Proof is central to mathematical practice, yet a perennial concern is that students of all ages struggle to understand the nature of proof. Mathematics education scholars have suggested that overreliance on examples to justify the truth of statements is a primary reason for students’ difficulties. While example-based reasoning has typically been viewed as a stumbling block to learning, the researchers of this project 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.
Mathematicians often spend significant time carefully considering and analyzing examples prior to attempting to develop a formal proof of a conjecture. This effort provides not only a deeper understanding of the conjecture but also insight into the development of a proof. In contrast, secondary school and undergraduate 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 dual 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. Thus, understanding both the nature of students’ thinking about the examples they use to develop, explore, and prove conjectures and the nature of instruction designed to facilitate strategic example use may be a critical step toward helping students successfully learn to prove.
The research seeks to address the following research questions:
- How do students’ use of examples and the nature of their thinking about the examples they use to develop, explore, understand, and prove mathematical conjectures develop as they gain mathematical expertise?
- What is the relationship and interplay among the purposes—to develop, to explore, to understand, or to prove a conjecture—and the nature of examples students use as they gain mathematical expertise?
- How do the various ways of thinking about and analyzing examples facilitate the need to prove and the development of secondary school mathematics and undergraduate mathematics students’ learning to prove, and are particular ways of thinking critical to learning to prove?
- How can instruction foster students’ need and ability to develop proofs by leveraging their strategic use of examples?
- What are the particular ways of reasoning with examples that are critical in supporting students’ transition to proof?