A Co-Publication of Routledge for the National Council of Teachers of Mathematics (NCTM)

In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students’ understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.

chapter |12 pages


ByDespina A. Stylianou, Maria L. Blanton, Eric J. Knuth

part I|51 pages

Theoretical Considerations on the Teaching and Learning of Proof

chapter 1|4 pages

What I Would Like My Students to Already Know About Proof

ByReuben Hersh

chapter 2|19 pages

Exploring Relationships Between Disciplinary Knowledge and School Mathematics

Implications for Understanding the Place of Reasoning and Proof in School Mathematics
ByDaniel Chazan, H. Michael Lueke

chapter 3|24 pages

Proving and Knowing in Public

The Nature of Proof in a Classroom
ByPatricio Herbst, Nicolas Balacheff

part II|82 pages

Teaching and Learning of Proof in the Elementary Grades

chapter 4|16 pages

Representation-based Proof in the Elementary Grades

ByDeborah Schifter

chapter 5|15 pages

Representations that Enable Children to Engage in Deductive Argument

ByAnne K. Morris

chapter 6|18 pages

Young Mathematicians at Work

The Role of Contexts and Models in the Emergence of Proof
ByCatherine Twomey Fosnot, Bill Jacob

chapter 7|13 pages

Children's Reasoning

Discovering the Idea of Mathematical Proof
ByCarolyn A. Maher

chapter 8|14 pages

Aspects of Teaching Proving in Upper Elementary School 1

ByDavid A. Reid, Vicki Zack

part III|122 pages

Teaching and Learning of Proof in Middle Grades and High School

chapter 9|18 pages

Middle School Students' Production of Mathematical Justifications 1

ByEric J. Knuth, Jeffrey M. Choppin, Kristen N. Bieda

chapter 10|20 pages

From Empirical to Structural Reasoning in Mathematics

Tracking Changes Over Time
ByDietmar Küchemann, Celia Hoyles

chapter 11|13 pages

Developing Argumentation and Proof Competencies in the Mathematics Classroom

ByAiso Heinze, Kristina Reiss

chapter 12|18 pages

Formal Proof in High School Geometry

Student Perceptions of Structure, Validity, and Purpose 1
BySharon M. Soucy McCrone, Tami S. Martin

chapter 13|13 pages

When is an Argument Just an Argument?

The Refinement of Mathematical Argumentation 1 , 2
ByKay McClain

chapter 14|15 pages

Reasoning-and-Proving in School Mathematics

The Case of Pattern Identification 1
ByGabriel J. Stylianides, Edward A. Silver

chapter 15|19 pages

“Doing Proofs” in Geometry Classrooms

ByPatricio Herbst, Chialing Chen, Michael Weiss, Gloriana González, Talli Nachlieli, Maria Hamlin, Catherine Brach

part VI|102 pages

Teaching and Learning of Proof in College

chapter 16|15 pages

College Instructors' Views of Students Vis-à-Vis Proof

ByGuershon Harel, Larry Sowder

chapter 17|17 pages

Understanding Instructional Scaffolding in Classroom Discourse on Proof 1

ByMaria L. Blanton, Despina A. Stylianou, M. Manuela David

chapter 18|16 pages

Building a Community of Inquiry in a Problem-based Undergraduate Number Theory Course

The Role of the Instructor
ByJennifer Christian Smith, Stephanie Ryan Nichols, Sera Yoo, Kurt Oehler

chapter 19|16 pages

Proof in Advanced Mathematics Classes

Semantic and Syntactic Reasoning in the Representation System of Proof
ByKeith Weber, Lara Alcock

chapter 20|16 pages

Teaching Proving by Coordinating Aspects of Proofs with Students' Abilities

ByJohn Selden, Annie Selden