Geometrical proof: the mediation of a microworld

Abstract. From a didactic point of view, the introduction of a deductive approach presents two interwoven aspects to be developed: on the one hand the need of justification and on the other hand the idea of a theoretical system within which that justification may becomes a proof. Proof makes sense in respect to a theory and vice versa; thus, the introduction of a deductive approach presents two problems of sense, which are interrelated: the sense of proof and the sense of theory. Thus, the first difficulty the teacher has to overcome, is related to developing the need of a justification, and this contrasts with the intuitive approach to which pupils are used, the second difficulty is related to the possible cognitive rupture between argumentation, i.e. a set of arguments supporting the acceptance of a statement, and mathematical proof, validating a statement within a theory. After analysing the nature of these difficulties, the author discusses how it is possible to face these crucial educational issues presenting the choice of a specific "field of experience" (Boero et al., 1995): geometrical constructions within a particular Dynamic Geometry Environment (Cabri- géomètre