Table Of ContentALGEBRA
Early Grades
in the
STUDIES IN MATHEMATICALTHINKING AND LEARNING
Alan H. Schoenfeld, Series Editor
Artzt/Armour-Thomas • Becoming a Reflective Mathematics Teacher: AGuide for Observation
and Self-Assessment
Baroody/Dowker (Eds.) • The Development of Arithmetic Concepts and Skills: Constructing
Adaptive Expertise
Boaler • Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and
Their Impact on Student Learning
Carpenter/Fennema/Romberg (Eds.) • Rational Numbers: An Integration of Research
Cobb/Bauersfeld (Eds.) • The Emergence of Mathematical Meaning: Interaction in Classroom
Cultures
Cohen • Teachers’Professional Development and the Elementary Mathematics Classroom: Bringing
Understandings to Light
Clements/Sarama/DiBiase (Eds.) • Engaging Young Children in Mathematics: Standards for
Early Childhood Mathematics Education
English (Ed.) • Mathematical and Analogical Reasoning of Young Learners
English (Ed.) • Mathematical Reasoning: Analogies, Metaphors, and Images
Fennema/Nelson (Eds.) • Mathematics Teachers in Transition
Fennema/Romberg (Eds.) • Mathematics Classrooms That Promote Understanding
Fernandez/Yoshida • Lesson Study: AJapanese Approach to Improving Mathematics Teaching and
Learning
Kaput/Carraher/Blanton (Eds.) • Algebra in the Early Grades
Lajoie • Reflections on Statistics: Learning, Teaching, and Assessment in Grades K-12
Lehrer/Chazan (Eds.) • Designing Learning Environments for Developing Understanding of
Geometry and Space
Ma • Knowing and Teaching Elementary Mathematics: Teachers’Understanding of Fundamental
Mathematics in China and the United States
Martin • Mathematics Success and Failure Among African-American Youth: The Roles of
Sociohistorical Context, Community Forces, School Influence, and Individual Agency
Reed • Word Problems: Research and Curriculum Reform
Romberg/Fennema/Carpenter (Eds.) • Integrating Research on the Graphical Representation of
Functions
Romberg/Carpenter/Dremock (Eds.) • Understanding Mathematics and Science Matters
Schliemann/Carraher/Brizuela (Eds.) • Bringing Out the Algebraic Character of Arithmetic:
From Children’s Ideas to Classroom Practice
Schoenfeld (Ed.) • Mathematical Thinking and Problem Solving
Senk/Thompson (Eds.) • Standards-Based School Mathematics Curricula: What Are They? What
Do Students Learn?
Sophian • The Origins of Mathematical Knowledge in Childhood
Sternberg/Ben-Zeev (Eds.)• The Nature of Mathematical Thinking
Watson • Statistical Literacy at School: Growth and Goals
Watson/Mason • Mathematics as a Constructive Activity: Learners Generating Examples
Wilcox/Lanier (Eds.) • Using Assessment to Reshape Mathematics Teaching: A Casebook for
Teachers and Teacher Educators, Curriculum and Staff Development Specialists
Wood/Nelson/Warfield (Eds.) • Beyond Classical Pedagogy: Teaching Elementary School
Mathematics
Zaskis/Campbell (Eds.) • Number Theory in Mathematics Education: Perspectives and Prospects
For additional information on titles in the Studies in Mathematical Thinking and Learning
series visit www.routledge.com
ALGEBRA
Early Grades
in the
Edited by
James J. Kaput s David W. Carraher s Maria L. Blanton
Lawrence Erlbaum Associates Lawrence Erlbaum Associates
Taylor & Francis Group Taylor & Francis Group
270 Madison Avenue 2 Park Square
New York, NY 10016 Milton Park, Abingdon
Oxon OX14 4RN
© 2008 by Taylor & Francis Group, LLC
Lawrence Erlbaum Associates is an imprint of Taylor & Francis Group, an Informa business
Transferred to Digital Printing 2008
International Standard Book Number-13: 978-0-8058-5473-2 (Softcover) 978-0-8058-5472-5 (Hardcover)
No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic,
mechanical, or other means, now known or hereafter invented, including photocopying, microfilming,
and recording, or in any information storage or retrieval system, without written permission from the
publishers.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
Dedication
It is a bittersweet experience for us to put the finishing touches on this
manuscript without our dear friend and colleague, Jim Kaput. Jim origi-
nated the volume and labored long and hard to turn the promise of early
algebra into reality. He found it rewarding to play a pivotal role in a book
that reflects the depth of children’s thinking while recognizing the chal-
lenges for educators and the importance of grounding the work in math-
ematical foundations.
We offer this volume as evidence of his vision and leadership.
—David Carraher and Maria Blanton
Contents
Preface ix
Acknowledgments xiii
About the Contributors xv
Skeptic’s Guide to Algebra in the Early Grades xvii
PART I: The Nature of Early Algebra 1
1 What Is Algebra? What Is Algebraic Reasoning? 5
James J. Kaput
2 Algebra From a Symbolization Point of View 19
James J. Kaput, Maria L. Blanton, and Luis Moreno
3 Making Use of Children’s Powers to Produce
Algebraic Thinking 57
John Mason
4 Quantitative Reasoning and the Development
of Algebraic Reasoning 95
John P. (Jack) Smith III and Patrick W. Thompson
5 Representational Thinking as a Framework for
Introducing Functions in the Elementary Curriculum 133
Erick Smith
PART II: Students’Capacity for Algebraic Thinking 161
6 Classroom Stories: Examples of Elementary
Students Engaged in Early Algebra 165
Virginia Bastable and Deborah Schifter
7 Children’s Reasoning About Change Over Time 185
Cornelia Tierney and Stephen Monk
8 What Is a Legitimate Arithmetic Number Sentence?
The Case of Kindergarten and First-Grade Children 201
Nitza Mark-Zigdon and Dina Tirosh
vii
viii CONTENTS
9 Visualizing Algebraic Reasoning 211
Timothy Boester and Richard Lehrer
10 Early Algebra Is Not the Same as Algebra Early 235
David W. Carraher, Analúcia D. Schliemann, and Judah
L. Schwartz
11 Multiple Notational Systems and Algebraic 273
Understandings: The Case of the “Best Deal” Problem
Bárbara M. Brizuela and Darrell Earnest
12 Signed Numbers and Algebraic Thinking 303
Irit Peled and David W. Carraher
PART III: Issues of Implementation: Taking
Early Algebra into the Classroom 329
13 Content Matters: Algebraic Reasoning
in Teacher Professional Development 333
Megan Loef Franke, Thomas P. Carpenter, and Dan Battey
14 Building District Capacity for Teacher Development in
Algebraic Reasoning 361
Maria L. Blanton and James J. Kaput
15 Measure Up: AQuantitative View of Early Algebra 389
Barbara Dougherty
16 Early Algebra: What Does Understanding the Laws
of Arithmetic Mean in the Elementary Grades? 413
Deborah Schifter, Stephen Monk, Susan Jo Russell,
and Virginia Bastable
17 Early Algebra: The Math Workshop Perspective 449
E. Paul Goldenberg and Nina Shteingold
Afterword
18 Early Algebra as Mathematical Sense Making 479
Alan H. Schoenfeld
Author Index 511
Subject Index 521
Preface
This book is the result of approximately 15 years of thinking, research,
and collective discussion about the role of algebra in school mathematics.
It originated in the context of the Algebra Working Group (AWG), an
early online discussion group led by the first editor and supported by the
National Center for Research in Mathematical Sciences Education
(NCRMSE) based at the University of Wisconsin Center for Education
Research. For 4 years, this group—which included approximately 40
researchers and leaders in mathematics education, some of whom were
members of the National Council of Teachers of Mathematics Standards
development committees—held vigorous discussions about the nature
and role of algebra in school mathematics. The question “What is alge-
braic reasoning?” was raised regularly, but it was never fully answered.
Nonetheless, many of the issues explored eventually became the basis of
research reported in the present volume, including our characterizations
of algebraic reasoning.
As the AWG continued to hold small conferences, produce discussion
documents, and sponsor symposia and papers at research conferences to
further the discussions of algebra learning, it became clear to an AWG
subgroup that a long-term solution to the challenges of school algebra
required reconceptions of school mathematics in the early grades. This, in
turn, would require a major research and development initiative. In the
early to mid-1990s, a smaller group known as the Early Algebra Research
Group (EARG) began to meet to focus on research issues concerning alge-
braic reasoning in the elementary grades. The National Center for Student
Learning and Achievement (NCISLA), a successor to NCRMSE, sup-
ported EARG for the next several years as it began to investigate these
issues and share the results with the math education community. Some of
these appear, in revised form, as chapters in this book.
Indeed, the present volume contains echoes of previous conversations
from the algebra working groups, and we have tried to recognize here the
origins of ideas and findings in prior work as much as possible. Even so,
researchers today are still defining and mapping the space of possibilities
for developing algebraic reasoning in the elementary grades. Replication
and data refinement represent ongoing and future work. Because the
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