In the previous sessions, we have studied the Communication and Problem Solving Standards. In this session, we will examine Reasoning and Proof.

"Reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied. In mathematically productive classroom environments, students should expect to explain and justify their conclusions. When questions such as, What are you doing? or Why does that make sense? are the norm in a mathematics classroom, students are able to clarify their thinking, to learn new ways to look at and think about situations, and to develop standards for high-quality mathematical reasoning." (NCTM, 2000, p. 342.)

Mathematics is often referred to as a "sense-making" endeavor. A key way in which we can understand this sense-making capacity is to consider the how mathematics prompts us to use ideas, symbols and procedures to make conjectures, and prove, or disprove, them using deductive reasoning. In fact, the notion of reasoning about a problem situation underlies all of mathematics; even the algorithms encountered in the earliest grades had to be derived and proven. Often however, we think of these aspects, particularly proof, as something apart, perhaps only a key part of the Geometry curriculum, or advanced topics, but less emphasized in other classes. In this session we will show how teaching and learning throughout all high school mathematics topics can benefit from the application of the reasoning and proof standard. When students and teachers recognize these ideas as fundamental to mathematics their skill and power in using this subject increases.

Learning Objectives

This session shows how to help students:

• 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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