**1. Introduction**

The aim of this text is to present part of a research project led by the team in mathematics didactics in Geneva. The overall project entitled: "Problem solving as an object or means of education at the heart of learning in the mathematics classroom", ResoPro for short, was piloted by Sylvie Coppé and Jean-Luc Dorier and funded by the Swiss National Science Foundation (SNSF – Grant n°100019\_173105/1 – Period from 31.08.2017 to 31.01.2022).

The context and main. Results of the whole project are about to be published in a collective book in French [1]. Here we quote a passage from the introduction of this book to help the reader better understand the background of the whole project:

"By presenting innovative activities, these various projects on the inquiry-based teaching approach aim to provide teachers with the tools they need, but few of them question what pupils actually learn, even if this question is always an underlying one. Indeed, the value of problem-solving is often a presupposition that is rarely, if ever, questioned. What's more, questioning the effects of practices and problem types on students' actual learning remains theoretically and methodologically complex. Since the assessments used in classrooms to evaluate learning depend heavily on the type of education provided, what elements should be used to evaluate learning? On what time scale? What are the effects linked to the teacher, in particular his or her skills in engaging and keeping students on task, and in sustaining motivation? That's why we've set ourselves the goal of investigating and evaluating the effects of problemsolving in the mathematics classroom on student learning, using various theoretical frameworks from mathematics didactics and assessment. We draw on the work of our team in which problem solving is used either as a means of teaching thematic knowledge, or as an object of education (to learn how to solve problems). Our aim is to assess how learning classical mathematical themes can be achieved primarily through problem solving, and to better determine what can be learned when the focus is on problem solving independently of mathematical content, then how students can identify this constructed knowledge and know-how (how institutionalization can be managed) and what aids can be provided". ([1], Introduction).

The main aim of the project was to show how the teaching of problem-solving is conceived and constructed, how it is implemented by teachers, and what learning is achieved by students, through various works. The project as a whole has been structured around three PhDs, co-supervised by Sylvie Coppé and Jean-Luc Dorier, and two other works, enabling us to question problem-solving from a variety of angles (school levels, teachers' and students' points of view). In this text, we present only two of the PhD and the new research projects that resulted from them.

Maud Chanudet's PhD [2], defended on October 22, 2019, entitled "Étude des pratiques évaluatives des enseignants dans le cadre d'un enseignement centré sur la résolution de problèmes en mathématiques" (study of teachers' assessment practices in a course centered on problem solving in mathematics), focuses on a course offered at lower secondary level in Geneva centered on problem solving in mathematics. Observation of the practices of teachers involved in teaching and assessing students' problem-solving skills aims to identify their assessment action logics, taking into account both the certificative and formative functions of assessment. The analyses provide a clearer picture of teachers' expectations of problem-solving but also reveal certain contradictions. The theoretical and methodological frameworks, which combine tools from mathematics didactics and the field of assessment, provide a general tool for further study.

Stéphane Favier's PhD [3] was defended on February 8, 2022, under the title "Étude des processus de résolution de problèmes par essais et ajustements en classe de mathématiques à Genève" (Study of trial-and-error problem solving processes in mathematics classes in Geneva). The aim of this work is to document the work of students at different school levels (primary and lower secondary) in problem-solving situations, through detailed analyses of the pupils' work and their interactions with each other and with the teacher. More specifically, this work enables us to characterize students' practices when solving mathematical problems, which may involve trials and adjustments under the usual classroom conditions, leading to a complexity that few other studies on the same subject have tackled, and which therefore constitutes one of its original features.
