Series Editor's Foreword: The Soul of Mathematics, *Alan H. Schoenfeld*

Preface

List of Contributors

Introduction

**Section I: Theoretical Considerations on the Teaching and Learning of Proof**

1. What I Would Like My Students to Already Know About Proof, *Reuben Hersh*

2. Exploring Relationships Between Disciplinary Knowledge and School Mathematics: Implications For Understanding the Place of Reasoning And Proof in School Mathematics, *Daniel Chazan and H. Michael Lueke*

3. Proving and Knowing In Public: The Nature of Proof in A Classroom, *Patricio Herbst and Nicolas Balacheff*

**Section II: Teaching and Learning of Proof in the Elementary Grades**

4. Representation-based Proof in the Elementary Grades, *Deborah Schifter*

5. Representations that Enable Children To Engage in Deductive Argument, *Anne K. Morris*

6. Young Mathematicians At Work: The Role of Contexts And Models in the Emergence of Proof, *Catherine Twomey Fosnot and Bill Jacob*

7. Children’s Reasoning: Discovering the Idea of Mathematical Proof, *Carolyn A. Maher*

8. Aspects of Teaching Proving In Upper Elementary School, *David A. Reid and Vicki Zack*

**Section III: Teaching and Learning of Proof in Middle Grades and High School**

9. Middle School Students’ Production of Mathematical Justifications, *Eric J. Knuth, Jeffrey M. Choppin and Kristen N. Bieda*

10. From Empirical to Structural Reasoning in Mathematics: Tracking Changes Over Time, *Dietmar Küchemann and Celia Hoyles*

11. Developing Argumentation and Proof Competencies in the Mathematics Classroom, *Aiso Heinze and Kristina Reiss*

12. Formal Proof in High School Geometry: Student Perceptions of Structure, Validity And Purpose, *Sharon M. Soucy McCrone and Tami S. Martin*

13. When is an Argument Just An Argument? The Refinement of Mathematical Argumentation, *Kay McClain*

14. Reasoning-and-Proving in School Mathematics: The Case of Pattern Identification, *Gabriel J. Stylianides and Edward A. Silver*

15. "Doing Proofs" in Geometry Classrooms, *Patricio Herbst, Chialing Chen, Michael Weiss, and Gloriana González, with Talli Nachlieli, Maria Hamlin and Catherine Brach*

**Section IV: Teaching and Learning of Proof in College**

16. College Instructors’ Views of Students Vis-á-Vis Proof, *Guershon Harel and Larry Sowder*

17. Understanding Instructional Scaffolding in Classroom Discourse on Proof, *Maria L. Blanton, Despina A. Stylianou and M. Manuela David*

18. Building a Community of Inquiry in a Problem-Based Undergraduate Number Theory Course: The Role of the Instructor, *Jennifer Christian Smith, Stephanie Ryan Nichols, Sera Yoo and Kurt Oehler*

19. Proof in Advanced Mathematics Classes: Semantic and Syntactic Reasoning in the Representation System of Proof, *Keith Weber and Lara Alcock*

20. Teaching Proving by Coordinating Aspects of Proofs with Students’ Abilities, *John Selden and Annie Selden*

21. Current Contributions toward Comprehensive Perspectives on the Learning and Teaching of Proof, *Guershon Harel & Evan Fuller*

References

Index

P>Index