**2.1 Theoretical approaches to problem-solving and learning assessment**

### *2.1.1 Problem-solving as an object of teaching and learning*

A number of approaches, such as "open problems" [4] and "research narratives" [5], have emerged in recent decades with the shared aim of introducing students to scientific practice and transferring mathematical research into the classroom. Georget [6], for his part, proposes to use the term Peer Research and Proof Activities (PRPA) to describe this type of activity "whose main objective is to train students in the mathematical research practice and peer exchanges in the manner of professional mathematicians1 " (p. 77). He highlights five dimensions that enable the specific features of these activities to be taken into account and characterized: research potential; resistance potential; dynamic resistance potential; debate potential and didactic potential. In our dissertation, we used these potentials to characterize problems designed to make students search, thus enriching the classic a priori analysis tools of mathematics didactics. Indeed, a priori determination of the potential of the problems proposed to students enables us to better understand the effect of the teacher's choices, interventions, and interactions with students during classroom sessions.

From our interest in the teaching and assessment of problem-solving, we came to question the possible learning associated with this object of knowledge. We got inspiration from Houdement [7], who looks at "problems for searching" in elementary schools in France, with a strong hypothesis that research and proof activities

<sup>1</sup> Our translation from French.

between peers can, among other things, lead students to develop learning linked to ways of reasoning and proving in mathematics. The need to characterize this learning in greater detail led us to take an interest in the work carried out by Jeannotte [8] on mathematical reasoning. She highlights the complementarity of two viewpoints: the structural and the processual aspects. The former allows us to take into account the structure of mathematical reasoning and "describe the constituent elements of a step or sequence of steps and the relationships they have with each other2 " ([8], p. 124). The second allows us to integrate the temporal dimension of reasoning and the goal pursued by the person carrying it out, in connection with the functions of mathematical reasoning. Taking these two points of view into account makes it possible to characterize different types of reasoning (according to the structural aspect of mathematical reasoning, such as hypothetico-deductive reasoning, reasoning by exhaustive study of cases, *etc.*), approaches (according to the processual aspect, such as the experimental approach or the process of adjusting successive trials) and proofs (by ostension, by counter-example, etc.) involved in solving mathematical problems. The variety of ways of searching, reasoning, and proving in mathematics led us to question the way teachers go about organizing and implementing such teaching.
