From examples to proof: Purposes, strategies, and affordances of example use

Examples can be a powerful tool for students to learn to prove, particularly if used purposefully and strategically, but there is a pressing need to better understand the nature of productive example use. Therefore, we examined the characteristics of the successful and unsuccessful cases of proving in the context of a number theory task across the three student populations (middle school, high school, undergraduate), where by successful case we mean the ability to develop a viable justification that accounts for why the conjecture must be true. We present the characteristics of the successful and unsuccessful provers regarding the purposes, strategies, and affordances of example use, offer detailed accounts of a few illustrative cases of students' proving processes, and highlight a couple factors that seemed to have hindered the unsuccessful provers' ability to gain greater affordances from examples. We conclude with a discussion of the instructional implications of the results. (C) 2017 Elsevier Inc. All rights reserved.