Problem-based learning and teacher training in mathematics: making sense of word problems

In the literature we can find many examples showing the difficulty of students in solving mathematical problems. In particular, according to many authors, the critical aspect is precisely the ability to "making sense" of the problems to be solved (e.g. see de Corte, Greer, Verschaffel, 2000). From our point of view, it is not likely to gain an improvement of students' skills if we do not act adequately on teacher training. A first necessary condition is that teachers must be able to adequately solve the tasks that are usually proposed to students. For this reason, in the "Mathematics teaching" courses for the primary school prospective teacher training program at the University of Milano-Bicocca we discuss and promote a PBL (Problem-Based learning) model. These courses combine theoretical lectures with practical activities (for example the didactic pedagogical laboratories described in previous IATED conferences). One of the main topics of such courses is precisely a discussion of the idea of "problem", as opposed to the idea of "exercise" (Schoenfeld, 1992), and of the didactic potential of a teaching practice that makes good use of problems. At the end of the course we tested prospective primary school teachers with a classic problem on proportionality combined with a typical examples of pseuso-proportional problems, for which the inadequacy of the proportional model should emerge. We proposed the task to 57 prospective teachers, asking them to give an explicit and detailed resolution of the problems and to carry out an analysis of the teaching potential of such problems for primary school students. In this paper, we present a qualitative analysis of the answers collected. From such an analysis, inconsistencies between the theoretical model described at lectures and the practical application emerge for many students. In some cases students explicitly admit to have adapted their answers to the "didactic contract". The observations collected allowed us to rethink some aspects of the course, and we will experiment some changes in the next academic year.