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We Reason & We Prove for ALL Mathematics

We Reason & We Prove for ALL Mathematics
Building Students’ Critical Thinking, Grades 6-12

August 2018 | 272 pages | Corwin

Sharpen concrete teaching strategies that empower students to reason-and-prove

How do teachers and students benefit from engaging in reasoning-and-proving? What strategies can teachers use to support students’ capacity to reason-and-prove? What does reasoning-and-proving instruction look like?

We Reason & We Prove for ALL Mathematics helps mathematics teachers in grades 6-12 engage in the critical practice of reasoning-and-proving and support the development of reasoning-and-proving in their students. The phrase “reasoning-and-proving” describes the processes of identifying patterns, making conjectures, and providing arguments that may or may not qualify as proofs – processes that reflect the work of mathematicians. Going beyond the idea of “formal proof” traditionally relegated only to geometry, this book transcends all mathematical content areas with a variety of activities for teachers to learn more about reasoning-and-proving and about how to support students’ capacities to engage in this mathematical thinking through:

  • Solving and discussing high-level mathematical tasks
  • Analyzing narrative cases that make the relationship between teaching and learning salient
  • Examining and interpreting student work that features a range of solution strategies, representations, and misconceptions
  • Modifying tasks from curriculum materials so that they better support students to reason-and-prove
  • Evaluating learning environments and making connections between key ideas about reasoning-and-proving and teaching strategies

 We Reason & We Prove for ALL Mathematics is designed as a learning tool for practicing and pre-service mathematics teachers and can be used individually or in a group. No other book tackles reasoning-and-proving with such breadth, depth, and practical applicability. Classroom examples, case studies, and sample problems help to sharpen concrete teaching strategies that empower students to reason-and-prove!

About the Authors
Chapter 1 Setting the Stage
Are Reasoning and Proving Really What You Think?

Supporting Background and Contents of This Book

What is Reasoning and Proving in Middle and High School Mathematics?

Realizing the Vision of Reasoning-and-Proving in Middle and High School Mathematics

Discussion Questions

Chapter 2 Convincing Students Why Proof Matters
Why Do We Need to Learn How To Prove?

The Three Task Sequence

Engaging in the Three Task Sequence, Part 1: The Squares Problem

Engaging in the Three Task Sequence, Part 2: Circle and Spots Problem

Engaging in the Three Task Sequence, Part 3: The Monstrous Counterexample

Analyzing Teaching Episodes of the Three Task Sequence: The Cases of Charlie Sanders and Gina Burrows

Connecting to Your Classroom

Discussion Questions

Chapter 3 Exploring the Nature of Reasoning-and-Proving
When is an Argument a Proof?

The Reasoning-and-Proving Analytic Framework

Developing Arguments

Developing a Proof

Reflecting on What You’ve Learned about Reasoning and Proving

Revisiting the Squares Problem from Chapter 2

Connecting to Your Classroom

Discussion Questions

Chapter 4 Helping Students Develop the Capacity to Reason-and-Prove
How Do You Help Students Reason and Prove?

A Framework for Examining Mathematics Classrooms

Determining How Student Learning is Supported: The Case of Vicky Mansfield

Determining How Student Learning is Supported: The Case of Nancy Edwards

Looking Across the Cases of Vicky Mansfield and Nancy Edwards

Connecting to Your Classroom

Discussion Questions

Chapter 5 Modifying Tasks to Increase the Reasoning-and-Proving Potential
How Do You Make Tasks Reasoning-and-Proving Worthy?

Returning to the Effective Mathematics Teaching Practices

Examining Textbooks or Curriculum Materials for Reasoning-and-Proving Opportunities

Revisiting the Case of Nancy Edwards

Continuing to Examine Tasks and Their Modifications

Re-Examining Modifications Made to Tasks Through a Different Lens

Comparing More Tasks with their Modifications

Strategies for Modifying a Task to Enhance Students’ Opportunities to Reason-and-Prove

Connecting to Your Classroom

Discussion Questions

Chapter 6 Using Context to Engage in Reasoning-and-Proving
How Does Context Affect Reasoning-and-Proving?

Considering Opportunities for Reasoning-and-Proving

Solving the Sticky Gum Problem

Analyzing Student Work from the Sticky Gum Problem

Analyzing Two Different Classroom Enactments of the Sticky Gum Problem

Connecting to Your Classroom

Discussion Questions

Chapter 7 Putting it All Together
Key Ideas at the Heart of this Book

Tools to Support the Teaching of Reasoning-and-Proving

Putting the Tools to Work

Moving Forward in Your PLC

Discussion Questions

Appendix A Developing a Need for Proof: The Case of Charlie Sanders
Appendix B Motivating the Need for Proof: The Case of Gina Burrows
Appendix C Writing and Critiquing Proofs: The Case of Vicky Mansfield
Appendix D Pressing Students to Prove It: The Case of Nancy Edwards
Appendix E Making Sure that All Students Understand: The Case of Calvin Jenson
Appendix G Helping Students Connect Pictorial and Symbolic Representations: The Case of Natalie Boyer

"Simply stated, this book is a must-have for preservice and inservice mathematics teachers and teacher leaders who are looking to enhance their understanding of how reasoning-and-proving are critical processes for increasing proficiency across all mathematics content domains. This expert author team has illustrated a clear vision and plan, supported by key strategies and exceptional tools, for guiding teacher teams as they help their students learn how to make conjectures and develop and judge the effectiveness of their arguments and proofs. This book is an exceptionally useful and timely resource for schools and districts that are looking to connect and deepen their professional focus with the Effective Mathematics Teaching Practices (NCTM, 2014) and other evidence-based practices."

Jonathan (Jon) Wray, Coordinator of Secondary Mathematics, Howard County Public Schools (MD)
Board of Directors, National Council of Teachers of Mathematics

"Reasoning-and-proving are central to investigating ideas, solving problems, and establishing mathematics knowledge at all levels. Built around rich classroom cases, this book provides research-supported frameworks and practical resources for teachers to deepen their understanding and develop practices to aid students in reasoning-and-proving as powerful mathematical thinkers."

Daniel Heck, Vice President
Horizon Research, Inc.

"We Reason & We Prove for ALL Mathematics provides an enlightening and engaging examination of reasoning-and-proving in secondary mathematics classrooms. Filled with carefully designed tasks and task sequences, along with illustrative classroom cases, it clearly articulates the nature of reasoning-and-proving, what students need to know and understand about it, and how teachers can support this learning. The thought-provoking discussion questions and recommended classroom activities support readers’ implementation of reasoning-and-proving activities into their own classrooms. We Reason & We Prove for ALL Mathematics is an outstanding resource for practice-based learning on this essential component of mathematics learning. I recommend it most highly."

Diane J. Briars, PhD, Mathematics Education Consultant
Past President, National Council of Teachers of Mathematics (NCTM)

"The authors of We Reason & We Prove for ALL Mathematics have taken aim at a long-standing challenge in mathematics education: helping students become proficient with mathematical reasoning and proof. In so doing they have produced a book that will be useful to teachers and scholars alike in addressing a topic that is both difficult to teach and difficult to learn. This volume blends knowledge obtained through rigorous research with practical wisdom derived from extensive experience. Building upon a solid foundation of prior research on students’ mathematical reasoning, the authors offer a collection of narrative cases and mathematics activities designed to deepen the understanding of teachers in ways that will enhance the teaching and learning of proof and reasoning."

Edward A. Silver, Senior Associate Dean for Research & Graduate Studies, William A. Brownell Collegiate Professor of Education, & Professor of Mathematics
University of Michigan

"Grounded in the research on effective mathematics teaching practices and connected to the mathematical content taught in middle and high school, We Reason & We Prove for ALL Mathematics offers exceptional guidance, superb exemplars, and important classroom discussion questions to support student reasoning-and-proving. The ideas in this book are what we need to move away from repeat-after-me mathematics toward a convince-me mathematics—totally transforming mathematics classrooms and increasing students’ opportunities to engage in doing authentic mathematics."

Jennifer Bay-Williams, PhD, Mathematics Educator & Professor, Co-Author of Teaching Student-Centered Mathematics: Developing Appropriate Instruction Series
University of Louisville
Key features


  • Vignettes, teaching takeaways, and key vocabulary terms
  • Pause and consider sections and discussion questions
  • In-depth analysis of student work
  • Companion website contains downloadable tasks and case studies for analysis

Sample Materials & Chapters

Table of Contents


Chapter 1

For instructors

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