An Alternative Approach to Proof in Dynamic Geometry
Michael de Villiers University of Durban-Westville
[There is an underlying] formalist dogma that the only function of proof is that of verification and that there can be no conviction without deductive proof. If this philosophical dogma goes unchallenged, the critic of the traditional approach to the teaching of proof in school geometry appears to be advocating a compromise in quality: he is a sort of pedagogic opportunist, who wants to offer the student less than the 'real thing.' The issue then, is not, what is the best way to teach proof, but what are the different roles and functions of proof in mathematics. (adapted from Hersh, 1979, p. 33)
The problems that students have with perceiving a need for proof are well known to all high-school teachers and have been identified without exception in all educational research as a major problem in the teaching of proof. Who has not yet experienced frustration when confronted by students asking "Why do we have to prove this?" Gonobolin ( 1954/ 1975) noted that "the students . . . do not . . . recognize the necessity of the logical proof of geometric theorems, especially when these proofs are of a visually obvious character or can easily be established empirically" (p. 61).
The recent development of powerful new technologies such as Cabri-Geometre and Geometer's Sketchpad with drag-mode capability has made possible the continuous variation of geometric configurations and allows students to quickly and easily investigate the truth of particular conjectures. What implications does the development of this new kind of software have for the teaching of proof? How can we still make proof meaningful to students?
In this chapter, a brief outline of the traditional approach to the teaching of proof in geometry is critiqued from a philosophical as well as a psychological point of view, and in its place an alternative approach to the teaching of proof (in a dynamic geometry environment) is proposed.