Developing Student Understanding: The Case of Proof by Contradiction

Proof is central to the curriculum for undergraduate mathematics majors. Despite transition-to-proof courses designed to facilitate the transition from computation-based mathematics to proof-based mathematics, students continue to struggle with all aspects of mathematical proof. In particular, research suggests that proof by contradiction is an especially difficult proof method for students to construct and comprehend and yet, there are no satisfactory instructional models for how to teach the method. The purpose of this paper is to discuss preliminary results of a teaching experiment on student comprehension of proof by contradiction within a transition-to-proof course. Grounded in APOS Theory, this paper will illustrate that a student’s conception of mathematical logic, and quantification in particular, plays an important role in their comprehension of proof by contradiction.