A Theoretical Model for the Development of Mathematical Talent Through Mathematical Creativity

education sciences

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

Zeidy M. Barraza-García 1,* , Avenilde Romo-Vázquez 1 and Solange Roa-Fuentes 2 1 Programa de Matemática Educativa, CICATA-Instituto Politécnico Nacional, CDMX 11500, Mexico; [email protected] 2 Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga 680002, Colombia; [email protected] * Correspondence: [email protected]

Received: 31 March 2020; Accepted: 17 April 2020; Published: 23 April 2020

Abstract: This study was conducted from a perspective that adopts a broad vision of mathematical talent, defined as the potential that a subject manifests when confronting certain types of tasks, in a successful way, that generate creative mathematical activity. To analyse this, our study proposes a Praxeological Model of Mathematical Talent based on the Anthropological Theory of Didactics and the notion of mathematical creativity, which defines four technological functions: (1) producing new techniques; (2) optimizing those techniques (3); considering tasks from diverse angles; and (4) adapting techniques. Using this model, this study analyses the creative mathematical activity of students aged 10–12 years displayed as they sought to solve a series of infinite succession tasks proposed to encourage the construction of generalization processes. The setting is a Mathematics Club (a talent-promoting institution). The evaluation of results shows that the Praxeological Model of Mathematical Talent allows the emergence and analysis of mathematical creativity and, therefore, encourages the development of mathematical talent.

Keywords: mathematical potential; mathematical talent; mathematical creativity; generalization

1. Introduction For several decades, diverse models and definitions of talent have been proposed in the literature. Singer, Sheffield, Freiman and Brandl point out that most of those models and definitions describe talent as the potential to successfully perform a certain activity, and indicate that while a broad range of models of mathematical talent exist, they are still limited [1]. Research devoted specifically to defining mathematical talent emphasize three areas of study: (1) the characteristics of subjects with talent [2–4]; (2) differences in the characteristics of “talented” vs. “good” students [5,6]; and (3) perspectives on creativity as a fundamental element of talent [7,8]. Generally-speaking, however, mathematical talent has been defined on the basis of the subjects’ abilities. Krutetskii specified that:

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)

This perspective on talent stresses the importance of creative dominion in mathematical activity. In contrast to other postures that, as Leikin points out, associate mathematical talent with the innate characteristics of certain individuals, our work seeks to demonstrate that talent is not a “given” quality, but can be developed under certain circumstances and with adequate accompaniment [10,11]. This leads to the following deﬁnition of potential talent:

Educ. Sci. 2020, 10, 118; doi:10.3390/educsci10040118 www.mdpi.com/journal/education Educ. Sci. 2020, 10, 118 2 of 21

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)

This definition underscores a transcendental aspect of talent; namely, that “it is developable” [11,13], though it is not clear how this development can be achieved. Studies have revealed certain characteristics strongly-related to talent, such as creativity, which is defined as a person’s capacity to produce new ideas, focuses or actions and then manifest them by moving from thinking to reality [14]. At the professional level, mathematical talent is understood as outstanding performance that leads to discoveries and so is closely related to mathematical creativity [10]. Research has demonstrated that developing creative thinking in mathematics is closely connected to the progress of society [15]. In schools, then, mathematical creativity should be included when developing mathematical talent to prepare students to be innovative and reflexive leaders in diverse fields of knowledge that can promote new methods and atypical perspectives in the evolution of science [1]. Leikin sustains that research on the education of subjects with talent and creativity should be oriented by two interrelated aspects: one theoretical, the other applied [16]. The former seeks to understand the nature of creativity and mathematical talent, while the second seeks to foster mathematical creativity and develop mathematical talent. This requires, on the one hand, confronting open, challenging, novel (non-routine) problems [2,4,9,17–20] and tasks that lead students to problem posing [20], and, on the other, determining the conditions necessary to develop creative thinking, such as forging connections with previously-learned material [10], taking distinct paths to solve the same task [8], or discovering the multiple solutions that a task may admit [21]. The latter entail the development of flexibility, a trait long recognized as an essential component of mathematical talent [2,6,8,9,22–26]. These elements are related to the medium of students’ development and their teachers’ or tutors’ didactic and disciplinary knowledge [17,27]. Two current issues stand out on how to nurture talent: differentiated attention, which entails characterizing and identifying subjects with talent [28,29], and an inclusive focus that develops didactic proposals for heterogeneous groups [13,30–33]. These antecedents lead to the following affirmations: (1) mathematical creativity is closely-related to talent; (2) talent is developable; and (3) theoretical models need to be produced to understand the conditions that will make such development possible. As a result, the research questions that guided our study were:

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?

To examine these issues, this work established relations between elements of the Anthropological Theory of Didactics (ATD) and the concept of mathematical creativity that led this study to propose a theoretical approach called the Praxeological Model of Mathematical Talent (PMMT). This instrument is elucidated in the following section.

2. Theoretical Framework This section ﬁrst presents elements of the ATD, an approach that analyses the institutional dimension of human activity [34]. Second, it clariﬁes the notion of creativity [8,35–37]. These are the basic elements this study utilized to propose a theoretical model of mathematical talent.

2.1. Definitions: Institution, Praxeology, and Levels of Complexity An institution is defined as “a stable social organization that makes it possible to efficaciously confront problematic tasks thanks to the material and intellectual resources it provides to subjects, and the restrictions defined for executing those tasks” [38] (p. 85, translation by the author). In Educ. Sci. 2020, 10, 118 3 of 21 this sense, a program of attention to talent (e.g., a science fair or Olympic club) can be considered institutions because they offer subjects conditions and resources to confront problems that are framed as elements of human activity. The ATD analyses this using the four components of praxeology, that is, task type, T; technique, τ; technology, θ; and theory, Θ. Task type (T) refers to “what is done”; technique (τ) is “how it is done”; technologies (θ) are “the discourses that produce, explain, and validate the techniques”; and theory (Θ) refers to “broader discourses that produce, explain, validate, and justify the technologies” [39]. Praxeologies have different levels of complexity. The first, and most elemental, is called “punctual” [T, τ, θ, Θ], which is defined as having a task type, T; a technique, τ; a technology, θ; and a theory, Θ. The second is called “local”, and includes various punctual praxeologies [Ti, τi, θ, Θ], but the same technology, θ; for example, finding the joint solution of a system of lineal equations using only the Gauss-Jordan technique that corresponds to a punctual praxeology, while considering all available techniques (equalization, elimination, substitution, iterative methods, etc.) to find the joint solution of that system would constitute a local praxeology. In this case, the regional praxeology would be Lineal Algebra, the global praxeology is Algebra per se, and the disciplinary praxeology, the field of mathematics. One postulate of the ATD is that producing mathematical activity entails constructing, or reconstructing, at least one local praxeology [39], so any proposal for developing mathematical talent must include this element of praxeology; that is, a praxeology in which tasks can be performed using techniques produced by the same mathematical technology. In order to identify the characteristics of creative mathematical praxeologies—that is, those in which creativity plays a specific role—the following section analyses the notion of mathematical creativity.

2.2. Mathematical Creativity In Mathematical Education, the characterization and identification of creativity is often related to four components: fluency, flexibility, originality [8,36,37], and elaboration [35]. ‘Creativity’ is a term widely used to refer to the capacity to produce new ideas, focuses, or actions, and then manifest them by transiting from thought to reality [14]. Sriraman, one of the author/editors of the book, Creativity and Giftedness, proposes a significant differentiation between creativity at the professional and school levels [40]. There, professional mathematical creativity is defined as the “(a) the ability to produce original work that significantly extends the body of knowledge, and/or (b) the ability to open avenues of new questions for other mathematicians” (p. 23), while school-level mathematical creativity refers to:

“(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)

Our approach adopts the second conceptualization to deﬁne the theoretical model of mathematical talent that is described in detail in the next section.

2.3. Praxeological Model of Mathematical Talent (PMMT) The PMMT considers the aforementioned, classic components of praxeology: task type, T; technique, τ; technology, θ; and theory, Θ; but distinguishes two elements of technology, one mathematical, the other creative. This study associates the technological mathematical component, θm, with what other conceptual frameworks classify as mathematical abilities (reasoning mathematically, economizing thought, expressing logical and sequential thought, abstraction, and generalization, among others); elements associated mainly with mathematical institutions. The creative technological component θc, in contrast, relates to the production of unique, unusual, ﬂexible or perspicacious techniques that integrate elements from experiences in various institutions, such as family, school, the street, and clubs, among others. m The PMMT is represented as a creative praxeology in which Pc = [T, τ, θc , Θ]. In order to analyse m the creative mathematical activity, in this model, the technology (θc ) is divided by two components: Educ. Sci. 2020, 10, 118 4 of 21

The mathematical component (θm) corresponds to the mathematical technology from the classic model [T, τ, θ, Θ] sustained by mathematical theory, while the second (θc) reﬂects creativity understood through the following four technological functions:

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.

These functions are closely interrelated during the solving process; for example, the function of producing unique techniques (F1) may emerge when adapting a technique (F4), while considering tasks from diverse angles (F3) may generate unique techniques (F1). Hence, these functions do not follow a ﬁxed order during their development, but are assumed as key elements that permit the analysis of creative mathematical activity.

m 2.4. Implementing the Praxeological Model Pc = [T, τ, θc , Θ] This investigation examines a local-level praxeology called “inﬁnite successions”. The task type assigned, T, is “to determine the nth stage of an inﬁnite succession”. The technique, τ, consists in:

1. Studying the ﬁrst cases or stages and determining the relation between ai and i = 1, 2, ... , n. 2. Constructing a general rule for the nth stage. 3. Implementing the rule constructed in (2) for speciﬁc close and distant stages of the succession (n = 10, n = 20, n = 100).

The creative technology, θc, is associated with the four aforementioned technological functions, while the mathematical technology, θm, is determined by the process of recursive and/or algebraic generalization: Recursive generalization appears when coincidences are observed in the base stages (perceptual field). On this basis, a relation of dependence is considered between an and an+1. Later, a rule is constructed for the stage an+1 based on the stage an (inferential field). This makes it possible to determine “close” values [41–44]. Algebraic generalization appears when a verbal rule or expression is constructed on the basis of the stages of the perceptual field or recursive rule that determines a relation of dependence between any stage n and the stage of the sequence an (with no need to obtain the previous term, an 1). This permits − the determination of both “close” and “distant” values [41–44]. Generalization is taken as the mathematical technology, θm, because it is determinant in constructing a general rule for a succession, be it figural, numeric, or figural with a tabular aid. In other praxeologies, technology may be associated with logical-sequential thought or abstraction. Finally, the theory, Θ, in this model is defined by Combinatory Theory. Educ. Sci. 2020, 10, 118 5 of 21

3. The Study

3.1. The Setting and Characteristics of the Institution This is a qualitative investigation that adopted a case study methodology to analyse creative mathematical activity at an institution that promotes mathematical talent. The institution is called the Math Club (MC), and it was formed with the aim of implementing a series of tasks of the same type—local praxeology—that together enable creative mathematical activity. Institution, the MC. The MC was conceived and constituted by one of the authors of this article with the support of the government of Baja California (México) during the development of her doctoral thesis. The participants were children from 15 public schools in the state, their parents, and an expert in mathematical didactics. The MC held 10 weekly 90-min sessions that were videotaped and transcribed for a deeper analysis of the creative mathematical activity that was potentiated using the PMMT proposed herein. Students. The students were selected by applying a written test that presented four math problems. In addition to the results of the test, the authors took into account the recommendations of math teachers regarding the interest of diﬀerent students in participating in the MC. Table1 summarizes the characteristics of the group.

Table 1. Characteristics of the participants in the Math Club.

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

The role of the instructor. The instructor’s participation consisted of constructing the didactic proposal presented below, after considering previously-analysed didactic variables that were transformed according to the students’ needs. Throughout the study, the instructor emphasized that all ideas, first explorations, and embryos of techniques were important, and motivated the students to explore cases, explain their techniques, and then determine their validity. The instructor monitored the students’ work, guided group meetings, and promoted the collective validation or refutation of the different techniques proposed. The instructor’s participation was indispensable in establishing a climate of respect and collective analysis. The role of parents. Though the parents only participated actively in the first (presentation of the MC) and last sessions (when parents and children played a mathematical game called “double domino”), their role in accompanying their children and ensuring their regular participation in the MC was fundamental to the success of the study. The inputs obtained as the basis for analysing the participants’ creative mathematical activity consisted of the following: (1) their worksheets; and (2) transcriptions of selected “episodes” from the videotaped sessions. Episodes are defined as “work intervals” in which pairs of students sought to construct or present the mathematical and creative technological elements that were analysed using the PMMT, as shown in the Results section.

3.2. Didactic Design of the MC The personal characteristics of a subject such as good memory, commitment to the task, and a positive attitude towards mathematics, all increase the possibilities of enhancing mathematical talent. One way to ensure that these characteristics participate in developing such talent is to have students confront a didactic design that consists of a set of challenging, interrelated mathematics tasks in an institutional setting that propitiates the development of creativity. The present study proposed Educ. Sci. 2020, 10, 118 6 of 21 a didactic design based on an “inﬁnite successions” praxeology with six challenging problem situations adapted from earlier studies of mathematical talent. Table2 brieﬂy describes the tasks included in the six situations.

Table 2. Tasks presented in all the MC sessions.

Problem Situation Tasks Task 1.1: Determine the change in the number of triangles while building Sierpiński’striangle. Task 1.2: Determine changes in the area while building Sierpiński’s 1. Sierpiński’striangle triangle. Task 1.3: Determine changes of the perimeter while building Sierpiński’s triangle. Task 2.1: Determine the number of cubes required to construct any stage of the sequence (different sequences of origami cubes were presented). 2. Origami cubes Task 2.2: Construct three stages using the origami cubes following a pattern. 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 3. Tables and chairs of the sequence. Task 3.3: Determine the number of shaded squares required to construct any image. 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. 4. Intersections and regions 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. Task 5.1: Determine the minimum number of movements with any 5. Tower of Hanoi number of disks. Task 6.1: Determine the total thickness of the folded sheet for any number of folds. 6. Folding paper 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.

The tasks defined for each situation (Table2) permitted constructing lineal, quadratic, or exponential general rules. Task 6.3 has the most challenging algebraic generalization rule: n 1 an = 32/ 2 − for all n N. It is important to clarify that this study did not expect the students to ∈ construct expressions with alphanumeric algebraic syntaxes, but generalizations of algebraic type, understood from the perspective of Radford as factual, contextual, or symbolic [40]. Though all six situations shared the goal of reaching an algebraic expression, certain specific elements were not common: for example, the use of didactic material (situations 2, 4, 5), problem posing (situation 2), and managing concepts foreign to traditional instruction (situation 1). These problem situations were organized with the expectation that they would foster the emergence of a creative component that could be analysed using the four technological functions outlined above. As elucidated below, each situation was accompanied by several tasks designed to generate a mathematical activity. While the analysis of the mathematical activity generated by a specific task may not necessarily discern all four creative functions, together these tasks fomented creative mathematical activity and, hence, the development of mathematical talent. The next section focuses on situations 2 and 4, which were used in this study to evidence the creative component of the students’ activity and how it can be analysed using the PMMT. Educ. Sci. 2020, 10, 118 7 of 21

Educ. Sci. 2020, 10, x FOR PEER REVIEW 7 of 21 4. Results 4. Results The results presented in this section reflect our analysis of the mathematical activity of three The results presented in this section reflect our analysis of the mathematical activity of three pairs of students. This study chose situations 2, “Origami Cubes”, and 4, “Intersections and Regions”, pairs of students. This study chose situations 2, ”Origami Cubes”, and 4, ”Intersections and Regions”, which were developed in sessions 2 and 4–5, respectively, because they allowed the authors to analyse which were developed in sessions 2 and 4-5, respectively, because they allowed the authors to analyse the evolution of creative mathematical activity using the PMMT model. the evolution of creative mathematical activity using the PMMT model. 4.1. Problem Situation 2: Origami Cubes 4.1. Problem Situation 2: Origami Cubes This problem included two tasks: This problem included two tasks: Task 2.1: Determine the number of cubes required to construct any stage of the sequence (different • • Task 2.1: Determine the number of cubes required to construct any stage of the sequences of origami cubes were presented). sequence (different sequences of origami cubes were presented). Task 2.2: Construct three stages using the origami cubes following a pattern. • • Task 2.2: Construct three stages using the origami cubes following a pattern. Task 2.1 consisted in determining, based on the first three stages of a sequence, a general rule that wouldTask define 2.1 allconsisted the stages in ofdetermining, the sequence. based Task on 2.2 the involved first three building, stages with of tangiblea sequence, materials a general (origami rule cubes),that would three define stages all of the any stages sequence. of the sequence. To perform Task Task 2.2 2.1, involved the students building, were with shown tangible four materials different construction(origami cubes), sequences three stages of the cubes,of any some sequence. with linealTo perform generalization Task 2.1, rules, the othersstudents with were quadratic shown rules.four Onedifferent of the construction quadratic rules sequences is shown of belowthe cubes, (Figure some1). with lineal generalization rules, others with quadratic rules. One of the quadratic rules is shown below (Figure 1).

Figure 1. Construction of origami cubes presented to the students of the MC.

The technique required to solve this task entails constructing a relation between the number of cubes in each figurefigure generatedgenerated forfor eacheach term,term, followingfollowing threethree mainmain steps:steps: (1)1) studying the first first three stages to construct an initial rule that determines a relation between the number of cubes and the stage of the sequence (Figure2 2);); (2)2) constructing a rule for the ݊ݐ݄nth stage stage (Table(Table3 );3); and and (3) 3)verifying verifying the general rule for the stage n݊൅ͳ+ 1,, whichwhich cancan bebe demonstrateddemonstrated using the induction principle and, and, for these students, proving thatthat rulerule forfor didifferentfferent stages.stages.

.TableTable 3. 3. ArrangementsArrangements of of the the cubes cubes required required to to construct construct the the nth݊ݐ݄ stage stage of of situation situation 2. 2 Increase of CombinationCombination CombinationCombination TotalTotal Increase of cubes Term Term Cubes between pink/orange Pink/Orange pink/bluePink/ Blue cubesCubes between stages Stages 1 2 ʹ Ͷ 1 2 2 4 2 2 ͸ ͺ ൅Ͷ 2 2 6 8 +4 3 32 2ͳʹ 12ͳͶ 14 +൅͸6 4 42 2 20 20 22 22 +൅ͺ8 ڭ. . ڭ . ڭ . ڭ . ڭ ݊ . ʹ . ݊ሺ݊ ൅ ͳሻ . ݊ሺ݊൅ͳሻ ൅ʹ. ൅ʹ݊. n 2 n(n + 1) n(n + 1) + 2 +2n The cubes and the table containing the combinations of colours and totals (Figure 2) constitute two ostensives, that is, palpable records according to Bosch and Chevallard [45]. They allow the developmentThe cubes of and the thetechnique table containing and technology. the combinations Specifically, of the colours table and(Figure totals 2) (Figureallowed2) students constitute to two“decompose” ostensives, [43] that (p. is,66) palpable the task down records into according sub-tasks to and Bosch so control and Chevallard the technique. [45]. They allow the development of the technique and technology. Speciﬁcally, the table (Figure2) allowed students to “decompose” [43] (p. 66) the task down into sub-tasks and so control the technique. Educ. Sci. 2020, 10, x FOR PEER REVIEW 8 of 21

Educ. Sci. 2020, 10, 118 8 of 21 Educ. Sci. 2020, 10, x FOR PEER REVIEW 8 of 21

Figure 2. Table to determine the number of cubes according to the combinations of colours.

Tables 4, 5 and 6 show the techniques and technologies produced by the three pairs of students that were representative and contrasting in terms of the mathematical and creative technologies in Figure 2. Table to determine the number of cubes according to the combinations of colours. play. Figure 2. Table to determine the number of cubes according to the combinations of colours. Tables4–6 show the techniques and technologies produced by the three pairs of students that Tables 4, 5 and 6Table show 4. thePair techniques A’s techniques and and technologies technologies produced related to byTask the 2.1. three pairs of students were representative and contrasting in terms of the mathematical and creative technologies in play. that were representative and contrasting in terms of the mathematical and creative technologies࢓ in Technique Technologies ࣂ and ࣂ࡯ play. Step 1. Determine Tablethat the 4. Pair first A’s combination techniques and(orange/pink) technologies related to Task 2.1. ૚ remains unchanged, and place the number 2 in the first ࣂmǣ2QWKHILUVWWKUHHVWDJHVWKH\ TechniqueTable 4. Pair A’s techniques and technologies Technologies relatedθ and to TaskθC 2.1. combination up to stage 100 (Figure 2). H[SORUHGRWKHUFORVHRQHVXVLQJ Step 1. Determine that the first combination (orange/pink) ࢓ remainsStep unchanged, 2. Count and the place cubesTechnique the in number the second 2 in the combination first TechnologiesGUDZLQJVEXWFRXOGQRW ࣂ and ࣂ࡯ combination(blue/pink)Step 1. Determine up and to stage write that 100 the (Figurethe numbers first2). combination 2, 6 and 12, (orange/pink) respectively, GHWHUPLQHDUHFXUVLYHUXOHper se Stepremains 2. Count unchanged, the cubesin in the the and secondfirst place three combination the cells. number 2 in the first ࣂ૚ǣ2QWKHILUVWWKUHHVWDJHVWKH\ (blue/pink) and write the numbers 2, 6 and 12, respectively, θ1 : On the first threeࣂ stages,ǣ They they optimized explored the other Stepin the 3. first Identify threecombination cells. a pattern up based to stage on the 100 stages (Figure given 2). and make H[SORUHGRWKHUFORVHRQHVXVLQJ૚ close ones using drawings, but could not determine a Stepdrawings 3.Step Identify 2.with Count a pattern squares the based cubes to represent on in the the stages second the given cubes combination and for stages 4, 5 techniqueGUDZLQJVEXWFRXOGQRW by changing from recursive rule per se. make drawings with squares to represent the cubes for GHWHUPLQHDUHFXUVLYHUXOHdrawing the complete figureper tose (blue/pink)and and 6 in write the two the numberscombinations 2, 6 and (Figure 12, respectively, 3). θ : They optimized the technique by changing from stages 4, 5 and 6 in the two combinations (Figure3). 1 Step 4: Based on thein drawing the first of three stage cells. 6, they made drawingsdrawing the completedrawings figure that to drawings show only that showthe Step 4: Based on the drawing of stage 6, they made only the increments (F2). drawingsStepof the 3. incrementsIdentify of the increments a pattern for n for = based7,n =n7, = n8,on= n 8,the =n 9= stagesand9 and nn =given= 1010 (Figure and make 3). ࣂ૚ǣ incrementsThey optimized (F2). the θ : They tried to optimize (F2) by adapting a school (Figure3). Next they counted and made the sums. 2 technique by changing from drawings Nextwith squaresthey counted to represent and made the thecubes sums. for stages technique 4, 5 called ‘the rule of 3’ to determine case 100, Step 5: To obtain the result for stage 100, they multiplied the Step 5: Toand obtain 6 in the the result two combinations for stage 100, (Figurethey multiplied 3). but failed the to solveࣂdrawing૛ǣ the They task. thetried complete to optimize figure (F2) to number of cubes of stage 10 by 10. The instructor asked numberthemStep to4: proveBased of cubes this on ‘rule theof stage ofdrawing 3’ for 10 stage by of 10. 9stage based The 6, oninstructor they stage made 3. asked drawings them bydrawings adapting that a school show technique only the Afterofto the prove doing incrementsso, this they ‘rule abandoned for of n3’ = for 7, this n stage = technique. 8, n9 =based 9 and Later, on n = theystage 10 (Figure 3. After 3). called ‘theincrements rule of 3’ to(F2). determine returneddoing to theirso,Next they attempt they abandoned counted to count the andthis number madetechnique. of the cubes sums. Later, they case 100, but failed to solve the using drawings. returnedStep 5: To to obtain their attempt the result to forcount stage the 100, number they ofmultiplied cubes using the ࣂ૛ǣ They triedtask. to optimize (F2) number of cubes of stage drawings.10 by 10. The instructor asked them by adapting a school technique to prove this ‘rule of 3’ for stage 9 based on stage 3. After called ‘the rule of 3’ to determine doing so, they abandoned this technique. Later, they case 100, but failed to solve the returned to their attempt to count the number of cubes using task. drawings.

Figure 3. Pair A’s drawings to solve Task 2.1. Figure 3. Pair A’s drawings to solve Task 2.1. Table4 shows a first approximation towards the creative mathematical activity defined by F2 Table 4 shows a first approximation towards the creative mathematical activity defined by F2 and F4. To determine the stage n = 100, pair A optimized the drawing technique of each figure by and F4. To determine the stage ݊ ൌ ͳͲͲ, pair A optimized the drawing technique of each figure by considering only the increments. They then sought to optimize utilizing the school technique called considering only the increments. They then sought to optimize utilizing the school technique called the rule of 3. Their return to theFigure tangible 3. Pair material A’s drawings led the to authors solve Task to suppose 2.1. that they tried to obey the the rule of 3. Their return to the tangible material led the authors to suppose that they tried to obey rules that govern the institution of the CM. Their adherence to the physical material of the cubes and Table 4 shows a first approximation towards the creative mathematical activity defined by F2 drawings did not allow them to reach a rule of recursive generalization. The development of creativity and F4. To determine the stage ݊ ൌ ͳͲͲ, pair A optimized the drawing technique of each figure by is associated with the task type and the implicit rules established to execute it, as shown below. considering only the increments. They then sought to optimize utilizing the school technique called the rule of 3. Their return to the tangible material led the authors to suppose that they tried to obey Educ. Sci. 2020, 10, 118 9 of 21

In what follows (Table5), the study presents the technique used by pair B and its associated technologies.

Table 5. Pair B’s techniques and technologies related to Task 2.1.

m Technique Technologies θ and θC Step 1. Same as for pair A. Step 2. Same as for pair A. Step 3. Based on the initial stages of the second θ1 : Based on the initial stages, they constructed an combination, they determined that stage 1 has 2 cubes, algebraic rule for distant stages as follows: stage 2 has 2 + 4 = 6 cubes, and stage 3 has 2 + 4 + 6 = 12 an = 2 + 4 + 6 + + 2n. cubes. They then departed from the material and made a ··· θ : They optimized the technique (F2) by setting the sum to determine the number of cubes in stage 4: 1 cubes aside and focusing on the sum required to 2 + 4 + 6 + 8 = 20 cubes. determine the number of cubes in the second Step 4. Based on this algebraic rule, they attempted to combination. implement this for n = 100, proposing the operation: 2 + 4 + 6 + 8 + + 200; however, they were unsuccessful ··· in calculating the operation or optimizing their technique.

The creative mathematical activity in this pair is manifested in their optimization of the technique (F2). These students set the tangible material aside and used the data in the Table to construct a technique and then implement it for the stages n = 4 and n = 100. Though this technique is uneconomical due to the arithmetic calculations required to reach the result, it did solve the task. Table6 shows pair C’s technique and associated technologies.

Table 6. Pair C’s techniques and technologies related to Task 2.1.

m Technique Technologies θ and θC 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 θ1 : Based on the initial stages, they determined an completed the row of combination two up to stage 100: 101 algebraic rule with two equivalent patterns of columns by 100 floors: 101 100 = 10, 100. × construction: Step 4. They noted that in the table (Figure2) only 2 cubes a = n(n + 1) + 2 must be added to the total number of cubes in combination n bn = n n + n + 2. two (Figure4). They then calculated the sum that × θ : To optimize the technique, they generated a novel corresponds to the total up to stage 100, which is 10, 100 + 2. 1 technique using a known reference; namely, a Step 5. Upon the instructor’s indication, they determined building to represent the cubes (F1, F2, F4). another technique to solve the task by identifying that θ : They produced two different techniques to ‘squares’ are formed in combination two and calculating 2 perform the same task (F3). their respective area; that is, for the first stage, a 1 1 × square, for the second, a 2 2 square, and for the third, a × 3 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 (Figure5). 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’. Educ. Sci. 2020, 10, x FOR PEER REVIEW 10 of 21

Figure 4. Pair C’s reconﬁgurationreconfiguration of the sequence in Figure1 1..

Pair C manifested creative mathematical activity by optimizing a technique based on counting and generating a novel technique; that consisted of determining the number of cubes by multiplying the number of the stage by the number of the stage +1. This entailed relating and adapting the rows of cubes to the floors of a building. This relation did not emerge from just their knowledge of math, but involved aspects of real life (F1, F2, F4). These students adapted the decomposition technique presented in Task 2.1 (based on identifying the stages of the sequence by separating the two colour combinations) to deconstruct [43] each stage into three distinct organizations (Figure 5): a constant (2), a square (݊ൈ݊ሻ, and a line (݊). This enabled them to propose a new technique to solve the task that evidences the development of F1, F3 and F4.

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

When pair C finished, the instructor asked them to explain their techniques to the group, and then continue with Task 2.2, which involved constructing three figures of a sequence with the origami cubes following a pattern similar to the previous ones. They were informed that other teams would have to determine the number of cubes required to construct stage 100 of the sequences posited. Figure 6 shows some of the sequences they constructed.

Figure 6. Construction with cubes by the MC students for Task 2.2. Educ. Sci. 2020, 10, x FOR PEER REVIEW 10 of 21

Educ. Sci. 2020, 10, 118 10 of 21 Figure 4. Pair C’s reconfiguration of the sequence in Figure 1. Pair C manifested creative mathematical activity by optimizing a technique based on counting Pair C manifested creative mathematical activity by optimizing a technique based on counting and generating a novel technique; that consisted of determining the number of cubes by multiplying and generating a novel technique; that consisted of determining the number of cubes by multiplying the number of the stage by the number of the stage +1. This entailed relating and adapting the rows the number of the stage by the number of the stage +1. This entailed relating and adapting the rows of cubes to the ﬂoors of a building. This relation did not emerge from just their knowledge of math, of cubes to the floors of a building. This relation did not emerge from just their knowledge of math, but involved aspects of real life (F1, F2, F4). These students adapted the decomposition technique but involved aspects of real life (F1, F2, F4). These students adapted the decomposition technique presented in Task 2.1 (based on identifying the stages of the sequence by separating the two colour presented in Task 2.1 (based on identifying the stages of the sequence by separating the two colour combinations) to deconstruct [43] each stage into three distinct organizations (Figure5): a constant (2), combinations) to deconstruct [43] each stage into three distinct organizations (Figure 5): a constant a square (n n), and a line (n). This enabled them to propose a new technique to solve the task that (2), a square× (݊ൈ݊ሻ, and a line (݊). This enabled them to propose a new technique to solve the task evidences the development of F1, F3 and F4. that evidences the development of F1, F3 and F4.

Figure 5. Pair C’s second reconfiguration of the sequence of Figure1. Figure 5. Pair C’s second reconfiguration of the sequence of Figure 1. When pair C finished, the instructor asked them to explain their techniques to the group, and then continueWhen with pair Task C finished, 2.2, which the involved instructor constructing asked them three to figuresexplain of their a sequence techniques with to the the origami group, cubes and followingthen continue a pattern with Task similar 2.2, to which the previous involved ones. constructing They were three informed figures of that a sequence other teams with would the origami have tocubes determine following the a number pattern ofsimilar cubes to required the previous to construct ones. They stage were 100 of informed the sequences that other posited. teams Figure would6 showshave to some determine of the sequencesthe number they of constructed.cubes required to construct stage 100 of the sequences posited. Figure 6 shows some of the sequences they constructed.

Figure 6. Construction with cubes by the MC students for Task 2.2.

Creative mathematicalFigure 6. Construction activity was with observed cubes by when the MC all students the students for Task proposed2.2. tasks of the same type; that is, tasks that can be solved using techniques associated with the same technology. The generalization rules that determine the nth stage of the constructions of cubes created by the students were of two types: lineal and quadratic. For the first five sequences constructed, the pairs of students found a rule of algebraic generalization associated with the nth term. Sequence F, which they called “pyramids” (Figure6), was presented at a first moment to one pair of students, but they were unable to determine the rule of algebraic generalization, so the instructor presented the rule to all the students at the end of the session so they were able to solve the task as a group. This task was:

Educ. Sci. 2020, 10, x FOR PEER REVIEW 11 of 21 Educ. Sci. 2020, 10, x FOR PEER REVIEW 11 of 21 Creative mathematical activity was observed when all the students proposed tasks of the same Creative mathematical activity was observed when all the students proposed tasks of the same type; that is, tasks that can be solved using techniques associated with the same technology. The type; that is, tasks that can be solved using techniques associated with the same technology. The generalization rules that determine the ݊ݐ݄ stage of the constructions of cubes created by the generalization rules that determine the ݊ݐ݄ stage of the constructions of cubes created by the students were of two types: lineal and quadratic. For the first five sequences constructed, the pairs of students were of two types: lineal and quadratic. For the first five sequences constructed, the pairs of students found a rule of algebraic generalization associated with the ݊ݐ݄term. Sequence F, which students found a rule of algebraic generalization associated with the ݊ݐ݄term. Sequence F, which they called ”pyramids” (Figure 6), was presented at a first moment to one pair of students, but they they called ”pyramids” (Figure 6), was presented at a first moment to one pair of students, but they were unable to determine the rule of algebraic generalization, so the instructor presented the rule to wereEduc. Sci.unable2020, to10, determine 118 the rule of algebraic generalization, so the instructor presented the rule11 of to 21 all the students at the end of the session so they were able to solve the task as a group. This task was: all the students at the end of the session so they were able to solve the task as a group. This task was: how many cubes are needed to construct stage 100 of the sequence of pyramids? Table 7 shows the how many cubes are needed to construct stage 100 of the sequence of pyramids? Table 7 shows the howtechniques many cubesand technologies are needed that to construct emerged stage from 100the ofgroup the sequencemathematical of pyramids? activity. Table7 shows the techniques and technologies that emerged from the group mathematical activity. techniques and technologies that emerged from the group mathematical activity. Table 7. Group techniques and technologies related to Task 2.2. Table 7. Group techniques and technologies related to Task 2.2. Table 7. Group techniques and technologies related to Task 2.2. ࢓ Technique Technologies ࣂ࢓ and ࣂ࡯ Technique Technologies ࣂ and ࣂ࡯ StepTechnique 1. By manipulating the material, they determined that Technologies a row of θ3m and θ Step 1. By manipulating the material, they determined that a row of 3 C cubesStep was 1. By added manipulating between the the material, stage they 1 and determined stage 2 pyramids, that and that cubes was added between the stage 1 and stage 2 pyramids, and that ࣂ૚ǣBased on the initial stages, a rowa row of 4 of cubes 3 cubes was was added added between between the the stage stage 1 and 2 stageand 2stage 3 pyramids ࣂ૚ǣBased on the initial stages, a row of 4 cubes was added between the stage 2 and stage 3 pyramids they determined the following pyramids, and that a row of 4 cubes(Figure was 7). added between the they determined the following stage 2 and stage 3 pyramids (Figure(Figure7). 7). θ1 : Based on the initialrecursive stages, they generalization: determined the StepStep 2. Next, they they determined determined that that a row a of row ﬁve of cubes five had cubes had to be recursive generalization: Step 2. Next, they determined that a row of five cubesfollowing had to be recursive generalization:ܽ௡ାଵ ൌܽ௡ ൅ ሺ݊൅ͳሻǤ addedto be below added belowpyramid pyramid 4 and, 4 and, similarly, similarly, a a row row of sixsix cubes had to be ܽ௡ାଵ ൌܽ௡ ൅ ሺ݊൅ͳሻǤ added below pyramid 4 and, similarly, a row of six cubesan+ had1 = aton + be(n + 1). cubes had to be added below pyramid 5. added below pyramid 5. θ1: They optimized the technique of counting into a Step 3. They determinedadded that below calculating pyramid the number 5. of ࣂ૚ǣ They optimized the Step 3. They determined that calculating the number of recursivecubes of technique any (F2).ࣂ૚ǣ They optimized the Stepcubes 3. They of any determined pyramid of thisthat sequence calculating required the totalling number the of cubes of any technique of counting into a pyramidnumber of of cubesthis sequence of the previous required pyramid totalling and the the stage number+ 1 of cubes of technique of counting into a pyramid of this sequence required totalling the number of cubes of recursive technique (F2). the(Figure previous8). Using pyramid this rule, and they the constructed stage + 1 a (Figure list up to 8). Using this rule, recursive technique (F2). thestage previousn = 15. pyramid and the stage + 1 (Figure 8). Using this rule, they constructed a list up to stage ݊ൌͳͷ. they constructed a list up to stage ݊ൌͳͷ.

Figure 7. Representation of the technique for the pyramid task. Figure 7. Representation of the technique for the pyramid task.

Figure 8. List of the first 15 stages for Task 2.2. FigureFigure 8. 8. ListList of of the the first first 15 15 stages stages for for Task Task 2.2. 2.2. The students’ creativity in this activity was observed when they optimized the counting TheThe students’ creativitycreativity in in this this activity activity was observedwas observed when theywhen optimized they optimized the counting the techniquecounting technique (F2). Beyond that, their positing of these constructions evidenced embryos of creative technique(F2). Beyond (F2). that, Beyond their positingthat, their of positing these constructions of these constructions evidenced embryosevidenced of creativeembryos technologies of creative technologies associated with their mathematical activity; namely, recognizing the characteristics of technologiesassociated with associated their mathematical with their mathematical activity; namely, activity; recognizing namely, recognizing the characteristics the characteristics of tasks of theof tasks of the same type with their associated technology. Though no pair of students determined a taskssame of type the with same their type associated with their technology. associated Though technology. no pair Though of students no pair determined of students a rule determined of algebraic a generalizationrule of algebraic that generalization correspondsto that the corresponds stages of the sequenceto the stages where of the numbersequence of where cubes ofthe the numbernth stage of rule of algebraic generalization that corresponds to the stages of the sequenceଵ ଷ where the number of 1 2 3 ଶ ofcubes the sequenceof the ݊ݐ݄ of stage pyramids of the issequence defined of by pyramidsan = n +is definedn + 1, itby was ܽ௡ suggestedൌଵ ݊ଶ ൅ଷ ݊൅ͳ to keep, it thinkingwas suggested about cubes of the ݊ݐ݄ stage of the sequence of pyramids2 is defined2 by ܽ௡ ൌ ଶ݊ ൅ ଶ݊൅ͳ, it was suggested thisto keep task thinking after the about session. this Membertask after 2the of session. pair C constructed Member 2 of a pair table C thatconstructedଶ includedଶ a alltable the that stages included from to keep thinking about this task after the session. Member 2 of pair C constructed a table that included nall= the1 to stagesn = 100frombased ݊ൌͳ on to the ݊ recursive ൌ ͳͲͲ based rule, ona +the= recursivea + (n +rule,1), whichܽ ൌܽ turned൅ ሺ݊൅ͳ out toሻ, be which very turned useful all the stages from ݊ൌͳ to ݊ ൌ ͳͲͲ based onn the1 recursiven rule, ܽ ௡ାଵൌܽ௡൅ ሺ݊൅ͳሻ, which turned forout performingto be very useful Task 4.1, for as performing shown below. Task In 4.1, general, as shown the students’ below. In experience ௡ାଵgeneral,௡ the with, students’ and reflections experience on, out to be very useful for performing Task 4.1, as shown below. In general, the students’ experience constructingwith, and reflections the techniques on, constructing for tasks 2.1 andthe 2.2techniques became afor creative tasks technological2.1 and 2.2 elementbecame thata creative allows with, and reflections on, constructing the techniques for tasks 2.1 and 2.2 became a creative the development of other techniques for this task and their adaptation to perform other types of tasks.

4.2. Problem Situation 4: Intersections and Regions This situation, or a variant of it, has been used in other studies, especially programs of attention to talent (for example, in ESTALMAT; see http://www.estalmat.org).

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. Educ. Sci. 2020, 10, x FOR PEER REVIEW 12 of 21 Educ. Sci. 2020, 10, x FOR PEER REVIEW 12 of 21 technological element that allows the development of other techniques for this task and their technologicaladaptation to elementperform thatother allows types ofthe tasks. development of other techniques for this task and their adaptation to perform other types of tasks. 4.2. Problem Situation 4: Intersections and Regions 4.2. Problem Situation 4: Intersections and Regions This situation, or a variant of it, has been used in other studies, especially programs of attention to talentThis (forsituation, example, or a invariant ESTALM of it,AT; has see been http://www.estalmat.org). used in other studies, especially programs of attention to talent (for example, in ESTALMAT; see http://www.estalmat.org). • Determine the number of regions into which a plane is divided by ݊ non-parallel lines, Educ. Sci. 2020, 10, 118 12 of 21 •where Determine an intersection the number can only of regions be formed into which by two a lines.plane is divided by ݊ non-parallel lines, where an intersection can only be formed by two lines. The technique required to solve this task entailed identifying the regions generated by the The technique required to solve this task entailed identifying the regions generated by the intersectionsThe technique of a family required of ݊ to non-parallel solve this tasklines, entailed where thereidentifying is no common the regions point generated in three different by the intersections of a family of n non-parallel lines, where there is no common point in three diﬀerent intersectionslines. The tasks of a assigned family of involved ݊ non-parallel determining lines, the where number there of is intersections no common and point regions. in three To different this end, lines. The tasks assigned involved determining the number of intersections and regions. To this end, lines.the students The tasks were assigned initially involved given six determining pieces of pasteboard the number as of in intersections Figure 9. and regions. To this end, thethe students students were were initially initially given given six six pieces pieces of of pasteboard pasteboard as as in in Figure Figure 99..

Figure 9. Evidence of the ostensive delivered to the students for situation 4. Figure 9. Evidence of the ostensive delivered to the students for situation 4. Figure 9. Evidence of the ostensive delivered to the students for situation 4. The arrow in Figure 9 at ܽ signals an intersection that is not visible, but that would appear if The arrow in Figure9 at a5 signalsହ an intersection that is not visible, but that would appear if the ܽ two-linethe two-lineThe arrow segments segments in Figure were were prolonged.9 at prolonged. ହ signals While While an studyingintersection studying these these that cases, iscases, not then, visible,then, it it was wasbut important thatimportant would toto appear considerconsider if thethethe two-line infinityinfinity segmentsproperty property ofwere of lines lines prolonged. on on a a plane plane While so so as studying as to to correctly correctly these count cases, count all then, all the the itintersections was intersections important and and to regions. consider regions. In theInaddition additioninfinity to property tothe the tangible tangible of lines material, material, on a plane the the instructor so instructor as to correctly posed posed several severalcount allquestions questions the intersections designed designed and to mobilize regions. theIn additiontechnologicaltechnological to the elements elementstangible associated material,associated withthe with instructor the the two two tasks; posed tasks; that several is,that parallelism questionsis, parallelism and designed non-parallelism and tonon-parallelism mobilize between the technologicallines,betweenthe infinitylines, elements theproperty infinity associated of property the lines with of and thethe plane, linestwo the andtasks; intersection plane, that theis, of intersectionparallelism two lines,and andof thetwo non-parallelism regions lines, and formed the betweenbyregions those formedlines, intersections. the by infinitythose Both intersections. property the questions of Boththe lines andthe questions theand material plane, and the functionedthe intersection material to functioned remind of two thelines, to studentsremind and the the to regionsconsider,students formed toas consider, well by, thethose as intersections well,intersections. the intersections that Both did the not that questions appear did not visually andappear the on visuallymaterial the plane on functioned the by prolongingplane toby remind prolonging the lines the studentstothe generate lines to to consider, thegenerate first cases.as thewell, first In the the cases.intersections first stepIn the of that thefirst technique,did step not of appear the a ruletechnique, visually was constructed on a therule plane was to byconstructed determine prolonging the to therelationdetermine lines betweento thegenerate relation the numberthe between first of cases. linesthe number andIn the regions firstof lines step generated. and of regionsthe Figuretechnique, generated. 10 represents a rule Figure was the 10 constructed first represents five cases. theto determinefirst five cases. the relation between the number of lines and regions generated. Figure 10 represents the first five cases.

Figure 10. Construction of the ﬁvefive ﬁrstfirst cases of the intersections task. Figure 10. Construction of the five first cases of the intersections task. nth In a second step, studying the the ݊ݐ݄ case, case,students students constructedconstructed thethe relationrelation betweenbetween thethe numbernumber n of linesIn a andsecond regions step, generatedstudying the when ݊ݐ݄ ݊ case, lineslines students werewere placedplaced constructed inin accordanceaccordance the relation withwith between thethe tasktask the conditionsconditions number n + of(Table(Table lines8 7).).and TheThe regions thirdthird step stepgenerated consistedconsisted when inin verifying verifying݊ lines were thatthat thetheplaced rulerule in positedposited accordance inin stepstep with twotwo functionedfunctionedthe task conditions for ݊൅ͳ1 (Tablelines.lines. In7). In this The case, third the step students consisted were in verifying able to prove that the that rule the posited rulerule functionedfunctioned in step two for functioned anyany numbernumber for ofof ݊൅ͳ lineslines = = = ) lines.((n݊ ൌ 5, ͷǡIn ݊nthis ൌ ʹͲǡ20,case, ݊n ൌ the ͳͲͲሻ100 students. . were able to prove that the rule functioned for any number of lines (݊ ൌ ͷǡ ݊ ൌ ʹͲǡ ݊ ൌ ͳͲͲሻ. Table 8. Construction of regions on the plane.

Lines a(n) Increase of Regions 1 2 2 4 +2 3 7 +3 4 11 +4 ...... n n(n+1) +n 1 + 2 Educ. Sci. 2020, 10, 118 13 of 21

This problem situation included the following two tasks:

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.

As the session progressed, but before these tasks were presented, the instructor posed a series of questions that included: How many regions are formed if 100 lines are placed but with only one intersection? The aim of these questions was to mobilize the technological elements associated with the two tasks; that is, parallelism and non-parallelism between lines, the inﬁnity property of the line and plane, the number of intersections between lines, and the regions formed by those intersections. Here, this study discusses the techniques and technologies produced by pair A in relation to Task 4.1 (Table9).

Table 9. Pair A’s techniques and technologies related to Task 4.1.

M Technique Technologies θ and θC 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 θ1 : Taking the first 6 stages as the base, they (Figure 11) to count all the intersections more systematically. considered a relation of dependence between a and This allowed them to count all the intersections formed by n a , and, later, the recursive rule: drawing seven lines. n+1 an = an 1 + (n 1). Step 3. They constructed a recursive process for the first 8 − − θ : They optimized their technique by passing from lines: 6 lines generate 15 intersections (10 + 5); 7 lines 1 the drawings to constructing a recursive rule (F2). produce 21 lines (15 + 6). Hence, 8 lines should give θ : They attempted to optimize their recursive 21 + 7 = 28 intersections. They verified the number of 2 technique and find a rule of algebraic generalization intersections by following the conditions on the list (F2). They also applied the verification technique (Figure 11) and then used this technique to reach 20 lines. presented previously (F4). 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 Educ.in establishingSci. 2020, 10, anx FOR algebraic PEER rule.REVIEW 14 of 21

FigureFigure 11. 11. GraphicGraphic representation representation of of pair pair A’s A’s technique for the intersections task.

In contrast contrast to to the the previous previous task task,, this this case case requires requires a a construction construction based based on on a a process process of of exploring exploring the ﬁrst stages that deﬁne the relation between the number of intersections (ai) and lines (i). For this the first stages that define the relation between the number of intersections (ܽ௜ሻ and lines ሺ݅ሻ. For thisreason, reason, the the students students changed changed the the technique technique of of using using the the material material to to one one basedbased on ”drawing “drawing lines”. lines”. This allowed them to count all the intersections (F3). Creative mathematical activity also appeared when they proposed a recursive process to determine the number of lines, thus optimizing their technique (F1, F2). Upon contrasting step 5 of Task 2.1 with step 4 of this one, this study observed that the students adapted the technique (F4) when they proposed diverse rules of algebraic generalization, and when they verified those rules for close and distant stages in the sequence. Table 9 displays the creative mathematical activity based on pair B’s techniques and technologies with Task 4.1.

Table 9. Pair B’s techniques and technologies related to Task 4.1.

࢓ Technique Technologies ࣂ and ࣂ࡯

Step 1. Using the tangible material, they counted the number ࣂ૚ǣBased on the first 5 cases, they of intersections for the first 5 stages, including the considered a relation of dependence intersections that do not appear visually (Figure 5). between ܽ௡ and ܽ௡ାଵ that led to the Step 2. They constructed a recursive process for the first 8 recursive rule: lines based on the differences between stages ܽ and ௡ ܽ௡ ൌܽ௡ିଵ ൅ሺ݊െͳሻ. ܽ௡ାଵǢthat is: with 2 lines, one intersection; with 3 lines, ͳ൅ ʹൌ͵intersections; with 4 lines, ͵൅͵ൌ͸intersections; with ࣂ૛ǣThey constructed two rules of 5 lines, ͸൅ͶൌͳͲ intersections; and with 6 lines ͳͲ ൅ ͷ ൌ algebraic generalization for݊ lines: ͳͷ ൅ሺ݊െͳሻڮintersections; so with 7 lines the number of intersections ܽ௡ ൌͳ൅ʹ൅ will be the same as with 6 lines plus 6: ͳͷ ൅ ͸ ൌ ͳ ൈ ሺ݊ሻ ൅ ݊Ǥ ۂʹሺ݊ െ ͳሻȀہ ൌ ܽ ʹͳintersections, and with 8 lines the number will be the ௡ ʹ same as with 7 lines plus 7: ʹͳ ൅ ͹ ൌ ʹͺ intersections (Figure 12). ࣂ૚ǣ They optimized their technique by Step 3. They identified that the technique associated with the passing from the tangible material to a pyramid task (Figure 4) can be applied to solve the recursive rule. intersection task, as well, though they recalled that the earlier task was not solved. ࣂ૛ǣ They recognized the applicability Step 4. They determined which task is solved by adding up of a technique based on considering three types of tasks: the intersection consecutive positive whole numbers, such that ܽଵ଴଴ ൌͳ൅ ൅ͻͻ. task, the pyramid task, and the task ofڮ൅͵൅ʹ Step 5. They related a famous technique from Carl Friedrich the sum of the first 100 positive whole Gauss (Hayes, 2006) to add up the first consecutive positive numbers (F3, F4).

Educ. Sci. 2020, 10, 118 14 of 21

This allowed them to count all the intersections (F3). Creative mathematical activity also appeared when they proposed a recursive process to determine the number of lines, thus optimizing their technique (F1, F2). Upon contrasting step 5 of Task 2.1 with step 4 of this one, this study observed that the students adapted the technique (F4) when they proposed diverse rules of algebraic generalization, and when they veriﬁed those rules for close and distant stages in the sequence. Table 10 displays the creative mathematical activity based on pair B’s techniques and technologies with Task 4.1.

Table 10. Pair B’s techniques and technologies related to Task 4.1.

m Technique Technologies θ and θC 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 (Figure5). Step 2. They constructed a recursive process for the first 8 lines based on the differences between stages an and an+1; that is: with 2 lines, one intersection; with 3 lines, 1 + 2 = 3 intersections; with 4 lines, 3 + 3 = 6 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 intersections, and with 8 lines the number will be the same as with 7 lines plus 7: 21 + 7 = 28 intersections (Figure 12). θ1 : Based on the first 5 cases, they considered a Step 3. They identified that the technique associated with the relation of dependence between an and an+1 that pyramid task (Figure4) can be applied to solve the intersection led to the recursive rule: task, as well, though they recalled that the earlier task was not an = an 1 + (n 1). − − solved.Educ. Sci. 2020, 10, x FOR PEER REVIEW θ2 : They constructed two rules of algebraic15 of 21 Step 4. They determined which task is solved by adding up generalization for n lines: consecutivenumbers positive (Figure whole 13) numbers, that the such instructor that had mentioned an =ࣂ1ǣ+ They2 + adapted+ (n 1Gauss’) technique for ૜ ··· − 1 a = 1 + 2 + 3 + + 99. an = [(n 1)/2] (n) + n. 100 ··· briefly in the first session. the first− 10 whole× positive2 numbers for Step 5. They related a famous technique from Carl Friedrich θ : They optimized their technique by passing ͳ൅ʹ൅͵൅ 1 GaussStep (Hayes, 6. They 2006) determined to add up the the first value consecutive of the sum positive from the tangiblesum of the material first 99 to numbers a recursive (F4). rule. ൅ͻͻǡ by forming pairs between the 10 initial and 10 finalڮ numbers (Figure 13) that the instructor had mentioned briefly in θ2 : They recognized the applicability of a the first session. numbers, as shown here: techniqueࣂ૝ǣ They based optimized on considering the technique three types of of Step 6. Theyͳ ൅ determined ͻͻ ൌ ͳͲͲǢ ʹ the ൅ valueͻͺ ൌ ͳͲͲǢof the ͵ sum ൅ ͻ͹1 + ൌ2 ͳͲͲǢ+ 3 + Ͷ ൅ ͻ͸+ 99, ൌ tasks:adding the intersection pairs and task,constructed the pyramid a rule task, of and ··· by formingͳͲͲǢ ͷ pairs ൅ ͻͷ between ൌ ͳͲͲǢ the ͸ ൅ 10 ͻͶ initial ൌ ͳͲͲǢ and ͹ 10 ൅ ͻ͵final ൌ numbers, ͳͲͲǢ ͺ ൅ as ͻʹ ൌ the task ofalgebraic the sum generalization of the first 100 positive(F2). whole shown here:ͳͲͲǢ ͻ ൅ ͻͳ ൌ ͳͲͲǢ ͳͲ ൅ ͻͲ ൌ ͳͲͲǢ until they obtained numbers (F3, F4). 1 + 99 = 100; 2 + 98 = 100;ͳͲͲ 3 + ൈ97 ͳͲ= ൌ100; ͳǡͲͲͲǤ 4 + 96 = 100; 5 + 95 = θ3 : They adapted Gauss’ technique for the first 10

100; 6Finally,+ 94 = they100; 7stated+ 93 = that100; this 8 + process92 = 100; must 9 + be91 repeated= up towhole positive numbers for the sum of the first 99 100; 10 + 90 = 100; until they obtained 100 10 = 1000. numbers (F4). the number 50, but that this technique× is not effective for any Finally, they stated that this process must be repeated up to the θ4 : They optimized the technique of adding pairs numbernumber 50, but of thatlines this stipulated, technique so is they not e ffcontinuedective for anytrying number to comeand constructed a rule of algebraic generalization of lines stipulated, so theyup with continued an algebraic trying to rule. come up with an (F2). algebraicStep rule. 7. The instructor asked them to perform the same Step 7. Theactivity, instructor but asked with them20 lines. to perform This led the to samethe following activity, but withconclusion: 20 lines. Thistaking led away to the one following (leaving conclusion: 19), dividing taking by two away one (leaving 19), dividing by two (leaving 9.5), taking only (leaving 9.5), taking only the 9 (the whole part) and the 9 (the whole part) and multiplying 9 20 (making 180) plus ͻൈʹͲ × the onemultiplying in the half, leaves 190 (making intersections. 180) plus the one in the half, Step 8: They verified thisleaves technique 190 intersections. with one member’s results for 20 linesStep and then8: They implemented verified this it for technique 100 lines: with99/2 one= 49.5 member’s, taking 49, multiplyingresults for it by20 100,lines and and adding then implemented 50 from the half it tofor obtain 100 lines: the resultͻͻȀʹ of 4,950 ൌ ͶͻǤͷ intersections., taking 49, multiplying it by 100, and adding 50 from the half to obtain the result of 4,950 intersections.

FigureFigure 12. 12.Record Record of pair pair B’s B’s technique technique

Figure 13. Pair B’s adaptation of Gauss’ technique.

The development of creativity in this task became apparent when the students identified that the three different tasks, approached in three sessions, could be solved using the same technique (F3, F4). Unlike the case of the technique applied on the previous task (Table 5), on this one, they produced three types of generalizations—one recursive, two algebraic—that solved it (F1, F3), then chose the ”optimal” one as a function of the arithmetical calculations required to solve it (F2). Table 10 presents pair C’s creative mathematical activity on Task 4.1 Educ. Sci. 2020, 10, x FOR PEER REVIEW 15 of 21

numbers (Figure 13) that the instructor had mentioned ࣂ૜ǣ They adapted Gauss’ technique for briefly in the first session. the first 10 whole positive numbers for Step 6. They determined the value of the sum ͳ൅ʹ൅͵൅ the sum of the first 99 numbers (F4). ൅ͻͻǡ by forming pairs between the 10 initial and 10 finalڮ

numbers, as shown here: ࣂ૝ǣ They optimized the technique of ͳ ൅ ͻͻ ൌ ͳͲͲǢ ʹ ൅ ͻͺ ൌ ͳͲͲǢ ͵ ൅ ͻ͹ ൌ ͳͲͲǢ Ͷ ൅ ͻ͸ ൌ adding pairs and constructed a rule of ͳͲͲǢ ͷ ൅ ͻͷ ൌ ͳͲͲǢ ͸ ൅ ͻͶ ൌ ͳͲͲǢ ͹ ൅ ͻ͵ ൌ ͳͲͲǢ ͺ ൅ ͻʹ ൌ algebraic generalization (F2). ͳͲͲǢ ͻ ൅ ͻͳ ൌ ͳͲͲǢ ͳͲ ൅ ͻͲ ൌ ͳͲͲǢ until they obtained ͳͲͲ ൈ ͳͲ ൌ ͳǡͲͲͲǤ

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 ͻൈʹͲ (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: ͻͻȀʹ ൌ ͶͻǤͷ, taking 49, multiplying it by 100, and adding 50 from the half to obtain the result of 4,950 intersections.

Educ. Sci. 2020, 10, 118 15 of 21 Figure 12. Record of pair B’s technique

FigureFigure 13. 13.Pair Pair B’s B’s adaptation adaptation of of Gauss’ Gauss’ technique. technique.

The developmentThe development of creativity of creativity in thisin this task task became became apparent apparent when the the students students identified identiﬁed that that the three ditheﬀerent three tasks, different approached tasks, approached in three in three sessions, sessions, could could be be solved solved using using the same same technique technique (F3, (F3, F4). Unlike theF4). Unlike case of the the case technique of the technique applied applied on theon the previous previous task (Table (Table 5),5 ),on on this this one, one,they produced they produced three typesthree oftypes generalizations—one of generalizations—one recursive, recursive, two two algebraic—that algebraic—that solved solved it it(F1, (F1, F3), F3), then then chose chose the the ”optimal” one as a function of the arithmetical calculations required to solve it (F2). “optimal” oneTable as 10 a functionpresents pair of theC’s arithmeticalcreative mathematical calculations activity required on Task 4.1 to solve it (F2). Table 11 presents pair C’s creative mathematical activity on Task 4.1.

Table 11. Pair C’s techniques and technologies related to Task 4.1.

m Technique Technologies θ and θC 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. θ1 : Based on the first 6 cases, they structured a Step 3. (Member 1). With the goal of detecting a pattern that relation of dependence between an and an+1, and would allow her/him to construct an algebraic rule, he continued later constructed the recursive rule: to apply this recursive rule up to 55 lines (Figure 14). an = an 1 + (n 1). − − Step 3. (Member 2). This student related the technique associated θ2 : They constructed a rule of algebraic with the pyramid cube task (Figure4), realizing that it could be generalization to find the number of intersections used to solve the intersection task. This student showed the of n lines: n n instructor a sheet where –on his own initiative– he had resolved an = n . 2 − 2 the pyramid cube task using the recursive rule θ1 : They optimized their technique by passing an+1 = an + (n + 1) one-by-one up to n = 100. He mentioned that from the tangible material to a recursive rule (F2). the numbers are the same, but ‘go two behind’; that is, ‘If the θ2 : Member 2 recognized the applicability of a number of lines here is five, there it would be pyramid 3’. Though technique with respect to two task types: pyramid this student recognized that this new task is solved using pyramid cubes and line intersections. He identified the 98, he continued to search for a more direct way. similarities and differences between the tasks in Step 4. (Member 1). Once this student had a sufficient number of order to implement the technique (F3, F4). cases (55 lines), he identified a pattern based on ‘tens’ (10, 20, 30, θ3 : Member 1 optimized his recursive technique 40 ... ). He explained the process on the blackboard (Figure 15): and produced a unique technique based on ‘tens’. ‘The way in which it is possible to reach 45 is by multiplying 10 He verified that this new technique worked for × 5—the half– and subtracting 5 to reach 45. I made it to 50 this way, other cases (F1, F2). 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”. Educ. Sci. 2020, 10, x FOR PEER REVIEW 16 of 21

Table 10. Pair C’s techniques and technologies related to Task 4.1.

࢓ Technique Technologies ࣂ and ࣂ࡯ 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. ࣂ ǣ Based on the first 6 cases, they Based on this step, they proposed their own techniques, structured a relation of dependence which were discussed at a couple different moments. between ܽ௡ and ܽ௡ାଵ, and later Step 3. (Member 1). With the goal of detecting a pattern constructed the recursive rule: that would allow her/him to construct an algebraic rule, ܽ௡ ൌܽ௡ିଵ ൅ሺ݊െͳሻ. he continued to apply this recursive rule up to 55 lines ૛ (Figure 14). ࣂ ǣ They constructed a rule of algebraic Step 3. (Member 2). This student related the technique generalization to find the number of associated with the pyramid cube task (Figure 4), intersections of ݊ lines: ௡ ௡ realizing that it could be used to solve the intersection ܽ ൌ݊ቀ ቁ െ . ௡ ଶ ଶ task. This student showed the instructor a sheet where –

on his own initiative– he had resolved the pyramid cube ࣂ૚ǣ They optimized their technique by task using the recursive rule ܽ௡ାଵ ൌܽ௡ ൅ ሺ݊൅ͳሻone-by- passing from the tangible material to a one up to ݊ ൌ ͳͲͲ. He mentioned that the numbers are recursive rule (F2). the same, but ‘go two behind’; that is, ‘If the number of

lines here is five, there it would be pyramid 3’. Though ࣂ૛ǣMember 2 recognized the applicability this student recognized that this new task is solved using of a technique with respect to two task pyramid 98, he continued to search for a more direct types: pyramid cubes and line way. intersections. He identified the similarities Step 4. (Member 1). Once this student had a sufficient and differences between the tasks in order number of cases (55 lines), he identified a pattern based to implement the technique (F3, F4). on ‘tens’ (10, 20, 30, 40…). He explained the process on

the blackboard (Figure 15): ‘The way in which it is ࣂ૜ǣ Member 1 optimized his recursive possible to reach 45 is by multiplying 10 × 5 –the half– technique and produced a unique and subtracting 5 to reach 45. I made it to 50 this way, technique based on ‘tens’. He verified that one-by-one and verified it [algebraic rule] for 20, 30 and this new technique worked for other cases 40, and it worked; for example, for 20 I obtained 190, (F1, F2). 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 Educ.did was Sci. 2020 100, 10× 50, 118 minus 50, which is half. That gave me 16 of 21 4,950”.

Educ. Sci. 2020, 10Figure, x FOR 14. PEERRecord REVIEW of the recursive rule for the ﬁrst 55 stages by member 1 of pair C. 17 of 21 Figure 14. Record of the recursive rule for the first 55 stages by member 1 of pair C.

Figure 15. Written record of the algebraic rule by member 1 of pair C.

As withwith pairpair B,B, membermember 2 2 of of pair pair C C evidenced evidenced his his creative creative mathematical mathematical activity activity by identifyingby identifying the similaritiesthe similarities between between the pyramidthe pyramid task task with with and theand intersection the intersection task. task. In addition In addition to identifying to identifying these regularities,these regularities, he recognized he recognized the diﬀ theerences differences in the initialin thestage initial of stage each of task each and task how and they how would theyneed would to beneed implemented to be implemented in the new in one the (F3,new F4). one The (F3, creative F4). The mathematical creative mathematical activity was activity clear in memberwas clear 1 ofin pairmember C when 1 of hepair proposed C when ahe peripheral proposed taska peripheral that economized task that hiseconomized technique. his Based technique. on the Based stages on in the tablestages (Figure in the 14table), this (Figure student 14), focused this student on multiples focused of on 10 multiples to relate the of number10 to relate of lines the withnumber the numberof lines ofwith intersections the number (Figure of intersections 14). This evidenced(Figure 14). functions This evidenced F1, F2, F3functions and F4. F1, F2, F3 and F4. The series of tasks proposed in this investigationinvestigation is well-known and has been utilized in early (even(even pre-algebra)pre-algebra) algebra; that is, a sector of the ﬁeldfield thatthat isis addressedaddressed duringduring the transitiontransition from arithmetic to Algebra per se. Th Thisis has often been related to thethe generalization process [[41–44].41–44].

5. Discussion Most of the studies or models that set out to analyse mathematical talent and its association with creativity [[13,46]13,46] tend to centre their gaze on mathematical mathematical abilities, types of problems, problems, topics topics in in math, math, or the mathematicalmathematical theorytheory that that stimulates stimulates students’ students’ potential. potential. Our Our PMMT PMMT model, model, in in contrast, contrast, takes takes all m theseall these elements elements into into account account (task (task type, type, techniques techniques produced produced by students,by students, and and the mathematical,the mathematical,θ , and࢓ creative, θC, technologies that come into play), together with their interrelations during the creative ࣂ , and creative, ࣂ࡯, technologies that come into play), together with their interrelations during the mathematical activity. It then characterizes this activity on the basis of the creative technological creative mathematical activity. It then characterizes this activity on the basis of the creative functions that emerge as protagonists during the analysis of results. technological functions that emerge as protagonists during the analysis of results. The infinite successions tasks presented herein facilitated the production of diverse techniques for The infinite successions tasks presented herein facilitated the production of diverse techniques finding recursive generalization and algebraic rules while performing the same task (F3). This approach for finding recursive generalization and algebraic rules while performing the same task (F3). This can thus be compared to flexible thinking (number of focuses of a solution) in the sense elucidated by approach can thus be compared to flexible thinking (number of focuses of a solution) in the sense Kattou, Kontoyianni, Pitta-Pantazi and Christou [8]. The case of pair C revealed creative technologies elucidated by Kattou, Kontoyianni, Pitta-Pantazi and Christou [8]. The case of pair C revealed from the first session (F1, F2, F3, F4), but it was the global analysis that generated evidence of the creative technologies from the first session (F1, F2, F3, F4), but it was the global analysis that developmentgenerated evidence of their of creativity the development and of how of thetheir other creativity functions and became of how clearer. the other However, functions confronting became oneclearer. sole However, task, as in confronting the context one of this sole investigation, task, as in the doescontext not of allow this investigation, the authors to does “characterize not allow the flexibility”authors to “characterize of all the students, the flexibility” as illustrated of all by the the students, creative as mathematical illustrated by activity the creative reported mathematical for pairs A andactivity B, who reported began for developing pairs A and creativity B, who duringbegan developing the second task,creativity but did during not provide the second clear task, indications but did ofnot this provide until Taskclear 4.1.indications Thus, presenting of this until a seriesTask 4.1. of tasks Thus, of presenting the same typea series (not of identical, tasks of the but same ones type that share(not identical, the same but technology), ones that isshare what the made same it possibletechnology), to clearly is what evidence made theit possible systematic to clearly development evidence of the systematic development of creativity and, in this study specifically, the way in which most of the students constructed recursive generalization and algebraic rules by recognizing, modifying, adapting, and applying the same technique to several different tasks. In this sense, proposing a local praxeology based on various tasks of the same type associated with the same technology allowed those techniques to be optimized or adapted to other conditions; for example, the techniques that pairs B and C produced on the pyramid task were implemented to generate the technique for Task 4.1 of the intersections task. The conditions of proposing a local praxeology—tasks of the same type—aligns with the condition of proposing ”challenging” tasks, as has been defined in other research on mathematical talent and creativity [4,14,17–19]. In particular, our work revealed that these MC students developed mathematical creativity through the implementation of a didactic design that emphasized task type, Educ. Sci. 2020, 10, 118 17 of 21 creativity and, in this study specifically, the way in which most of the students constructed recursive generalization and algebraic rules by recognizing, modifying, adapting, and applying the same technique to several different tasks. In this sense, proposing a local praxeology based on various tasks of the same type associated with the same technology allowed those techniques to be optimized or adapted to other conditions; for example, the techniques that pairs B and C produced on the pyramid task were implemented to generate the technique for Task 4.1 of the intersections task. The conditions of proposing a local praxeology—tasks of the same type—aligns with the condition of proposing “challenging” tasks, as has been defined in other research on mathematical talent and creativity [4,14,17–19]. In particular, our work revealed that these MC students developed mathematical creativity through the implementation of a didactic design that emphasized task type, T: “determine the nth stage of an infinite succession” and, on that basis, developed talent by means of a mathematical activity based on generalization. In addition, this study observed that certain steps of the technique produced by the students were key to developing mathematical technologies. The study of the first three stages (step one of the technique), for example, made it possible to generate a technological element; search for the pattern associated with the general rule (stage two of the technique). This was evidenced specifically by pair C’s activity, as they recognized that determining any stage of the sequence using a recursive rule is an uneconomical technique, though it allowed them to find a pattern and an algebraic rule that generated any stage of the sequence. In particular, member 1 of pair C showed an embryo of a mathematical technology that led him to elaborate a list with a considerable number of stages that was not limited by a dependency between an and an+1 but, rather, by choosing the specific stages (a10, a20, a30, ... ) that were determinant in optimizing the technique. This reveals a general vision of recurrence that, for this student, seemed to occupy a “first level” of generalization; one that he then applied to construct a rule of algebraic generalization to solve the task. A similar situation occurred with member 2 of pair C, who constructed a table containing a set of a data that, once analysed, allowed him to solve the pyramid task for the first 100 stages. This student established a direct relation between the pyramid and intersections tasks, pointing out that: “it goes two behind”. Though this student found a100 and recognized that this new task can be solved with a98, the technique was guided by the objective of the task; that is, determine any stage of the sequence. Hence, the implicit non-dependence task with the previous stage (an a + ) on Task 2.1 was very clear for these two students. → n 1 One advantage of using this model, together with generalization as the mathematical component (θm), is the flexibility it allows regarding the objectives of each task; that is, the students have the possibility to propose preliminary conclusions for tasks—e.g., “obtaining a recursive rule”—although they do not succeed in constructing an algebraic rule. Moreover, this model allows them to produce diverse techniques for the same task (F3), either on their own initiative or following an instructor’s cue; thus promoting creative mathematical activity. This advantage is determinant in the setting of the “regular classroom”, where the characteristics of the students are more diverse than in the MC. The conditions established for the MC included an institutional setting in which the students worked in an environment propitious for generating creative mathematical activity, and where the roles assigned to the instructor and parents enhanced the conditions in which the didactic design—based on a local praxeology—was implemented. In addition to ensuring the regular participation of their children, the parents participated actively in the MC in the first and last sessions. As Bicknell [47] and Feldhusen [48] proposed, the parents’ participation is an important element in developing mathematical talent. It is important to emphasize that the instructor’s participation in every situation included in the didactic design was fundamental as well, as illustrated by the selection of the sub-task that entailed determining the number of intersections for 20 lines that the instructor proposed to pair B. This allowed those students to recognize that the clearly-stated pattern for the first ten numbers was easily verified for the 20-line case, and so could be generalized to 100 lines. Without doubt, planning these sessions entailed great complexity, for the activities proposed in the didactic design could be performed using diverse techniques and entailed generating proposals for “sub-tasks” that allowed the students to Educ. Sci. 2020, 10, 118 18 of 21 approach the task situations presented. This leads the authors to recommend elaborating a guide for instructors. In this regard, it is sustained that elaborating a guide for teachers—even holding workshops to professionalize teaching—is necessary for the implementation of such a didactic design. This paper could provide a basis for producing both.

6. Conclusions and Future Research One consequence of this theoretical proposal is that it considers the students in relation to a certain institution through a didactic focus that, instead of emphasizing individual characteristics, analyses how praxeological equipment is provided to develop mathematical talent; specifically, by generating proposals for activities that allow students to construct and reconstruct creative, local-level mathematical praxeologies. This report highlights the solidity of the didactic design for developing the mathematical talent of all the students involved. For example, the students who performed all the tasks successfully (pair C) from situation 2 onwards adapted their techniques to new tasks (F4) and proposed diverse techniques for solving the same task (F3). The students who did not succeed in solving the tasks in situation 2 (pair A) did, however, demonstrate creative mathematical activity in later sessions of the MC, as is clearly evidenced in situation 4. One of the postulates of this proposal is that creativity is an important element of talent, so developing and analysing it is a fundamental principle for potentiating and observing the evolution of mathematical talent. As a result, the foundation of the PMMT model is creative mathematical activity developed through a solid didactic design in a propitious institutional context. As this creative activity is a necessary, but insufficient, element for realizing the development of mathematical talent, additional elements must be incorporated into the model; for example, considering students’ attitudes. This leads to the emergence of another key question for research: How can the PMMT theoretical model be enriched so as to take into account the relations among these additional elements, creative mathematical activity, and the development of mathematical talent? The didactic design conceived, implemented, and analysed herein constitutes an example of the experimental validation of the PMMT model that, together with the theoretical validation provided by the ATD [49], demonstrates the importance of the model for designing studies of creative mathematical activity, together with its impact on the development of mathematical talent. Nonetheless, it is necessary to implement this model in the context of regular classrooms and/or with various types of tasks. The PMMT model can be implemented in regular classrooms, but this requires reorganizing study programs to reflect flexible didactic designs based on local mathematical praxeologies. This process demands arduous labors by math teachers due to current institutional conditions (curricular rigidity, evaluation centered on handling algorithms, teaching based on presenting concepts, and administrative obligations, among others), and requires considering students as producers of knowledge. Clearly, this is extremely complex.

Author Contributions: Z.M.B.-G., A.R.-V. and S.R.-F. produced the theoretical model presented in this research and supports the didactic design that was implemented in a mathematics club. The execution of the didactic design was in charge of Z.M.B.-G. The three authors, A.R.-V., S.R.-F. and Z.M.B.-G., analysed the transcripts individually and subsequently there was a triangulation, which methodologically supports the analysis and the results shown in this contribution. The three authors wrote this article equally. All authors have read and agreed to the published version of the manuscript. Acknowledgments: The third author was partially supported by the Project C-2018-1 of Vicerrectoría de Investigación y Extensión (VIE - UIS) and also thanks Programa de Movilidad of Universidad Industrial de Santander for financial support. Conflicts of Interest: The authors declare no conflict of interest. Educ. Sci. 2020, 10, 118 19 of 21

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