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DIDACTICS OF MATHEMATICS AS A SCIENTIFIC DISCIPLINE

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Mathematics Education Library
VOLUME 13
Managing Editor
A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board
H. Bauersfeld, Bielefeld, Germany
J. Kilpatrick, Athens, U.S.A.
G. Leder, Melbourne, Australia
S. Turnau, Krakow, Poland
G. Vergnaud, Paris, France
The titles published in this series are listed at the end of this volume.

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DIDACTIC
S OF MATHEMATICS
AS A SCIENTIFIC DISCIPLINE
Edited by
ROLF BIEHLER
ROLAND W. SCHOLZ
RUDOLF STRÄSSER
BERNARD WINKELMANN
Institute for Didactics of Mathematics,
University of Bielefeld, Germany
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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eBook ISBN: 0-306-47204-X
Print ISBN:
0-7923-2613-X
©2002 Kluwer Academic Publishers
New York, Boston, Dordrecht, London, Moscow
Print ©1994 Kluwer Academic Publishers
All rights reserved
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Visit Kluwer Online at:
http://kluweronline.com
and Kluwer's eBookstore at:
http://ebooks.kluweronline.com
Dordrecht

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Dedicated to Hans-Georg Steiner.
R.
B.,
R. W.
S.,
R.
S.,
B. W.

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Introduction
Bernard Winkelmann
Eclectic approaches to elementarization: Cases of curriculum
construction in the United States
James T. Fey
Didactical engineering as a framework for the conception
of teaching products
Michèle Artigue
Mathematical curricula and the underlying goals
Uwe-Peter Tietze
TABLE OF CONTENTS
Preface
103
117
121
133
15
27
41
55
61
73
89
1
9
1. PREPARING MATHEMATICS FOR STUDENTS
2. TEACHER EDUCATION AND RESEARCH ON TEACHING
Introduction
Rolf Biehler
Reflections on mathematical concepts as starting points
for didactical thinking
Hans-Joachim Vollrath
Beyond subject matter: A psychological topology of teachers'
professional knowledge
Rainer Bromme
Dialogue between theory and practice in mathematics education
Heinz Steinbring
On the application of science to teaching and teacher education
Thomas J. Cooney
3. INTERACTION IN THE CLASSROOM
Introduction
Rudolf Sträßer
Theoretical and empirical approaches to classroom interaction
Maria G. Bartolini Bussi
Theoretical perspectives on interaction in the mathematics classroom
Heinrich Bauersfeld

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TABLE OF CONTENTS
5. PSYCHOLOGY OF MATHEMATICAL THINKING
Introduction
Roland W. Scholz
The interaction between the formal, the algorithmic, and the intuitive
components in a mathematical activity
Efraim Fischbein
From Piaget's constructivism to semantic network theory:
Applications to mathematics education - A microanalysis
Gerhard Steiner
The Sociohistorical School and the acquisition of mathematics
Joachim Lompscher
Action-theoretic and phenomenological approaches to research in
mathematics education: Studies of continually developing experts
Richard Lesh and Anthony E. Kelly
VIII
147
159
171
177
189
201
213
225
231
247
263
277
287
291
6. DIFFERENTIAL DIDACTICS
Introduction
Roland W. Scholz
Mathematically retarded and gifted students
Jens Holger Lorenz
4. TECHNOLOGY AND MATHEMATICS EDUCATION
Introduction
Bernard Winkelmann
The role of programming: Towards experimental mathematics
Rosamund Sutherland
Computer environments for the learning of mathematics
David Tall
The role of cognitive tools in mathematics education
Tommy Dreyfus
Intelligent tutorial systems
Gerhard Holland
Working in small groups: A learning situation?
Colette Laborde
Mathematics classroom language: Form, function and force
David Pimm

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TABLE OF CONTENTS
7. HISTORY AND EPISTEMOLOGY OF MATHEMATICS AND
MATHEMATICS EDUCATION
Introduction
Rolf Biehler
The philosophy of mathematics and the didactics of mathematics
Paul Ernest
The human subject in mathematics education and in the history of
mathematics
Michael Otte and Falk Seeger
Mathematics in society
Mogens Niss
The representational roles of technology in connecting mathematics
with authentic experience
James J. Kaput
LIST OF AUTHORS
SUBJECT INDEX
IX
Should girls and boys be taught differently?
Gila Hanna
From "mathematics for some" to "mathematics for all"
Zalman Usiskin
303
315
8. CULTURAL FRAMING OF TEACHING AND LEARNING
MATHEMATICS
Introduction
Rudolf Sträßer
Comparative international research in mathematics education
David Robitaille and Cynthia Nicol
Cultural influences on mathematics teaching: The ambiguous role of
applications in nineteenth-century Germany
Hans Niels Jahnke
Mathematics and ideology
Richard Noss
Cultural framing of mathematics teaching and learning
Ubiratan D'Ambrosio
399
403
415
431
443
457
461
327
335
351
367
379

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PREFACE
DIDACTICS OF MATHEMATICS AS A SCIENTIFIC DISCIPLINE
Since the work of the International Commission for Mathematics Instruction
(ICMI) at the beginning of this century, nobody can challenge the fact that
scientific work has been done in the field of teaching and learning mathe-
matics. This research work has been carried out by mathematicians, psy-
chologists, educational scientists, mathematics teacher trainers, and mathe-
matics teachers themselves. However, scientific communication on these is-
sues long remained in its infancy, particularly on an international level;
much work was done in isolation; and it was rare to find people who con-
sidered that they belonged to a separate scientific discipline, independent
from mathematics or educational science.
In the late 1960s, a societal debate on the values and organization of a
large number of industrialized countries (such as Germany, France, and the
United States of America) stimulated a new concern for education and for
the related educational sciences. In the 1970s and 1980s, these develop-
ments led to a certain breakthrough for research in mathematics education.
The revival of international organizations such as ICMI and regular global
conferences known as ICMEs (since 1969) has led to the formation of an
international community of mathematics educators. We call the scientific
discipline related to this research and the research-based development work
didactics of mathematics – a notion that is common at least in German- and
French-speaking countries and has become increasingly popular in the
English-speaking world. Didactics of mathematics certainly exists as a dis-
cipline, at least in a social sense, as can be seen from journals, research and
doctorate programs, scientific organizations, and conferences. However, di-
dactics of mathematics is fairly young compared to other sciences such as
mathematics or psychology. As a fairly young discipline, its system of ob-
jects, methodologies, and criteria for valid knowledge exhibits more vari-
1

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ability and less consensus. Its role among other sciences at the university is
still disputed.
This book has been written for the international scientific community of
researchers in mathematics education. It provides a state-of-the-art portrait
of a new branch of science. The reader will find a structured sample of orig-
inal contributions from researchers in the field of didactics of mathematics.
The book will be of interest to all researchers in the field. However,
mathematics educators who are interested in the theory of their practice and
teacher trainers will also appreciate this survey and the diverse stimulations
and reflections it provides. Prospective and practicing teachers of mathemat-
ics will find a variety of interesting spotlights on their practice that focus on
different age groups and ability ranges among their students. In addition to
persons directly engaged in mathematics education, the book as a whole
and/or individual papers should be of interest to researchers from neighbor-
ing disciplines, such as mathematics, general education, educational psy-
chology, and cognitive science.
The basic idea was to start from a general perspective on didactics of
mathematics, to identify certain subdisciplines, and to suggest an overall
structure of its field of research. This book should provide a structured
view, or a "topology," of the breadth and variety of current research in di-
dactics of mathematics by presenting authentic and vivid contributions of
individual authors on their current research in certain subdisciplines. The
subdisciplines are represented by the chapters of this book. The volume
provides a sample of 30 contributions from 10 countries. The authors were
asked to present an example of their research in a way that would also make
the broader research fields represented by the individual contributions ac-
cessible for other colleagues in didactics of mathematics.
We use chapter introductions to provide a synthesis and an orientation
for the research domain represented by the contributions. The individual
contributions are related to the overall idea of the chapter, and the readers'
attention is focused on relations and differences between the different pa-
pers in a chapter as well as their relation to other chapters. This makes it
clear that our aim is not to provide a handbook of didactics of mathematics
with authoratively written subchapters synthesizing research from one au-
thor's point of view. The organization of the book places more emphasis on
a variety and multiplicity of perspectives. It is through the readers' (re-) con-
struction and rethinking of our discipline – which we hope to stimulate with
this book – that we can contribute to further reflection on and interest in our
discipline.
The reader will find the following chapters:
2
PREFACE

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The first five chapters are widely accepted as subdisciplines in the sense of
the existence of many cross-references, intensive communication, and a
common object of study. The other three "subdisciplines" seem to be less
well-structured up to now. We include them because we regard them as im-
portant. This may be a certain bias due to our involvement with the IDM
and its research tradition. We invented the concept of "Differential
Didactics" in analogy to "Differential Psychology" in order to create a focus
for research on gender, cultural minorities, and different groups of learners
in contrast to what may be considered as "mathematics for all."
Didactics of mathematics is an applied area of activity: As in engineering,
(applied) psychology, and medicine, the boundary between scientific work
and (constructive) practice is – to say the least – "fuzzy." Didactics of math-
ematics shares a certain type of (social) problem with the above-mentioned
disciplines, namely mathematics education; and it uses a multiplicity of
methods. The topics of the first four chapters are often conceived of as
practical concerns requiring constructive work, namely, the preparation of
curricula and textbooks, the development of programs in teacher education,
the formulation of guidelines for classroom interaction and learning, and the
development of software. A major recent development has been the attempt
to establish a rationalization, theorization, and reflection of these practical
activities. Rationalization is understood in the twin sense of reflecting on the
rationality of goals as well as improving instrumental efficiency. Sometimes
this has led to work that is more comparable to basic science than applied
science, because researchers felt that it was necessary to deepen theory and
methodological reflection in order to improve our understanding of practical
problems. Research on teachers' cognition and on classroom interaction pre-
sents an example of this trend.
We can also group the chapters into those that are closer to classroom
teaching and learning (chapters 1 to 4) and those that reflect and analyze
Preparing Mathematics for Students
Teacher Education and Research on Teaching
Interaction in the Classroom
Technology and Mathematics Education
Psychology of Mathematical Thinking
Differential Didactics
History and Epistemology of Mathematics and Mathematics Education
Cultural Framing of Teaching and Learning Mathematics
PREFACE
3
1.
2.
3.
4.
5.
6.
7.
8.

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problems of learning, thinking, knowledge, and culture from a more general
perspective, though still related to problems in mathematics education
(chapters 5 to 8). In the first four chapters, the reader will find papers rang-
ing from a mere analytical stance to papers with research-based constructive
implications. Chapters 5 to 8 place more emphasis on analytical aspects.
Didactics of mathematics has to be structured from a systemic point of
view. Even work on subsystems such as the learner or the teacher have to
bear in mind the relation to other components. The chapters concentrate
mostly on subsystems in this sense. Starting from the knowledge to be
taught, namely mathematics, we first try to assemble research on the didac-
tical system in a strict sense: the "didactical triangle" of mathematics –
teacher – learner.
Chapter 1 discusses principles of preparing mathematics for students.
Concepts like "didactical transposition," "elementarization" of mathematics,
and "didactical engineering" are analyzed. Consequently, the focus of the
chapter is on the content of teaching, on knowledge to be taught.
Nonetheless, the influence of other factors and institutions is revealed.
Chapter 2 concentrates on teacher education and research on teaching. Its
link to the preceding chapter obviously is the knowledge to be taught. Its
main topic is the knowledge a teacher has or should have, the structure of
this knowledge, and ways to influence and develop the teachers' knowledge.
Chapter 3 on interaction in the classroom focuses on research that analyzes
the complex "social interaction" of teachers and learners in the classroom
and in small groups. The analysis of language and discourse in the class-
room is an important issue. Chapter 4 on technology and mathematics edu-
cation can be viewed from a systematic point of view as "educational tech-
nology" including textbooks and assessment schemes. These form an impor-
tant product of the didactics of mathematics that is handed on to the practice
of teaching. The design and use of such "products" is an important research
topic. The focus on problems and potentials of the use of computers and
software was chosen because this technology represents a critical issue in
the current development of the teaching and learning of mathematics as well
as an important research field in didactics. Chapter 5 on psychology of