# A Theoretical Model for the Development of Mathematical Talent through Mathematical Creativity

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## Abstract

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## 1. Introduction

Mathematical giftedness is the name we shall give to a unique aggregate of mathematical abilities that opens up the possibility of successful performance in mathematical activity (or with school children in mind, the possibility of a creative mastery of the subject).[9] (p. 77)

Talent that has not yet been developed or evidenced; that is, the potential exists to develop and demonstrate it, but due to one or more factors have been unable to manifest it in action schemes.[12] (p. 27, translation by the author)

- What type of activities foster developing creativity and, hence, mathematical talent?
- What theoretical model allows us to best analyse creative mathematical activity?
- What institutional conditions propel the development of creative mathematical activity?

## 2. Theoretical Framework

#### 2.1. Definitions: Institution, Praxeology, and Levels of Complexity

#### 2.2. Mathematical Creativity

“(a) the process that results in unusual (novel) and/or insightful solution(s) to a given problem or analogous problems, and/or (b) the formulation of new questions and/or possibilities that allow an old problem to be regarded from a new angle requiring imagination”.(p. 24)

#### 2.3. Praxeological Model of Mathematical Talent (PMMT)

- F1. Producing unique techniques: Faced with a novel task, distinct steps are produced without following a specific routine. Exploring the task presented in relation to that which is already known generates new ideas and leads to constructing new techniques, all guided by the assignment.
- F2. Optimizing the technique: Evaluating a range of possible routes that allow performance of the task, then choosing the “optimal” one as a function of the number of steps and mathematical knowledge involved. This may mean constructing a general rule (verbal, iconic, or alphanumeric) instead of using a series of drawings to determine the properties of an unknown stage in a figural sequence.
- F3. Considering tasks from diverse angles: This means analysing the task without restricting it to a certain domain (such as algebra, geometry, discrete mathematics, etc.), or discipline (physics, chemistry, visual arts, etc.), either by producing steps that permit task execution (recognizing the knowledge that motivated the subject to follow a certain path or to change direction), or generating diverse techniques to perform the same task.
- F4. Adapting a technique: This entails identifying the functioning, scope, and limits of a technique produced in order to implement it on another task after certain modifications. First, the technique must be validated (that is, verifying that its steps make it possible to do what is proposed); second, it needs to be adapted and, perhaps, improved while solving another task.

#### 2.4. Implementing the Praxeological Model ${P}_{c}=\left[T,\tau ,{\theta}_{c}^{m},\Theta \right]$

- Studying the first cases or stages and determining the relation between ${a}_{i}$and $i=1,2,\dots ,n.$
- Constructing a general rule for the $nth$ stage.
- Implementing the rule constructed in (2) for specific close and distant stages of the succession ($n=10,n=20,n=100)$.

## 3. The Study

#### 3.1. The Setting and Characteristics of the Institution

#### 3.2. Didactic Design of the MC

## 4. Results

#### 4.1. Problem Situation 2: Origami Cubes

- Task 2.1: Determine the number of cubes required to construct any stage of the sequence (different sequences of origami cubes were presented).
- Task 2.2: Construct three stages using the origami cubes following a pattern.

Technique | $\mathbf{Technologies}\text{}{\mathit{\theta}}^{\mathit{m}}$$\mathbf{and}\text{}{\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. Determine that the first combination (orange/pink) remains unchanged, and place the number 2 in the first combination up to stage 100 (Figure 2). Step 2. Count the cubes in the second combination (blue/pink) and write the numbers 2, 6 and 12, respectively, in the first three cells. Step 3. Identify a pattern based on the stages given and make drawings with squares to represent the cubes for stages 4, 5 and 6 in the two combinations (Figure 3). Step 4: Based on the drawing of stage 6, they made drawings of the increments for n = 7, n = 8, n = 9 and n = 10 (Figure 3). Next they counted and made the sums. Step 5: To obtain the result for stage 100, they multiplied the number of cubes of stage 10 by 10. The instructor asked them to prove this ‘rule of 3’ for stage 9 based on stage 3. After doing so, they abandoned this technique. Later, they returned to their attempt to count the number of cubes using drawings. | ${\theta}^{1}:$On the first three stages, they explored other close ones using drawings, but could not determine a recursive rule per se. ${\theta}_{1}:$ They optimized the technique by changing from drawing the complete figure to drawings that show only the increments (F2). ${\theta}_{2}:$ They tried to optimize (F2) by adapting a school technique called ‘the rule of 3’ to determine case 100, but failed to solve the task. |

Technique | Technologies ${\mathit{\theta}}^{\mathit{m}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. Same as for pair A. Step 2. Same as for pair A. Step 3. Based on the initial stages of the second combination, they determined that stage 1 has 2 cubes, stage 2 has $2+4=6$ cubes, and stage 3 has $2+4+6=12$ cubes. They then departed from the material and made a sum to determine the number of cubes in stage 4: $2+4+6+8=20$ cubes. Step 4. Based on this algebraic rule, they attempted to implement this for $n=100$, proposing the operation: $2+4+6+8+\cdots +200;$ however, they were unsuccessful in calculating the operation or optimizing their technique. | ${\theta}^{1}:$Based on the initial stages, they constructed an algebraic rule for distant stages as follows: ${a}_{n}=2+4+6+\cdots +2n.$ ${\theta}_{1}:$They optimized the technique (F2) by setting the cubes aside and focusing on the sum required to determine the number of cubes in the second combination. |

Technique | Technologies ${\mathit{\theta}}^{\mathit{m}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. Same as for pair A. Step 2. They related the origami cubes in combination two to the floors of a building, observing that ‘In stage 1 there are 2 columns of 1 floor, in stage 2 there are 3 columns of 2 floors, in stage 3 there are 4 columns of 3 floors’. Step 3. They determined that stage 4 of combination two has 5 columns of 4 floors. Using this recursive rule, they completed the row of combination two up to stage 100: 101 columns by 100 floors: $101\times 100=10,100$. Step 4. They noted that in the table (Figure 2) only 2 cubes must be added to the total number of cubes in combination two (Figure 4). They then calculated the sum that corresponds to the total up to stage 100, which is $10,100+2.$ Step 5. Upon the instructor’s indication, they determined another technique to solve the task by identifying that ‘squares’ are formed in combination two and calculating their respective area; that is, for the first stage, a $1\times 1$ square, for the second, a $2\times 2$ square, and for the third, a $3\times 3$ square. They then proposed that to each calculation of the product they needed to add 1 to the first term, 2 to the second, and 3 to the third (Figure 5). They concluded that: ‘the number of the stage is multiplied twice, the new stage is added, and then 2 from the other combination is added’. | ${\theta}^{1}:$Based on the initial stages, they determined an algebraic rule with two equivalent patterns of construction: ${a}_{n}=n\left(n+1\right)+2$ ${b}_{n}=n\times n+n+2$. ${\theta}_{1}:$ To optimize the technique, they generated a novel technique using a known reference; namely, a building to represent the cubes (F1, F2, F4). ${\theta}_{2}:$ They produced two different techniques to perform the same task (F3). |

Technique | Technologies ${\mathit{\theta}}^{\mathit{m}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. By manipulating the material, they determined that a row of 3 cubes was added between the stage 1 and stage 2 pyramids, and that a row of 4 cubes was added between the stage 2 and stage 3 pyramids (Figure 7). Step 2. Next, they determined that a row of five cubes had to be added below pyramid 4 and, similarly, a row of six cubes had to be added below pyramid 5. Step 3. They determined that calculating the number of cubes of any pyramid of this sequence required totalling the number of cubes of the previous pyramid and the stage + 1 (Figure 8). Using this rule, they constructed a list up to stage $n=15$. | ${\theta}^{1}:$Based on the initial stages, they determined the following recursive generalization: ${a}_{n+1}={a}_{n}+\left(n+1\right).$ θ _{1}: They optimized the technique of counting into a recursive technique (F2). |

#### 4.2. Problem Situation 4: Intersections and Regions

- Determine the number of regions into which a plane is divided by $n$ non-parallel lines, where an intersection can only be formed by two lines.

- Task 4.1: Determine the number of intersections formed on the plane when any number of non-parallel lines are placed and only two lines can form an intersection.
- Task 4.2: Determine the number of regions formed on the plane when any number of non-parallel lines are placed and only two lines can form an intersection.

Technique | Technologies ${\mathit{\theta}}^{\mathit{M}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. They used the tangible material to count the intersections for the first 6 lines, but failed to consider the non-visible intersections. The instructor pointed out this error, which led to step 2. Step 2. They made drawings of the lines on a blank sheet (Figure 11) to count all the intersections more systematically. This allowed them to count all the intersections formed by drawing seven lines. Step 3. They constructed a recursive process for the first 8 lines: 6 lines generate 15 intersections $\left(10+5\right)$; 7 lines produce 21 lines $\left(15+6\right)$. Hence, 8 lines should give $21+7=28$ intersections. They verified the number of intersections by following the conditions on the list (Figure 11) and then used this technique to reach 20 lines. Step 4. They made several attempts to find a rule based on the relation between the number of lines and the number of intersections of the stage $n=20$, and tested their proposed rules with the information for stage 4, but did not succeed in establishing an algebraic rule. | ${\theta}^{1}:$Taking the first 6 stages as the base, they considered a relation of dependence between ${a}_{n}$ and ${a}_{n+1}$, and, later, the recursive rule: ${a}_{n}={a}_{n-1}+\left(n-1\right)$. ${\theta}_{1}:$ They optimized their technique by passing from the drawings to constructing a recursive rule (F2). ${\theta}_{2}:$ They attempted to optimize their recursive technique and find a rule of algebraic generalization (F2). They also applied the verification technique presented previously (F4). |

Technique | $\mathbf{Technologies}\text{}{\mathit{\theta}}^{\mathit{m}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. Using the tangible material, they counted the number of intersections for the first 5 stages, including the intersections that do not appear visually (Figure 5). Step 2. They constructed a recursive process for the first 8 lines based on the differences between stages ${a}_{n}$ and ${a}_{n+1};$that is: with 2 lines, one intersection; with 3 lines, $1+2=3\text{}$intersections; with 4 lines, $3+3=6\text{}$intersections; with 5 lines, $6+4=10$ intersections; and with 6 lines $10+5=15$ intersections; so with 7 lines the number of intersections will be the same as with 6 lines plus 6: $15+6=21\text{}$intersections, and with 8 lines the number will be the same as with 7 lines plus 7: $21+7=28$ intersections (Figure 12). Step 3. They identified that the technique associated with the pyramid task (Figure 4) can be applied to solve the intersection task, as well, though they recalled that the earlier task was not solved. Step 4. They determined which task is solved by adding up consecutive positive whole numbers, such that ${a}_{100}=1+2+3+\cdots +99$. Step 5. They related a famous technique from Carl Friedrich Gauss (Hayes, 2006) to add up the first consecutive positive numbers (Figure 13) that the instructor had mentioned briefly in the first session. Step 6. They determined the value of the sum $1+2+3+\cdots +99,$ by forming pairs between the 10 initial and 10 final numbers, as shown here: $1+99=100;\text{}2+98=100;\text{}3+97=100;4+96=100;\text{}5+95=100;\text{}6+94=100;7+93=100;\text{}8+92=100;\text{}9+91=100;\text{}10+90=100;$ until they obtained $100\times 10=1000.$ Finally, they stated that this process must be repeated up to the number 50, but that this technique is not effective for any number of lines stipulated, so they continued trying to come up with an algebraic rule. Step 7. The instructor asked them to perform the same activity, but with 20 lines. This led to the following conclusion: taking away one (leaving 19), dividing by two (leaving 9.5), taking only the 9 (the whole part) and multiplying $9\times 20$ (making 180) plus the one in the half, leaves 190 intersections. Step 8: They verified this technique with one member’s results for 20 lines and then implemented it for 100 lines: $99/2=49.5$, taking 49, multiplying it by 100, and adding 50 from the half to obtain the result of 4,950 intersections. | ${\theta}^{1}:$Based on the first 5 cases, they considered a relation of dependence between ${a}_{n}$ and ${a}_{n+1}$ that led to the recursive rule: ${a}_{n}={a}_{n-1}+\left(n-1\right)$. ${\theta}^{2}:$They constructed two rules of algebraic generalization for $n$ lines: ${a}_{n}=1+2+\cdots +\left(n-1\right)$ ${a}_{n}=[\left(n-1\right)/2]\times \left(n\right)+\frac{1}{2}n.$ ${\theta}_{1}:$ They optimized their technique by passing from the tangible material to a recursive rule. ${\theta}_{2}:$ They recognized the applicability of a technique based on considering three types of tasks: the intersection task, the pyramid task, and the task of the sum of the first 100 positive whole numbers (F3, F4). ${\theta}_{3}:$ They adapted Gauss’ technique for the first 10 whole positive numbers for the sum of the first 99 numbers (F4). ${\theta}_{4}:$ They optimized the technique of adding pairs and constructed a rule of algebraic generalization (F2). |

Technique | Technologies ${\mathit{\theta}}^{\mathit{m}}$and ${\mathit{\theta}}_{\mathit{C}}$ |
---|---|

Step 1. Using the tangible material, they counted the number of intersections for the first 6 lines, after arranging them so that all intersections were visible. Step 2. Same as pair B, but followed this rule up to 18 lines. Based on this step, they proposed their own techniques, which were discussed at a couple different moments. Step 3. (Member 1). With the goal of detecting a pattern that would allow her/him to construct an algebraic rule, he continued to apply this recursive rule up to 55 lines (Figure 14). Step 3. (Member 2). This student related the technique associated with the pyramid cube task (Figure 4), realizing that it could be used to solve the intersection task. This student showed the instructor a sheet where –on his own initiative– he had resolved the pyramid cube task using the recursive rule ${a}_{n+1}={a}_{n}+\left(n+1\right)$one-by-one up to $n=100$. He mentioned that the numbers are the same, but ‘go two behind’; that is, ‘If the number of lines here is five, there it would be pyramid 3’. Though this student recognized that this new task is solved using pyramid 98, he continued to search for a more direct way. Step 4. (Member 1). Once this student had a sufficient number of cases (55 lines), he identified a pattern based on ‘tens’ (10, 20, 30, 40…). He explained the process on the blackboard (Figure 15): ‘The way in which it is possible to reach 45 is by multiplying 10 × 5—the half– and subtracting 5 to reach 45. I made it to 50 this way, one-by-one and verified it [algebraic rule] for 20, 30 and 40, and it worked; for example, for 20 I obtained 190, because it’s 20 × 10, which is half of 20 minus 10, and that gives 190. I did the same with 30, 40… up to 100. What I did was 100 × 50 minus 50, which is half. That gave me 4950”. | ${\theta}^{1}:$ Based on the first 6 cases, they structured a relation of dependence between ${a}_{n}$ and ${a}_{n+1}$, and later constructed the recursive rule: ${a}_{n}={a}_{n-1}+\left(n-1\right)$. ${\theta}^{2}:$ They constructed a rule of algebraic generalization to find the number of intersections of $n$ lines: ${a}_{n}=n\left(\frac{n}{2}\right)-\frac{n}{2}$. ${\theta}_{1}:$ They optimized their technique by passing from the tangible material to a recursive rule (F2). ${\theta}_{2}:$Member 2 recognized the applicability of a technique with respect to two task types: pyramid cubes and line intersections. He identified the similarities and differences between the tasks in order to implement the technique (F3, F4). ${\theta}_{3}:$ Member 1 optimized his recursive technique and produced a unique technique based on ‘tens’. He verified that this new technique worked for other cases (F1, F2). |

## 5. Discussion

**,**technologies that come into play), together with their interrelations during the creative mathematical activity. It then characterizes this activity on the basis of the creative technological functions that emerge as protagonists during the analysis of results.

## 6. Conclusions and Future Research

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Pair C’s reconfiguration of the sequence in Figure 1.

**Figure 5.**Pair C’s second reconfiguration of the sequence of Figure 1.

5th Grade | 6th Grade | |||
---|---|---|---|---|

Sex | 10 Years | 11 Years | 12 Years | Total |

Female | 2 | 8 | 2 | 12 |

Male | 3 | 8 | 3 | 14 |

Total | 5 | 16 | 5 | 26 |

Problem Situation | Tasks |
---|---|

1. Sierpiński’s triangle | Task 1.1: Determine the change in the number of triangles while building Sierpiński’s triangle. Task 1.2: Determine changes in the area while building Sierpiński’s triangle. Task 1.3: Determine changes of the perimeter while building Sierpiński’s triangle. |

2. Origami cubes | Task 2.1: Determine the number of cubes required to construct any stage of the sequence (different sequences of origami cubes were presented). Task 2.2: Construct three stages using the origami cubes following a pattern. |

3. Tables and chairs | Task 3.1: Determine the number of chairs that can be arranged at any number of tables. Task 3.2: Determine the number of squares required to build any stage of the sequence. Task 3.3: Determine the number of shaded squares required to construct any image. |

4. Intersections and regions | Task 4.1: Determine the number of intersections formed on a plane when any number of non-parallel lines are placed but only two lines can form an intersection. Task 4.2: Determine the number of regions formed on a plane when any number of non-parallel lines are placed but only two lines can form an intersection. |

5. Tower of Hanoi | Task 5.1: Determine the minimum number of movements with any number of disks. |

6. Folding paper | Task 6.1: Determine the total thickness of the folded sheet for any number of folds. Task 6.2: Determine the surface area of the sheet for any number of folds. Task 6.3: Determine the perimeter of the surface of the sheet for any number of folds. |

Term | Combination Pink/Orange | Combination Pink/Blue | Total Cubes | Increase of Cubes between Stages |
---|---|---|---|---|

1 | 2 | $2$ | $4$ | |

2 | 2 | $6$ | $8$ | $+4$ |

3 | 2 | $12$ | $14$ | $+6$ |

4 | 2 | 20 | 22 | $+8$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$n$ | $2$ | $n\left(n+1\right)$ | $n\left(n+1\right)+2$ | $+2n$ |

Lines | $\mathit{a}\left(\mathit{n}\right)$ | Increase of Regions |
---|---|---|

1 | $2$ | |

2 | $4$ | $+2$ |

3 | $7$ | $+3$ |

4 | 11 | $+4$ |

$\vdots $ | $\vdots $ | $\vdots $ |

$n$ | $1+\frac{n\left(n+1\right)}{2}$ | $+n$ |

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**MDPI and ACS Style**

Barraza-García, Z.M.; Romo-Vázquez, A.; Roa-Fuentes, S.
A Theoretical Model for the Development of Mathematical Talent through Mathematical Creativity. *Educ. Sci.* **2020**, *10*, 118.
https://doi.org/10.3390/educsci10040118

**AMA Style**

Barraza-García ZM, Romo-Vázquez A, Roa-Fuentes S.
A Theoretical Model for the Development of Mathematical Talent through Mathematical Creativity. *Education Sciences*. 2020; 10(4):118.
https://doi.org/10.3390/educsci10040118

**Chicago/Turabian Style**

Barraza-García, Zeidy M., Avenilde Romo-Vázquez, and Solange Roa-Fuentes.
2020. "A Theoretical Model for the Development of Mathematical Talent through Mathematical Creativity" *Education Sciences* 10, no. 4: 118.
https://doi.org/10.3390/educsci10040118