Developing proof comprehension and proof by contradiction through logical outlines

Proof is central to the curriculum for undergraduate mathematics majors. Despite transition-to-proof courses designed to facilitate the shift from computation-based mathematics to proof-based mathematics, students continue to struggle with mathematical proof. In particular, there are few tasks beyond writing proofs that are specifically designed to develop students’ understanding of the proofs they read and the proof methods they utilize. The purpose of this paper is to introduce and discuss the merits of two such tasks: constructing and comparing logical outlines of presented proofs. Grounded in APOS Theory, this paper will illustrate a case study that suggests students can improve their understanding of the proofs they read as well as a particular proof method - proof by contradiction – through these two tasks.