Abstract:

Solving problems by different methods, identifying the knowledge put at stake in each case, anticipating learning difficulties and using this information to state variants of the problem are fundamental aspects of the competence of analysis of mathematical knowledge for teaching. The aim of this paper is to describe, by means of a case study, the design and implementation of a formative action with prospective primary school teachers, which aims to develop this competence. Some theoretical and methodological tools of the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction are used.The teacher’s Didactic-Mathematical Knowledge and Competence (DMKC) model proposed by the OSA interprets and extends other teacher’s knowledge models used in mathematics education, such as the MKT model. In the DMKC model, it is assumed that teachers should have a common mathematical knowledge regarding the educational level where they teach, and an extended mathematical knowledge that allows them to articulate it with higher levels. In addition, for each mathematical content, the teacher should have didactic-mathematical knowledge of the different facets that affect the educational process. This allows the teacher, when dealing with a certain mathematical situation, to recognize the diversity of meanings that are put into play, to be able to solve the task using different procedures and showing different justifications (epistemic facet), as well as to be competent to modify it according to the students’ learning needs (instructional and cognitive facets). The system of categories for the analysis of the mathematics teacher’s knowledge and competences proposed by the DMKC model is related to the type of analysis tools elaborated in the core of the OSA, in particular, the ontosemiotic configuration of practices, objects and processes. In the intervention, the prospective teachers first solve, in various ways, a mathematics problem taken from a primary school mathematics textbook. Then they analyse the objects and processes involved in their proposed solutions and identify both the didactic objectives pursued by the task and the potential difficulties pupils may encounter. Taking this analysis into account, they should create new problems by variation of the information (data that are related in the problem), requirement (what is requested to determine, examine or conclude), or context (intra-mathematical or extra-mathematical) in the initial problem. Finally, they are asked to describe again the objects as well as the difficulties associated with the new problems.

Keywords:

Ontosemiotic analysis, Problem-posing, Problem-solving, Teacher training.