Proof and proving in Algebra

Introduction The theme of proof has been at the core of a number of studies, different aspects have been discussed and different perspectives have been proposed. Although not always researchers found an agreement, certainly some basic ideas can be considered as shared and among others the assumption that proving constitutes an important part of mathematical activity, so that education to mathematical thinking, even if not reduced to formal proving, cannot ignore it. This shared assumption should have and in some cases had (Hoyles, 1997) important consequences on school practice, both in terms of curricula and in terms of classroom organization. Still the didactical problem seems far to be solved. Teachers have troubles introducing pupils to proof and, generally speaking, to a theoretical perspective. The analogy, but also possible discrepancies between a " natural " approach and mathematical approach to argumentation has been widely discussed. Clear evidence has been provided of the difficulties related to the distance between the way of spontaneously supporting a statement and the specific way mathematicians have elaborated in order to make a statement acceptable within a specific theoretical In this perspective, the paper present and discuss on the pupilsìntroduction to proof and it will do it considering an unusual math context for proving, Algebra, and the specific support of a microworld. This contribution is based on a long-term study, carried out in collaboration with Michele Cerulli, who in particular developed a prototype microworld, L'Algebrista (Cerulli, 2004). The aim of this paper is that of discuss on some aspects related to the design of the specific technological environment; in particular, I'd like to discuss how the design of the microworld has been influenced by the basic assumption about its functioning as a tool of semiotic mediation for proving in Algebra, and generally speaking for the idea of Theory. The notion of semiotic mediation, introduced by Vygotsky (1978), has been elaborated in the last years in the field of mathematics education, in particular let me refer to a previous discussion, that I published not long ago (Mariotti, 2002), where the process of semiotic mediation is explained referring to the use of computational tools, and microworlds in particular. The key element on which the process of semiotic mediation is based concerns on the one hand, the link between tools and meanings emerging from their use in classroom activities and on the other hand the mathematical notions, which are the …