Fran Arbaugh & Margaret (Peg) S. Smith 
We Reason & We Prove for ALL Mathematics 
Building Students’ Critical Thinking, Grades 6-12

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!

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