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## Customer Reviews

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### Summary

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 recent 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.

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 understand 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.

"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**

### Author Biography

### Table of Contents

Author's Preface to the Anniversary Edition | p. xi |

Series Editor's Introduction to the Anniversary Edition | p. xvi |

A Note about the Anniversary Edition | p. xviii |

Foreword | p. xix |

Acknowledgments | p. xxiii |

Introduction | p. xxvi |

Subtraction With Regrouping: Approaches To Teaching A Topic | p. 1 |

The U. S. Teachers' Approach: Borrowing Versus Regrouping | p. 2 |

The Chinese Teachers' Approach: "Decomposing a Higher Value Unit" | p. 7 |

Discussion | p. 21 |

Summary | p. 26 |

Multidigit Number Multiplication: Dealing with Students' Mistakes | p. 28 |

The U.S. Teachers' Approach: Lining Up Versus Separating Into Three Problems | p. 29 |

The Chinese Teachers' Approach: Elaborating the Concept of Place Value | p. 38 |

Discussion | p. 52 |

Summary | p. 54 |

Generating Representations: Division By Fractions | p. 55 |

The U.S. Teachers' Performance on Calculation | p. 56 |

The Chinese Teachers' Performance on Calculation | p. 58 |

The U.S. Teachers' Representations of Division by Fractions | p. 64 |

The Chinese Teachers' Approach to the Meaning of Division by Fractions | p. 72 |

Discussion | p. 80 |

Summary | p. 82 |

Exploring New Knowledge: The Relationship Between Perimeter And Area | p. 84 |

How the U. S. Teachers Explored the New Idea | p. 85 |

How the Chinese Teachers Explored the New Idea | p. 90 |

Discussion | p. 103 |

Summary | p. 106 |

Teachers' Subject Matter Knowledge: Profound Understanding of Fundamental Mathematics | p. 107 |

A Cross-Topic Picture of the Chinese Teachers' Knowledge: What Is Its Mathematical Substance? | p. 108 |

Knowledge Packages and Their Key Pieces: Understanding Longitudinal Coherence in Learning | p. 113 |

Elementary Mathematics as Fundamental Mathematics | p. 116 |

Profound Understanding of Fundamental Mathematics | p. 118 |

Summary | p. 123 |

Profound Understanding Of Fundamental Mathematics: When And How Is It Attained? | p. 125 |

When Is Profound Understanding of Fundamental Mathematics Attained?: What the Preteaching Groups Knew About the Four Topics | p. 126 |

Profound Understanding of Fundamental Mathematics: How It Is Attained | p. 129 |

Summary | p. 142 |

Conclusion | p. 144 |

Address Teacher Knowledge and Student Learning at the Same Time | p. 146 |

Enhance the Interaction Between Teachers' Study of School Mathematics and How to Teach It | p. 147 |

Refocus Teacher Preparation | p. 149 |

Understand the Role That Curricular Materials, Including Textbooks, Might Play in Reform | p. 150 |

Understand the Key to Reform: Whatever the Form of Classroom Interactions Might Be, They Must Focus on Substantive Mathematics | p. 151 |

Appendix | p. 154 |

References | p. 156 |

New to the Anniversary Edition: Fang and Paine's "Bridging Polarities: How Liping Ma's Knowing and Teaching Mathematics Entered the U. S. Mathematics and Mathematics Education Discourses" | p. 161 |

New to the Anniversary Edition: Ma's Response to "Bridging Polarities" | p. 186 |

Author Index | p. 189 |

Subject Index | p. 191 |

Table of Contents provided by Ingram. All Rights Reserved. |