. However, they would not like them to be
too binding and admitted they were not really sure they would make use of these
instructions.

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Using video recordings made it possible to create and cultivate skills in
assessment of students' answers, to diagnose students' mistakes and their sources.
Given the support of joint reflection, participants learned to exert deeper insight
into the content taught and into how content was grasped by the students. Although
the aim of the project was not to support professional development of the
participating teachers, significant changes in their perception of teaching of
mathematics could be observed, as well as changes of the teachers' reflections
(Ticha & HoSpesova, 2006), and the teaching itself. We have to stress that the
changes of the teachers differed depending on their personalities, education,
experience, age and so on. We illustrate this idea with the following brief accounts
of the change made by two teachers participating in the project, Anna and Cecily:

Anna was entering the project with the stated intention to change her teaching of
mathematics. The video recordings of her lessons taken at the beginning, reveal her
considerable mastery of methodology and her effort to prepare for students such
problems that they would be able to solve them without greater difficulties.
However, she always tried to ensure that her students would also understand the
process of numerical operations that they were taught.

After a few meetings of our team, Anna stopped looking for merely effective
methodological approaches. In her lessons, she began to create problem situations
for which the solution would promote understanding and lead to building of
concepts. For example, she found her inspiration in German textbooks and
prepared the teaching experiment for her second grade students dealing with
nonstandard procedures of subtraction. She asked her students to decide which of a
set of given calculations were correct and also to explain why. As for the
justifications of the students, she was not satisfied merely with calculation of the
result. She insisted that the explanation should be comprehensible to all other
students. The goal of teaching in the second grade is usually a practice of addition
and subtraction of numbers up to 100. It is by no means usual to ask the students to
develop their own procedures or to explain unusual procedures. In the discussion
on the issue she stressed that she wanted to promote understanding of the counting
procedures and realized how her students would cope with the problem. In the joint
reflection, she stressed that she had no experience with such a type of problem; and
her satisfaction with the students' performance was visible; the lesson is analysed
in greater detail in Ticha and HoSpesova (2006). After that experience, Anna
returned to the problem several times, modified it and followed her pupil's
development. Her action research served her actual practical needs.

Our second example tells of Cecily, who entered the project as a regular teacher
with a solid knowledge about teaching and mathematics and who was interested in
learning more about different approaches and representations and in further
developing of her own abilities. For her, participation in discussions was a stimulus
for further studies. Gradually, her interest moved from the effort to "show
something", to a search for problems, misunderstandings and their sources and
causes. She started to propose content and methods of investigations, sometimes to
suggest modifications of the conditions of performed research. For example, she
carried out her own instruction experiment with an entire class. Immediately after

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the lesson, she suggested a repetition of this experiment with a group of six
students, which allowed studying individual processes. Another time, she
suggested a modification of the research performed with teachers for her students.
After a reflection of this teaching episode, she suggested to jointly reflect on the
same episode with other participants (Ticha & HoSpesova, 2006). During the
cooperation, Cecily entered her PhD study.

Cecily constitutes an example for the claim that those teachers who are strongly
invested in action research or reflective inquiry are prone to go on for further study
to become a "true researcher" - in which they might leave the non-academic
teaching profession (this has not happened in this case - yet). For this teacher, the
participation and collaboration in the Comenius team meant a great challenge for
her further education (Ticha & HoSpesova, 2006). This is also attested in her
written self-reflections.

After the first year of the project, Cecily wrote:

Before starting the project I didn't know what to expect at all. I just knew that
it was something about mathematics. 1 wasn't able to imagine the reason or
the aim of the project. I told myself that I would see.

After three years, she commented:

[...] for me it was very interesting to meet teachers from other countries and
compare our teaching experiences and opinions. I started to think about what
we should have to prepare for our presentation. [...]. I found that it was very
useful to see myself as another person during discussions with my colleagues
because it is one of the ways of improving my work and thinking about
myself differently.

And after beginning her doctoral degree programme, she reflected:

When watching the video, it is not easy to separate the outer and inner view
of my activity during the lesson (compare this with the above). I think that it
is impossible to see oneself as another person. When watching the recorded
lesson, I relive the whole lesson step by step again and again. I can see what I
could not see during the lesson itself, I can find out why the students
misunderstood, but sometimes, despite all my effort, I do not know why
something happened the way it happened and why some misunderstandings
both of the students and myself could not be solved. I am not sure whether
my conjecture is correct; it seems probable to me. It is really useful in this
position to have a view of another person who can see the matter in a
different way, and the subsequent discussion can lead to finding the solution.
When analysing our own lessons and behaviour, we can discover many
things. At certain moments, I might think that there is nothing more to
analyse, but the view of other people and the discussion with them shows that
there are matters that I haven't noticed.

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SUMMARY - THE CONTRIBUTIONS OF ACTION RESEARCH AND OUR

EXPERIENCE

Looking back, we find ourselves still very much at the beginning. Practitioners and
educational researchers working with action research are generally positive about
their experiences, even though the objectives of researchers and teachers seemingly
so similar - to improve teaching and achieve higher standard of education - are
vastly different. In general, researchers look for answers to theoretical questions
while teachers deal with practical problems. For teachers doing action research, it
brings about a change of understanding and assessing of (a) their own role, (b)
what is essential for their work, (c) their own professional knowledge, and it brings
about (d) a new way of reflecting their own practice. To researchers, it means (a) a
deepening of understanding of processes, which take place at mathematics
teaching, (b) improvement of quality of teacher competence assessment (from
intuition to junior researcher), (c) possibility to influence teacher competences, and
(d) improvement of didactic research.

But as reflected above, the strong and theoretically well supported tenet to work
with problems located at the chosen sites, precludes any easy systematic evaluation
of the overall impact of action research and the differential contribution of specific
implementations. Moreover, the practice-theory problem, that is the question of
how to communicate insights from practice and insights from educational research
at large - in particular the community of mathematics didactitians - between the
communities of practitioners and researchers, is very much at the heart of effective
action research.

In brief, action research remains a promising strand of professional development
and education research, which still has to prove itself. However, regardless of the
specific contributions of action research to professional development, engaging in
action research forces the educational researcher to consider many questions very
much at the heart of educational research for teacher development and learning in
general; for example: How can we examine changes in teachers' beliefs and
knowledge? How can university people support teachers to reflect more deeply?
What criteria are characteristic for a "reflective teacher"? A teacher has different
tensions and priorities than a researcher. How does a teacher-researcher balance
these tensions? How can we examine the benefit of joint reflection? And ultimately
following Dewey's vision: How can we support teachers' growth to become truer
to themselves as teachers and persons?

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Weick, K. E. ( 1 995). Sensemaking in organizations. Thousand Oaks, CA: Sage.
Wenger, E. ( 1 998). Communities of practice. New York: Cambridge University Press.

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Gertraitd Benke

Institute of Instructional and School Development

University ofKlagenfurt

Austria

Alena Hospesovd

Faculty of Education

University of South Bohemia Ceske Budejovice

Czech Republic

Marie Tichd

Institute of Mathematics, v. v. i.

Academy of Sciences of the Czech Republic

Czech Republic

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13. BUILDING AND SUSTAINING INQUIRY

COMMUNITIES IN MATHEMATICS TEACHING

DEVELOPMENT

Teachers and Didacticians in Collaboration

Teachers and didacticians both bring areas of expertise, forms of knowing and
relevant experience to collaboration in mathematics teaching development. The
notion of inquiry community, provides a theoretical and practical foundation for
development. Within an inquiry community all participants are researchers (taking
a broad definition). With reference to a research and development project in
Norway (Learning Communities in Mathematics - LCM) this chapter explains the
theoretical notions, discusses how one community was conceived and emerged in
practice and addresses the issues contingent on emergence and sustaining of
inquiry practices. In doing so it provides examples of collaborative activity and the
reciprocal forms of expertise, knowing and experience that have contributed to
community building. It illuminates issues and tensions that have been central to the
developmental process and shows haw an activity theory analysis can help to
navigate the complexity in characterizing development.

INTRODUCTION

This chapter focuses on co-learning inquiry, a mode of developmental research in
which knowledge and practice develop through the inquiry activity of the people
engaged (Jaworski, 2004a, 2006). This involves the creation of inquiry
communities between didacticians and teachers to explore ways of improving
learning environments for students in mathematics classrooms. Research both
charts the developmental process and is a tool for development. The chapter draws
on a research and development project in Norway, 1 Learning Communities in
Mathematics (LCM), for which co-learning inquiry and communities of practice
have formed a theoretical basis. The nature of inquiry, development and research in
the project is used as a basis for extracting more general principles and issues.

The LCM project focused on how learners of mathematics at any level of
schooling can develop conceptual understanding of mathematics that is reflected in
nationally and internationally measured success. The project was rooted in

' The LCM project was funded by the Research Council of Norway (RCN) in their advertised
programme Kunnskap, Utdanning og Laering (Knowledge, Education and Learning - KUL): Project
number 157949/S20.

BARBARA JAWORSKI

established systems and communities in which education is formalised and
mathematics learning and teaching take place. 2

The chapter weaves theory and practice to address meanings and roots of co-
learning inquiry and inquiry community and issues in creating and sustaining
inquiry communities for development of learning and teaching mathematics.

THEORETICAL BACKGROUND

Knowledge in Sociocultural Settings

Knowledge is seen to be both brought by people engaged in the educational
process and embedded in the practices and ways of being of these people -
students in classrooms, teachers of mathematics in schools, and mathematics
didacticians in a university.

According to Lave and Wenger (1991), knowledge is in participation in the
practice or activity, and not in the individual consciousness of the participants.
"The unit of analysis is thus not the individual, nor the environment, but a relation
between the two" (Nardi, 1996, p. 71). So, the practice, or activity, in which
participants engage is crucial to a situated (social practice theory) perspective.
Wenger (1998) talks of belonging to a community of practice involving
engagement, imagination and alignment. The terms participation, belonging,
engagement and alignment all point towards the situatedness of activity and the
growth of knowledge in practice.

Within the communities of our project we recognize both individuals and
groups: that is we ascribe identity to both. Holland, Lachicotte, Skinner, and Cain
(1998, p. 5) write, "Identity is a concept that figuratively combines the intimate or
personal world with the collective space of cultural forms and social relations".
Identity refers to ways of being (Holland et al., 1998). We talk about ways of being
in the LCM project community and in the other various communities of which
project members are a part, leading to a concept of inquiry as a way of being
(Jaworski, 2004a). Inquiry is first of all a tool used by participants in a community
of practice in consideration and development of the practice, that of mathematics
learning and teaching in classrooms. Inquiry mediates between the activity of the
classroom and the developmental goals of participants. Participants engage in
action that involves inquiry and learn from the outcomes of their action relative to
established ways of being. Relationships between individuals and the communities
in which they are participants are complex with respect to the forms of knowledge
they encompass and growth of knowledge within the communities.

Wertsch (1991, p. 12) emphasises that "the relationship between action and
mediational means is so fundamental that it is more appropriate, when referring to
the agent involved, to speak of 'individual(s)-acting-with-mediational-means' than

2 A copy of the project proposal can be obtained from the author by direct communication.

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INQUIRY COMMUNITIES IN MATHEMATICS TEACHING

to speak simply of 'individual(s)'". Wertsch refers to Vygotsky's (1978, p. 57;
emphasis in original) well known law of cultural development which states:

Every function of a child's cultural development appears twice: first, on the
social level, and later, on the individual level; first between people
(interpsychologicat), and then inside the child (intrapsychological). This
applies equally to voluntary attention, to logical memory, and to the
formation of concepts. All the higher functions originate as actual relations
between human individuals.

Such a perspective sees learning as participation in social practice or activity.
As we participate we "take part" in the practices or activities involved, grow into
those practices or activities, and learn through our doing and acting. We engage
mentally and physically, and communicate with those around us. We use the
language, words or gestures, of the practice or activity to engage and communicate.
Different social groups use language in different ways and within any group we
speak or learn to speak the group language.

Leont'ev (1979, pp. 47-^8) writes,

in a society, humans do not simply find external conditions to which they
must adapt their activity. Rather these social conditions bear with them the
motives and goals of their activity, its means and modes. In a word, society
produces the activity of the individuals it forms.

Thus, activity is necessarily motivated; actions have explicit goals, and
individuals engage in activity with goal-directed action leading to integral
formation of "the intramental plane". Mediation is central to this formation, with
the mediational means (tools, signs or other) a key focus in activity theory
(Leont'ev, 1979; Wertsch, 1991).

Thus, starting from identity as meaning belonging in practice, with knowledge
firmly rooted in practice (Wenger, 1998), we move to identity as the mediational
formation of the intramental plane through goal-directed action (Wertsch, 1991).
This extension of belonging through goal-directed action offers a theoretical
grounding for the extension of alignment to critical alignment through processes of
inquiry. I shall return to this below.

Co-Learning Inquiry

Co-learning inquiry means people learning together through inquiry; inquiry being
a mediational tool as indicated above. The term "co-learning" comes from Wagner
(1997, p. 16) who writes

In a co-learning agreement, researchers and practitioners are both participants
in processes of education and systems of schooling. Both are engaged in
action and reflection. By working together, each might learn something about
the world of the other. Of equal importance, however, each may learn

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something more about his or her own world and its connections to institutions
and schooling.

An aim of the LCM project was that didacticians from the university and
teachers from schools would work together to explore and develop mathematics
learning and teaching in classrooms. In such collaboration, both groups are
practitioners and, since both engage in exploration and inquiry, both are
researchers. We thus adapted slightly the words from Wagner (1997, p. 16) to read:
"teachers and didacticians are both practitioners and researchers in processes of
education and systems of schooling". The simple aim, that didacticians and
teachers would work together as both practitioners and researchers, was both a
guiding force for LCM and a source of tension in relation to power and hierarchy.
Didacticians conceptualized the project, gained the funding, invited participation
from schools, and set up the basic project design. Given such clear "ownership" of
the project, could it be possible to redress the obvious hierarchy and create some
kind of sharing of power and responsibility? This question will be addressed
throughout the chapter with relation to the developmental project (LCM) and the
theoretical perspectives outlined above.

Inquiry Community

Inquiry community was part of didacticians' vision for the LCM project; a
theoretical concept rooted in wide previous experience and a number of key
sources. According to Chambers Dictionary, inquiry means to ask a question; to
make an investigation; to acquire information; to search for knowledge. Wells
(1999, p. 122) speaks of "dialogic inquiry" as "a willingness to wonder, to ask
questions, and to seek to understand by collaborating with others in the attempt to
make answers to them". He emphasizes the importance of dialogue to the inquiry
process in which questioning, exploring, investigating, and researching are key
activities or roles of teachers and didacticians (and ultimately students). These
activities can be discerned through the analysis of dialogue in interactions within
the community.

Didacticians had distinguished between use of inquiry as a tool in teaching and
learning, and developing inquiry as a way of being, so that the identity of an
individual or group within an inquiry community would be rooted in inquiry
(Jaworski, 2004a). Developing inquiry as a way of being involves becoming, or
taking the role of, an inquirer; becoming a person who questions, explores,
investigates and researches within everyday, normal practice. The vision has much
in common with what Cochran-Smith and Lytle (1999) speak of as "inquiry as
stance" - the stance of teachers who engage in an inquiry way of being.
Participants in a community of inquiry aspire to develop an inquiry way of being,
an inquiry identity, in engagement in practice. A focus of the LCM project was to
explore what inquiry could mean in mathematics classrooms and in the activity of
teachers and didacticians trying to explore development of mathematics teaching
and learning.

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INQUIRY COMMUNITIES IN MATHEMATICS TEACHING

These words suggest that we do not necessarily have inquiry ways of being in
"normal" practice. Brown and Mclntyre (1993), researching teaching in classrooms
from observation of classroom activity and interviews with teachers, suggested that
teaching and (earning in classrooms develops "normal desirable states". Teachers
and students find ways of working together that fit as well as possible with
expectations of educational and social systems and groups and allow a workable
environment. The workable environment comes from an implicit agreement
between teachers and students about what is expected, and what is acceptable in
classroom activity - a sort of didactic contract (Brousseau, 1984). Such ways of
working and being in classrooms might be characterized as communities of
practice 3 (Lave & Wenger, 1991), in which participants align themselves with the
normal desirable state. However, the normal desirable state does not necessarily
foster the kinds of mathematical achievement didacticians, and society more
broadly, would like to see. 4

In terms of Wenger's (1998) theory, that belonging to a community of practice
involves engagement, imagination and alignment, we might see the normal
desirable state as engaging students and teachers in forms of practice and ways of
being in practice with which they align their actions and conform to expectations.
Imagination ensures comfortable existence within the broader social expectations
and acceptable or desirable patterns of activity.

One of the reasons for introducing inquiry as a tool - for example, in designing
inquiry tasks to stimulate inquiry in the classroom - is to challenge the normal
(desirable) state and question what it is achieving. For example, if students are
learning mathematics through text book exercises, in which the goal is to practise
skills and become fluent with operations, we might ask questions about the degree
of conceptual understanding that is afforded by this practice. If the normal
desirable state is to be sure that students can do what is required, and not to worry
too much about understanding, then it could be that we are denying students an
important opportunity - to understand the mathematics they are learning, and to
relate particular ideas more widely, both in mathematics and in real world
applications. So, we might ask, what can we do in classrooms to enable students to
understand better the mathematics they meet in text book exercises? This is a
developmental question. As soon as we strive to address such a question, we enter
an inquiry or a research process.

In an inquiry community, we are not satisfied with the normal (desirable) state,
but we approach our practice with a questioning attitude, not to change everything
overnight, but to start to explore what else is possible; to wonder, to ask questions,

3 The practice is that of engaging in classroom activity according to the norms and expectations of the
particular setting in which activity takes place. Such practice is often referred to as mathematics
teaching and/or teaming.

4 The TIMSS and PISA studies provide ample evidence of this, for Norway and for many other
countries. See, for example, Kjaernsli, Lie, Olsen, and Turmo, 2004; Gronmo, Bergem, Kjaernsli, Lie,
and Turmo (2004); Mullis, Martin, Gonzalez, and Chrostowski (2004); Mullis, Martin, Beaton,
Gonzalez, Kelly, and Smith ( 1 998).

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and to seek to understand by collaborating with others in the attempt to provide
answers to them (Wells, 1999). In this activity, if our questioning is systematic and
we set out purposefully to inquire into our practices, we become researchers.

The community of the LCM project, set up to generate a community of inquiry,
had to learn, to grow into, to come to know what it could mean to work in inquiry
ways, to develop questioning attitudes, to design inquiry tasks and to foster
students' own inquiry. Thus the community of inquiry was an emergent rather than
an established form of practice. Inquiry practices in schools bring new elements to
established practices. Thus, in order to move from a community of practice to a
community of inquiry, participants will engage in existing practices, aligning to
some extent with those practices, but in a questioning or inquiry mode. This has
been termed "critical alignment" (Jaworski, 2006). It involves a recognition that
within existing practices, alignment (in Wenger's terms) is essential, but if we
bring a critical attitude to alignment - that is we question, we explore, we seek
alternatives while engaging - then we have possibilities to develop and change the
normal states.

Activity Theory as an Analytical Tool

The theoretical ideas outlined above have allowed us to conceptualise the roots of
inquiry communities; we have found, however, that they do not go far enough in
allowing us to analyse the various forms of data we have generated in order to cut
through complexities in the various communities in which the LCM project has
been embedded. For this reason we have turned to activity theory which has
allowed us to inter-relate concepts of community, inquiry and critical alignment in
seeking to explain issues and tensions in the project and emergent growth of
knowledge.

We start here from transitions between intermental and intramental planes
(Vygotsky, 1978; Wertsch, 1991) and the roles of didacticians and teachers in
promoting development in mathematics classrooms. As I shall explain below,
practices within the LCM project, although goal-directed, were not pre-designated.
It is one thing to propose creation of a community of inquiry and quite another to
realize it. A major part of our developmental activity and associated research
involved exploring the creation and nature of an inquiry community. The inquiry
community was emergent in the project as were the knowledge and learning
associated with it. As the people of the project engaged with the activity of the
project within the project community, also working simultaneously in other
communities of practice (schools or university), people learned and knowledge
grew. From knowledge and activity within existing communities of practice, and
activity within the project, new understandings, and new ways of being and acting,
emerged. In Wertsch's (1991) terms, people acting with mediational means within
their respective communities, with goals relating to developing mathematics
learning and teaching in classrooms, form, as part of their communicative
interaction, their /wtermental plane. We see the intermental plane to be the learning
and knowing that occurs within the community as a whole, with the formation of

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intramental planes as individuals participate in mediated action. Leont'ev's (1979)
concepts of motivated activity and goal-directed action have been employed in
analysis of data to chart learning in LCM (Goodchild & Jaworski, 2005; Jaworski
& Goodchild, 2006), along with EngestrSm's (1999) mediational triangle and
concept of expansive learning (see below).

CREATING AN INQUIRY COMMUNITY

Starting Points: From Motives to Goal-Directed Action

The term "community" designates a group of people identifiable by who they are
in terms of how they relate to each other, their common activities and ways of
thinking, beliefs and values. Activities are likely to be explicit, whereas ways of
thinking, beliefs and values are more implicit. Wenger (1998, p. 5) describes
community as "a way of talking about the social configurations in which our
enterprises are defined as worth pursuing and our participation is recognisable as
competence".

According to Rogoff, Matusov, and White (1996, p. 388), in a learning
community, "learning involves transformation of participation in collaborative
endeavour". The idea of inquiry community makes the nature of transformation
more explicit: didacticians and teachers (and ultimately students) will engage
together in inquiry activity. What such activity should or could consist of, and how
it should or could relate to activity in existing communities of practice, the
classrooms, schools and university settings was a focus of research in LCM.

LCM was motivated by developmental aims from which project activity was
designed. In submitting a proposal to seek funding, didacticians proposed certain
forms of action which would give shape to the project. These included workshops
for teachers and didacticians in university settings, design of tasks for workshops
and classrooms, teacher teams in schools for design of classroom activity, and
collection of data from all activity. Thus, realization, or operationalization of the
project required activity in which this design was implemented into project
practice. We proposed engagement in an inquiry cycle (plan, act & observe, reflect
and analyse, feedback) in the design process as the basis for our practical
realization of a developmental research paradigm - more of this below.

The nature of the inquiry cycle was something that emerged in project activity.
The proposed practices set out in the initial design were what engaged us initially
along with the philosophy of co-learning inquiry. We (didacticians) wished to
collaborate with teachers as partners in developing and researching mathematics
teaching in classrooms (Jaworski, 1999). We wanted to try to avoid positions of
offering teachers models of practice and supporting their implementation, or of
bringing teachers into developmental practice after the design stage and including
them only then in the action (Jaworski, 2004b). Nevertheless, the project had been
conceived by didacticians: the philosophical basis of the project (in co-learning
inquiry) was not negotiable but was clearly open to interpretation; the more
practical aspects of project design could be negotiated but award of funding

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brought with it a responsibility for didacticians to achieve what had been set out, so
at some levels it was not possible to start to (re)negotiate the ground with teachers.
So> the initial position was that motivation for the project was in place together
with some designated action and goals. A major developmental question at this
stage was how to bring teachers into the project.

Action and Inquiry

In creating an inquiry community, the participants have to come together in goal-
directed action. Establishing goals within a community is itself a developmental
task, and goes with initial action. In their invitation to schools and teachers to
participate, didacticians set out the principles of the project and outlined its
operation based on workshops at the university and innovation in schools and
classrooms. Schools were recruited for two years, with the possibility of a third
year. 5

So, with regard to action and goals some things were taken as basic (e.g.,
workshops and co-learning inquiry) and (many) others were open to negotiation
and experimentation in project activity. The motivating principle on which we all
agreed (didacticians and teachers) was our desire to develop better learning
environments for students in mathematics at the levels of schooling with which we
were associated. Unsurprisingly, the ways of thinking about this principle were
deeply related to the communities of practice from which we came, and these
varied across the schools and between schools and university. The knowledge we
brought to the project initially was also deeply embedded in our established
communities with sociohistorical precedents and cultural practices forming
identities in the project. 6

Two examples from LCM illustrate the initial position. Didacticians' planning
for workshops was rooted in their philosophy for the project and their knowledge
of educational literature and research relating to teacher education and
developmental practice in mathematics classrooms. Some had pioneered small
group problem solving in mathematics teaching at the university (Borgersen, 1994)
and all believed strongly in investigative approaches to teaching mathematics at
any educational level. This embedded knowledge was highly motivational in the
activity that didacticians planned for workshops. Teachers came from an
educational system in which initial teacher education was provided in a university
with practice in schools, and continuing teacher education was provided through
workshops and seminars led by teachers from the university. There was
expectation by at least some teachers that didacticians would lead the way in
proposing developmental activity. A quotation from a teacher Agnes, in a focus

5 Eight schools from early primary to upper secondary joined the project and 40 teachers participated
during three years. Some funding was given to schools to support teachers in attending workshops in
school time. Further details can be found in Jaworski (2005).

6 Norwegian culture and society along with educational values and systems were in common for most
project participants (teachers and didacticians).

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group interview at the end of two years of classroom activity indicates that she
struggled in the beginning because didacticians did not seem to show teachers what
to do.

Agnes: [...] in the beginning I struggled, had a bit of a problem with this
because then I thought very much about you should come and tell us how we
should run the mathematics teaching. This was how I thought, you are the
great teachers [...] (FG060313. Translated from the Norwegian by Espen
Daland)

Thus teachers found it difficult in the initial stages to initiate innovative activity
in their schools because it had not been their custom to think of doing so and they
expected a clear lead from didacticians. There were other barriers as well, which I
shall come to.

In LCM, particularly in the first year, it was the workshops which led the way in
bringing participants together to build community and create an inquiry approach
to thinking about mathematics, teaching and learning. Didacticians planned tasks to
stimulate thinking and action: these were increasingly influenced over three years
by teachers' comments, suggestions and requests for particular forms of activity. In
any workshop, teachers and didacticians worked side by side on tasks, usually in
small groups, in a mode intended to create genuine collaboration in doing
mathematics and talking about associated classroom issues. Groups were organized
sometimes to cross school levels, at other times to align with levels more or less
finely. Plenary sessions allowed input on relevant topics, presentations from school
activity (often using video recordings from classrooms), and feedback from group
activity and discussion.

In all workshops, mathematical tasks were chosen or designed carefully (mainly
by didacticians - discussed further below) for their mathematical or didactical
appropriateness for the stage of the project. In the first workshops, problems were
chosen which had rich potential for stimulating mathematical thinking and which
were accessible to people with widely different mathematical experience. Later,
problems or tasks were designed related to curriculum topics. All work on tasks led
to discussions, in both small group and plenary, around the didactics and pedagogy
of creating tasks for classrooms and the associated issues. 7

The workshops were spaced throughout the school year so that, between
workshops, school activity and innovation could take place. Two forms of activity
in school emerged from this opportunity. In some cases, teachers took tasks from
the workshops and, with suitable modification, used them in their classrooms with
students. Frequently, reporting at a workshop included presentations from such
student activity. In other cases, the teacher team in a school designed a task or set
of tasks to bring an inquiry approach to a curriculum topic. Varying degrees of
collaboration between teachers and didacticians were involved in designing and

A special issue of the Journal of Mathematics Teacher Education (JMTE 4-6, 2007) is devoted to
research into the design and use of mathematical related tasks in teacher education.

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BARBARA JAWORSKI

planning such tasks. Didacticians often recorded classroom activity on video when
such innovation took place, and video became an important medium in the project
for sharing experience of task design and use in schools. We see here clear
examples of mediation between inter- and intra-mental planes.

Thus early action took place in workshops in the university and in school
activity stimulated by the workshops. Inquiry was evident in the planning process,
in ways in which teachers took workshop ideas back to schools and tried out ideas
in classrooms and in the developing relationships between the participants as
activity progressed. I shall talk later about the outcomes of such activity in terms of
participants' learning and issues and tensions which arose.

An Inquiry Cycle in the Design Process

From the beginning of the projects, design was a central factor in creating
workshop or classroom activity and innovation. Didacticians followed loosely a
design research approach to creating activity in workshops (Kelly, 2003; Wood &
Berry, 2003). The approach was inquiry-based and iterative (plan, act and observe,
reflect and analyse, feedback to planning) and was in Kelly's terms "generative and
transformative" (2003, p. 3). Typically, following an initial planning meeting, a
small team of didacticians took on the design of tasks according to agreed criteria
for the coming workshop: for example, tasks relating to algebra at a range of levels
including opportunities for generalization and justification of conjectures. The
small team circulated the outcomes of their design process and these were
discussed in a subsequent meeting. After the workshop, one meeting of
didacticians was dedicated to reflecting on the workshop activity including
outcomes from the use of tasks; these reflections feeding into subsequent decision
making and planning. This inquiry process was centrally important in sharing
knowledge and expertise among didacticians, stimulating creativity, generating a
group outcome in terms of tasks for a workshop, and building new knowledge
within the didactician community.

Didacticians envisaged a similar process for school teams planning for the
classroom. Unsurprisingly, the outcomes were very variable, and related to
particular school circumstances. While, in at least one school, the design cycle, in
planning and implementing tasks and reflecting on their use by students, was
exemplary (Fuglestad, Goodchild, & Jaworski, 2006; Hundeland, Erfjord,
Grevholm, & Breiteig, 2007) in other schools planning was more ad hoc, often
individual and relating to one class only (Daland, 2007). The most common
practice observed was teachers' use of workshop tasks, modified for classrooms.
Teachers reported in subsequent workshops from their classroom activity and the
engagement of their students, and video extracts showed evidence of classroom
innovation. The words of Agnes, continuing from the quotation above, testify to
teacher growth through this process:

[...] but now I see that my view has gradually changed because I see that you
are participants in this as much as we are even though it is you that organise.

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INQUIRY COMMUNITIES IN MATHEMATICS TEACHING

Nevertheless I experience that you are participating and are just as interested
as we are to solve the tasks on our level and find possibilities, find tasks, that
may be appropriate for the students, and that I think is very nice. So I have
changed my view during this time. (FG_060313. Translated from the
Norwegian by Espen Daland)

Project activity in schools proved a major learning experience for didacticians as
I discuss below.

A Developmental Research Paradigm

The central use of design, as in design of tasks and activity for workshops and
classrooms suggested a design research approach to the project. Didacticians would
design for workshops, albeit taking into account strongly the views and suggestions
of teachers. Teachers would design for the classroom, drawing on experiences in
workshops and inviting didacticians' contributions as appropriate. However, the
theory of design research (see Wood & Berry, 2003) proved too "clinical". The
design cycle, even in the activity of didacticians, was rarely conceived "up front",
and emerged largely from human interactivity around the aims of the project. In
schools, it was often hard to recognize clearly the elements of a cycle, intertwined
as they were with the multitude of factors that make up teachers' lives.

Here we recognize the developmental nature of the projects - activity emerged
from engagement. Action, observation, reflection and analysis in the inquiry/design
cycle led to growing awareness of the nature of co-learning in the projects. This
inquiry cycle was overtly a learning process for all participants who acted as
insider researchers, inquiring into their own practices and feeding back what they
learned into future action (Bassey, 1995; Jaworski, 2004b; Goodchild, 2007). The
systematic nature of such inquiry varied considerably across the project.

From the beginning, didacticians collected data as far as possible from all
activity - all meetings at which didacticians were present were recorded on audio,
all workshop activity on video or audio, photographs were taken and documents
carefully stored. Some school meetings were audio recorded and some classroom
activity video recorded. A large data bank was organized to which all didacticians
had access. These data were not related to particular research questions; rather
research questions evolved through activity and data was used according to need.
As didacticians followed up initial research questions in analysis of data and
writing of papers, more refined questions emerged which then fed into future
activity and further research. In this way, the emergent nature of research in the
project became centrally visible, and it was possible to trace links between research
activity and developmental progress. 8

See Gravemeijer (1994) and Goodchild (Volume 4 of this Handbook) for related and extended
accounts of developmental research.

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INQUIRY COMMUNITY AND OUTCOMES IN THE LCM PROJECT

The essence of an inquiry community is that, through goal-directed action in
communities of practice, participants explore, inquire into, their own practice with
the motive of learning how to improve the practice (see also Benke, HoSpesova, &
Ticha, this volume). All participants engaged in the project community, but they
were also a part of other communities which made demands on their work and
lives, and the inquiry process resulted in a more critical scrutiny of the range of
practices and possibilities they afforded. Thus, didacticians and teachers, in their
respective established communities both aligned with the practices of those
communities and looked critically at their engagement. Teachers participated in the
day to day life of their schools and, integrally, explored the use of inquiry-based
tasks in their classrooms and observed their students' mathematical activity and
learning. Didacticians collected and analysed data and wrote research papers, as
expected of university academics and, integrally, explored the design of tasks for
workshops and their work with teachers in school environments to support teachers
in their project activity. Activity in the project community emerged from action.
Action in the form of task design led to action in a workshop, which led in its turn
to action in schools, each of these feeding back to inform succeeding stages of
activity. The inquiry community of the project could not be separated from the
established communities of which project members were a part. Interaction
between established communities, their joint enterprise, mutual engagement and
shared repertoire (Eriksen 2007; Wenger, 1998), and the emerging project
community led to recognition of a complexity of inter-relations, issues and tensions
as the project progressed. As indicated earlier, didacticians used activity theory to
try to make sense of the complexity and address issues and tensions.

Mediated Action and Engestrom 's Triangle

Relating to Vygotsky's (1978) law of cultural development and ideas from
Leont'ev (1979) and Wertsch (1991), expressed above, a simple triangle (see
Figure 1) expresses the mediational process as individuals or groups (the subject of
activity) engage in action to achieve goals (the object of activity).

In all cases, according to the theory, activity is mediated: for example, activity
in workshops is mediated by the tasks in which teachers and didacticians engage;
activity in schools is mediated by the ideas teachers' bring from workshops related
to tasks for the classroom and approaches to working with students.

However, according to Engestrom (1998) this "simple" mediational triangle
ignores the "hidden curriculum", the factors in education in schools that influence
fundamentally what is possible for teachers and their students, and ultimately for
didacticians in a developmental project such as LCM. Engestrom (1998, 1999)
extended the simple triangle to the more complex version (see Figure 2).

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INQUIRY COMMUNITIES IN MATHEMATICS TEACHING

MEDIATING ARTEFACTS

SUBJECT

"=>

OBJECT r-A OUTCOME

Based on Vygotsky's model of a complex mediated act

Figure 1. The simple mediational triangle.

TOOLS
sir

SUBJECT

4 OBJECT | — K

OUTCOME

RULES

COMMUNITY

DIVISION OF
LABOUR

Engestrom's 'complex model of an activity system'

Figure 2. Engestrdm 's mediational triangle including the hidden curriculum.

The progression from subject to object can be achieved in mediation through
any of the paths indicated. Rules include the curriculum and its assessment, the
ways in which school and educational systems operate, the societal and political

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BARBARA JAWORSKI

expectations of schools and teachers. Community includes the established
communities discussed above, as well as the project community in LCM. Division
of labour includes the roles of participants, teachers in their school system,
didacticians within a university setting; new roles developing through the project.

Issues and tensions arise when elements of the hidden curriculum challenge the
achievement of goals. I use the theory of mediated action within communities of
inquiry and the hidden curriculum expressed in Engestrom's triangle to present
some of the outcomes of the LCM project (in the rest of this section) and lead to
more general observations concerning development in communities of inquiry (in
the final section of this chapter).

Didacticians ' Roles

A tension with which didacticians have grappled since the beginning of the project
concerns a didactician's role in working with teachers, either in a workshop or in a
school environment. To what extent were we to offer our own thinking, viewpoint
or expertise? In one early meeting, considering our role in a workshop small group,
the term "coordinator" was used and rejected. Someone equated it with being "the
boss". The words "facilitator" was preferred (Cestari, Daland, Eriksen, & Jaworski,
2006). It was clear that the didactician in a group had some responsibility to ensure
the smooth working of the group according to the declared task. This might mean
ensuring that all participants were included in dialogue and activity. It might mean
helping to keep the group focused. It might mean taking initiative to suggest roles
for participants. It was agreed that it should not mean explaining the mathematics,
or giving the solution of a problem. However, to what extent should a didactician
participate in the mathematics? To what extent should he or she present a personal
point of view in discussion? We had no clear answers to such questions. It
remained for us to work according to broadly agreed principles and respond to
particular circumstances. Activity was mediated through workshops tasks,
experience from our activity in other communities, responses from the community
of teachers present and so on. Mediation through subsequent sharing of experience
with the didactician team enabled our awareness to grow and strengthened our
ability to act knowledgeably according to agreed principles. For example, after one
workshop, a didactician praised the actions of one colleague in enabling discussion
in a small group. A further meeting was planned to watch a video recording of this
group and to synthesise from the praised actions. From such interactivity over time,
we learned both to live with uncertainty and to recognize the nature of growth in
being a didactician in such a project. Despite saying these things so simply, this
was not always a comfortable process.

The Locus of Power and Control

This issue of the didactician's role in activity with teachers adumbrates a
fundamental issue that underpinned much of the LCM project - that of where the
power and responsibility in the project was located, and its implications.

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Undeniably, the project originated with the didacticians; they were responsible to
the research council, owned both conceptualization and operational isation to a high
degree, and controlled funding. Schools had volunteered to be in the project and
signed a contract with the university regarding their participation (Jaworski, 2005).
Teachers participated with willingness and enthusiasm, and there was also much
evidence of enjoyment. Teachers were also critical of what they experienced, and
expressed points of view that were not always in accord with didacticians'
concepts of events.

For example, although workshop activity in small groups which crossed school
levels was presented by didacticians as valuable for understanding students'
experience beyond one's own level, teachers preferred overwhelmingly to work
with colleagues at the same school level, and said so! After the very early months,
small groups were usually same-level (and sometimes same-school) groups. Some
teachers were critical of mathematical problems that were not clearly related to a
topic in their own curriculum. They indicated that demands of curriculum and
available time meant there was no possibility for them to use such problems, even
though the problems were interesting and often fun to engage with. One teacher
expressed this point of view after having chosen himself to engage with a 'fun'
problem in a workshop. The implication was that in his lessons there was no time
for 'fun'. Didacticians responded to such comments by designing mathematical
tasks which could be seen as clearly curriculum-related, but nevertheless might be
fun to engage with. Teachers responded that such tasks could be seen as valuable,
but were much more time consuming than the text book tasks they used. However,
the teachers expressing this point of view in one school invited didacticians to
engage with them in designing more open tasks that could engage students
conceptually. This resulted in a set of lessons, according to the teachers, quite
different from those they held normally. They reported that students had seemed to
have a better understanding of the mathematical concepts than earlier groups.
Nevertheless, they were clear that they could not afford generally the amount of
time demanded by these tasks (Fuglestad et al., 2006; Hundeland et al., 2007).

We see here clear examples of critical alignment by both didacticians and
teachers - a complex set of actions and reactions in and to project activity closely
related to school activity. On the one hand, activity was led by design of tasks and
group organization designed by didacticians. On the other, teachers' responses and
perspectives led to reconceptualization and redesign; for example, groups became
mainly same-level groups; tasks were increasingly curriculum-related. Teachers
spoke from their own experiences and perspectives rooted in their normal activity
in school communities and from the demands of the rules of schooling, for
example the pressure of needing to "cover" the national curriculum. Rules and
communities mediating the thinking and actions of teachers impinged on the
project and mediated the design of tasks and workshop groupings. In order to
achieve project goals didacticians needed to recognize and respond to teachers'
concerns. Teachers surprised didacticians nevertheless by engaging in activity in
ways that showed workshop goals being achieved in classrooms. Thus, control

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shifted between didacticians and teachers in interesting ways showing a complex
division of labour in the project.

Mutual Adaptation and Learning

> 9

The first year of activity with schools constituted Phase 1 of the project. Before
the start of the second year (Phase 2), didacticians responded to teachers'
comments on workshops by holding a consultative meeting. Teachers were invited
to express frankly their views on workshops and to make suggestions for
workshops in the coming phase of activity. Many indicated that finding time in
school for the kinds of planning meetings they needed to design activity for
classrooms was extremely difficult. School structures militated against such
meetings and time was limited. They would like the opportunity to plan together
with colleagues from other schools at the same level, to produce classroom tasks
and to report on the classroom activity on a future occasion. These suggestions
were so strongly supported across school levels, that Phase 2 of the project became
structured accordingly. The Norwegian phrase "planlegge et opplegg" (devise the
lesson plan) became a watchword for Phase 2. Here didacticians could be seen
clearly to take on board teachers' perspectives and to build these into ongoing
activity in workshops. Increasingly in Phase 2, input from teachers relating to
activity in classrooms became a central feature of plenary sessions. Curriculum
topics were used explicitly as a focus for mathematical activity. Same- level groups
predominated. Feedback from a focus group interview with each school team at the
end of Phase 2 indicated that teachers had appreciated didacticians'
accommodation to their perspectives in a range of factors and showed
corresponding activity in classrooms. Invitations from teachers to didacticians to
videorecord innovative activity in classrooms resulted in a bank of videodata
charting development in classrooms. We might see, in retrospect the meeting
between Phases 1 and 2 as a watershed in project activity. EngestrSm's (1999)
theory of expansive learning might be seen to capture this watershed.

Expansive Learning

The outcome of tensions, such as those expressed above, in the LCM project was
that activity went on. We did not see a breakdown. Trust, good will and positive
intentions led to realization (both recognition and making-real) of ways of working
that enabled some achievement of some goals (on both sides) to some extent. In
this process, there was some event or initiative which acted as a force to resolve
tensions - expressed by EngestrSm (1999) as "expansive learning". For example,
during the first phase we had seen a build up of tension as teachers engaged with
activity, provided clear evidence of valuing the project and their participation, yet

9 The LCM project was funded for four years. During this time, there were three Phases of activity, each
of one school year, in which didacticians and teachers worked together as described here.

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at the same time increasingly expressed a wish for modified forms of action (such
as the nature of small groups or the kinds of mathematical tasks). The meeting
between the phases allowed overt expression of desire for alternative action and
clear suggestions for the form such action might take.

Expansive learning is rooted in the activity theory concepts expressed above -
notably goal directed mediated action, based in Vygotsky (1978) and Leont'ev
(1979). EngestrSm (1999, p. 382), following Leont'ev (1979), expresses it as a
dialectic of "ascending from the abstract to the concrete" and adds (pp. 382-383):

A method of grasping the essence of an object by tracing and reproducing
theoretically the logic of its development, of its historical formation through
the emergence and resolution of inner contradictions. [...] The initial simple
idea is transformed into a complex object, a new form of practice. [...] The
expansive cycle begins with individual subjects questioning the accepted
practice, and it gradually expands into a collective movement or institution.

Through complex interactions traceable to all three elements of the hidden
curriculum, participants in the project are able to recognize and isolate the inner
contradictions expressed by Engestrdm (1999). In the case above, the concerns
about groups and about tasks in workshops, through the between-phases meeting,
led to the emergence of the new idea of planlegge et opplegg through which
planning in homogenous groups in a workshop with didacticians' support could
lead to teachers having suitable activity for their classrooms leading to
development for students' learning of mathematics. What started as internal
rumblings within activity resulted in an external development explicit for all to
engage with. Such analysis enables us to trace our activity, noting its historical
development and becoming clearer about the issues, tensions or contradictions
inherent in the developmental process.

As a further example, I refer to an event with took place in Phase 3 of the
project. This phase was introduced (by didacticians) as focusing on declaring and
achieving school goals for development of mathematics learning and teaching
within a school. Activity in Phase 3 proceeded along familiar lines with
engagement in workshops and associated work in schools and with associated
issues and tensions acknowledged but not resolved. The focus on school goals was
elusive and progression towards school goals not achieved. Then one didactician
suggested a task that was to have important consequences for the goals of Phase 3.
The task was connected to a series of three workshops focusing on algebra. It
involved teachers in undertaking some focused observation of some of their own
students related to work on algebra. Teachers were asked to bring to the next
workshop some input from their observations. The workshop was organized to
develop a "red thread" through observations at different levels of students'
algebraic understanding across the range of school levels. In order to visualize
teachers' judgement on the quality of students' algebraic thinking, teachers were
asked to pin their written observations to a line which was strung across the
workshop's main room. The coffee break allowed all to view the line and think
about its contents. The quality of teachers' perceptions expressed in the final

325

BARBARA JAWORSKI

plenary discussion and comments received from several teachers after the
workshop indicated that this had been an important experience for teachers: most
significant had been their insights into the thinking and understanding of their own
students and recognition of the task as a research event with serious learning
outcomes for themselves. The task had provided the opportunity for expansion and
for a breakthrough in activity.

Teachers' participation and comments in and from this activity suggested to
didacticians that certain goals had been achieved. Teachers had engaged overtly in
a research task, conducting activity in their schools, findings the time to do so,
recognizing their learning, and valuing their insights into students'
perceptions/thinking/understandings of algebra. This signified for didacticians
strong developmental outcomes from their own activity and participation - with
evidence of both teachers' learning and didacticians' associated learning. For
example, teachers suddenly came to see, through their study of students' thinking
and activity in algebra, how they could explore in their school environment ways to
develop teaching and learning; didacticians saw the nature of a task that could lead
to teachers' effective recognition of the nature of school goals for students'
development and learning in mathematics.

Seeing the enterprise in terms of an activity system made it possible to pick out
elements in the complexity and trace developmental patterns for participants in the
project (see Goodchild & Jaworski, 2005; Jaworski & Goodchild, 2006). In this
process, tensions became evident as catalysts providing opportunity for learning.
We see the nature of community as central to this provision of opportunity. During
the three years, the members of the project community came to know each other as
colleagues, appreciating good intentions, trusting good will, recognizing
differences, respecting alternative points of view and becoming aware of
developing thinking and associated possibility for action. This is not to claim
hugely visible changes to the everyday practices in which established communities
were rooted, but rather to recognize a relationship between developmental aims
and the realities of normal working life. For example, the structure in a school
could not change to suit the aims of the project; nor should the project fail because
these aims could not be met. So, in what ways might we accommodate to achieve
the aims? Through such recognition also, the aims become more understandable
and perhaps more open to flexibility in their achievement - that is we were able to
relate the aims to real settings and work out alternative approaches compatible with
the aims.

A MORE GENERAL PERSPECTIVE

This chapter has interwoven complex aspects of theory and a specific
developmental research project to illuminate notions of the development of

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INQUIRY COMMUNITIES IN MATHEMATICS TEACHING

mathematics learning and teaching through developmental research in inquiry
communities involving teachers and didacticians. 10

In this final section of the chapter my purpose is to pull out to a more general
viewpoint on communities of inquiry and the associated theoretical perspectives.
Key areas of theory have been

• Communities of practice with notions of belonging through
engagement, imagination and alignment (Wenger, 1998) shifting to
critical alignment through inquiry (Jaworski, 2006);

• Mediated activity between people involving individuals acting with
mediational means (Vygotsky, 1978; Wertsch, 1991);

• The motivated nature of activity involving goal-directed action
(Leont'ev, 1979);

• Engestrom's expanded mediational triangle and the concept of
expansive learning (EngestrBm, 1998, 1999).

The concept of community is clearly central in all of these and needs no further
comment. The place of inquiry perhaps needs further elucidation. Inquiry brings
the critical element to community of practice through which participants can
inquire into existing practices with possibility to modify and improve. Inquiry can
be seen as a mediational tool in social settings enabling development of knowing
between people and hence of participative individuals. Inquiry as in the
design/inquiry cycle promotes goal-directed action leading to developmental
outcomes. Inquiry ways of being allow the possibility of contradictions emerging
as powerful motivators for expansion within an activity system.

The inquiry community starts with intentions to use inquiry as a tool for learning
and development. Through engagement with an inquiry cycle in the design of tasks
and opportunity for participation, a community grows into inquiry ways of being
which encourage mediation of complexity within the hidden curriculum of systems
and structures that constrain development. As compared to established
communities of practice, in which norms of practice nurture undesirable states, the
inquiry community is emergent. It does not avoid issues, tensions and
contradictions, but deals with them as part of emergent recognition and
understanding leading to possibilities for expansive learning. Inquiry ways of being
accept the unfinished nature of learning and development. There is not an end
point.

10 For those interested in knowing more about the LCM project, the website http://fag.hia.no/1cm/
contains a list of relevant publications and the book (Jaworski, Fuglestad, Bjuland, Breiteig, Goodchild,
& Grevholm, 2007) charts the project as a whole.

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EPILOGUE

LCM ended in December 2007. However, in 2006, an extension to LCM was
already started in the form of a new project, TBM, Teaching Better Mathematics,
funded again by the RCN. This new project involves a consortium including five
centres in different parts of Norway linking didacticians with schools and rooted in
a philosophy of inquiry communities. At Agder University, TBM is linked to LBM
(Learning Better Mathematics), a parallel project owned by schools. LBM and
TBM work in concert with a managing committee including school leaders and
didacticians. Schools pay for the work of one didactician based at the university
with a responsibility for liaising between the two projects and supporting teachers'
participation. Both schools and didacticians contribute to conceptualization,
planning and engagement in project activity in workshops and classrooms.

The consortium has come about through didacticians in institutions in the five
regions recognizing shared goals rooted in developing inquiry communities
between didacticians and teachers in their own region. Each regional group has
their own specific project with its own clear focus and goals, but all share the same
theoretical basis. The research council has seen value in such collaboration in
supporting the project. Its invitation to the Agder community to offer a dedicated
day conference in Oslo in October 2007 was a further indication of its support. We
see this as very positive encouragement from an important part of the
establishment to continue this developmental approach.

ACKNOWLEDGEMENTS

I should like to thank most sincerely Gertraud Benke, Simon Goodchild, and
Konrad Kramer for their kind but critical and extremely helpful comments on an
early draft of this chapter.

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Barbara Jaworski
Mathematics Education Centre
Loughborough University
UK

330

NANETTE SEAGO

14. MATHEMATICS TEACHING PROFESSION

Recommendations for the profess ionalization of teaching were made over two
decades ago. One of the most difficult problems facing the professionalization of
mathematics teachers has been the definition and description of the specialized
knowledge needed by teachers. Mathematical knowledge for teaching has been
given recent attention as a specialized knowledge needed for teaching by focusing
attention on the considerable mathematical demands that are placed on classroom
teachers. This chapter will show how professional development opportunities that
use records of classroom practice such as student work, classroom videotapes, or
lesson plans, can support the learning of mathematical knowledge for teaching and
provide the opportunity to further develop the mathematics teaching profession by
increasing teachers ' authority as reliable source of professional knowledge.

INTRODUCTION

Professionalization commonly refers to the process in which an occupation
engages in the improvement of the conditions or standards of their work, and is
based on the assumption that professions have distinctive characteristics that
distinguish them from other occupations. Professionalization involves the extent to
which members of that occupation has autonomy over the content of their work
and the degree to which society places value on this work.

Recommendations for the professionalization of teaching were made over two
decades ago (Carnegie Task Force, 1986; Holmes Group, 1986). These groups
used existing professions such as law and medicine to suggest graduate level
specialized education for teachers. In addition, they proposed that a board of
professionals be created to manage the testing and licensing of teaching candidates.
They sparked a groundswell of support to strengthen teaching as a profession - in
which teachers were perceived as professionals who made important decisions in a
complex setting. This movement was intended to upgrade the quality of teaching
and public education, increase public confidence in teachers, raise the status of
teachers and support their efforts to develop professionally (National Board of
Professional Teaching Standards, 1988).

Two decades have passed since the recommendations were made and during
that time technological advances have changed the world in dramatic ways. Of all
the occupations that are labelled as professions, it is teaching that is responsible for
creating the human skills and capacity that will enable individuals to survive and
succeed in today's information society. The kind of professionalism needed is a
more up-to-date version of the one recommended in the 1980s. Hargreaves (2003)

NANETTE SEAGO

reminds us in his book, Teaching in the Knowledge Society, "as catalysts of
successful knowledge societies, teachers must be able to build a special kind of
professionalism". He argued that this professionalism is not based in old paradigms
in which teachers had the autonomy to teach in ways that are most familiar to
them. Instead, Hargreaves (2003, p. 24) points to a new professionalism for the
information age that requires teachers as professionals to:

Promote deep cognitive learning; learn to teach in ways they were not taught;
commit to continuous professional learning; work and learn in collegial
teams; treat parents as partners in learning; develop and draw on collective
intelligence; build a capacity for change and risk; and foster trust in
processes.

Yet in the midst of a rapidly changing information society, many people still
cling to basic premises of a pre-professional age - that anyone who knows a
subject matter fairly well can teach it to secondary students and anyone who loves
children can teach primary school. Still others believe that people are born teachers
or that teaching is something you learn individually by trial and error, within the
confines of your own classroom. In a world that changes so quickly, it is
impossible for any one teacher to obtain enough knowledge alone to improve him
or herself. New professionalism requires teachers to engage in continual,
collaborative professional learning communities. Indeed, over the past two
decades, teachers in many countries have become more expert at and experienced
in working with their colleagues, yet more work is needed to truly professionalize
mathematics teaching.

In this chapter, I will first discuss the issue of the professional ization of
mathematics teaching - what it means and why it is important to the further
development of the mathematics teaching profession. Second, I will talk about how
professional development that holds certain design principles can provide
opportunities for teachers to develop as professionals. Then I will use an example
to illustrate the enactment of these design principles, and describe how they
provided teachers the opportunity to learn and professional expertise, judgement
and trust. Finally, I will summarize the issues, challenges and possibilities of the
further development of mathematics teaching in the information age.

PROFESSIONALIZATION AND MATHEMATICS TEACHING

With an assumption that professionalization was a necessary condition for teachers
to successfully implement the NCTM Standards (NCTM, 1989), the push for
professionalism continued when the National Council of Mathematics Teachers
published the Professional Standards for Teaching Mathematics (NCTM, 1991).
These standards assumed that the type of instruction they promoted required
professionalism - a high degree of autonomy, individual responsibility and
authority. This perspective acknowledges "the teacher as a part of a learning
community that continually fosters growth in knowledge, stature, and
responsibility" (NCTM, 1991, p. 6). To give guidance to the development of such

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MATHEMATICS TEACHING PROFESSION

professionalism in mathematics teaching, the Professional Standards for Teaching
Mathematics consisted of separate sections of specific standards: (1) Standards for
teaching mathematics, (2) Standards for the evaluation of the teaching of
mathematics, (3) Standards for the professional development of teachers of
mathematics, (4) Standards for the support and development of mathematics
teachers and teaching. These teaching standards were intended as a set of principles
accompanied by illustrations or indicators to give direction and guidance for
moving toward excellence in teaching mathematics.

One year after the publishing of the Professional Standards for Teaching
Mathematics, Noddings (1992) pointed out that a culture that downplays the
complexity involved in teaching works against the professional ization of teaching.
If one holds a pre-professional view that once you are qualified to teach, you know
the basics of teaching forever; that once you gain managerial control over your
students, teaching is easy; then the quest for the professional ization of teaching
does not make sense. However, if one believes teaching is a complex practice
involving specific skills and knowledge that require ongoing, collaborative
professional learning, it makes sense that professional ization should be coveted and
pursued. Noddings (1992, p. 202) argued the latter and suggested that one of the
biggest challenges facing the professionalization of mathematics teachers in
particular is the definition and description of the specialized body of knowledge
needed by teachers. She states:

Knowledge of mathematics cannot be sufficient to describe the professional
knowledge of teachers. What does a mathematics teacher know that someone
with similar mathematical preparation does not? What specialized knowledge
does the teacher have?

Over the past two decades, a number of studies have attempted to define the
nature and elements of knowledge specific to mathematics teaching. Most of this
research has used teachers' practice as the site for investigating this specialized
knowledge. These studies make the claim that the knowledge required for teaching
is rooted in the mathematical demands of teaching itself and that this differs from
the knowledge that a teacher gains from formal education (Ball & Bass, 2003).

Lee Shulman's pioneering notion of "pedagogical content knowledge" began to
address the kind of specialized knowledge that Noddings referred to by labeling
and describing a type of knowledge specific to teaching that teachers need to
employ within their practice (Shulman, 1986, 1987, 1989). Shulman distinguished
three categories of teacher content knowledge: subject matter knowledge,
pedagogical content knowledge and curricular knowledge. Although these
categories are not specific to mathematics teaching, many mathematics education
researchers have used this classification as a framework for their work. In
particular, the notion of pedagogical content knowledge (PCK) as "bundled"
mathematical, pedagogical, and cognitive/developmental knowledge, Shulman
argued could help teachers anticipate and address typical issues of students'
learning mathematics. His groundbreaking work opened educators' eyes to the
possibility that teaching requires specialized professional knowledge - a view he

333

NANETTE SEAGO

In this chapter, I will first discuss the issue of the professionalization of
mathematics teaching - what it means and why it is important to the further
development of the mathematics teaching profession. Second, I will talk about how
professional development that holds certain design principles can provide
opportunities for teachers to develop as professionals. Then I will use an example
to illustrate the enactment of these design principles, and describe how they
provided teachers the opportunity to learn and professional expertise, judgement
and trust. Finally, I will summarize the issues, challenges and possibilities of the
further development of mathematics teaching in the information age.

PROFESSIONALIZATION AND MATHEMATICS TEACHING

332

MATHEMATICS TEACHING PROFESSION

One year after the publishing of the Professional Standards for Teaching
Mathematics, Noddings (1992) pointed out that a culture that downplays the
complexity involved in teaching works against the professionalization of teaching.
If one holds a pre-professional view that once you are qualified to teach, you know
the basics of teaching forever; that once you gain managerial control over your
students, teaching is easy; then the quest for the professionalization of teaching
does not make sense. However, if one believes teaching is a complex practice
involving specific skills and knowledge that require ongoing, collaborative
professional learning, it makes sense that professionalization should be coveted and
pursued. Noddings (1992, p. 202) argued the latter and suggested that one of the
biggest challenges facing the professionalization of mathematics teachers in
particular is the definition and description of the specialized body of knowledge
needed by teachers. She states: