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.

A Note about the 20th Anniversary Edition; Author's Preface to the 20th Anniversary Edition (Liping Ma); Series Editor's Introduction to the 20th Anniversary Edition (Alan Schoenfeld); Foreword (Lee Shulman); Acknowledgments; Introduction; 1. Subtraction with Regrouping: Approaches to Teaching a Topic; 2. Multidigit Number Multiplication: Dealing with Students' Mistakes; 3. Generating Representations: Division by Fractions; 4. Exploring New Knowledge: The Relationship Between Perimeter and Area; 5. Teachers' Subject Matter Knowledge: Profound Understanding of Fundamental Mathematics; 6. Profound Understanding of Fundamental Mathematics: When and How Is it Attained?; 7. Conclusion; Appendix; References; "Bridging Polarities: How Liping Ma's Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses" (Yanping Fang and Lynn Paine, Pedagogies); Response to "Bridging Polarities" (Liping Ma, Pedagogies); New to the 20th Anniversary edition: "A Critique of the Structure of U.S. Elementary School Mathematics" (Liping Ma, Notices of the American Mathematical Society); Index