MATHEMATICS CLUB AND THE PROCESS OF UNDERSTANDING THE PEDAGOGICAL ACTIVITY: A LOOK AT THE REORGANIZATION OF AN ACTIVITY THAT APPROACHES THE CONCEPT OF EQUATION
18.01.2024
Maria Marta da Silva, Lukas Adriel Francisco Alves
State University of Goiás, Brazil
MATHEMATICS CLUB AND THE PROCESS OF UNDERSTANDING THE PEDAGOGICAL ACTIVITY: A LOOK AT THE REORGANIZATION OF AN ACTIVITY THAT APPROACHES
THE CONCEPT OF EQUATION
Abstract: The aim of this article is to investigate the actions of Mathematics teachers in initial training at the Mathematical Club, which demonstrate evidence of the contribution of a Learning Triggering Situation (LTS) regarding the concept of equation for the learning of pedagogical activity. The Mathematical Club of the State University of Goiás, Campus Sudoeste – Sede Quirinópolis is the context of the investigation and the subjects were 30 (thirty) undergraduates who participate in the Club. This space is based on the assumptions of the Historical-Cultural Theory and the Activity Theory. The methodology chosen was the formative experiment, and data were collected through audiovisual recordings throughout the meetings of the 2022 school year. The chosen data analysis structure was: unit, episode and flashes. The results demonstrate that the subjects understood the meaning of the pedagogical activity and attributed new meanings to the learning process they were experiencing.
Keywords: mathematics club; pedagogical activity; mathematics teachers.
Introduction. Teacher training implies different elements, specific objectives, and there are various theoretical perspectives that discuss this phenomenon in Brazil, however, in this article, the starting point to explain this training is the Historical-Cultural Theory, which has its genesis in the theoretical contributions of Vigotski [20], whose epistemological origin is found in the historical-dialectical materialism fr om the philosophy of Marx [10]. In this scenario, the formative process of Mathematics teachers is found, which is part of another process: the historical-cultural development of this subject-teacher [12], [16], [19]. This teacher, as the human subject that he is, generates needs to secure his existence and does this in many ways, among which there is the “[...] articulation between the elements that make up the pedagogical activity of the Mathematics teacher” [19, p. 1].
In line with these discussions, the objective is to investigate the actions of Mathematics teachers in initial training at the Mathematical Club, which demonstrate the contribution of an activity about the concept of equation, for the understanding of what would really be the pedagogical activity. In this way, the methodological choice was made by the formative experiment, lasting a school year with 30 (thirty) future Mathematics teachers who were participating in the project. In order for the data analysis process to be effective so that the research objective was achieved, all the encounters of the experiment were recorded in audiovisual format. As a data analysis structure, being in line with the chosen theoretical basis, was opted for analysis unit [20], episode [12] and flashes [16].
Seeking to contribute, as well as to elucidate the path taken by the investigation, this text begins with the discussion between the training of Mathematics teachers currently offered and the training proposal of the Mathematical Club, focusing on creative tension as an element precursor and omnipresent in the movement to re-elaborate the LTS in question. Subsequently, the research methodology is pointed out and explained. Afterwards, the data analysis is presented, finally, some particularities about the formative process are listed in view of the evidence of contributions arising from the movement of re-elaboration of an activity that addresses the mathematical concept of equation, for the understanding of the pedagogical activity.
The Mathematics Club and the formation of Mathematics teachers. Discussions surrounding teacher training processes have been intensifying over the last few decades in Brazil and worldwide. In this sense, among many reasons, there is the fact that it is configured as an independent field with a particular object of study, in addition to, recognizing it as a fundamental element for the change in educational quality.
According to Libâneo [9] and Silva & Cedro [18], teacher training processes, traditionally, follow two models, the first of which is based exclusively on mastering the specific contents of a given area of knowledge, that is other elements are disregarded that make up and guide the subjects when they become teachers. In another sense, in the second, the diversity of didactic-methodological proposals is valued and, therefore, the way that is taught is emphasized without worrying about the specific contents and other elements. In the opposite direction of such proposals, the Mathematical Club (CluMat) emerges as a space that facilitates the learning of teaching in Mathematics, connected to the understanding of the teaching processes of mathematical concepts arranged in the BE, that is, the CluMat is an interface between the models, as it is the place wh ere “[...] planning is understood as an interdependent element between teaching-learning and content-evaluation” [18, p. 353].
In this way, authors such as [11], [16] and [18], among others, point out that the interface between such models is the one that approaches the formative processes based on the assumptions of the Historical-Cultural Theory, because as stated by Silva & Cedro [18, p. 353] “[...] both models oppose and complement each other”. In such assumptions, the formative processes should allow the subjects to appropriate the historically produced culture and, in this sense, the appropriation of concepts occurs through the movement of collective activities to individual ones [11]. Thus, it is argued that “Man is not evidently removed from the field of action of biological laws. What is true is that hereditary biological modifications do not determine the socio-historical development of man and humanity” [8, p. 264].
However, understanding how such processes are carried out in a formative-collaborative space requires appropriating the way in which such actions and operations can stress the subjects to put in motion the various elements that make up the pedagogical activity. These subjects, when immersed in the construction and re-elaboration of teaching activities, assume the role of creators, therefore, they have the freedom to share their ideas with colleagues, as well as to express desires and expectations [6, 2005]. During such interactions, the subjects are stressed in order to solve the problems presented there, so such tension sets in motion “[...] a complexity of other psychological functions that contribute, together with the cognitive function, to meaningful learning. of scientific concepts” [7, p. 37]. In this way, Vigotski [20, p. 59] states that "human learning presupposes a specific social nature", that is, it is through the interactions of human beings with their peers and with the environment, which leads them to develop their functions higher psychological functions, which enable the conscious and intentional control of actions, since these are mediated by an instrument or sign.
Therefore, in this movement, the activities planned at the Club become the result of a collective work, since they have the “contribution of the individual experience of each one, the result of their own history” [14, p. 15]. It is these individual elements, that can provide the formation of contradictions, which through the need for discussion and argumentation allow future teachers to share and appropriate meanings that are relevant to themselves.
In the CluMat-UEG space, through the theoretical perspective that underpins its teaching activity proposals, an activity is not defined as completed, it is understood that these are in motion in the same way as mathematical concepts. Through various reasons, desires, anxieties and expectations that pushes these subjects to organize an activity that allows Basic Education students to experience part of the needs that led humanity to develop such a concept, the subjects are being led to the possibility of organizing the teaching of Mathematics in a way that contemplates the essence of concepts, that is, these are formed in their subjectivity, “as subjects who learn a concept” [6, p. 1].
Such actions place subjects in front of the needs that put them into activity, that is, in a movement of transformation regarding the organizational processes of teaching actions, which have a movement of a social nature. However, for such actions to materialize, a methodology was needed to anchor such a proposal. In the next topic, the chosen methodological path is discussed and how it supported such choices.
The Mathematics Club and the formation of Mathematics teachers. Among the countless questions that led to the choice of the formative experiment as a methodological option for the research, answers can be found in the studies of Davýdov & Markova [3], which understand that such a structure allows an investigation based on the assumptions that underlie Vigotskian works, therefore it is an analysis of the development of the psyche of the human being. According to Vigotski [20], Silva & Cedro [17] and other authors, the quest to unveil the existing essence in the process of development of the human psyche takes place through “experimental analysis” [17, p. 476 apud 20), therefore, an investigation of this type could be the “key to understanding the process by which concept formation develops in real human life” [20, p. 86].
Seeking to understand how the process of re-elaboration of this activity regarding the concept of equations could provide contributions to the understanding of the pedagogical activity, the meetings were recorded in an audiovisual way, as well as through the application of questionnaires at the beginning and at the end of the investigation. In this sense, for a better organization and explanation of the course of the experiment, it was organized in moments as shown in Chart 1.
In order to understand the evidence of contribution of this experiment, an analytical structure is needed so that the phenomenon under study can be apprehended in its entirety, in this way, the structure that will subsidize the analysis of the experiment will be composed of unit, episode and flashes as will be discussed soon.
Data collection analysis. In order to understand the contributions that the movement of re-elaboration of an activity about the concept of equations brought to the training process of future Mathematics teachers, and to their understanding of the pedagogical activity, it was decided to carry out an analytical process that highlights the internal relationships of the phenomena studied. In this way, the analysis proposed by Vigotski [20] enables a “[...] study of the development, functioning and structure of the units, in which thought and speech are interrelated. [...] combines the advantages of analysis and synthesis, and allows the adequate study of complex wholes” [20, p. 4]. In line with such ideas, Moura [11, p. 276] emphasizes that the episodes make up the units proposed by Vigotski [20] and these are the “moments in which a conflict situation becomes evident that can lead to learning the new concept”, because in these are arranged the “revealing actions of the training process of the participating subjects” [14, p. 272].
Contained in the episode are the flashes, which according to Silva [16] are the “[...] most significant moments in which action and thought are unified in the appropriation of the objective world” [16, p. 145]. These, according to Silva [16], don’t present themselves in an akinetic way and even less in one-dimensionality, as they understand subjects not only as existence, “[...] but as convenience, as they should be as a result of their practical activity [...]. The existing is apprehended by the convenient, but this is based on the knowledge of the objective reality, of the laws of its movement” [5, p. 62]. Below, in Table 2, the organizational structure of data analysis is observed.
The understanding about the organizational structure of the pedagogical activity, comes from assumptions that little value the logical path traced by the concepts throughout the development of humanity. In this sense, “the way teaching in schools is organized often seeks to link scientific knowledge to experience, but this is a foundation that limits thinking” [4, p. 4], that is, by presenting students with only the most advanced stage of a concept, it is "[...] depriving them of really attributing meaning to their operations” [4, p. 4].
It is a fact that by keeping the teaching of Mathematics based only on “[...] memorization and repetition, the empiricist perspective ends up limiting the students' thinking process and, consequently, human development” [2, p. 432]. However, in order to carry out such actions and operations, it is crucial to offer spaces with theoretical-methodological foundations capable of leading future teachers to overcome the fossilized understandings over the last decades about the structure of pedagogical activity. Let's see the following flashes that meet such conceptions: 'I had to break everything I thought to be able to rebuild again' (Episode 1, Flash 1); ‘I saw the pedagogical activity as an unfinished mechanical cycle: Planning the class with pre-defined structures, teaching the class objectively and evaluating the students. Through my participation in the club and the re-elaboration of the LTS, I could see that the pedagogical activity is much more’ (Episode 1, Flash 2). Therefore, in this activity of re-elaboration movement, both the teacher and the student are in activity, the teacher, specifically in study activity that, “[...] in search of the improvement of the means of work [...]” [13, p. 423].
The questions that emerged from the development of LTS in schools required teachers in training to change the Virtual History script. The following flash indicates that they understand that the re-elaboration movement is connected to the understanding of the teacher's role: 'The re-elaboration process makes us identify the necessary changes so that the activity contemplates its objective, which is the appropriation of knowledge' (Episode 1, Flash 3). Thus, during the re-elaboration process, there are moments to discuss the teacher's role in the organization of their pedagogical activity, "more specifically, seeking to reflect on the subjective involvement with learning" [17, p. 101].
In this path, planning is seen as an initial condition for the development of those who are immersed, that is, the starting point for understanding the pedagogical activity is based on planning. In this bias, we have the following flashes: 'Initially, it was believed that the Pedagogical Activity was restricted to the application of content in the classroom, however, during the process of re-elaboration of the activity, it was possible to understand that the Pedagogical Activity started whether with lesson planning' (Episode 1, Flash 4); ‘Through the club meetings, I could see that the teacher’s “work” is not just at school, inside the classroom. Everything happens through planning. Everything is planned, the classes, the content to be addressed, the way this content will be passed on to the students' (Episode 1, Flash 5). In this sense, when they understand the meaning of pedagogical activity, of their role in becoming teachers, they simultaneously conceive “[...] the teacher in training as a product and subject of planned circumstances and, later, are developed in the school context [...]” [16, p. 193].
Imbued with such understandings, it is assumed that, through planning, it becomes possible that “[...] the student is not only the object of the teacher's activity, but mainly the subject, constituting himself as such in the activity of teaching and learning, as it participates actively and intentionally in the process of appropriating knowledge [...]” [1, p. 114]. Let's see the following flashes that endorse such findings: 'The teacher's pedagogical activity is a constant creative cycle, it is a process focused on the effective learning of the student and also the teacher (Episode 1, Flash 6); The Pedagogical Activity starts from the emergence of how a content will be worked, its planning, development and evaluation. Through this activity I realized that the act of discussing a concept, an idea for an activity, the search for a reference, learning, this moment that occurs outside the classroom, constitute the pedagogical activity, everything is related to teaching (Episode 1, Flash 7). Thus, motivated to provide a teaching that allows students to appropriate the elements that structure a mathematical concept, teachers in training become aware that the objective of pedagogical activity is to lead students to engage in learning activity and, for that, it is essential that “[...] in the teaching process, the object to be taught is understood by students as a learning object” [11, p. 92].
The data discussed here show that the subjects, by being immersed in a collective movement of re-elaboration of an intentionally organized LTS, could be led through the movement of analysis and synthesis to understand the compositional structure of the pedagogical activity, that is, involved in a collective and tensioned in a single objective that were able to strip away pre-defined conceptions and attribute new meanings to the idea of pedagogical activity.
Some final thoughts. Based on the discussions carried out, the existence of formative processes of Mathematics teachers that carry in themselves archaic models of teaching and learning about the mathematical concepts developed in Brazilian Basic Education. However, thinking of actions that are sufficiently capable of contributing to change in the formation of these subjects is not an easy or simple action. Thus, the (re)organization of such processes in spaces such as the CluMat – UEG has the potential to contribute to the assumption of a pedagogical activity based on the logical-historical movement of mathematical concepts.
Thus, through the actions performed, the change of meanings carried out by the subjects during the movement of re-elaboration of the presented activity becomes perceptible. Therefore, such movements made it possible for the subjects to enter the abstraction processes and, through these, to appropriate the elements that were significant to them and, in this way, to constitute new conceptions about the pedagogical activity to be developed.
Thus, it is argued that, in the context of teacher training, one should adopt as a guiding principle the promotion of planning, development and re-elaboration of situations that trigger learning within the constitution of the pedagogical activity. This defense is based on the understanding that the teacher's main activity is essential for promoting the qualitative leaps necessary for the development of individuals in different periods of their lives.
References
1. Asbahr, F. S. F. (2005). Sentido pessoal e Projeto político pedagógico: análise da atividade pedagógica a partir da Psicologia Histórico-Cultural. Dissertação de mestrado, Universidade de São Paulo, São Paulo-SP.
2. Cedro, W. L., Moraes, S. R. A., & Rosa, J. E. (2010). A atividade de ensino e o desenvolvimento do pensamento teórico em matemática. Ciência & Educação, 16(2), 427–445. https://doi.org/10.1590/s1516-73132010000200011
3. Davýdov, V. V., & Markova, A. K. (1987). La concepción de la actividad de estudio en los escolares. In: M. Shuare (orgs), La Psicología Evolutiva en la URSS: Antología (pp. 156-178). Moscú: Editorial Progreso.
4. Ferreira, C. A., Silva, M. M., & Cedro, W. L. (2021). Professores de matemática em formação inicial elaborando uma situação desencadeadora de aprendizagem no Clube de Matemática. Revemop, 3, Artigo e202121. https://doi.org/10.33532/revemop.e202121
5. Kopnin, P. V. (1978). A dialética como lógica e teoria do conhecimento. Rio de Janeiro: Civilização Brasileira.
6. Lanner de Moura, A. R., & Ferreira, E. S. (2005). A ressignificação da linguagem formal do sistema de numeração decimal [Paper presentation]. Congresso de Leitura do Brasil, 15, Associação de Leitura do Brasil, Campinas.
7. Lanner de Moura, A. R., & Lorenzato, S. (2007). O medir de crianças pré-escolares. Zetetiké, 1-2(9), 7–42.
8. Leontiev, A. N. (1978). O desenvolvimento do psiquismo. Lisboa: Livros Horizonte.
9. Libâneo, J. C. (2014). A integração entre o conhecimento disciplinar e o conhecimento pedagógico na formação de professores e a contribuição da teoria do ensino de Vasili Davídov. In: J. C. Libâneo (orgs), Adeus professor, adeus professora. São Paulo: Cortez.
10. Marx, K. (1978). Para a crítica da economia política. São Paulo: Abril. (Col. Os pensadores)
11. Moura, M. O. & et al. (2010). A atividade orientadora de ensino: unidade entre ensino e aprendizagem. In: M. O. Moura (orgs), Educação Escolar e Pesquisa na Teoria Histórico-Cultural (pp. 93-126). Brasília, DF: Liber Livro.
12. Moura, M. O. (2017). Educação Escolar e Pesquisa na Teoria Histórico-Cultural. São Paulo: Edições Loyla.
13. Moura, M. O., Araujo, E. S., & Serrão, P. (2019). Atividade Orientadora de Ensino. Linhas Críticas, 24. https://doi.org/10.26512/lc.v24i0.19817
14. Moura, M. O. (2021). Atividade de formação em espaço de aprendizagem da docência. Revista Iberoamericana Do Patrimônio Histórico-Educativo, 7. https://doi.org/10.20888/ridpher.v7i00.16028
15. Piotto, D. C., Asbahr, F. S. F., & Furlanetto, F. R. (2017). Significação e sentido na psicologia histórico-cultural: implicações para a educação escolar. In: M. O. Moura (orgs), Educação Escolar e Pesquisa na Teoria Histórico-Cultural (pp. 101-124). São Paulo: Edições Loyla.
16. Silva, M. M. (2018). A apropriação dos aspectos constituintes da atividade pedagógica por professores de matemática em formação inicial. Tese de Doutorado, Programa de Pós-Graduação em Ciências e Matemática, Universidade Federal de Goiás.
17. Silva, M. M., & Cedro, W. L. (2019). Discutindo as Operações de Adição e Subtração com Futuros Professores dos Anos Iniciais. Bolema. https://doi.org/10.1590/1980-4415v33n64a02
18. Silva, M. M,, & Cedro, W. L. (2021). Planejar para quê? Professores de matemática, em formação inicial aprendendo sobre o planejamento. Revista Paranaense De Educação Matemática. https://doi.org/10.33871/22385800.2020.9.20.351-374
19. Silva, M. M., & Cedro, W. L. (2021). Understanding Mathematics Contents as a way to Change Pedagogical Activities of Preservice Teachers. Ciência & Educação, 27. https://doi.org/10.1590/1516-731320210019
20. Vigotski, L. S. (2003). Obras Escogidas II: Historia del desarrollo de las funciones psíquicas superiores. España: Visor.
State University of Goiás, Brazil
MATHEMATICS CLUB AND THE PROCESS OF UNDERSTANDING THE PEDAGOGICAL ACTIVITY: A LOOK AT THE REORGANIZATION OF AN ACTIVITY THAT APPROACHES
THE CONCEPT OF EQUATION
Abstract: The aim of this article is to investigate the actions of Mathematics teachers in initial training at the Mathematical Club, which demonstrate evidence of the contribution of a Learning Triggering Situation (LTS) regarding the concept of equation for the learning of pedagogical activity. The Mathematical Club of the State University of Goiás, Campus Sudoeste – Sede Quirinópolis is the context of the investigation and the subjects were 30 (thirty) undergraduates who participate in the Club. This space is based on the assumptions of the Historical-Cultural Theory and the Activity Theory. The methodology chosen was the formative experiment, and data were collected through audiovisual recordings throughout the meetings of the 2022 school year. The chosen data analysis structure was: unit, episode and flashes. The results demonstrate that the subjects understood the meaning of the pedagogical activity and attributed new meanings to the learning process they were experiencing.
Keywords: mathematics club; pedagogical activity; mathematics teachers.
Introduction. Teacher training implies different elements, specific objectives, and there are various theoretical perspectives that discuss this phenomenon in Brazil, however, in this article, the starting point to explain this training is the Historical-Cultural Theory, which has its genesis in the theoretical contributions of Vigotski [20], whose epistemological origin is found in the historical-dialectical materialism fr om the philosophy of Marx [10]. In this scenario, the formative process of Mathematics teachers is found, which is part of another process: the historical-cultural development of this subject-teacher [12], [16], [19]. This teacher, as the human subject that he is, generates needs to secure his existence and does this in many ways, among which there is the “[...] articulation between the elements that make up the pedagogical activity of the Mathematics teacher” [19, p. 1].
In line with these discussions, the objective is to investigate the actions of Mathematics teachers in initial training at the Mathematical Club, which demonstrate the contribution of an activity about the concept of equation, for the understanding of what would really be the pedagogical activity. In this way, the methodological choice was made by the formative experiment, lasting a school year with 30 (thirty) future Mathematics teachers who were participating in the project. In order for the data analysis process to be effective so that the research objective was achieved, all the encounters of the experiment were recorded in audiovisual format. As a data analysis structure, being in line with the chosen theoretical basis, was opted for analysis unit [20], episode [12] and flashes [16].
Seeking to contribute, as well as to elucidate the path taken by the investigation, this text begins with the discussion between the training of Mathematics teachers currently offered and the training proposal of the Mathematical Club, focusing on creative tension as an element precursor and omnipresent in the movement to re-elaborate the LTS in question. Subsequently, the research methodology is pointed out and explained. Afterwards, the data analysis is presented, finally, some particularities about the formative process are listed in view of the evidence of contributions arising from the movement of re-elaboration of an activity that addresses the mathematical concept of equation, for the understanding of the pedagogical activity.
The Mathematics Club and the formation of Mathematics teachers. Discussions surrounding teacher training processes have been intensifying over the last few decades in Brazil and worldwide. In this sense, among many reasons, there is the fact that it is configured as an independent field with a particular object of study, in addition to, recognizing it as a fundamental element for the change in educational quality.
According to Libâneo [9] and Silva & Cedro [18], teacher training processes, traditionally, follow two models, the first of which is based exclusively on mastering the specific contents of a given area of knowledge, that is other elements are disregarded that make up and guide the subjects when they become teachers. In another sense, in the second, the diversity of didactic-methodological proposals is valued and, therefore, the way that is taught is emphasized without worrying about the specific contents and other elements. In the opposite direction of such proposals, the Mathematical Club (CluMat) emerges as a space that facilitates the learning of teaching in Mathematics, connected to the understanding of the teaching processes of mathematical concepts arranged in the BE, that is, the CluMat is an interface between the models, as it is the place wh ere “[...] planning is understood as an interdependent element between teaching-learning and content-evaluation” [18, p. 353].
In this way, authors such as [11], [16] and [18], among others, point out that the interface between such models is the one that approaches the formative processes based on the assumptions of the Historical-Cultural Theory, because as stated by Silva & Cedro [18, p. 353] “[...] both models oppose and complement each other”. In such assumptions, the formative processes should allow the subjects to appropriate the historically produced culture and, in this sense, the appropriation of concepts occurs through the movement of collective activities to individual ones [11]. Thus, it is argued that “Man is not evidently removed from the field of action of biological laws. What is true is that hereditary biological modifications do not determine the socio-historical development of man and humanity” [8, p. 264].
However, understanding how such processes are carried out in a formative-collaborative space requires appropriating the way in which such actions and operations can stress the subjects to put in motion the various elements that make up the pedagogical activity. These subjects, when immersed in the construction and re-elaboration of teaching activities, assume the role of creators, therefore, they have the freedom to share their ideas with colleagues, as well as to express desires and expectations [6, 2005]. During such interactions, the subjects are stressed in order to solve the problems presented there, so such tension sets in motion “[...] a complexity of other psychological functions that contribute, together with the cognitive function, to meaningful learning. of scientific concepts” [7, p. 37]. In this way, Vigotski [20, p. 59] states that "human learning presupposes a specific social nature", that is, it is through the interactions of human beings with their peers and with the environment, which leads them to develop their functions higher psychological functions, which enable the conscious and intentional control of actions, since these are mediated by an instrument or sign.
Therefore, in this movement, the activities planned at the Club become the result of a collective work, since they have the “contribution of the individual experience of each one, the result of their own history” [14, p. 15]. It is these individual elements, that can provide the formation of contradictions, which through the need for discussion and argumentation allow future teachers to share and appropriate meanings that are relevant to themselves.
In the CluMat-UEG space, through the theoretical perspective that underpins its teaching activity proposals, an activity is not defined as completed, it is understood that these are in motion in the same way as mathematical concepts. Through various reasons, desires, anxieties and expectations that pushes these subjects to organize an activity that allows Basic Education students to experience part of the needs that led humanity to develop such a concept, the subjects are being led to the possibility of organizing the teaching of Mathematics in a way that contemplates the essence of concepts, that is, these are formed in their subjectivity, “as subjects who learn a concept” [6, p. 1].
Such actions place subjects in front of the needs that put them into activity, that is, in a movement of transformation regarding the organizational processes of teaching actions, which have a movement of a social nature. However, for such actions to materialize, a methodology was needed to anchor such a proposal. In the next topic, the chosen methodological path is discussed and how it supported such choices.
The Mathematics Club and the formation of Mathematics teachers. Among the countless questions that led to the choice of the formative experiment as a methodological option for the research, answers can be found in the studies of Davýdov & Markova [3], which understand that such a structure allows an investigation based on the assumptions that underlie Vigotskian works, therefore it is an analysis of the development of the psyche of the human being. According to Vigotski [20], Silva & Cedro [17] and other authors, the quest to unveil the existing essence in the process of development of the human psyche takes place through “experimental analysis” [17, p. 476 apud 20), therefore, an investigation of this type could be the “key to understanding the process by which concept formation develops in real human life” [20, p. 86].
Seeking to understand how the process of re-elaboration of this activity regarding the concept of equations could provide contributions to the understanding of the pedagogical activity, the meetings were recorded in an audiovisual way, as well as through the application of questionnaires at the beginning and at the end of the investigation. In this sense, for a better organization and explanation of the course of the experiment, it was organized in moments as shown in Chart 1.
In order to understand the evidence of contribution of this experiment, an analytical structure is needed so that the phenomenon under study can be apprehended in its entirety, in this way, the structure that will subsidize the analysis of the experiment will be composed of unit, episode and flashes as will be discussed soon.
Data collection analysis. In order to understand the contributions that the movement of re-elaboration of an activity about the concept of equations brought to the training process of future Mathematics teachers, and to their understanding of the pedagogical activity, it was decided to carry out an analytical process that highlights the internal relationships of the phenomena studied. In this way, the analysis proposed by Vigotski [20] enables a “[...] study of the development, functioning and structure of the units, in which thought and speech are interrelated. [...] combines the advantages of analysis and synthesis, and allows the adequate study of complex wholes” [20, p. 4]. In line with such ideas, Moura [11, p. 276] emphasizes that the episodes make up the units proposed by Vigotski [20] and these are the “moments in which a conflict situation becomes evident that can lead to learning the new concept”, because in these are arranged the “revealing actions of the training process of the participating subjects” [14, p. 272].
Contained in the episode are the flashes, which according to Silva [16] are the “[...] most significant moments in which action and thought are unified in the appropriation of the objective world” [16, p. 145]. These, according to Silva [16], don’t present themselves in an akinetic way and even less in one-dimensionality, as they understand subjects not only as existence, “[...] but as convenience, as they should be as a result of their practical activity [...]. The existing is apprehended by the convenient, but this is based on the knowledge of the objective reality, of the laws of its movement” [5, p. 62]. Below, in Table 2, the organizational structure of data analysis is observed.
The understanding about the organizational structure of the pedagogical activity, comes from assumptions that little value the logical path traced by the concepts throughout the development of humanity. In this sense, “the way teaching in schools is organized often seeks to link scientific knowledge to experience, but this is a foundation that limits thinking” [4, p. 4], that is, by presenting students with only the most advanced stage of a concept, it is "[...] depriving them of really attributing meaning to their operations” [4, p. 4].
It is a fact that by keeping the teaching of Mathematics based only on “[...] memorization and repetition, the empiricist perspective ends up limiting the students' thinking process and, consequently, human development” [2, p. 432]. However, in order to carry out such actions and operations, it is crucial to offer spaces with theoretical-methodological foundations capable of leading future teachers to overcome the fossilized understandings over the last decades about the structure of pedagogical activity. Let's see the following flashes that meet such conceptions: 'I had to break everything I thought to be able to rebuild again' (Episode 1, Flash 1); ‘I saw the pedagogical activity as an unfinished mechanical cycle: Planning the class with pre-defined structures, teaching the class objectively and evaluating the students. Through my participation in the club and the re-elaboration of the LTS, I could see that the pedagogical activity is much more’ (Episode 1, Flash 2). Therefore, in this activity of re-elaboration movement, both the teacher and the student are in activity, the teacher, specifically in study activity that, “[...] in search of the improvement of the means of work [...]” [13, p. 423].
The questions that emerged from the development of LTS in schools required teachers in training to change the Virtual History script. The following flash indicates that they understand that the re-elaboration movement is connected to the understanding of the teacher's role: 'The re-elaboration process makes us identify the necessary changes so that the activity contemplates its objective, which is the appropriation of knowledge' (Episode 1, Flash 3). Thus, during the re-elaboration process, there are moments to discuss the teacher's role in the organization of their pedagogical activity, "more specifically, seeking to reflect on the subjective involvement with learning" [17, p. 101].
In this path, planning is seen as an initial condition for the development of those who are immersed, that is, the starting point for understanding the pedagogical activity is based on planning. In this bias, we have the following flashes: 'Initially, it was believed that the Pedagogical Activity was restricted to the application of content in the classroom, however, during the process of re-elaboration of the activity, it was possible to understand that the Pedagogical Activity started whether with lesson planning' (Episode 1, Flash 4); ‘Through the club meetings, I could see that the teacher’s “work” is not just at school, inside the classroom. Everything happens through planning. Everything is planned, the classes, the content to be addressed, the way this content will be passed on to the students' (Episode 1, Flash 5). In this sense, when they understand the meaning of pedagogical activity, of their role in becoming teachers, they simultaneously conceive “[...] the teacher in training as a product and subject of planned circumstances and, later, are developed in the school context [...]” [16, p. 193].
Imbued with such understandings, it is assumed that, through planning, it becomes possible that “[...] the student is not only the object of the teacher's activity, but mainly the subject, constituting himself as such in the activity of teaching and learning, as it participates actively and intentionally in the process of appropriating knowledge [...]” [1, p. 114]. Let's see the following flashes that endorse such findings: 'The teacher's pedagogical activity is a constant creative cycle, it is a process focused on the effective learning of the student and also the teacher (Episode 1, Flash 6); The Pedagogical Activity starts from the emergence of how a content will be worked, its planning, development and evaluation. Through this activity I realized that the act of discussing a concept, an idea for an activity, the search for a reference, learning, this moment that occurs outside the classroom, constitute the pedagogical activity, everything is related to teaching (Episode 1, Flash 7). Thus, motivated to provide a teaching that allows students to appropriate the elements that structure a mathematical concept, teachers in training become aware that the objective of pedagogical activity is to lead students to engage in learning activity and, for that, it is essential that “[...] in the teaching process, the object to be taught is understood by students as a learning object” [11, p. 92].
The data discussed here show that the subjects, by being immersed in a collective movement of re-elaboration of an intentionally organized LTS, could be led through the movement of analysis and synthesis to understand the compositional structure of the pedagogical activity, that is, involved in a collective and tensioned in a single objective that were able to strip away pre-defined conceptions and attribute new meanings to the idea of pedagogical activity.
Some final thoughts. Based on the discussions carried out, the existence of formative processes of Mathematics teachers that carry in themselves archaic models of teaching and learning about the mathematical concepts developed in Brazilian Basic Education. However, thinking of actions that are sufficiently capable of contributing to change in the formation of these subjects is not an easy or simple action. Thus, the (re)organization of such processes in spaces such as the CluMat – UEG has the potential to contribute to the assumption of a pedagogical activity based on the logical-historical movement of mathematical concepts.
Thus, through the actions performed, the change of meanings carried out by the subjects during the movement of re-elaboration of the presented activity becomes perceptible. Therefore, such movements made it possible for the subjects to enter the abstraction processes and, through these, to appropriate the elements that were significant to them and, in this way, to constitute new conceptions about the pedagogical activity to be developed.
Thus, it is argued that, in the context of teacher training, one should adopt as a guiding principle the promotion of planning, development and re-elaboration of situations that trigger learning within the constitution of the pedagogical activity. This defense is based on the understanding that the teacher's main activity is essential for promoting the qualitative leaps necessary for the development of individuals in different periods of their lives.
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