Proof and proving in mathematics education.

Abstract. Since two decades, mathematical proof has been at the core of an active debate in the community of mathematics educators: often blamed as responsible of pupils’ difficulties, but also recognised as a crucial aspect of mathematics activity. In math education that on proof is a very active field of research and this chapter aims at providing a panorama of the state of art in this field, as it emerges form the numerous contributions presented at the PME conferences. The complex and sometimes controversial relationship between argumentation and proof will be discussed, especially in the perspective of its relevance in the educational field. Different epistemological assumptions differently direct research studies, and lead to different approaches to proof as an the educational issue: a short account of the main streams will be given. A number of studies have been devoted to describe and analyse students’ difficulties, in different school context. The rich collection of data and the living discussion accompanying them have provided a solid base for a number of research projects aiming to experiment new approaches to proof. Different experiences have been carries out, at different age levels in different countries. Especially interesting, appears the proposal of introducing a “proof culture” at the primary school level. Actually, an early exposition to a practice of argumentation seems fundamental for establishing a correct relationship between different components of mathematics activity. The discussion is organized according to three main streams of research, identified by three main categories of research questions, which I summarize as follows: Proof at school. What is the status of proof at school? This quite general question is formulated differently in different studies, but the general characteristic aim consists in searching for a global view that captures widespread phenomena and possible correlations between them. Students’ Difficulties. The general issue concerns the study of students’ difficulties, and it refers to two main questions, roughly corresponding to describing and to interpreting students’ behaviours in proving tasks. What are the main difficulties that students face in relation to proof? Which might be the origin of such difficulties? Teaching Interventions. Is it possible to overcome the difficulties that students meet in relation to proof? How can teaching interventions be designed? What general suggestions can be given? A final remark will be devoted to the special issues raised by the introduction of new technologies and in particular of microworlds. For instance, Geometry, traditionally related to the first approach to proof, has seen a dramatic change in ideas and practice, after the introduction of dynamic Geometry environments