Table Of ContentKnowing and Teaching
Elementary Mathematics
Anniversary Edition
“Both sides in the math wars claim Dr. Ma as their own. This book’s broad appeal offers some
hope for common ground in math education…. We will continue fights over whether children
should be taught arithmetic rules or theory. What Dr. Ma shows is that we need both.”
—New York Times
“Liping Ma’s work has given me hope about what can be done to improve mathematics
education.”
—Richard Askey, Professor of Mathematics, University of Wisconsin-Madison
“A stealth hit for math junkies on both sides of the ‘math wars’, and a must read for anyone
interested in solving the problems of public schools.”
—Christian Science Monitor
“This book supports the need for, and indeed the educational benefits of, changing profes-
sional teaching conditions for U.S. teachers.… It provides some food for thought for every-
one involved in improving mathematics education. And it supports the necessity that even
at the elementary school level, students can, and should, learn challenging mathematics.”
—National Council of Teachers of Mathematics News Bulletin
Studies of teachers in the United States often document insufficient subject matter knowl-
edge in mathematics. Yet, these studies give few examples of the knowledge teachers need
to support teaching, particularly the kind of teaching demanded by recent reforms in math-
ematics education. Knowing and Teaching Elementary Mathematics describes the nature
and development of the knowledge that elementary teachers need to become accomplished
mathematics teachers, and suggests why such knowledge seems more common in China
than in the United States, despite the fact that Chinese teachers have less formal education
than their U.S. counterparts.
The anniversary edition of this bestselling volume includes the original studies that
compare U.S. and Chinese elementary school teachers’ mathematical understanding and
offers a powerful framework for grasping the mathematical content necessary to under-
stand and develop the thinking of school children. Highlighting notable changes in the field
and the author’s work, this new edition includes an updated preface, introduction, and key
journal articles that frame and contextualize this seminal work.
Liping Ma earned a Ph.D. from Stanford University, following a masters degree in educa-
tion from East China Normal University. After a term as a senior scientist at the Carnegie
Foundation for the Advancement of Teaching, she is now an independent scholar.
STUDIES IN MATHEMATICAL
THINKING AND LEARNING
Alan H.Schoenfeld, Series Editor
Artzt/Armour-Thomas/Curcio • Becoming a Reflective Mathematics Teacher: A Guide
for Observation and Self-Assessment, Second Edition
Baroody/Dowker (Eds.) • The Development of Arithmetic Concepts and Skills: C onstructing
Adaptive Expertise
Boaler • Experiencing School Mathematics: Traditional and Reform Approaches to
T eaching and Their Impact on Student Learning
Carpenter/Fennema/Romberg (Eds.) • Rational Numbers: An Integration of Research
Chazan/Callis/Lehman (Eds.) • Embracing Reason: Egalitarian Ideals and the Teaching
of High School Mathematics
Cobb/Bauersfeld (Eds.) • The Emergence of Mathematical Meaning: Interaction in
C lassroom 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: A Japanese Approach to Improving Mathematics
Teaching and Learning
Greer/Mukhopadhyay/Powell/Nelson-Barber (Eds.) • Culturally Responsive
M athematics Education
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
F undamental Mathematics in China and the United States, Anniversary Edition
Martin • Mathematics Success and Failure Among African-American Youth: The Roles of
Sociohistorical Context, Community Forces, School Influence, and Individual Agency
Martin (Ed.) • Mathematics Teaching, Learning, and Liberation in the Lives of Black
Children
Petit/Laird/Marsden • A Focus on Fractions: Bringing Research to the Classroom
Reed • Word Problems: Research and Curriculum Reform
Romberg/Fennema/Carpenter (Eds.) • Integrating Research on the Graphical
R epresentation of Functions
Romberg/Carpenter/Dremock (Eds.) • Understanding Mathematics and Science
M atters
Romberg/Shafer • The Impact of Reform Instruction on Mathematics Achievement: An
Example of a Summative Evaluation of a Standards-Based Curriculum
Sarama/Clements • Early Childhood Mathematics Education Research: Learning
T rajectories for Young Children
Schliemann/Carraher/Brizuela (Eds.) • Bringing Out the Algebraic Character of
A rithmetic: 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?
Solomon • Mathematical Literacy: Developing Identities of Inclusion
Sophian • The Origins of Mathematical Knowledge in Childhood
Sternberg/Ben-Zeev (Eds.) • The Nature of Mathematical Thinking
Stylianou/Blanton/Knuth (Eds.) • Teaching and Learning Proof Across the Grades: A
K-16 Perspective
Sultan/Artzt • The Mathematics That Every Secondary Mathematics Teacher Needs to
Know
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
Knowing and Teaching
Elementary Mathematics
Teachers’ Understanding of Fundamental
Mathematics in China and the United States
Anniversary Edition
Liping Ma
NEW YORK AND LONDON
First edition published 1999
by Lawrence Erlbaum Associates, Inc.
This anniversary edition published 2010
by Routledge
270 Madison Avenue, New York, NY 10016
Simultaneously published in the UK
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
Routledge is an imprint of the Taylor & Francis Group, an informa business
This edition published in the Taylor & Francis e-Library, 2010.
To purchase your own copy of this or any of
Taylor & Francis or Routledge’s collection of thousands of eBooks
please go to www.eBookstore.tandf.co.uk.
© 1999 Lawrence Erlbaum Associates, Inc.
© 2010 Taylor & Francis
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 identification and
explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Ma, Liping.
Knowing and teaching elementary mathematics: teachers’ understanding of
fundamental mathematics in China and the United States/Liping Ma.
p. cm—(Studies in mathematical thinking and learning)
Includes bibliographical references and index.
1. Mathematics—Study and teaching (Elementary)—United States.
2. Mathematics—Study and teaching (Elementary)—China. 3. Comparative
education. I. Title
QA135.5.M22 2010
372.70973–dc22 2009049390
ISBN 0-203-85634-1 Master e-book ISBN
ISBN 10: (hbk) 0-415-87383-5
ISBN 10: (pbk) 0-415-87384-3
ISBN 10: (ebk) 0-203-85634-1
ISBN 13: (hbk) 978-0-415-87383-3
ISBN 13: (pbk) 978-0-415-87384-0
ISBN 13: (ebk) 978-0-203-85634-5
To Jianfeng, Sushu, and John,
and to Cathy Kessel,
with deepest love and appreciation
Contents
Author’s Preface to the Anniversary Edition ix
Series Editor’s Introduction to the Anniversary Edition xiii
A Note about the Anniversary Edition xv
Foreword xvi
Acknowledgments xx
Introduction xxiii
1. Subtraction With Regrouping: Approaches To Teaching A Topic 1
The U.S. Teachers’ Approach: Borrowing Versus Regrouping 2
The Chinese Teachers’ Approach: “Decomposing a Higher Value Unit” 6
Discussion 18
Summary 22
2. Multidigit Number Multiplication: Dealing With Students’ Mistakes 24
The U.S. Teachers’ Approach: Lining Up Versus Separating Into 25
Three Problems
The Chinese Teachers’ Approach: Elaborating the Concept of 33
Place Value
Discussion 44
Summary 46
3. Generating Representations: Division By Fractions 47
The U.S. Teachers’ Performance on Calculation 47
The Chinese Teachers’ Performance on Calculation 49
The U.S. Teachers’ Representations of Division by Fractions 55
The Chinese Teachers’ Approach to the Meaning of Division 61
by Fractions
Discussion 69
Summary 71
4. Exploring New Knowledge: The Relationship Between Perimeter 72
And Area
How the U.S. Teachers Explored the New Idea 73
How the Chinese Teachers Explored the New Idea 77
Discussion 89
Summary 91
viii Contents
5. Teachers’ Subject Matter Knowledge: Profound Understanding Of 92
F undamental Mathematics
A Cross-Topic Picture of the Chinese Teachers’ Knowledge: What Is 92
Its Mathematical Substance?
Knowledge Packages and Their Key Pieces: Understanding 97
L ongitudinal Coherence in Learning
Elementary Mathematics as Fundamental Mathematics 99
Profound Understanding of Fundamental Mathematics 101
Summary 105
6. Profound Understanding Of Fundamental Mathematics: When And 107
How Is It Attained?
When Is Profound Understanding of Fundamental Mathematics 107
Attained?: What the Preteaching Groups Knew About the Four Topics
Profound Understanding of Fundamental Mathematics: How It 110
Is Attained
Summary 121
7. Conclusion 123
Address Teacher Knowledge and Student Learning at the Same Time 124
Enhance the Interaction Between Teachers’ Study of School 125
M athematics and How to Teach It
Refocus Teacher Preparation 127
Understand the Role That Curricular Materials, Including Textbooks, 127
Might Play in Reform
Understand the Key to Reform: Whatever the Form of Classroom 128
Interactions Might Be, They Must Focus on Substantive Mathematics
Appendix 131
References 133
New to the Anniversary Edition: Fang and Paine’s “Bridging P olarities: 138
How Liping Ma’s Knowing and Teaching Mathematics Entered the U.S.
Mathematics and Mathematics Education Discourses”
New to the Anniversary Edition: Ma’s Response to “Bridging P olarities” 158
Author Index 160
Subject Index 164
Author’s Preface to the Anniversary Edition
I must admit that I did not expect this book to be a success. About twelve years ago, when I
decided to devote my effort to turning my doctoral dissertation into a book, my motivation
was very simple. I did not feel good thinking that my dissertation, which had taken so much
of my passion and work, would sleep forever on the shelf in the Stanford School of Educa-
tion Library. My dissertation should not merely be a “test” for me in earning a degree; it
should serve others, in particular, classroom teachers. Indeed, this simple feeling pushed
me to undertake the creation of this book. Now, ten years after its publication in 1999, over
70,000 copies have been sold.
A few years ago, I paid a visit to a beautiful east coast city. On Saturday, in the middle
of my morning walk, I was stopped by a blocked street. The excited people standing on the
sidewalk told me that they were waiting for the annual cross-city footrace to pass by. A few
minutes later, the procession began. Runners—male, female, young, old, smiling, serious,
alone, accompanied by friends and family—covered the wide street in front of me. They
kept on coming. It took about twenty minutes for all the runners to pass the place where
I stood. “There are so many people!,” I exclaimed. “This year we have about 40,000 run-
ners,” someone replied. The number struck me—it was the number of copies of my book
that had been sold. Only when I physically saw 40,000 people, was I aware of its signifi-
cance. I was moved—if the book benefited that many people, it was worthwhile work.
In the fall of 2003, I received a package mailed from Korea—two copies of my book in
Korean were included. Last year, mathematics education scholars in China and Chile began
to translate the book into Chinese and Spanish. I am told that before the end of 2009, the
Chinese translation of the book will have been published.
It is not a surprise that the book has drawn attention from different countries—after all,
it is based on a cross-national comparative study. Moreover, as my doctoral adviser Lee
Shulman pointed out, I myself, as a scholar in education, have been nurtured by Chinese
and American learning environments. Indeed, a cross-national perspective may help us to
see what might be otherwise unnoticed. Now, I invite you to look at two pictures of math-
ematics education in the U.S. and China. Both illustrate aspects of education in the two
countries, but the scenes they illustrate are one hundred years apart.
The first picture is sketched in this book, which was published in 1999. A group of
American teachers were interviewed and asked questions about four topics of elementary
school mathematics. The same questions were asked of a group of Chinese teachers. The
comparison of their responses constitutes the main content of this book.
Compared with their U.S. counterparts, the Chinese interviewees were significantly more
proficient in the procedural aspect of all four topics, particularly in the more advanced top-
ics of division by fractions and area and perimeter. All the Chinese teachers attained the
correct answer when computing , while only about half of the U.S. teachers did. Not
a single Chinese teacher showed any hesitation in using formulas to find the perimeter and
