The Role and Use of Examples in Learning to Prove

This research focuses on the discipline of mathematics education and on students at the middle school, high school, and undergraduate levels.

Proof is central to mathematical practice, yet a perennial concern is that students of all ages struggle to understand the nature of proof. 

We view example-based reasoning as an important object of study and posit that examples play both a foundational and essential role in the development, exploration, and understanding of conjectures, as well as in subsequent attempts to develop proofs of those conjectures.

Students typically spend little time carefully considering and analyzing examples prior to attempting to develop a formal proof. Moreover, students typically receive very little, if any, explicit instruction on how to strategically think about and analyze examples in developing, exploring, understanding, and proving conjectures. The hypothesis of this project is that these twin factors—little time spent thinking about and analyzing examples, and little explicit instructional time on the strategic use of examples—underlie students’ difficulties in learning to prove.

This project will

(a) investigate the nature of middle school and high school (hereafter, secondary school) students’, undergraduate students’, and mathematicians’ thinking about the examples they use when developing, exploring, and proving conjectures;

(b) investigate ways in which thinking about and analyzing examples may facilitate the development of students’ learning to prove; and

(c) develop instructional materials designed to help teachers and university instructors foster the development of their students’ thinking about and use of examples in learning to prove.