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Reasoning and ProofSession 04 Overviewtab atab btab ctab dtab eReference



Reasoning and proof are two important elements to consider in the upper elementary classroom. This may come as a surprise, as many of us associate mathematical reasoning and proof most closely with high school and college mathematics. However, even very young children engage in reasoning and informal proof. At the upper elementary level, engaging in reasoning enables students to make sense of new ideas and to develop habits that will be of lifelong importance, both in and out of the classroom. Students' work with informal proofs helps establish a strong basis for formal proofs in later grades.

Although reasoning and proof are often discussed together, in the upper elementary grades students primarily engage in reasoning. Students also engage in informal proof when they build a logical argument based on known facts. Formal mathematical proofs generally involve a logical chain of reasoning that establishes the truth of a general statement; such proofs are important to older, more sophisticated mathematical thinkers. In this session, we emphasize the aspects of reasoning and proof that are most relevant to the development of upper elementary students.

Learning Objectives

This session shows you how to help students do the following:

  • Reason about a problem
  • Make and test a conjecture
  • Justify their ideas and solutions
  • Provide a convincing argument
  • Refine their thinking

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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