Relationships between proof and examples: Comments arising from the papers in this issue

Abstract

The papers in this issue provide an excellent example of how pertinent distinctions which emerge from analysis of significant data sets can be communicated to colleagues and used to make sense of observations. The distinctions are vividly described, illustrated and exemplified, and used to chart maturation in student use of examples from middle school, through high school and into university. Curiously, no corresponding maturation was detected in students’ choice of examples to use for themselves. Contrast was detected between the way students used the same examples for comprehending a conjecture and testing its validity, and the way that some mathematicians used specifically constructed examples for locating underlying structure.What emerged for me from reading these papers is confirmation of the central role of attention, the productiveness of engaging students at the earliest ages and throughout their schooling with the detection and expression of generality, and the importance of being explicit about the scope of generality encompassed by a claim. Purposeful use of examples by students could be expected to develop as teachers explicitly use examples purposefully with and for students. A reasonable conjecture might be that getting students to be explicit about the scope of generality, of possible variation of which they are aware, might shed light on obstacles and barriers to reasoning which might otherwise lead to proof.