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!

### Setting the Stage

Chapter 1 setting the stage

Proof has historically been considered a topic first encountered in high school geometry class, featuring a two-column format of “statements” and “reasons.” In this chapter, we present reasoning-and-proving as a set of activities that transcends content and format and is accessible to all middle and high school students. Thinking about reasoning-and-proving more broadly will help you to enhance students’ understanding of the mathematics they are learning and their ability to construct valid mathematical arguments. While reading this chapter, we encourage you to consider

• how this broader perspective on reasoning-and-proving could benefit students’ understandings of mathematics; and
• what it would take to support students in building the capacity to engage in reasoning-and-proving.

### Is Reasoning-and-Proving Really What You Think?

Over the past few ...