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The learning and teaching of algebra: ideas, insights, and activities
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The Learning and Teaching of Algebra provides a pedagogical framework for the
teaching and learning of algebra grounded in theory and research.
Areas covered include:
• Algebra: Setting the Scene
• Some Lessons From History
• Seeing Algebra Through the Eyes of a Learner
• Emphases in Algebra Teaching
• Algebra Education in the Digital Era
This guide will be essential reading for trainee and quali ed teachers of mathematics,
graduate students, curriculum developers, researchers and all those who are
interested in the “problématique” of teaching and learning algebra. It allows you
to get involved in the wealth of knowledge that teachers can draw upon to assist
learners, helping you gain the insights that mastering algebra provides.
Abraham Arcavi holds the Lester B. Pearson Professorial Chair at the Weizmann
Institute of Science, Israel. He has written about the teaching and learning of
algebra for researchers and teachers, led large curriculum development projects,
and has been involved in teacher professional development for more than 30 years.
Paul Drijvers is Professor of Mathematics Education at the Freudenthal Institute,
Utrecht University, The Netherlands. His research interests include the role of
ICT in mathematics education, the teaching and learning of algebra, and teachers’
professional development.
Kaye Stacey is Professor Emeritus at the University of Melbourne, Australia,
having held the Foundation Chair of Mathematics Education there for 20 years.
She has made major contributions to understanding students’ early learning of
formal algebra and discovering how information technology can enhance the
teaching of algebra and functions throughout secondary school.
THE LEARNING AND
TEACHING OF ALGEBRA
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IMPACT (Interweaving Mathematics Pedagogy
and Content for Teaching)
IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an
exciting new series of texts for teacher education which aims to advance the
learning and teaching of mathematics by integrating mathematics content with the
broader research and theoretical base of mathematics education.
The Learning and Teaching of Algebra
Ideas, Insights, and Activities
Abraham Arcavi, Paul Drijvers, and Kaye Stacey
www.Ebook777.com
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THE LEARNING AND
TEACHING OF ALGEBRA
Ideas, Insights, and Activities
Abraham Arcavi, Paul Drijvers,
and Kaye Stacey
www.Ebook777.com
Free ebooks ==> www.Ebook777.com
First published 2017
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
711 Third Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2017 Abraham Arcavi, Paul Drijvers, and Kaye Stacey
The right of Abraham Arcavi, Paul Drijvers, and Kaye Stacey to be
identi ed as authors of this work has been asserted by them in accordance
with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilized in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing
from the publishers.
Trademark notice: Product or corporate names may be trademarks or
registered trademarks, and are used only for identi cation and explanation
without intent to infringe.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Names: Arcavi, Abraham. | Drijvers, Paul (Paulus Hendrikus Maria),
1958- | Stacey, Kaye, 1948-
Title: The learning and teaching of algebra : ideas, insights. and activities /
Abraham Arcavi, Paul Drijvers, and Kaye Stacey.
Description: Abingdon, Oxon ; New York, NY : Routledge, 2017. |
Includes bibliographical references.
Identi ers: LCCN 2016005953| ISBN 9780415743693 (hardback) |
ISBN 9780415743723 (pbk.) | ISBN 9781315545189 (ebook)
Subjects: LCSH: Algebra--Study and teaching.
Classi cation: LCC QA152.3 .A27 2017 | DDC 512.9071--dc23
LC record available at https://lccn.loc.gov/2016005953
ISBN: 978-0-415-74369-3 (hbk)
ISBN: 978-0-415-74372-3 (pbk)
ISBN: 978-1-315-54518-9 (ebk)
Typeset in Bembo and Stone Sans
by Saxon Graphics Ltd, Derby
To students, teachers, and teacher educators, hoping that
this book will contribute to making the learning of algebra
productive, enjoyable, and accessible to all.
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CONTENTS
Acknowledgments ix
IMPACT – Series Foreword xi
Preface xiii
1 Algebra—Setting the Scene 1
1.1 Introduction1
1.2 Algebra—Aims, Actions, and Entities 1
1.3 Why Algebra?16
1.4 Chapter Summary19
1.5 Thinking Further20
1.6 References22
2 Some Lessons From History 25
2.1 Introduction25
2.2 Linear Equations in Ancient Egypt 26
2.3 Quadratic Equations in Ancient Babylonia 31
2.4 A Geometric View of Algebra From Arabic Mathematics 33
2.5 Beyond Solving Equations: The Emergence of Algebra in Europe 37
2.6 Chapter Summary41
2.7 Thinking Further42
2.8 References47
3 Seeing Algebra Through the Eyes of a Learner 48
3.1 Introduction—Putting on Teachers’ Bifocal Spectacles 48
3.2 What Do Algebraic Letters Represent? 50
3.3 The Process–Object Duality 53
viii Contents
3.4 The Meaning of the Equals Sign 55
3.5 Algebra for Recording and Revealing Mathematical Structure 56
3.6 Transitions From Learning Arithmetic to Learning Algebra 58
3.7 The Procedures of Equation Solving 64
3.8 Functions as Processes and Objects 69
3.9 Chapter Summary72
3.10 Thinking Further73
3.11 References77
4 Emphases in Algebra Teaching 80
4.1 Introduction80
4.2 Teaching Algebra in Context 81
4.3 Productive Practice 87
4.4 The Reconciliation of Routine and Insight 90
4.5 Exploiting Student Mistakes 95
4.6 Proofs in Algebra Teaching 99
4.7 Chapter Summary101
4.8 Thinking Further102
4.9 References104
5 Algebra Education in the Digital Era 106
5.1 Introduction106
5.2 Digital Tools for Algebra 108
5.3 Core Algebra Entities With Digital Means 118
5.4 Teaching and Learning Algebra With Digital Means 127
5.5 Chapter Summary130
5.6 Thinking Further132
5.7 References134
Epilogue 136
Index 140
We acknowledge with gratitude the support of the Weizmann Institute of Science,
the Freudenthal Institute at Utrecht University, and the University of Melbourne.
We thank Nathalie Kuijpers for her careful and thorough assistance with editing of
the manuscript. We were fortunate that the IMPACT series editors, Tommy
Dreyfus, Frank K. Lester, and Günter Törner, invited us to work together on this
project, thereby establishing for us an enjoyable and instructive international
collaboration and exchange of ideas. The nal manuscript bene tted from the
insightful comments of two reviewers.
ACKNOWLEDGMENTS
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IMPACT, an acronym for Interweaving Mathematics Pedagogy And Content for
Teaching, is a series of textbooks dedicated to mathematics education and suitable
for teacher education. The leading principle of the series is the integration of
mathematics content with topics from research on mathematics learning and
teaching. Elements from the history and the philosophy of mathematics, as well as
curricular issues are integrated as appropriate.
Whereas in mathematics there are many textbook series representing
internationally accepted canonical curricula, such a series has so far been lacking in
mathematics education. It is the intention of IMPACT, to ll this gap.
The books in the series will focus on fundamental conceptual understanding of
the central ideas and relationships, while often compromising on the breadth of
coverage. These central ideas and relationships will serve as organizers for the
structure of each book. Beyond being an integrated presentation of the central
ideas of mathematics and their learning and teaching, the volumes will serve as
guides to further resources.
The rst volume in the series treats Algebra, a central topic in any high school
mathematics curriculum around the world, and a topic with inherent complexities
due to factors such as the increasing numbers of students who are expected to learn
algebra, and to opportunities for new ways of doing algebra provided by
technological change. Hence a coherent view of the central ideas and relationships
that integrates algebra content with the main issues and results from research
appeared particularly crucial, leading to the choice of Algebra as topic of the rst
volume in the series.
Series editors
Tommy Dreyfus (Israel), Frank K. Lester (USA),
Günter Törner (Germany)
IMPACT – SERIES FOREWORD
xii IMPACT – Series Foreword
Series Advisory Board
Abraham Arcavi (Israel), Michèle Artigue (France), Jo Boaler (USA), Hugh
Burkhardt (Great Britain), Willi Dör er (Austria), Koeno Gravemeijer (The
Netherlands), Angel Gutierrez (Spain), Gabriele Kaiser (Germany), Carolyn Kieran
(Canada), Jeremy Kilpatrick (USA), Jürg Kramer (Germany), Fou-Lai Lin
(Republic of China - Taiwan), John Monaghan (Great Britain/Norway),
Mogens Niss (Denmark), Alan H. Schoenfeld (USA), Peter Sullivan (Australia),
Michael O. J. Thomas (New Zealand) and Patrick W. Thompson (USA)
This book addresses the problématique of the teaching and learning of school
algebra.
“Problématique” is a word borrowed from the French and it describes not just
problems in isolation, but comprehensive challenges posed by a certain large-scale
theme. In other words, it consists of articulating questions and dilemmas arising in
a certain area. The problématique addressed by this book can be posed in general
terms as follows. On the one hand, algebra is considered a central subject to be
studied in junior high and high schools in almost all educational systems around the
world. The reasons for this centrality may vary considerably (see Section 1.3) but
the consensus is that time and e ort should be devoted to its teaching and learning.
On the other hand, students have serious cognitive and a ective di culties with
algebra, that is, they have di culties in becoming competent at it and even if they
succeed, many fail to see the point of studying it.
This book addresses the problématique of teaching and learning school algebra
on the basis of knowledge accumulated by decades of research and development
work by mathematicians, mathematics educators and mathematics education
researchers during the last ve decades. That work will be succinctly evoked as a
springboard to propose additional ideas and suggestions illustrated by thoughtprovoking annotated examples.
This book is intended as a useful and lively resource for mathematics teachers,
teacher educators, student teachers (including those in career shifting programs),
curriculum developers, and graduate students, and it can also be of interest to
parents and students.
Chapter 1 describes the problématique and the essence of algebra: its aims,
actions, and entities and dwells upon the question of why all students should learn
algebra.
PREFACE
xiv Preface
Chapter 2 provides brief historical snapshots describing how the main ideas and
entities of school algebra came into being, and how algebraic symbolism (which is
relatively recent as compared to Euclidean geometry) facilitated the solution of
problems and accelerated the development of all areas of mathematics. Some
lessons for mathematics education are discussed.
Chapter 3 centers on the learning of algebra by bringing in the students’
perspective, their readiness (or lack of it) to cope with the main ideas of algebra,
their di culties, and sense making of algebraic problems.
Chapter 4 addresses teaching dilemmas and concerns such as: how to teach
algebra in context, how to practice, how to reconcile procedural uency and
algebraic insight, how to tackle well-known student di culties, and how to deal
with proofs in algebra.
Chapter 5 addresses the a ordances, opportunities, and challenges posed by new
technologies and their roles in the wise use of the unprecedented computational
power, in re-prioritizing goals and in re-sequencing the curriculum. Di erent
approaches to how these technologies can be harnessed are described.
The Epilogue presents conclusions and recommendations.
Throughout the book, annotated examples are presented to illustrate the main
claims made. A section entitled “Thinking Further” is included at the end of each
chapter. It includes suggestions for discussions and activities in pre-service or inservice teacher education workshops, forums, and teacher deliberations in
departmental or professional meetings. This section also includes some mathematical
tasks related to the contents of the chapter in order to enhance and extend the
points made therein.