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.
This joint effort is housed within the Wisconsin Center for Education Research at the School of Education, University of Wisconsin-Madison. Copyright ©2012, The Board of Regents of the University of Wisconsin System.