Defining Reasoning and Proof
 Introduction | Investigating Conjectures | Reasoning and Justification | Additional Methods | Your Journal
 "Conjecture -- that is, informed guessing -- is a major pathway to discovery. Teachers and researchers agree that students can learn to make, refine, and test conjectures in elementary school. Beginning in the earliest years, teachers can help students learn to make conjectures by asking questions: What do you think will happen next? What is the pattern? Is this true always? Sometimes? Simple shifts in how tasks are posed can help students learn to conjecture" (NCTM, 2000, p. 57). While working on projects or investigations, students can be encouraged to make and investigate conjectures. For example, in Part A, we saw an open-ended assignment that encouraged students to search for patterns and relationships in an organized chart of n + n and n x n values. One classmate, Robert, made what might seem at first an obvious conjecture about n + n always being even. The teacher posed Robert's conjecture to all the students to provide them with experience in using reasoning and proving a conjecture. Frequent exposure to students' examples, or a teacher's modeling of conjectures, advances the idea that conjectures are made by actual people and are based on observation. Review the teacher's questions from Sums of Numbers in Part A. Notice that the teacher's questions guide and encourage the class to use increasingly sophisticated levels of reasoning and a variety of methods of verification, including proof. Models, manipulatives, drawings, and the systematic organization of observations can help reveal patterns and pinpoint mathematical relationships. These are all useful tools for reasoning and for testing conjectures.
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