EXPLORATION, EXPLANATION, FORMALIZATION: A THREE-STEP APPROACH TO PROOF

Proof writing is a multi-faceted, complex skill. As such, there are many potential explanations for students' difficulties with proof. In a study of their initial efforts to write proofs, Moore [3] suggests that one source of error is the abrupt transition to proof in upper-division courses. According to Moore, students often enter these courses with extensive computational experience, yet they are ill-prepared to use the language of mathematics to craft mathematical arguments. Moreover, their initial conceptions of proof are often narrow and restrictive. In their 1982 analysis of mathematical thinking, Mason, Burton, and Stacey [2] theorize that the use of a three-step approach, in which students initially engage in exploration and construct informal, intuitive arguments, facilitates an understanding of formal proof. Our paper outlines the three-step approach and provides a detailed example of its use. Additionally, the paper offers reflections on this approach to proof and our own adaptations.