Generic Proving: Reflections on Scope and Method.

Abstract

For the Learning of Mathematics 33, 3 (November, 2013) FLM Publishing Association, Fredericton, New Brunswick, Canada A generic proof is, roughly, a proof carried out on a generic example. We introduce the term generic proving to denote any mathematical or educational activity surrounding a generic proof. The notions of generic example, generic proof, and proof by generic example have been discussed by a number of scholars (e.g., Balacheff, 1988; Mason & Pimm, 1984; Rowland, 1998; Malek & MovshovitzHadar, 2011). All acknowledge the role of proof not only in terms of validating the conclusion of a theorem but, just as importantly, as a means to gain insights to why the theorem is true. In particular, we support and extend the argument made by Rowland (1998) that a generic proof does carry a substantial “proof power”, and may in fact lie on the same continuum as the working mathematician’s proof. In the same vein, we analyze possible ways that generic proof and proving may help in unpacking and making accessible to students at all levels the main ideas of a proof [1]. The article is organized as a reflection on three examples, or “mathematical case studies”, which reveal increasingly more subtle facets of generic proving. The first mathematical case study is a simple and elementary theorem of numbers (also discussed in Rowland, 1998). The second example, a decomposition theorem on permutations, is still elementary in the sense of not requiring subject-matter knowledge beyond high school mathematics, but is more sophisticated in terms of the proof techniques required. The third example, Lagrange’s theorem from elementary group theory, is more sophisticated both in terms of the proof techniques and the subject matter knowledge required. All the examples are introduced in a self-contained manner and all the terminology is explained and exemplified. In the second part of the article, we reflect in more depth on the mathematical case studies of the first part, in an attempt to explicate some of the general features of generic proofs. For example, in an attempt to characterize the mathematical content of generic proofs, we look for commonalities with professional mathematicians’ proofs as they appear in research journals and in university-level textbooks and lectures. For another example, we ask—and try to give some partial answers—about the scope of generic proving: what kind of proofs can be more or less helpfully approached via a generic version? The article has been written in the form of a thought experiment. It is, however, solidly based in the experience of the authors in running many workshops with students and in international conferences on exactly these examples and ideas. Several researchers have previously discussed the more theoretical aspects of generic proofs. This research, while relevant to the topic at hand, would take us away from our mathematical and pedagogical focus [2].