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Teaching and Learning Proof Across the Grades: A K–16 Perspective

Despina A. Stylianou, Maria L. Blanton, and Eric J. Knuth, editors
Publication Date: 
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[Reviewed by
Elizabeth A. Burroughs
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Teaching and Learning Proof Across the Grades is a compilation of 21 articles, each presented as a chapter, all of which address some aspect of reasoning and proof in grades K–16. The first section is an introduction, offering three chapters on the theory of the teaching and learning of proof. The second section focuses on elementary grades, the third on middle grades and high school, and the final section on proof in college. Some of the chapters report on case studies; others present theoretical analysis. A theme which emerges quite strongly throughout the chapters is students’ confusing a collection of examples with a proof.

As a mathematics educator who prepares mathematics teachers to teach grades K–16, I found all of the chapters to provide useful information. A chapter by Patricio Herbst, Chialing Chen, Michael Weiss, and Gloriana Gonzàlez, “‘Doing Proofs’ in Geometry Classrooms,” analyzes the norms of doing proofs in high school geometry and would be particularly useful for engaging pre-service high school teachers in reflecting on the role of proof in high school geometry classes. I also found the chapter by Guershon Harel and Larry Sowder, “College Instructors’ Views of Students Vis-à-vis Proof,” to resonate with many of the issues I find important in teaching undergraduate mathematics courses, particularly with regard to developing my students’ understanding of the role of definitions.

Those who prepare mathematics teachers at the K–16 level will find this book very useful, both for their own reading and in providing articles for their students to read. Those who teach undergraduate mathematics courses will find the final section quite relevant to their work, and may find that reading the earlier chapters increases their understanding of why students come to college with the views of proof that they do.

Elizabeth A. Burroughs, is an assistant professor of mathematics education in the Department of Mathematical Sciences at Montana State University.

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


List of Contributors



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