Approaches to Algebra Perspectives for Research and Teaching
In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magn
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Mathematics Education Library VOLUME 18
Managing Editor A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board H. Bauersfeld, Bielefeld, Germany J. Kilpatrick, Athens, U.S.A. C. Laborde, Grenoble, France G. Leder, Melbourne, Australia S. Tumau, Krakow. Poland
The titles published in this series are listed at the end a/this volume.
APPROACHES TO ALGEBRA Perspectives for Research and Teaching Edited by NADINE BEDNARZ, CAROLYN KIERAN
and
LESLEY LEE Departement de Mathimatiques. Universiti du Quebec a Montreal
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-0-7923-4168-0 e-ISBN: 978-94-009-1732-3 DOl: 10.1007/978-94-009-1732-3
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk: and MfP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved © 1996 Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS ACKNOWLEOOMENTS ......................................................................... xv
INTRODUCTION CHAP1ER 1.
APPROACHES TO ALGEBRA: RESEARCH AND TEACHING
PERSPECTIVES FOR
Nadine Bednarz, Carolyn Kieran, Lesley Lee ........................ 3
1. 2. 3. 4. 5.
Historical perspectives in the development of algebra A generalization perspective on the introduction of algebra A problem-solving perspective on the introduction of algebra A modeling perspective on the introduction of algebra A functional perspective on the introduction of algebra
PART I.
HISTORICAL PERSPECTIVES DEVELOPMENT OF ALGEBRA
CHAPTER 2.
FROM EUCLID TO DESCARTES: ALGEBRA AND ITS RELATION TO GEOMETRY Louis Charbonneau ....................................................... 15
1. 2.
3.
4. 5. 6.
IN
THE
Introduction Algebraic reasoning in Greek geometry 2.1. Equality in Greek geometry 2.1.1. Equality of magnitude 2.1.2. Equality of ratios: the Eudoxian theory of proportions 2.1.3. Apollonius and the Conics 2.1.4. Conclusion 2.2. Analysis (as opposed to synthesis) 2.3. Summary Geometric proofs of algebraic rules 3.1. The geometric puzzle of AI-Kbwarizmi (x 2 + lOx =39) 3.2. Cardano's resolution of x2 = ax + cxy 3.3. Cardano's resolution of x3 + 6x =20 3.4. Conclusion Algebraic solution of geometric problems 4.1. Chuquet 4.2. Regiomontanus Toward a homogeneous algebra 5.1. Viete and the analytic art
