In this section, we show a characterization of the backward reasoning in the whole group, followed by a case study that illustrates the model.
4.1. Heuristics and Background Mathematical Knowledge in the Group
Backward reasoning and the relationships with heuristics and background mathematical knowledge in the whole group were studied. We split the students into two groups according to their mathematical background: Students who had a minimum mathematical background, attending the first year of their university courses (Group 1 with 99 students) and students who had a higher mathematical background, such as those who attend the upper courses of the degree or PhDs (Group 2 with 86 students).
Analyzing the students’ epistemic actions and processes of thought in which backward reasoning appeared, we identified that backward reasoning was used during the mathematical formula development (see Figure 2
) and in other four resolution moments:
Analyzing geometric properties of winning lines.
Studying the geometric positions in which most of the lines pass through.
Searching for the final geometric configuration that leads a player to win the game: This consisted of the configuration in which the winner had two half-finished winning lines at the same time. This forced the opponent to put a token in a specific position, blocking only one of the two lines.
Studying the opponent’s configuration of winning lines and predicting his/her possible movements.
Developing the mathematical formula.
According to backward reasoning features (BR column in Table 1
) while analyzing winning lines, studying favorable positions, and developing a mathematical formula, the majority of epistemic actions that were involved backward reasoning were classified as breakdowns (D); meanwhile, during searching for the final movements and blocking the opponent, the majority of epistemic actions were classified as the search for cause–effect relationships (E). In all cases, some epistemic actions were classified as the introduction of auxiliary elements (X).
From a heuristic point of view, we identified six heuristics or strategies that students developed during the resolution (Table 2
). Among them, those based on backward reasoning are called backtracking heuristics.
The data indicated that the mathematical background affected the resolution of the task. Only 63 students (10 students from Group 1 and 53 from Group 2) developed the mathematical formula (34%). To do it, they analyzed the geometric properties of the board and used their algebraic knowledge. Its development has a higher percentage of realization in Group 2 (62%) than in Group 1 (10%). A similar result appeared in the heuristics analysis when using a mathematical language, visualizing the geometrical properties of 3D board, and geometrically reducing the board size, three heuristic processes in which background mathematical knowledge is deeply involved.
4.2. Patterns of Epistemic Actions
The whole group analysis showed that 80% of the students used backtracking heuristics (see Table 2
), and five moments in which backward reasoning was developed could be identified in their thought processes (Table 1
). The first four moments belonged to the resolution discovering phase. Regarding the mathematical formula development, it was split in two phases: discovering and verifying. In the first one, the students elaborated the formula, and they justified it in the second. Backward reasoning occurred only in discovering phases.
The use of the RBC model for the analysis of the protocols allowed us to identify two common recurring sequences of epistemic actions: B–R–C and B–C sequences.
When B–R–C sequences took place, the students’ process of thought proceed in this manner:
They did an exploration/manipulation of the elements/data/concepts given by the problem.
They recognize an element/concept useful for the resolution.
They manipulated the objects that appeared in previous point in order to build a new concept.
When B–C sequences occurred, the students’ processes of thought proceed in this way:
B–R–C sequences characterized all the backward reasoning moments, while B–C sequences appeared alongside the first ones in developing a mathematical formula. In fact, B–R–C sequences were produced when students constructed the formula, while B–C sequences were produced when they manipulated it to obtain the final expression. B–C sequences also took place in the mathematical formula verifying phase where backward reasoning did not occur. Table 3
summarizes the recurring pattern of reasoning sequences according to the phases for the 185 students.
Finally, in abstraction processes, a recurring behavior (see Figure 3
) was identified in the reasoning movements throughout three different representational contexts: one related to the game, the second in which mathematical particularization was developed, and the third in which mathematical concepts were generalized. In the discovering phase, the reasoning developed back and forth through the contexts, while in the verifying phase, it had a more linear trend. In the discovering phase, the analysis showed that, since the final abstraction concept was unknown, the epistemic actions content changed according to the three contexts. Through processes of exploration and progressions–regressions between contexts, the students constructed and understood the mathematical structure underlying the concept. In the verifying phase, since the final concept was known, the reasoning ran through the concept mathematical structure which was previously built.
In Section 4.3
, through a case study, we illustrate how the microscopic analysis was carried out to reach these results. The reasoning trends in backward reasoning moments are shown through a significant example concerning the mathematical formula development and its justification.
4.3. Epistemic Actions Model: Case Study
In this section, we illustrate how the epistemic actions sequences can concretely develop through a descriptive case-study [24
]. We show the protocol of Student-A: He was key informant of the PhD students’ group (consisting of eight students). He is an expert student who solved the task by investigating the mathematical relationships that were at the basis of the game. He wrote a highly elaborate protocol, structuring it according to 35 points. We identified some breaking elements in his reasoning sequences that triggered his processes of thought. Two excerpts of his protocol are shown. They belonged to a discovering phase (mathematical formula development) and a verifying phase (formula verification) of his resolution.
The student began the game resolution by solving the 2D version of the game (Three-in-a-Skate game). First, he played while trying to remember the winning strategy, and then he started to calculate the number of winning lines. Later, he moved on to the 3D version of the game, (4,3)-hypercube, where he continued to reason out the number of winning lines until he obtained a general formula. Then, he showed that the formula he had found was valid for any cube of (n,d) dimensions, and, finally, he reasoned again about the winning strategy, this time for the 3D case.
The first excerpt (Figure 4
) refers to the discovering process that led the student to create the mathematical formula. Backward reasoning strongly characterized this process.
The student’s first objective was to count the winning lines on the board. To do so, he divided the board into planes and counted the winning lines on each of them. Then, he started to think about the found number. Breaking it down, he tried to identify some game elements (number of tokens for each winning line, size, number of diagonals) in its formulation. Later, he analyzed and broke down each plane in the same way. At this point, he introduced a recursive auxiliary pattern and conjectured the existence of a general recursive formula that related the number of winning lines with the board size. Finally, by analyzing the raw formula (line 16), he looked for a mathematical expression for each part of it. Thus, he obtained the general recursive formula. The winning lines analysis that led to the conjecture was a basic step for the formula development.
By analyzing the excerpt, it was possible to identify different epistemic actions performed by the student. Each of them could be characterized according to the backward reasoning features and the RBC model. In the following table (Table 4
), the excerpt analysis is summarized: the epistemic actions, the features of backward reasoning (BR column), and the RBC model classification (RBC column) are identified. Where backward reasoning did not take place, a hyphen is shown.
From backward reasoning point of view, the student broke down the problem and inserted auxiliary elements in an alternating way. He did it to conjecture the existence of a general recursive formula and to mathematically represent it. The actions involved in breakdown moments (D) supposed a longer time of realization, while the introduction of auxiliary elements (X) was instantaneous. Line 15 corresponds to a structural analogy; it broke Student-A’s chain of thought, thus triggering the following epistemic actions. From the RBC model point of view, a certain regularity was noticed (see last column in Table 4
): Two B–R–C sequences (B–R–C–B–R–C) characterized the formulation of the conjecture, while two B–C sequences (B–C–B–C) characterized the general formula expression.
Recognizing actions changes the nature of the reasoning. Through them, the student introduced new elements in the resolution, whether they were specific to the problem (the scheme) or mathematical constructs that were structurally analogous (the recursive pattern). These actions were of two kinds: In the former, the student created a new construct processing already encountered knowledge in the resolution, and in the latter, the new construct emerged from a structural analogy (line 15). During this analogy, the student remembered geometrical concepts on finite fields that helped him identify patterns.
When the conjecture was formulated and the raw mathematical formula had been obtained, the reasoning changed. The student focused on the conjectured formula and analyzed it element by element until the desired mathematical expression was obtained. The reasoning proceeded within B–C sequences. No recognizing actions appeared, as the reasoning developed through the elaboration of notions that were already clear in his mind. The breakdown feature that portrayed this process was not continuous, but was interspersed with moments of forward reasoning. In fact, the student derived the expression of the factor and the diagonals from the notions previously learned in the resolution and then made them explicit in a forward way.
The second excerpt (Figure 5
) refers to the verifying phase, the moment in which the student checked the just obtained mathematical formula. Backward reasoning was absent in this part, which was characterized only by synthesis processes.
The student started to test the formula using small n and d values, so that, with his acquired game knowledge and counting the lines case by case, he could easily verify the formula. Then, he thought about proving the correctness of the general formula. He began thinking about proving it by induction, but he had difficulties representing diagonal values and abandoned the idea. Later, reasoning about the 4D case, Student-A recognized an analogy in the analytical structure of the problem and associated each winning line to a direction vector. Through the winning lines, represented as direction vectors, the student was able to rigorously construct the mathematical formula found previously. He made a continuous control on his actions and realized that there was something wrong in the reasoning (line 26) He managed to overcome his difficulty thanks to the introduction of another element from previous knowledge: the action of .
As in the previous case, the different epistemic actions performed by the student were identified and classified according to the backward reasoning features and the RBC model. In the following table (Table 5
), the excerpt analysis is summarized.
As anticipated before, backward reasoning was absent in this part. The excerpt was characterized by a series of B–C sequences. They were interrupted in lines 20, 22, and 26 by recognizing actions. There were two crucial structural analogies: The analytical representation led to the justification of the mathematical formula, and the action of
allowed for the identification and the overcoming of difficulties that emerged in the resolution. The reasoning developed through the elaboration of notions deriving from the game resolution experience and the student’s background. In the following diagram (Figure 6
), the sequences of actions identified in the previous excerpts are highlighted.
If we look at the whole process of the first excerpt (Figure 7
), we can see how the student passed through different contexts in order to achieve the general mathematical formulation. He began working within the strict context of the proposed game (informal), and then he moved to a mathematical context to interpret the example; afterwards, he went forward and explained the game in a more general mathematical context. The transition between the three contexts happened with a complex back and forth process, where the different contexts were repeatedly activated. The analogies that came into play are highlighted.
Likewise, if we observe the whole process of the second excerpt (Figure 8
), we can see how the student also passed through different contexts in order to achieve the general mathematical formulation and to justify his precedent achievement also in this case. The progression was linear, from the informal context to the mathematical generalization passing through the mathematical particularization. Additionally in this case, the analogies appear in the diagram.
In both cases, analogies occurred and broke the chains of thought. They were crucial because they triggered the subsequent actions, leading Student-A to construct more general mathematical concepts.