Knowing and Teaching Elementary Mathematics : Teachers' Understanding of Fundamental Mathematics in China and the United States

The 20th 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 20th 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 The 20th anniversary edition of this groundbreaking and bestselling volume offers powerful examples of the mathematics that can develop the thinking of elementary school children. Cover 1 Half Title 4 Series Page 5 Title Page 8 Copyright Page 9 Dedication 10 Table of Contents 12 A Note about the 20th Anniversary Edition 15 Author’s Preface to the 20th Anniversary Edition 16 Series Editor’s Introduction to the 20th Anniversary Edition: What Makes a Classic? 21 Author’s Preface to the 2010 Edition 23 Series Editor’s Introduction to the 2010 Edition 28 Foreword 30 Acknowledgments 34 Introduction 38 1. Subtraction With Regrouping: Approaches To Teaching A Topic 46 The U.S. Teachers’ Approach: Borrowing Versus Regrouping 47 The Chinese Teachers’ Approach: “Decomposing a Higher Value Unit” 51 Discussion 65 Summary 70 2. Multidigit Number Multiplication: Dealing With Students’ Mistakes 72 The U.S. Teachers’ Approach: Lining Up Versus Separating Into Three Problems 73 The Chinese Teachers’ Approach: Elaborating the Concept of Place Value 82 Discussion 95 Summary 98 3. Generating Representations: Division By Fractions 99 The U.S. Teachers’ Performance on Calculation 99 The Chinese Teachers’ Performance on Calculation 102 The U.S. Teachers’ Representations of Division by Fractions 108 The Chinese Teachers’ Approach to the Meaning of Division by Fractions 115 Discussion 124 Summary 126 4. Exploring New Knowledge: The Relationship Between Perimeter And Area 128 How the U.S. Teachers Explored the New Idea 129 How the Chinese Teachers Explored the New Idea 134 Discussion 147 Summary 150 5. Teachers’ Subject Matter Knowledge: Profound Understanding Of Fundamental Mathematics 151 A Cross-Topic Picture of the Chinese Teachers’ Knowledge: What Is Its Mathematical Substance? 151 Knowledge Packages and Their Key Pieces: Understanding Longitudinal Coherence in Learning 156 Elementary Mathematics as Fundamental Mathematics 158 Profound Understanding of Fundamental Mathematics 160 Summary 165 6. Profound Understanding Of Fundamental Mathematics: When And How Is It Attained? 168 When Is Profound Understanding of Fundamental Mathematics Attained?: What the Preteaching Groups Knew About the Four Topics 168 Profound Understanding of Fundamental Mathematics: How It Is Attained 172 Summary 184 7. Conclusion 187 Address Teacher Knowledge and Student Learning at the Same Time 189 Enhance the Interaction Between Teachers’ Study of School Mathematics and How to Teach It 190 Refocus Teacher Preparation 192 Understand the Role That Curricular Materials, Including Textbooks, Might Play in Reform 192 Understand the Key to Reform: Whatever the Form of Classroom Interactions Might Be, They Must Focus on Substantive Mathematics 193 Appendix 197 References 199 Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses 205 Response to “Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses” 229 New to the 20th Anniversary Edition: A Critique of the Structure of U.S. Elementary School Mathematics 231 Index 261 The 20th anniversary edition of this groundbreaking and bestselling volume offers powerful examples for grasping the mathematical content necessary to understand and develop the thinking of school children.