Knowing and Teaching Mathematics describes the nature and development of the "profound understanding of fundamental mathematics" that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

"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"

-- From the Foreword by Lee Shulman, President, Carnegie Foundation for the Advancement of Teaching

"Liping Ma's work has given me hope about what can be done to improve mathematics education. She illustrates what 'profound understanding of fundamental mathematics' is for elementary school teachers and has some useful suggestions about what we can do to help more teachers acquire this knowledge"

-- Richard Askey, John Bascom Professor of Mathematics, Univeristy of Wisconsin-Madison

Above is an extract from Knowing and Teaching Elementary Mathematics

Knowing and Teaching Elementary Mathematics

Author's Preface to the Anniversary Edition

Series Editor's Introduction to the Anniversary Edition

A Note about the Anniversary 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 Is It Attained?

Summary

7. Conclusion

Address Teacher Knowledge and Student Learning at the Same Time

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

Author Index

Subject Index

Samples

Sample 1

Sample 2

Sample 3

Sample 4