**2.4 Some results and perspectives**

In this research, Chanudet examined how teachers can help regulate student activity [13], in particular through their informal verbal interactions with students [11, 16] during problem-solving sessions. A typology of the way in which these interactions take place and the subject matter they may address, from the dual viewpoint of information gathered and feedback given by the teacher, has thus been developed. What emerges is that, while all teachers organize discussions that enable them to gather clues about students' activities and difficulties, the extent to which they do so varies greatly from one teacher to another. Moreover, these interactive episodes do not all seem to positively support student activity and, a fortiori, learning. In particular, teachers' interactions with students lead the latter to be more or less active in taking account of the teacher's feedback on their work, results, strategies used, errors, etc. This is in line with the work of Black and Wiliam [17], who question the link between feedback and student involvement in classroom discussions. Moreover, the fact that the teacher seeks to involve students in the feedback he or she gives them does not always seem to be enough to encourage student regulation. In fact, it also seems important that the teacher draws on the students' actual activity and ensures that they are involved in the way this feedback is taken into account, particularly in view of their knowledge. This question of the proximity between students' available knowledge and the relevance of the feedback given by the teacher is also questioned by Robert and Vandebrouck [18]. It thus shows that regulating student activity during a session is not self-evident. This leads us to believe that it would be worthwhile to do specific work with teachers in training (pre- or in-service) to enable them to develop such professional gestures.

Furthermore, this research has identified difficulties when, on the one hand, teachers have a wide margin of freedom when it comes to organizing their problemsolving teaching and, on the other, official instructions in terms of learning objectives remain vague. These difficulties relate to the determination of learning objectives associated with problem-solving, the determination of criteria for selecting problems and organizing their articulation, and finally to assessment. These findings echo the research carried out by Choquet-Pineau [19] on the practices of primary school teachers in France proposing open-ended problems to their students, which led her to identify two teacher profiles, associated with different learning objectives: the first aiming primarily to get students to search and find the solution; the second profile aiming to get students to learn mathematics through investigation. This raises questions about the nature of the learning that can be achieved through the practice of problem-solving and revives the question posed by Hersant [20] about the existence of associated institutional knowledge.

At another level, a second part of this PhD research, based on a large-scale study of teachers' declared practices, reveals that, when selecting problems, teachers often use non-mathematical criteria, sometimes even in opposition to official prescriptions. Complementing this, a detailed study of the actual practices of three teachers in the context of an MSA course focused exclusively on problem-solving shows that, while one of the teachers was careful to choose the problems, she proposed to students by drawing on the strategies involved in solving them, the other two did not seek to articulate the problems selected with possible institutionalizations. The choice of problems and the articulation of these problems over the long term, with a view to learning how to solve problems, are therefore crucial issues for the profession.

Finally, this research has highlighted the difficulties encountered by teachers in institutionalizing knowledge associated with problem-solving. In fact, in the three classes we observed, we found no evidence of institutionalization, either in written or oral knowledge texts. However, this finding seems to go beyond the scope of this *Improving the Teaching of Mathematical Problem Solving – A Collaborative Research Based… DOI: http://dx.doi.org/10.5772/intechopen.113258*

research and shows the importance of working on the link between what is institutionalized and what is assessed when problem-solving is taught for its own sake.

In our view, identifying the different types of reasoning, practice, and evidence involved in problem-solving is an interesting way of organizing the choice of problems to be proposed to students, their articulation, but also taking into account the question of learning over a long period of time, or the articulation between institutionalization and assessment.
