The ability to reason allows us to use mathematical knowledge and to generate and solidify mathematical ideas that are new to us. As part of the reasoning and proof process, students develop their mathematical ideas by making, testing, and refining conjectures. In the middle grades, students sharpen and extend their reasoning skills by analyzing their assertions, using both inductive and deductive reasoning.

How do we help students reason mathematically? In this session, we look at some of the specific steps of mathematical reasoning and proof, examine examples of student reasoning and proof as students solve problems, describe the teacher's role in setting the classroom environment to encourage student reasoning, and reflect on classroom practice in employing the Reasoning and Proof Standard.

Learning Objectives

This session shows how to help students do the following

• Extend what they already know
• Make a conjecture
• Provide a convincing argument
• Refine their thinking
• Defend or modify their arguments

NCTM Reasoning and Proof Standard

Instructional programs . . . should enable all students to --

• Recognize reasoning and proof as fundamental aspects of mathematics
• Make and investigate mathematical conjectures
• Develop and evaluate mathematical arguments and proofs
• Select and use various types of reasoning and methods of proof

## Principles and Standards of School Mathematics (NCTM, 2000.) For more information on this process standard, see the NCTM Web site.

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