Educational Algebra A Theoretical and Empirical Approach
Given its abstract nature and the highly syntactical competence required by the use of symbolic algebra, research on its teaching and learning must rely on approaches that include semiotic concepts and analyses that recall the history of algebraic ideas,
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Mathematics Education Library VOLUME 43 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board J.P. Becker, Illinois, U.S.A. C. Keitel, Berlin, Germany F. Leung, Hong Kong, China G. Leder, Melbourne, Australia D. Pimm, Edmonton, Canada A. Sfard, Haifa, Israel O. Skovsmose, Aalborg, Denmark
The titles published in this series are listed at the end of this volume.
Eugenio Filloy Luis Puig Teresa Rojano (Authors)
Educational Algebra A Theoretical and Empirical Approach
Eugenio Filloy Centre for Research and Advanced Studies Mexico City Mexico [email protected]
Luis Puig University of Valencia Valencia Spain [email protected]
Teresa Rojano Centre for Research and Advanced Studies Mexico City Mexico [email protected] Series Editor: Alan Bishop Monash University Melboure 3800 Australia [email protected]
Library of Congress Control Number: 2007930472 ISBN -13: 978-0-387-71253-6
e-ISBN-13: 978-0-387-71254-3
Printed on acid-free paper. © 2008 Springer Science+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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CONTENTS
Chapter 1 INTRODUCTION OVERVIEW 1. On the local character of the theoretical formulation and its components 2. Mathematical Sign Systems 2.1. Sign 2.2. Mathematical Sign Systems 3. Different answers to same questions? 4. The transition from arithmetic to algebra. Pre-algebra. Some observations about cognition 4.1. The role of historical analysis 4.2. Mathematics lessons at the beginning of secondary education 4.2.1. The reverse of multiplication syndrome 4.2.2. Different uses of the notion of equality. Polysemy of x 4.2.3. Difficulties in translations 4.3. Algebraic and natural language translations 5. Algebra as a language: Approaches from linguistic, semiotic, and historical perspectives SUMMARY Chapter 2 CURRICULUM DESIGN AND DEVELOPMENT FOR STUDENS, TEACHERS AND RESEARCHERS OVERVIEW 1. Introduction 1.1. Partial and eclectic points of view 2. Local theoretical models 2.1. Four characteristics of LTMs 2.2. Semantics and pragmatics 2.3. The components of LTMs 2.4. Local versus general, the reason for the local in our theoretical models 2.5. The component of formal competence 3. A general framework for curriculum development for the study of an LTM 3.1. Concepts 3.2. The relation with reality. Teaching mathematization 3.3. Practical
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