Forms of Mathematical Knowledge Learning and Teaching with Understan
What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elemen
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Edited by
DINA TIROSH School of Education, Tel Aviv University, Israel
Reprinted from Educational Studies in Mathematics 38 (l-3), 1999
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
A. C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-90-481-5330-5 ISBN 978-94-017-1584-3 (eBook) DOI 10.1007/978-94-017-1584-3
Printed on acid-free paper
AII Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 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 information storage and retrieval system, without written permis sion from the copyright owner.
TABLE OF CONTENTS
DINA TIROSH I Introduction EFRAIM FISCHBEIN I Intuitions and Schemata m Mathematical Reasoning
11
DINA TIROSH and RUTH STAVY I Intuitive Rules: A Way to Explain and Predict Students' Reasoning
51
PAUL ERNEST I Forms of Knowledge in Mathematics and Mathematics Education: Philosophical and Rhetorical Perspectives
67
TOMMY DREYFUS I Why Johnny Can't Prove
85
EDDIE GRAY, MARCIA PINTO, DEMETRA PITTA and DAVID TALL I Knowledge Construction and Diverging Thinking in Elementary & Advanced Mathematics
111
JOHN MASON and MARY SPENCE I Beyond Mere Knowledge of Mathematics: The Importance of Knowing-To Act in the Moment
135
THOMAS J. COONEY I Conceptualizing Teachers' Ways of Knowing
163
ANNA 0. GRAEBER I Forms of Knowing Mathematics: What Preservice Teachers Should Learn
189
PESSIA TSAMIR I The Transition from Comparison of Finite to the Comparison of Infinite Sets: Teaching Prospective Teachers
209
RUHAMA EVEN I Integrating Academic and Practical Knowledge in a Teacher Leaders' Development Program
235
FORMS OF MATHEMATICAL KNOWLEDGE: LEARNING AND TEACHING WITH UNDERSTANDING
This volume is dedicated to the memory of Professor Efraim Fischbein, a distinguished member of the mathematics education community
This volume has its origin in the presentations and discussions that occurred in the working group on 'Forms of Mathematical Knowledge' of the Eight International Conference on Mathematical Education (ICME8) held in Seville, Spain, in July 1996. The working group was mainly devoted to defining, discussing and contrasting psychological and philosophical issues related to various types of knowledge involved in mathematics learning and teaching (e.g., knowing that, knowing how, knowing why and knowing to; intuitive and analytical; explicit and tacit; elementary and advanced). After the conference the participants felt that issues raised in the working group were significant for the mathematics education community and decided to submit a proposal for a special issue on forms of mathematical knowledge to Educational Studies in Mathematics. This introduction attempts to provide the reader with a preliminary flavor of the volume. It describes its structure and prese
