Mathematical Knowledge: Its Growth Through Teaching
In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A. G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was
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Mathematics Education Library VOLUME 10
Managing Editor
A.J. Bishop, Cambridge, u.K.
Editorial Board H. Bauersfeld, Bielefeld, Germany J. Kilpatrick, Athens, U.S.A. G. Leder, Melbourne, Australia S. Tumau, Krakow, Poland G. Vergnaud, Paris, France
The titles published in this series are listed at the end of this volume.
MATHEMATICAL KNOWLEDGE: ITS GROWTH THROUGH TEACHING Edited by
ALAN J. BISHOP, STIEG MELLIN-OLSEN and
JOOP VAN DORMOLEN for the BACOMET group
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Mathematical knowledge; its growth through teaching I by Alan J. BishOp, Stieg Mell in-Olsen, and Joop van Dormolen (eds.l. p. cm. -- (Mathematics education library Volume 10') Inc 1udes index. ISBN 978-90-481-4095-4 DOI 10.1007/978-94-017-2195-0
ISBN 978-94-017-2195-0 (eBook)
1. Mathematics--Study and teach1ng.
II. Mellin-Olsen, Stieg, 1939OA 11 . M3753 1991
I. Bishop, Alan J. III. Dormolen, J. van. IV. Series.
510' .71--dc20
91-23234
ISBN 978-90-481-4095-4
Print~d on
acid-free paper
All Rights Reserved © 1991 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 Softcover reprint of the hardcover Ist edition 1991 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any ~nformation storage and retrieval system, without written permission from the copyright owner.
CONTENTS
Introduction
13
Entr'acte 2 The Double Bind as a Didactical Trap
39
G. BROUSSEAU AND M. OITE /
S. MELLIN-OLSEN /
W. D6RFLER /
1
Entr'acte 1 The Fragility of Knowledge
Entr'acte 3 Forms and Means of Generalization in Mathematics
63
Entr'acte 4 Metaphors Mediating the Teaching and Understanding of Mathematics
J. VAN DORMOLEN /
89
Entr'acte 5 Tool, Object, Setting, Window: Elements for Analysing and Constructing Didactical Situations in Mathematics
R. DOUADY /
109
Entr'acte 6 T. WERNER /
Observing Conceptual Complexity
Entr'acte 7 Developing Mathematical Knowledge Through Microworlds
C. HOYLES /
133
147
Entr'acte 8 The Benefits and Limits of Social Interaction: The Case of Mathematical Proof
N. BALACHEFF /
Entr'acte 9 A. J. BISHOP /
Mathematical Values in the Teaching Process
175 195
Index of Names
215
Index of Subjects
219
v
INTRODUCTION In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A.G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was the fundamental problem area of the relationships between mathematical knowledge and the teaching and learning processes. The subsequent project BACOMET 2, whose outcomes are presented in this book, continued this work, especially by focusing on the genesis of mathematical knowledge in the classroom. The book developed over the period 1985-9 through several meetings, much discussion and
