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!

# Exploring the Nature of Reasoning-and-Proving

### Exploring the Nature of Reasoning-and-Proving

Chapter 3 exploring the nature of reasoning-and-proving

There are many different and valid ways to construct a proof. For example, there are more than 350 different proofs of the Pythagorean Theorem. In order to qualify as a proof, however, certain criteria must be met and the proof must be accepted as valid by one’s peers. In 1993, when Andrew Wiles first presented his proof of Fermat’s Last Theorem, his peers critiqued it and identified flaws in his reasoning. He then revised his proof and resubmitted it for others to critique before it was accepted as valid in 1994. In this chapter, you will engage in activities that will help you articulate answers to the questions, “What kinds of activities constitute ...