# Backward Reasoning and Epistemic Actions in Discovering Processes of Strategic Games Problems

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## Abstract

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## 1. Introduction

## 2. Theoretical Elements

#### 2.1. Backward Reasoning

- Direction vs. cause–effect: In Pappus’ definition, the backward direction of reasoning is highlighted. By applying the method, the premises of a certain idea are sought. In the 17th and 18th centuries, authors such as Arnauld and Nicole interpreted the method as a backward search for cause–effect relationships between ideas. By these, the connection between the notions in background and the problem can be identified [11,17].
- Breakdown: According to Plato and Pappus, this kind of reasoning allows for the reduction of the problem to its simplest components. The properties that define the assignment and the relationships between the objects involved in it are identified by extracting and investigating the principles that are at the base of the task. Aristotle, for example, underlined the fact that “sometimes, to solve a geometrical problem, you can only analyze a figure,” breaking it down into its basic components and understanding the different parts of it [17].
- Introduction of auxiliary elements: Kant, Polya, and Hintikka focused their attention on a fundamental part of the process: the introduction of new elements (a well know practice in geometry with the auxiliary constructions). In progressive and deductive processes, all the bases are given, and from these, consequences are elaborated. Instead, in backward reasoning new notions appear and develop throughout the resolution according to the needs of the solver [1,17].

#### 2.2. Epistemic Actions

- The actions that correspond to backward reasoning features from the mathematical point of view. These actions are not only cognitive but also involve the qualification provided by mathematical knowledge (Section 2.1).
- The actions identify as problem-solving heuristics that are inherent to strategy games [18].
- The actions as defined in AiC theory, the RBC model. These actions give priority of cognitive processes involved in thinking processes (Section 2.3).

#### 2.3. RBC Model

- Recognizing: Recognizing some previously learned knowledge as relevant for the resolution of the problem.
- Building-with: Combining a set of knowledge with the aim to implement a strategy, to meet a conjecture or justification, or to find a solution to the problem.
- Constructing: Assembling and integrating the previous knowledge with the aim to produce a new construct.

## 3. Research Objective and Methodology

#### 3.1. Objective

- What epistemic actions, in terms of heuristics use and backward reasoning features, are involved solving the strategy game?
- What epistemic actions, in terms of chain of thought, are implied in the abstract process of mathematical formula development?
- How do epistemic actions generate a recurring model in discovering processes following the RBC model?

#### 3.2. Study Group and Presentation of the Task

**Proof**.

#### 3.3. Methodology and Data Analysis

#### 3.4. Reliability and Validation of Data Analysis

## 4. Results

#### 4.1. Heuristics and Background Mathematical Knowledge in the Group

- 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.

#### 4.2. Patterns of Epistemic Actions

- 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.

- They did an exploration/manipulation of the elements/data/concepts given by the problem.
- They manipulated the objects given by the problem in order to build a new concept.

#### 4.3. Epistemic Actions Model: Case Study

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) (3,2)-hypercube with its winning lines and (

**b**) (5,2)-hypercube with the (3,2)-hypercube extended winning lines.

**Table 1.**Backward reasoning and academic level groups. BR: backward reasoning; D: breakdown; X: auxiliary elements; E: cause–effect relationship.

BR | Whole Group | Group 1 | Group 2 | |
---|---|---|---|---|

Analyzing winning lines | D and X | 55% | 44% | 67% |

Studying favorable positions | D and X | 81% | 72% | 91% |

Searching for the final movements | E and X | 56% | 58% | 53% |

Blocking the opponent | E and X | 75% | 71% | 79% |

Developing a mathematical formula | D and X | 34% | 10% | 62% |

Heuristics | Whole Group | Group 1 | Group 2 |
---|---|---|---|

Making a systematic study of all geometric positions | 81% | 70% | 93% |

Using a mathematical language | 56% | 31% | 85% |

Visualizing the geometrical properties of 3D board | 43% | 29% | 58% |

Reducing geometrically the board size | 56% | 55% | 58% |

Making diagrams, drawings, graphic representations | 73% | 67% | 80% |

Using backtracking heuristics | 80% | 76% | 85% |

**Table 3.**Pattern of reasoning sequences in the whole group according to the phases. C: constructing.

Whole Group | Discovering Phase | Verifying Phase | |
---|---|---|---|

Analyzing winning lines | 55% | B–R–C | - |

Studying favorable positions | 81% | B–R–C | - |

Searching for the final movements | 56% | B–R–C | - |

Blocking the opponent | 75% | B–R–C | - |

Developing a mathematical formula | 34% | B–R–C and B–C | B–C |

**Table 4.**First excerpt of Student-A protocol: analysis of epistemic actions. RBC: recognizing, building-with, and constructing; FS: backward steps.

Protocols | Epistemic Action | BR | RBC |
---|---|---|---|

12 | Splitting the board into planes | D | B |

Counting the winning lines in each plan | D | B | |

Grouping winning lines into a scheme | X | R | |

13 | Mathematically breaking down a number | D | B |

Identifying each element of the decomposition | X | C | |

14 | Mathematically breaking down a number | D | B |

15 | Analogy/break motion | - | R–B |

16 | Introducing a recursive pattern | X | R |

Conjecturing: general recursive formula | FS | C | |

17 | Breaking down the formula into its elements | D | B |

Analyzing the factor element | D | B | |

Representing the factor in relation to the board dimensions | D | C | |

18 | Analyzing the diagonal element | D | B |

19 | Representing the diagonals in relation to the vertices of the hypercube | D | C |

Expressing the general formula | FS | C |

Protocols | Epistemic Action | BR | RBC |
---|---|---|---|

20 | Checking the formula | - | B |

Identifying possible proof path | - | R | |

21 | Checking the formula in 4D case | - | B |

Representing the 4D case | - | C | |

22 | Identifying a structural analogy with analytical representation | - | R |

Reasoning about possible coordinates | - | B | |

23 | Identifying the direction vector | - | C |

24 | Analyzing ${\epsilon}_{i}=0$ cases Introducing a recursive pattern | - | B |

Justifying “$nd\xb7L\left(n,d-1\right)$” | - | C | |

25 | Analyzing ${\epsilon}_{i}=\pm 1$ cases | - | B |

Constructing diagonals value: ${2}^{d}$ | - | C | |

26 | Identifying an action of ${\mathbb{Z}}_{2}$ | - | R |

Constructing diagonals value: ${2}^{d-1}$ | - | C |

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**MDPI and ACS Style**

Barbero, M.; Gómez-Chacón, I.M.; Arzarello, F.
Backward Reasoning and Epistemic Actions in Discovering Processes of Strategic Games Problems. *Mathematics* **2020**, *8*, 989.
https://doi.org/10.3390/math8060989

**AMA Style**

Barbero M, Gómez-Chacón IM, Arzarello F.
Backward Reasoning and Epistemic Actions in Discovering Processes of Strategic Games Problems. *Mathematics*. 2020; 8(6):989.
https://doi.org/10.3390/math8060989

**Chicago/Turabian Style**

Barbero, Marta, Inés M. Gómez-Chacón, and Ferdinando Arzarello.
2020. "Backward Reasoning and Epistemic Actions in Discovering Processes of Strategic Games Problems" *Mathematics* 8, no. 6: 989.
https://doi.org/10.3390/math8060989