The role and use of examples in learning to prove

Proof is central to mathematical practice, yet a perennial concern is that students of all ages struggle in learning to prove. Mathematics education scholars have suggested that overreliance on examples to justify the truth of statements is a contributing factor for students’ difficulties. While example-based reasoning has typically been viewed as a stumbling block to learning to prove, we view example-based reasoning as an important object of study and posit that examples play both a foundational and an essential role in the development, exploration, and understanding of conjectures, as well as in subsequent attempts to develop proofs of those conjectures. In this paper, we provide an overview of our project whose goals were to (a) investigate the nature of middle school and high school students’, undergraduate students’, and mathematicians’ thinking about the examples they use when developing, exploring, and proving conjectures; and (b) investigate ways in which thinking about and analyzing examples may facilitate the development of students’ learning to prove.