The Role of Proof in Mathematics Education

Abstract

To many–perhaps most–mathematicians, the idea of proof is the very essence of mathematics. In fact, discovering and proving theorems is as good a description as any of what mathematicians do. It is therefore not surprising that efforts to reform mathematics education, such as the “new math,” heavily emphasized a more “rigorous” treatment of the subject, especially when these efforts were strongly influenced by professional mathematicians. It is equally unsurprising that such efforts, for the most part, were total failures. The fact is, the world of mathematics as viewed by the professional mathematician is a far different world than that seen by the neophyte. Yet the idea of proof is, perhaps, the quintessential theme of mathematics. The question is, how does the teacher bridge the gap between the professional mathematician and, if you will, “finished” mathematics, and the learner who sees, at best, a small part of it? What follows provides partial answers to this question. It contains, among other things, some suggestions about basic rules of inference (which, surprisingly, are often ignored, even in “rigorous” mathematics texts), and a few elementary examples of simple proofs using these rules.