Improving the Teaching of Mathematical Problem Solving – A Collaborative Research Based on Theoretical Findings from Two PhD Dissertations

Abstract

This text briefly presents two PhDs achieved as part of a larger research project led by the team in mathematics didactics in Geneva on mathematical problem-solving at different school levels. The first PhD, by Maud Chanudet, examines teachers’ assessment practices in the context of a mathematics problem-solving course offered at a lower secondary level in Geneva. The second PhD, by Stéphane Favier, aims to document the work of students at different school levels (primary and lower secondary) in problem-solving situations, through detailed analyses of students’ work and interactions with each other and with the teacher. These two studies revealed ways to help teachers support students’ problem-solving activities. This led us to set up a collaborative project with teachers to study two complementary processes—devolution and institutionalization—with a view to developing a training resource. We’ll briefly introduce this new stage of our research.

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 problem-solving 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.

These two PhDs have led to questions and calls for further research into how to devolve problem-solving and what can be institutionalized at the end of such sessions. However, this work has also revealed ways of supporting students’ problem-solving activities. This has led us to develop a new research project, with one strand at the primary level, and the other at lower secondary level, both aimed at setting up collaborative work with teachers to study two complementary processes—devolution and institutionalization—with a view to developing a resource for training, based on the results of our previous research. We will briefly outline the reasons that led us to this new project, and what we are currently doing to implement it.

2. Study of assessment practices in problem-solving

In the canton of Geneva, in the French-speaking part of Switzerland, grade 8 students (13–14 years) in the science stream have a 45-minute period per week dedicated to mathematical problem-solving. As part of this course, entitled “Démarches mathématiques et scientifiques” (mathematical and scientifical approaches (MSA), teachers are required to assess their students’ problem-solving skills on a certifying basis. However, the instructions governing this course are vague, leaving teachers a great deal of freedom as to how they wish to organize and manage their teaching and the associated assessment. It is in this context that Chanudet has studied the ordinary practices of teachers, focusing in particular on their assessment practices. Her work focuses on an analysis of the mathematics teachers make their students encounter, and the way they organize and manage such sessions dedicated to problem-solving, with a focus on the assessment dimension of their practices.

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 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.

2.1.2 Assessing student learning

The study of the assessment dimension of teachers’ practices is based here on a characterization of assessment activity according to four invariants [9]:

• Defining the object of assessment, with particular reference to school curricula;

• Gathering information related to what the teacher is seeking to assess, formally or informally, instrumentally or not, in writing or orally;

• Interpretation of this information, usually using assessment criteria;

• Decision-making and communication, are in line with the function assigned to the assessment.

Research focuses on two functions of assessment: certifying, i.e., whether or not the student has acquired the targeted learning at the end of the associated teaching, and formative, i.e., assessment that serves student learning and aims to advance ongoing learning.

The broader conception of formative assessment emphasizes the possibility of informally gathering information about students’ activity, for example during discussions with them [10]. Informal formative assessment practices thus aim to provide information, generated in the course of everyday activities, on student learning [11]. The term interactive episodes, taken from Kiwan Zacka [12], characterizes moments of verbal interaction between students and teacher in the classroom, during which the teacher captures information about students’ activity, conceptions, and difficulties. They are an object of analysis in Chanudet’s PhD.

Moreover, regulation is a key concept in formative assessment. The term “formative assessment practices” used in this research refers to all teaching practices designed to support and encourage the regulation of student learning. The work of Allal [13] highlights various contextual elements that can be associated with such regulation: the structure of the learning situation, involving in particular the tasks proposed, their articulation and work methods; the teacher’s interventions and interactions with students; and interactions between students. The first two levels are examined in particular in the PhD.

2.2 The theoretical framework of the dual didactic-ergonomic approach

The general theoretical framework of Chanudet’s PhD is based on the study of practices using the dual didactic-ergonomic approach developed by Robert and Rogalski [14, 15]. This framework invites us to take into account both the generic, by looking at the constraints weighing on the teacher as he or she exercises his or her profession, and the individual, by considering the margins of freedom remaining to him or her and the way in which he or she invests them. Five components are highlighted by the authors to study teachers’ practices. The first two, cognitive and mediative, are linked to observations of classroom sessions and possible student activities. The cognitive component relates to the mathematical content presented to students (tasks, their articulation, etc.) and provides information on what the authors call the cognitive itinerary proposed to students. The mediative component focuses on the progress of classroom sessions (forms of work, assistance provided by the teacher, exchanges, and interactions with students). The combination of these two components enables us to describe the mathematics that teachers introduce their students to and to identify the logic of their actions. To integrate the fact that teachers’ practices are the expression of their work, three other components need to be taken into account: social, institutional, and personal. The social component takes into account the social dimension of the teacher’s work (students, parents, colleagues). The institutional component concerns common constraints (programs, timetables, textbooks). The personal component incorporates everything that is specific to the teacher as an individual (his or her representations of the profession, mathematics, knowledge, experience, etc.).

2.3 Research question and methodological guidelines

The various theoretical references mentioned above make it possible to formulate the macro-research question: what evaluative logics of action emerge from teachers’ practices in the context of problem-solving education in mathematics?

The analyses carried out during the PhD work to answer this question took place on the scale of a school year, with the study of education projects, the study of problems proposed for assessment, their articulation with other problems, and the assessment criteria grids used by teachers; but also on the scale of a sequence and, even more finely, of a session, targeting the actual implementation of problems in the classroom, in connection with informal formative assessment and the study of interactive episodes.

Below are some of the results and contributions of the PhD, opening up new perspectives for research and work with teachers.

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 problem-solving 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 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.

3. Study of problem-solving processes through trials and adjustments

This PhD work [3] set out to study and characterize the mathematical problem-solving practices used by primary and lower secondary school students. In the scientific literature, various characterizations of problem-solving processes can be found. The seminal work is that of Pólya [21], who proposes a linear model based on four successive stages:

• Understanding the problem;

• Devising a plan;

• Implement the plan;

• Examine the resulting solution.

The linearity of this model was challenged by Schoenfeld (see Figure 1) [22]. The latter added an exploration phase to account for the part of the search that moves away from problem appropriation, but which does not yet constitute a plan. In his view, this exploration phase in particular accounts for the unstructured part of the search. This model thus presents a cyclical characteristic in the sequence: Appropriation – Planning – Exploration or Planning – Exploration – Planning.

Figure 1.

Schoenfeld’s model ([22], p. 110).

Recently, Rott [23] has put these models to the test by analyzing the work of 10–12-year-old volunteer problem-solvers in his research laboratory. His results show that Pólya’s and Schoenfeld’s models are not sufficient to account for the complexity of the phenomena, so he proposes to enrich these models by highlighting the greater complexity of the links between the different phases, as shown in Figure 2.

Figure 2.

Rott’s descriptive model of problem-solving processes ([23], p. 106).

The scientific literature also highlights a second important aspect of the processes involved in problem-solving, namely the role of heuristics. This concept is explored in the field of psychology [24, 25, 26, 27, 28], artificial intelligence [29, 30, 31], and mathematics education [22, 32, 33, 34, 35, 36]. We have adopted Rott’s definition, based on a wide-ranging review of the literature in these fields:

“Heuristics is a collective term for devices, methods, or (cognitive) tools, often based on experience. They are used under the assumption of being helpful when solving a problem (but do not guarantee a solution). There are general (e.g., ‘working backwards’) as well as domain-specific (e.g., ‘reduce fractions first’) heuristics. Heuristics being helpful regards all stages of working on a problem, the analysis of its initial state, its transformation as well as its evaluation. Heuristics foster problem solving by reducing effort (e.g., by narrowing the search space), by generating new ideas (e.g., by changing the problem’s way of representation or by widening the search space), or by structuring (e.g., by ordering the search space or by providing strategies for working on or evaluating a problem). Though their nature is cognitive, the application and evaluation of heuristics is operated by metacognition.” ([36], p. 190).

Thus, given the theoretical background we have just presented, we can divide our first research question: “How can we characterize the solving processes implemented by students in the usual conditions of the classroom?”, into the following two sub-questions: “To what extent do the various models enable us to describe the work of students solving mathematical problems in the ordinary context of the classroom?” and “What role do heuristics play in the dynamics of the problem-solving process?”

In what follows, we discuss methodological aspects before briefly presenting some of the results of our work.

3.1 Methodology

Our research focuses on three different grades 2, 6, and 8 of compulsory school in the canton of Geneva. For each grade, we proposed a problem that could be solved by trial and adjustment. For example, for two graders, we proposed the Card Game problem: “Each card in my deck represents either a triangle or a square. I pick 15 cards at random. I count all the sides of the figures drawn on the cards I’ve picked and find 49. How many triangles and squares do you think I picked?

In our research, we preferred that the students work in groups, after a few minutes of individual research, so as to be able to analyze their oral exchanges. However, a major difficulty lies in collecting data as close as possible to the student’s work. For this reason, we equipped one student per group with an individual head-mounted action camera. This method of data collection provides first-person images [37]. For his part, the teacher is equipped with a tie microphone, and his interventions are recorded using an external camera controlled by the researcher.

The audiovisual data collected for each group of students were coded using the analysis framework developed by Schoenfeld [22]. This consists of dividing the students’ work into macroscopic blocks that he calls episodes: “An episode is a period of time during which an individual or a problem-solving group is engaged in one large task or a closely related body of tasks in the service of the same goal.” (p. 292). Schoenfeld thus distinguishes several categories of episodes: reading, appropriation, exploration, planning/implementation, and verification. In addition, Rott [23, 38] has introduced two further episodes, which we have also taken up: writing and digression.

Furthermore, the fact that, in our study, students solved problems in class, and not in laboratory conditions, led us to introduce an additional episode that allows us to characterize the moments when teacher and student(s) interact together about problem-solving. We called such episodes: regulation.

With regard to analysis in terms of heuristics, we have drawn up a coding manual that lists the various heuristics from the scientific literature that comply with the definition we have adopted. An extract is given in Table 1.

Draw, diagram, figure, graph

Organize data and trials in a particular form (rows, columns, table)

Reformulate the problem

Introduce names or notations

Perform a trial

Generate new data systematically

Table 1.

An extract from the heuristics coding manual.

This initial list was put to the test by certain experimental data, which led us to supplement it with the following three heuristics (Table 2).

Recopy or highlight certain data

Introduce artifacts or materials

Search for information (textbook, Internet, etc.)

Table 2.

New heuristics resulting from analysis of our experimental data.

In our coding process, we have marked each occurrence of each heuristic as a type of occurrence code [39]. It is therefore possible for the same heuristic to be coded several times throughout the video or, on the contrary, never to appear at all. We also had to deal with a difficulty inherent in the duration of a heuristic, which has an impact on temporal coding. Indeed, some heuristics (e.g. copying or highlighting certain data) have a fairly wide span, while others (e.g. introducing names or notations) are very punctual. Moreover, for the same heuristic (e.g. Make a trial), we can be confronted with very different durations, depending on how the students implement them. In all cases, we have chosen to code as an occurrence the moment when the students evoke the heuristic, or the start of the action when they implement it without evoking it beforehand, so that duration is not taken into account.

The video data were coded independently by a research assistant and Favier. We analyzed the work of 33 groups in terms of episodes, and 17 groups (of these 33) for heuristics. For each aspect, we compared our coding results, and when these did not coincide, we reached a consensus by recording together. In terms of inter-coder agreement, we calculated a percent agreement (Jacobs et al., 2003) equal to 0.76 for the nature of episodes and 0.82 for time codes. For the coding of heuristics, the inter-coder agreement is 0.73 for their nature and 0.85 for time codes. As Tinsley and Weiss [40], referring to Guttman et al. [41], point out, “here was a ‘tacit’ consensus that 65% represented the minimum acceptable agreement”. Thus, the inter-coder agreement percentages calculated for these different encodings seem quite acceptable.

3.2 Results

In connection with our first research question, we compared the coding results with the linear (Pólya’s model) or cyclic (Schoenfeld’s model) or neither linear nor cyclic (Rott’s model) characteristics. It turns out that the work of only 7 of the 33 groups can be described with these characteristics (2 linear, 3 cyclic, and 2 neither linear nor cyclic).

For the other groups, our analyses reveal numerous episodes of regulation. During these episodes, problem-solving continues to progress from the students’ point of view, so it is necessary to take them into account when describing and characterizing their work. However, because of the teacher’s intervention, these moments cannot be considered to be of the same nature as the other phases of the model. We, therefore, propose to enrich Rott’s model [23] by adding an additional dimension called “Regulation”. The 3D representation (Figure 3) is an interesting way of symbolizing that this additional dimension lies on another plane. Our observations show that this Regulation can be connected to all the other phases of the model. Double arrows represent these various possible connections.

Figure 3.

Proposal for a model to describe students’ solving processes in the presence of teacher intervention.

The distribution of the various solving processes (Table 3) shows that the vast majority of solving processes are non-linear, for both primary and secondary school students.

Solving process	Linear	Cyclic	Other	total
Without regulation	2	3	2	7
With regulation(s)	3	6	17	26
Total	5	9	19	31

Table 3.

Distribution of different solving processes.

With regard to heuristic analysis, we have operationalized concepts linked to the idea of problem space [42], namely Julo’s representation construction processes [43] and the path through semantic spaces [44]. We cannot go into detail here, but we have been able to identify three student profiles. The first, which we have called the explorer, shows a certain persistence in the search before considering and changing tracks. Students in this profile have a good track record of problem-solving. In contrast, students in the butterflyer profile tend to carry out a more superficial search, with numerous avenues considered without really investigating them. Nevertheless, these students have a good capacity for imagining different ideas. However, these students are not really successful. Finally, a few groups of students combine characteristics of each of the two previous profiles, i.e., they consider different avenues without investigating them before pursuing one. We have labeled students in this profile as prospectors, they present a balanced rate of successes and failures.

3.3 Perspectives

These research results show the inadequacy and unsuitability of the models used in education, the main and best-known of which is Pólya’s linear model. Moreover, the significant role of heuristics in problem-solving is confirmed, since in our study they enabled us to discriminate between different student profiles. We therefore suggest that heuristics could be an interesting lever, available to teachers, to help students solve problems. Our new research project is a way to transpose the results of our two PhDs into a tool for teachers wishing to teach problem-solving. This is what we will now present.

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.

• Because of their complexity, problem-solving skills can only be learned over the long term. Similarly, the devolution and institutionalization processes associated with students’ research skills also need to be thought through and studied from a long-term perspective.

• For all the above reasons, the implementation of these research findings by teachers is a delicate matter. We hypothesize that only collaborative research can lead to the adoption of such “tools”, and their improvement and also enable the processes at play in the classroom to be documented.

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.

5. Conclusion

Problem-solving plays an important role in mathematics and in school curricula. However, studies show that its teaching is complex. The aim of all our work is to gain a better understanding of the interplay between teachers’ practices and students’ learning around problem-solving in mathematics. More specifically, we are looking at how teachers can help students better appropriate a problem, without overdoing it for them.

Chanudet’s dissertation has shed light on the practices of secondary school teachers when teaching mathematical problem-solving. We will take up this work again with a study of some primary school teachers. Favier’s dissertation has given us a better understanding of how students work when solving problems through trials and adjustments. We intend to extend his work to other types of problems.

Our collaborative project with secondary school mathematics teachers (with plans to collaborate with primary school teachers at a later date) aims to gain a better understanding of what students do in the classroom when they solve a problem, and what can be done to support and encourage their work.

We are using the design and implementation of sessions and the analysis of their progress to devise an in-service training system designed to equip teachers.

Our work is based on a collaborative research approach, which we believe is conducive to a better appropriation of the results obtained. Various theoretical frameworks from the didactics of mathematics are mobilized to construct the research devices and analyze the data collected.