Abstract:

While in the teaching and learning of mathematics the development of problem solving skills has been placed at the centre of curricula and educational practice, problem posing has received less attention. Problem invention is not only closely related to problem solving, but both should be seen as complementary approaches to increasing students' mathematical skills and developing positive attitudes towards their learning.

In order for teachers to be able to design appropriate problem posing tasks for their students and to manage difficulties in that context, they must also be able to problem-solve. However, even teachers with years of experience have difficulties in coming up with problems that are relevant to their students' learning and come up with statements that are incorrect, or not relevant to the level of education at which they teach.

Research on problem creation in mathematics teacher education considers that it should be seen both as a means of instruction, aimed at involving future teachers in learning activities that produce a deep understanding of mathematical content, and as an object of instruction, focused on the development of the competence to formulate problems for didactic purposes.

The purpose of this paper is to describe, by means of a case study, the design, implementation and results of a training intervention with prospective primary school teachers, which aims to develop the competence to create problems for didactic purposes, based on the Ontosemiotic Approach. Within this framework, a model of categories of Didactic-Mathematical Knowledge and Competences of the mathematics teacher has been developed that can guide the training of mathematics teachers. In this model, it is accepted that the teacher must have mathematical knowledge per se, i.e. common knowledge related to the educational level where he teaches, and extended knowledge that allows him to articulate it with higher levels. In addition, as mathematical content comes into play, the teacher must have didactic-mathematical knowledge of the different facets (epistemic, ecological, cognitive, affective, mediational and instructional) that affect the educational process. The creation of problems, their solution, the analysis of the objects and processes involved and the modification according to these elements or the difficulties for the students, constitute an essential part of the epistemic, cognitive and interactional facets of the Didactic-Mathematical Knowledge and Competences model as they allow teacher to graduate the complexity of the tasks proposed to his students, to understand the learning conflicts and to manage the institutionalisation of knowledge.

Keywords:

Problem posing, teacher training, didactic-matehmatical knowledege, ontosemiotic approach.