The nature of students’ productive and non-productive example-use for proving

Our work stems from the view that example-based reasoning has the potential of enhancing students’ mathematical thinking, and in particular can be helpful in engaging in proving and learning to prove. We aimed at better understanding the nature of example-use across grade levels, and in particular, how judicious example-use may support students’ ability to reason and prove. The paper builds on individual task-based interviews that were conducted with 12 middle school students, 16 high school students, and 10 undergraduate students, whose majors were mathematics or mathematics related. The tasks called for conjecturing and proving. In our analysis we distinguish between empirical example-use and generic example-use, and examine whether the example-uses that we identified were productive for proving, in terms of developing a proof, a deductive argument, or a sound justification that may lead to a proof. We illustrate these distinctions through ten cases drawn from the data. Our findings indicate a relatively strong tendency of students to use examples generically. They also suggest a strong, though not surprising, connection between treating examples generically and productively. Implications for practice and further research are discussed.