Arzarello F. (1997) For an ecology of proof in the classroom. Tempus Project seminar, Budapest 31.1. - 2.2.1997.
© F. Arzarello

My presentation will focus on the following topics, all concerning geometry: 1. The epistemological status of proof has deeply changed during the centuries (generally the conception about what a proof is depends strongly on the status given to mathematical objects, as well as on one's ideas about the nature of mathematical knowledges); there are (at least) three main revolutions: (i) the greek hypothetical-deductive system of geometry vs/ the empirical approach of Egyptians and Babylonians; (ii) the modern methods started in the XVIIth century, in particular by Descartes (with the contrast between tools for convincing and ascertaining the truth [= synthetic means], with respect to tools for finding it and for illuminating one's mind about that [= analytic means]), as a new powerful mean to build up mathematics; (iii) the axiomatic method, such it has developed in the course of this century. Comment. It is a complex history, which has not yet been completely written from the pont of view of the didactics of mathematics; curiously, some pieces of it are (sometimes) missing in the analysis given in the literature. To teach properly the proof it is necessary to know some cornerstones of this history: otherwise one may be unable to see some important didactical phenomenon which happen in her/his class as well as to own some possible strategies of attacking the problem of proof teaching in the class. 2. Transformational schemes of reasoning seem crucial in the transition from more crude ways of supporting mathematical arguments (see the paper by Simon and by Harel & Sowder) to a more analytical one. Such schemes entail the overcoming of the big gap between arguing upon purely factual arguments and reasoning because of the operational status of the mathematical sentences, which constitute a proof. Two related major problems must be pointed out for an a-priori analysis of the didactics of proof (they are presented separately for the analysis'sake, but are intimately connected), namely: 2a. The dynamics between internal and external facts, focusing particularly its psychological aspects: for ex. the description of how arguments based upon factual evidence can develop into hypothetical forms of reasoning, based on chains of relationships between sentences, and how such a dynamic can be developed in the class, looking particularly to the transition from interpersonal to intrapersonal ways of thinking; in particular how to coach the individual subject's thoughts and actions, which seem essential to make possible for students to develope "rational schemes of knowledge". (for this point see the presentation of Paolo Boero) 2b. The dynamics between internal and external facts, focusing particularly its epistemological aspects: (i) which didactical situations are conceivable with respect to the main epistemological obstacles (generally corresponding to the big revolutions sketched in 1)? How and in which sense such variables as the uncertainty of the situation (i.e. where the result is not known in advance) or as the difficulty of a situation can be managed? Which are the main difficulties met, when approaching the proof in such a way (for ex., how to manage the fact that proofs do not grow up spontaneously in the class?) [see the paper by Balacheff] (ii) how to develop in the students more the suitable attention to the methods of research and proof than to the results? For ex., how to develop and coach uncertainty situations, where the role of conjecturing, putting forward conditions, changing them etc. becomes crucial? Is it possible to engage effectively (i.e. not at a purely formal level) students on such a game? Is it possible to develop for them an effective structured method, which can overcome the intrinsic difficulties of complex situations? How to avoid the paradox of the idiot savant, namely how to have in the class proofs as explicit objects of didactics and not only as implicit tools for finding results, but in the same time not reduced to their pure formal aspect, whose meanings students cannot rule? Comment. For the whole point 2, but particularly for 2a, the discussion of the topics in 1 is essential; in fact some big epistemological jumps of the first point are dramatically missing in the class; for ex. the geometry of coordinates of Descartes becomes a pure stack of computations: its status of method for solving geometric problems is often left to the individual expertise of single pupils, while the deep links between the cartesian method as a tool for solving problems [Analysis] and the proof of the result, found as a solution to the problem [Synthesis], are a missing link in our schools (as well in a lot of existing literature on the subject). During the presentation I shall try to illustrate with some concrete example how to fill up the gap; the examples will also show how the existing technology incorporate the mathematical objects at a high level, so that it makes it available to the students in a factual way some significant examples of the historical drama which has happened many centuries ago. The examples are taken from a joint project with people from Modena (M.Bartolini Bussi) and Genova (D.Paola). One of the examples will concern the following mathematical machine (someone saw it last year at the Exhibition in Turin); it is the pantograph of Sheiner; the parallelogramm is made of moving rods, which can rotate around the joints O, A, B, C; the rods ABP, BCP' form two (rigid) similar isosceles triangles; the point O is fixed to the plane; the others can move, according to the ties of the machine. The first task is to study the transformation which links P and P' (namely, if P describes a figure ¡, what happens to P' ?)