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

- Fran Arbaugh - Pennsylvania State University, USA
- Margaret (Peg) Smith - University of Pittsburgh, USA
- Justin Boyle - University of Alabama at Tuscaloosa
- Gabriel J. Stylianides - University of Oxford, Oxford, UK
- Michael Steele - University of Wisconsin-Milwaukee, USA

**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!

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 |

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 |

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 |

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 |

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 |

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 |

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 |

## Free resources

### We Reason & We Prove for ALL Mathematics Chapter 1: Setting the Stage

This chapter from *We Reason & We Prove for ALL Mathematics, "*Setting the Stage," will help students engage in mathematical activities that emphasize reasoning, justifying, and proving.

*"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*."

**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."*

**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."*

**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."*

**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."*

**University of Louisville**