**4. A new project for collaborative research**

As we have seen, Chanudet in her PhD [2] characterized various aspects of teachers' problem-solving practices, particularly assessment ones, in the context of a specific problem-solving course at a lower secondary level. On another side, Favier in his PhD [3] provided a detailed analysis of students' problem-solving activities at primary and lower secondary levels. Here, however, the problems proposed were chosen by the researcher, and the sessions observed were ad hoc, making it impossible to take account of problem-solving education over a long period of time. A complementary research work led both by Chanudet and Favier [45]) took up the question of identifying and characterizing possible learning features specific to problem-solving. The starting point for this work was the observation, shared by other authors [7, 46, 47], that in a teaching context designed to get students to practice problem-solving, as well as in the parts of the curricula dedicated to the subject, the targeted learning outcomes are not clearly expressed and do not refer to precise mathematical knowledge. Drawing on the work by Houdement [7], which highlights the fact that problem-solving in elementary school can lead students to develop, in particular, learning related to ways of reasoning and validating in mathematics, the reflection was extended to the secondary school level and led to the identification and characterization of possible learning in problem-solving as pertaining to mathematical practices and reasoning and modes of proof. Chanudet and Favier also drew on the work of Jeannotte [8] to establish a characterization of mathematical reasoning considered from the dual standpoint of its structure and the processes mobilized. It emerges that working on problem-solving can lead students to develop learning linked to trial-and-adjustment or experimental-type approach, hypothetico-deductive reasoning, logical implication, exhaustiveness of cases, disjunction of cases, and proofs by ostension, counter-example or associated with the correct implementation of deductive reasoning. This view of problems seems to us to be a relevant tool for teachers, enabling them to think about and organize problem-solving education.

This has led us to set up a collaborative research project with lower secondary school teachers to study the processes of devolution and institutionalization, and on this basis to develop a resource to equip teachers to help students in problem-solving, covering both the learning objectives targeted and the means for developing this learning. We also plan, through a new PhD, to complement our work with a study of the actual, ordinary practices of primary school teachers, in order to apprehend the professional gestures of these problem-solving teachers and better understand how they organize and manage such classroom sessions. This would be the counterpart to Chanudet's PhD for primary schools. The secondary school component will build on the results of our previous work and enable us to envisage their direct operationalization through collaborative work, while the primary school component will begin by targeting teachers' actual problem-solving practices, before analyzing their potential evolution through collaborative work. In what follows, we present only the secondary school component.

On the basis of the literature review and our previous research project, we put forward a number of hypotheses:

• The devolution and institutionalization processes are always complex for teachers to manage, and are even more so in the context of problem-solving because of the lack of precise identification of the knowledge involved.

*Improving the Teaching of Mathematical Problem Solving – A Collaborative Research Based… DOI: http://dx.doi.org/10.5772/intechopen.113258*


To describe and organize the project, we have adopted Desgagné's characterization of the stages involved in collaborative research (see [48]). The first step, before the actual start of the collaborative work, will be to present the project as initially conceived, and the questions that drive it, to the teachers. This "co-situation" stage, central to the collaborative process, may lead us to complete or refine our questioning and our objects of study, in order to take into account the teachers' concerns.

The actual start of the collaborative work will focus on the preparation of problem-solving sessions before they take place in the classroom. The first step will be to present and illustrate to teachers a number of theoretical references drawn from the literature, as well as from our first project. In particular, we will return to the non-linearity of the problem-solving processes highlighted and illustrated in detail in Favier [3], and to the types and articulation of the different episodes (in the sense of [22]) involved. We will then return to the different modes of reasoning, approach, and proof that can be involved in problem-solving. We will then identify these with the teachers in a variety of problems, in order to reflect collectively on the choice of problems to propose to the students, their articulation throughout the school year, and the possible traces of institutionalization. We refer in particular to Julo's work [43, 49] on memory and problem schemas to envisage long-term problem-solving education, and thus think about the articulation of the problems studied.

The following year, collaborative work will focus on the actual implementation of classroom problem-solving sessions. More specifically, the focus will be on the influence of what the teacher does on student activity, and on ways of encouraging devolution and institutionalization processes, with the aim of supporting student activity and learning. This is the "cooperation" stage. Teachers will experiment in their classrooms with the problems chosen above. On the basis of video recordings of the problem-solving sessions, the students' activity, their productions, interactions between students and teachers, and interactions between students during group work phases will be analyzed collectively and discussed during work sessions every two months, which will take place over half-days. It seems particularly interesting to us to be able to monitor students' actual activity in great detail, something that teachers do not usually have access to. This is why the data collection methodology will be based, among other things, on the use of onboard cameras attached to the students' heads. It also seems important to us to bring different types of perspectives to these classroom sessions, focusing specifically on the different issues raised in our previous work. If time allows, we'll try to share our analyses and findings during working meetings with teachers. The idea is to question the evolution of the devolution process, the initiatives taken and left to the students, their share of autonomy in the search for a solution to the problem, and the way in which the associated didactic contract is negotiated. At the same time, the institutionalization process will be studied via the dialectic between two scales of time: local institutionalization, on the scale of one

session, and the reinvestment over several sessions of what has been institutionalized previously, and the emergence of knowledge that can only be thought of over the long-term. To take this dynamic into account, we felt it wise to target problems requiring the same type of reasoning or practice. For several reasons, we have chosen to focus on problems involving an experimental approach. Firstly, these problems involve a wealth of mathematical activity on the part of the students (making trials, establishing conjectures, testing, proving). Secondly, we have observed that managing this type of problem is complex, especially given the diversity of student procedures. And lastly, these problems are rarely used in ordinary secondary school mathematics classes in Geneva.

In parallel, the training resource will be developed on the basis of the collaborative work carried out. Its aim will be to equip teachers to design, implement, and manage problem-solving sessions in their classrooms. Teachers who are interested will be able to continue collaborating with the researchers in the development phase of this resource. The content, which will be based on the collaborative work carried out and the results produced, as well as the form of the resource (video capsule or written document accompanied by videos of class sessions and their analyses in particular) will be discussed. The resource will then be distributed to other teachers for testing so that we can gather and analyze their feedback.

The aim of this project is to gain a better understanding of what happens between teaching practices and student activity when problem-solving is widely introduced into the classroom, through the study of the dual process of devolution/institutionalization. In addition, the work will lead to the production of a tool for initial and in-service teacher training. This practice should provide teachers with information on the knowledge involved in problem-solving, and on how to manage such sessions to support student learning.
