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

This PhD work [3] set out to study and characterize the mathematical problemsolving 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:


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.

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

**Figure 1.** *Schoenfeld's model ([22], p. 110).*

**Figure 2.** *Rott's descriptive model of problem-solving processes ([23], p. 106).*

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**.

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 [24–28], artificial intelligence [29–31], and mathematics education [22, 32–36]. 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 problemsolving process?"

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

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

#### **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**.

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**).

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


#### **Table 1.**

*An extract from the heuristics coding manual.*


#### **Table 2.**

*New heuristics resulting from analysis of our experimental data.*

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.

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.

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

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

#### **Figure 3.**

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


#### **Table 3.**

*Distribution of different solving processes.*

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.
