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Reasoning and Proof

 Instructional programs from prekindergarten through grade 12 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.

Systematic reasoning is a defining feature of mathematics. Exploring, justifying, and using mathematical conjectures are common to all content areas and, with different levels of rigor, all grade levels. Through the use of reasoning, students learn that mathematics makes sense. Reasoning and proof must be a consistent part of student's mathematical experiences in prekindergarten through grade 12.

Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts and from the earliest grades. At all levels, students reason inductively from patterns and specific cases. For example, even a first grader can use an informal proof by contradiction to argue that the number 0 is even: "If 0 were odd, then 0 and 1 would be two odd numbers in a row. But even and odd numbers alternate. So 0 must be even."

Increasingly over the grades, students should learn to make effective deductive arguments as well, using the mathematical truths they are establishing in class. By the end of secondary school, students should be able to understand and produce some mathematical proofs--logically rigorous deductions of conclusions from hypotheses--and should appreciate the value of such arguments.