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Knowing and Teaching Elementary Mathematics

Teachers' Understanding of Fundamental Mathematics in China and the United States

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## Book Description

The 20^{th }anniversary edition of this groundbreaking and bestselling volume offers powerful examples of the mathematics that can develop the thinking of elementary school children.

Studies of teachers in the U.S. often document insufficient subject matter knowledge in mathematics. Yet, these studies give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by reforms in mathematics 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.

Along with the original studies of U.S. and Chinese teachers’ mathematical understanding, this 20^{th }anniversary edition includes a new preface and a 2013 journal article by Ma, "A Critique of the Structure of U.S. Elementary School Mathematics" that describe differences in U.S. and Chinese elementary mathematics. These are augmented by a new series editor’s introduction and two key journal articles that frame and contextualize this seminal work.

## Table of Contents

A Note about the 20^{th} Anniversary Edition

Author's Preface to the 20^{th} Anniversary Edition

Series Editor's Introduction to the 20th Anniversary Edition: What Makes a Classic?

Author’s Preface to the 2010 Edition

Series Editor’s Introduction to the 2010 Edition

Foreword

Acknowledgments

Introduction

1.

Subtraction With Regrouping: Approaches To Teaching A Topic

The U.S. Teachers' Approach: Borrowing Versus Regrouping

The Chinese Teachers' Approach: "Decomposing a Higher Value Unit"

Discussion

Summary

2.

Multidigit Number Multiplication: Dealing With Students' Mistakes

The U.S. Teachers' Approach: Lining Up Versus Separating Into Three Problems

The Chinese Teachers' Approach: Elaborating the Concept of Place Value

Discussion

Summary

3.

Generating Representations: Division By Fractions

The U.S. Teachers' Performance on Calculation

The Chinese Teachers' Performance on Calculation

The U.S. Teachers' Representations of Division by Fractions

The Chinese Teachers' Approach to the Meaning of Division by Fractions

Discussion

Summary

4.

Exploring New Knowledge: The Relationship Between Perimeter And Area

How the U.S. Teachers Explored the New Idea

How the Chinese Teachers Explored the New Idea

Discussion

Summary

5.

Teachers' Subject Matter Knowledge: Profound Understanding Of Fundamental Mathematics

A Cross-Topic Picture of the Chinese Teachers' Knowledge: What Is Its Mathematical Substance?

Knowledge Packages and Their Key Pieces: Understanding Longitudinal Coherence in Learning

Elementary Mathematics as Fundamental Mathematics

Profound Understanding of Fundamental Mathematics

Summary

6.

Profound Understanding Of Fundamental Mathematics: When And How Is It Attained?

When Is Profound Understanding of Fundamental Mathematics Attained?: What the Preteaching Groups Knew About the Four Topics

Profound Understanding of Fundamental Mathematics: How It Is Attained

Summary

7.

Conclusion

Address Teacher Knowledge and Student Learning at the Same Time 146

Enhance the Interaction Between Teachers' Study of School Mathematics and How to Teach It

Refocus Teacher Preparation

Understand the Role That Curricular Materials, Including Textbooks, Might Play in Reform

Understand the Key to Reform: Whatever the Form of Classroom Interactions Might Be, They Must Focus on Substantive Mathematics

Appendix

References

Fang and Paine's "Bridging Polarities: How Liping Ma's Knowing and Teaching Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses"

Liping Ma's Response to "Bridging Polarities"

New to the 20^{th} Anniversary edition: A Critique of the Structure of U.S. Elementary School Mathematics

Index

## Author(s)

### Biography

**Liping Ma** earned a Ph.D. from Stanford University, following a masters degree in education from East China Normal University. After a term as a senior scholar at the Carnegie Foundation for the Advancement of Teaching, she is now an independent scholar. She served as a member of the National Mathematics Advisory Panel from 2006 to 2008.

## Reviews

"Both sides in the math wars claim Dr. Ma as their own. Districts have distributed her book to teachers. Its 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."

—The New York Times

"Ma's book is a significant contribution to mathematics education because it begins to tackle the important and complex question of 'what is mathematical knowledge for elementary teaching'. In doing so, she helps us to understand elementary mathematics as a complex and demanding subject that is to be taken seriously."—Contemporary Psychology

"Elementary school teachers need as deep an understanding of the mathematics they teach as high school teachers need of what they teach. Both need a deep knowledge of the mathematics which comes in later grades, at least three or four, for this knowledge should influence how topics are taught."—Mathematicians and Educational Reform

"Must reading for those who call for more mathematics and those who champion reform pedagogy in teacher education."—CHOICE

"…Ma has done a masterful job of showing how the conceptual approach of Chinese elementary school teachers succeeds where the procedural approach of their American counterparts flounders….I highly recommend this brief volume to elementary school teachers who wish to improve their teaching of mathematics. I also recommend it to all university teacher educators who want their students to develop that 'profound understanding of fundamental mathematics' that allows Chinese students to outscore their American counterparts in international assessments."—Mathematics Teaching in the Middle Schools

"The contributions of this book are multifaceted….This book is an excellent resource and will interest anyone involved in teaching preservice teachers, as well as researchers concerned with teachers' knowledge of content and methods."—Teaching Children Mathematics

"Even beyond education, the book supports the need for, and indeed the educational benefits of, changing professional teaching conditions for U.S. teachers….it provides some food for thought for everyone involved in improving mathematics education. And it supports the necessity, highlighted in NCTM's Standards documents, that even at the elementary school level, students can, and should, learn challenging mathematics."—NCTM News Bulletin

"For all who are concerned with mathematics education…Knowing and Teaching Elementary Mathematics is an important book. For those who are skeptical that mathematics education research can say much of value, it can serve as a counterexample. For those interested in improving precollege mathematics education in the U.S., it provides important clues to the nature of the problem. An added bonus is that, despite the somewhat forbidding educationese of its title, the book is quite readable….I recommend this book!"—Notices of the AMS

"Ma's work has been well received on both sides of the so-called math wars….Supporters of the National Council of Teachers of Mathematics (NCTM) reform agenda are pleased by her stress on real understanding as opposed to mere computational competence."—American Scientist

"…a book that is becoming 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."—The Christian Science Monitor Electronic Edition

"The book is earning praise both from some of those who support changes proposed in the NCTM Standards and from some of those who oppose them, and it is sparking discussion. It is also helping to unify some disparate forces in mathematics education on at least a few ideas for continuing positive changes….[the book] provides some food for thought for everyone involved in improving mathematics education."—NCTM News Bulletin

"Mathematical performance of children in countries in the west like the USA and the UK is a constant source of concern when comparisons are made with achievements in countries in the Far East and Eastern Europe. Liping Ma's book provides valuable insights into possible explanations for this disparity--and these have obvious implications for the training of mathematics teachers in any country."—British Journal of Educational Technology"Liping Ma's work has given me hope about what can be done to improve mathematics education."

—Richard Askey, University of Wisconsin-Madison

"This is indeed a valuable, enlightening book. It attests to the talent of its author, and to the Chinese and American learning environments that have nurtured that talent. It attests to the value of welcoming scholars from other nations to study in the United States. I urge all those who are seriously concerned about the quality of mathematics education in the United States to read this book, and to take its lessons seriously."—Lee Shulman, President, Carnegie Foundation for the Advancement of Teaching, From the Foreword

"For all who are concerned with mathematics education in the U.S., Liping Ma has written an important book. It provides valuable clues to the nature of the problem of improving our K-12 mathematics education. An added bonus is that, despite the somewhat foreboding educationese of its title, it is quite readable. I recommend this book."—Roger Howe, Yale University

"Must reading for those who call for more mathematics and those who champion reform pedagogy in teacher education."—Anna O. Graeber, University of Maryland-College Park"…both a graceful introduction (for mathematicians and other neophytes) to an important area of mathematics education and an interesting theoretical work in its own right. I recommend it highly."

—Judith Roitman, University of Kansas