Abstract:

Numerous research studies have focused their interest on problem-posing. These studies analyze the creation of problems from various perspectives, highlighting its significance as:
(i) a tool for teaching mathematics;
(ii) a goal of mathematics education;
(iii) a tool for researching the abilities of students and trainee teachers, or
(iv) a subject of research in its own right.

While most research on problem-posing focuses on the elaborated problems, i.e., the products, there is a recent interest in analyzing the actual process of problem creation, focusing on the cognitive processes involved in the sequence of actions taken during the formulation of new problems.

Regardless of how problem invention is viewed, a large number of researchers in Mathematics Education have emphasized the importance of incorporating problem creation into teacher education programs. On one hand, it is emphasized that this is the only way to solidly introduce problem-solving into mathematics curricula and classroom practices. On the other hand, problem creation for didactic purposes appears as a means to enhance teaching competencies. In this work, we analyze the processes that two trainee teachers undertake when approaching a task of problem creation with a specific didactic-mathematical purpose: to develop algebraic reasoning in a probabilistic context. The aim of the task is for trainee teachers to identify the potential of tasks that are not intentionally algebraic and modify them so that their solution involves algebraic objects and processes. We rely on the theoretical and methodological tools of the Ontosemiotic Approach for our research. Particularly:
a) the notions of mathematical practices, objects, and processes for the analysis of mathematical activity, and
b) the model of teacher's Didactic-Mathematical Knowledge and Competencies, to connect the knowledge and competencies required and developed through problem creation.

Keywords:

Problem-posing, teacher education, algebraic reasoning, probabilistic reasoning.