1 Introduction

The expansion of vacancies in Brazilian public schools, with regard to the first stage of Elementary Education (the period from the 1st to the 5th year), has brought to the fore among many elements the pressing need for more teachers for this level of schooling, a fact that, itself, makes discussion and research on teacher training in this period of school education urgent.

The proposal for the National Curriculum Reference for Early Childhood Education (^{Brasil, 1998}), which subsidizes teacher training in Pedagogy courses, is to develop different skills for teaching, which, certainly, demands extensive training. Therefore, the teacher, at the end of the course, must be able to teach specific content from different areas of knowledge, including Mathematics, in a way that suits the needs.

Such teaching aptitude is interconnected to the act of educating, understood as a dynamic process and linked to historical-cultural contexts. Therefore, to achieve this, the action of all subjects involved in the process is necessary, be they teachers, students, parents or institutions. In this sense, the training process of these teachers must, then, organize the establishment of a “school education that presents itself as the particularity that best allows, in our current society, the realization of the subject-human gender relationship” (^{Araujo; Moraes, 2017}, p. 52).

Among the studies on the training of teachers who teach Mathematics for the years that comprise Elementary School I, there are researchers such as: ^{Nacarato et al. (2023}); ^{Moretti et al. (2023}); ^{Gaio and Duarte (2003}); ^{Curi (2004}, ²⁰⁰⁵); ^{Carzola and Santana (2005}); ^{Cunha (2010}), among others. Thus, Curi (²⁰⁰⁴) highlights a worrying situation regarding the methodologies frequently used in Brazilian Pedagogy courses, in which they are materialized as expository classes and this option is in line with the indications presented in research on training of these teachers. Curi (2005) also highlights that the training of these teachers must have, among its objectives, the selection and the choice of content, the organization of training times and spaces, the methodological approach and evaluation.

The importance of research arising from the area of Mathematics Education aimed exclusively at the training of Elementary School I teachers, added to the increasing need to have training spaces that allow teaching learning, specifically focused on the teaching of Mathematics, justifies the primordiality of projects such as the Mathematics Club⁴ In this situation, researchers such as ^{Cedro (2008}); ^{Lopes (2004}); ^{Moura (2000}; ²⁰²¹); ^{Moretti (2014}), among others, show the possibility of overcoming the theoretical discourse about the formative difficulties of this teacher who teaches Mathematics.

Overcoming such challenges is possible, based on the demands established by MathClub, which allow teachers in training who participate in it to understand other conditions for planning and developing pedagogical activity, generating new learning, through understanding the importance of the participation of teachers during their training, whether initial or continued, in spaces marked by the assumption that “learning is a social process in which the interaction between those who carry out the pedagogical activity sets in motion the learning of those who carry it out: the teacher [...]; and the student” (^{Moura, 2021}, p. 18).

In meeting the need and the possibility of a space capable of solving it, the objective is to investigate the actions of teachers in training at the Mathematics Club that indicate an understanding of this context as a training space. In connection with this objective, it is intended to find answers to the following problematic question: How do teachers in training at the Mathematics Club appropriate a way of organizing the teaching and learning of Mathematics for the final grades of Elementary School I?

However, in order to understand the actions that preceded the elaboration of this article, so that the objective was achieved and answers were found, initially, it was sought to understand how training aimed at teaching Mathematics for teachers of Fundamental I Education has been carried out. Subsequently, the assumptions on which the Mathematics Club's training proposal is based will be explained.

Next, the formative experiment is presented as a methodological option for developing actions. Afterwards, data analysis will be discussed, based on a structure that privileges the movement of the phenomenon and to this end, the following will be used: unity, episodes and flashes. Finally, some considerations about the research carried out.

2 The training of teachers who teach mathematics in Elementary School I

Research on teacher training in Brazil has focused on understanding how and if learning to teach Mathematics in the years of schooling (1st to 5th) that comprise Elementary School I has occurred in Pedagogy courses (^{Nacarato et al., 2023}; ^{Gatti; Nunes, 2009}).

These authors emphasize, in their studies, that in the vast majority of these degrees approaches to specific Mathematics concepts are superficial. According to such research, this fact is correlated with the reduction in the workload allocated to subjects that address mathematical knowledge, leaving little space, which ends up in an overvaluation of some concepts (numbers and the four operations) and a neglect of others (Quantities and Measurements, Space and Shape, Information Processing). This ends up causing a deficit in knowledge ‘of and about Mathematics’.

In an analysis of the types of knowledge necessary for teaching Mathematics in Elementary School I, ^{Cunha (2010}) explains that the main ones would be content-specific and methodological-pedagogical knowledge. The author argues that there must be a unity between the two, so that they are interrelated, as the experience that students will have during this school period will certainly influence the way that these students, in the future, will view and relate mathematical knowledge with other types of human knowledge.

^{Gaio and Duarte (2003}) complete these understandings, conjecturing that the biggest mistake in the training of Elementary School I teachers is neglecting basic Mathematics, because the mathematical knowledge of this educational stage is evaluated as so simple that it does not require preparation for it.

In line with this discussion, ^{Nacarato et al. (2023}, p. 32-33) point out that the “generalist training and reduced workload allocated to the field of mathematics education” in Pedagogy courses are points that make it difficult for mathematics content to be worked on “in conjunction with discussions of assumptions of teaching and learning for children”, making it difficult to “adequately train multipurpose teachers for teaching mathematics” (^{Nacarato et al., 2023}, p. 88).

Along this path, they emphasize the perception that stands out among these teachers that Mathematics is a science that is already ready, which only studies numbers and shapes, ignoring other parts of it, as well as having little knowledge of the History of Mathematics.

As a result, teachers who attended these degrees probably finish their initial training without appropriate contact with the contents they will teach - specific knowledge of mathematics -, both with regard to how they will organize their teaching, methodological-pedagogical knowledge, as if “[...] the multipurpose teacher did not need to know mathematics [...]” (^{Curi, 2005}, p. 71).

In the wake of the discussions presented, ^{Gatti and Nunes (2009}, p. 22) emphasize that the majority of institutions holding the courses that train these teachers simply see “[...] the study of teaching contents associated with methodologies in a panoramic and in-depth way”, materializing a deficient way and allowing there to be no “more systemic approach to the characteristics of the concept”, making it difficult to have a “more in-depth view of a given mathematics contente” (^{Moretti et al., 2023}, p. 51).

However, it is urgent that another training must be offered to this teacher, in the sense that this subject is seen not only as an individual, but also as a social subject, and, as such, is formed based on a structure that allows the constitution of “its essence in providing students with the means to appropriate the objectifications of non-everyday spheres” (^{Dias; Souza, 2017}, p. 185).

To this end, the institutions responsible for their training must be “taken as means to materialize such an intention”, where specific learning can be achieved - such as Mathematics -, “coming from intentional training processes that are present in both undergraduate and undergraduate courses, during the exercise of the teaching profession” (^{Dias; Souza, 2017}, p. 185).

Therefore, the teacher's process of understanding that he needs to organize the teaching of the mathematical knowledge he learned in his degree in knowledge to the student is not a movement that is constituted “from linear processes, in which the subjects are limited to simple reproduction” (^{Dias; Souza, 2017}, p. 191).

Contrary to this, “it depends on conditions in which subjects interact, in activity, through feedback processes and resignifications of the individual- collective type, mediated by theoretical and practical issues that constitute the object of the activity” (^{Dias; Souza, 2017}, p. 192). It is in this movement of carrying out teaching and learning as an activity (^{Moura et al., 2010}) “that there is a change in the quality of subjects mobilized, motivated, by the need to better become subjects in the activities they carry out” (^{Moura, 2021}, p. 17).

To make this process possible, another training space is necessary that differs from that offered in undergraduate courses: the Mathematics Club would be one of these contexts. This space can act “voluntarily with the intention of making the student the object of the teaching activity and the subject of their learning activity” (^{Moura; Sforni; Lopes, 2017}, p. 85).

It represents an effort to capture singularities within interdependent movements of social and historical constitutions, highlighted by subjects in their objective reality. Therefore, MathClub represents an option in the face of criticism of the training of teachers who teach Mathematics in the years that make up Elementary School I.

The Club's methodological approach highlights the relationship between the individual and the collective, the logical and the historical, in an organization of Mathematics teaching that provides the learning of a concept, based on the appropriation of mathematical knowledge as a result of human activity in its historical development and culture.

3 MathClub's proposal for training teachers who teach mathematics

In the case of school education, it requires active subjects (^{Leontiev, 2001}) mobilized by reasons, which lead to the constitution of actions and psychic operations, which enable the appropriation of scientific concepts, that occur through the transformation of activity (Leontiev, 2001), what cannot, like any psychic activity, be taught, but only created.

This discussion, put forward by ^{Leontiev (2001}), guarantees that in any condition and form in which man's activity takes place, any structure he adopts cannot be considered outside of social relations, or life in society. In other words, “it is in the movement in which human life takes place that a system of activities takes place, some of which replace others” (^{Leontiev, 2001}, p. 67). Therefore, this means that the activity conceives the human action that mediates the interface of the human relationship, the subject of the activity, and the objects of objective reality.

However, why bring up such discussion? Because, it is understood that a space, which is considered to develop the teacher training process, must be structured as an Activity and, therefore, must be imbued with concerns for the provision of a pedagogical activity appropriate to the learning objectives, which can be “[...] organized in a way that generates a certain need in the individual. This need, in turn, will trigger the reason to act” (^{Piotto; Asbahr; Furnaletto, 2017}, p. 119).

The space to which reference is made is the Mathematics Club, a project initially conceived in 1999 at the Faculty of Education of the University of São Paulo (FE-USP), which is currently present in different regions of Brazil, establishing as a teaching training context in the following Higher Education Institutions (HEIs): Federal University of Goiás - Goiânia, Federal University of Santa Maria - Santa Maria, State University of Goiás, Southwest Campus - Headquarters: Quirinópolis, Federal University of Rio Grande do Norte - Natal, Federal Institute of Education, Science and Technology of Espírito Santo - Vitória, Federal University of the State of São Paulo - São Paulo and Federal University of Uberlândia.

However, what is special about the MathClub structure that has survived more than 20 years since its creation?

In this proposal of the Mathematics Club, as a teaching learning space, there is a new quality: the inauguration within the teacher training unit of the place where the pedagogical activity is carried out, similar to that to be experienced concretely upon leaving the degree (^{Moura, 2021}, p. 5).

Another reason that supports the permanence of the Club's structure in contemporary times is that it is configured as a proposal to think about learning the structure of pedagogical activity, which responds to ^{Vigotski's (2001}) call for the necessary and adequate organization of processes of educational-training, which develop in the subject the historical characteristics of the humankind.

As one of the responses to the primordiality of having a training space that could handle this proposal, “the emergence of the Club was guided by the search for answers to new questions encouraged by research, [...] which required conceptually and politically dimensioning the social role of the teaching profession [...]” (^{Moura, 2021}, p. 2). And over time, needs were transformed and other reasons were added. This movement has materialized in the following research: ^{Lopes (2004}); ^{Cedro (2008}); ^{Silva (2013}); ^{Borowsky (2013}); ^{Hundertmarck (2017}); ^{Ferreira (2019}); Silva (²⁰¹⁹); Silva (²⁰²⁰); among others.

Given the fruitful quantity of research on the Mathematics Club, the question once again arises: who is interested in an investigation into this space? To everyone who understands that the establishment of a teacher training process needs to start from the human dimension of training, which is socially determined in subjective and objective relationships, present in man's actions.

This understanding goes against the simplistic view of training processes for teachers who teach mathematics and the development of teaching practice. ^{Moura (2013}, p. 98) corroborates the discussion by stating that “Work in the dimension of praxis implies, for the worker, complete control over what he performs: planning, defining his instruments and choosing a set of actions that allow him to achieve the goal he envisioned”.

The training proposal of the Mathematics Club is anchored in the concept of Activity made by ^{Leontiev (2001}), in the Logical-Historical Movement of concepts (^{Kopnin, 1978}; ^{Sousa, 2018}; ^{Sousa, 2014}) and the Teaching Guiding Activity (TGA), defended by ^{Moura (2010}; ²⁰¹³), who highlights the need for the worker-teacher to be the subject of his main activity: teaching.

In this vein, there is the purpose of capturing the process of continued training that takes place in the Mathematics Club, in order to understand how teachers are appropriating a form of organization for the teaching and learning of Mathematics offered in Elementary School. To this end, the theoretical- methodological principles of TGA are adopted as a reference. The capture of the training development process will be explained in the next topic, which reports the main stages of a training experiment organized and developed with the research subjects that supported this article.

4 Methodological actions: choosing the trajectory

The development of the research actions, which supported this article, took place at the Mathematics Club of the State University of Goiás - Campus Sudoeste - Sede Quirinópolis (the aforementioned project has been under development since 2017). The organization of such an investigative context took place based on the structure of a formative experiment.

^{Davidov and Markova (1987}, p. 236) claim that the formative experiment “is a fundamental form of realizing the particularities of the general genetic- causal or genetic-modeling method, [...] being a structure for investigating the development of the human psyche [...]”.

The aforementioned experiment lasted in the years 2022 and 2023. The participating subjects were twenty-five teachers from the municipal public education network in the municipality that is the headquarters of the IES that houses the Mathematics Club, which was the context of the research. The meetings took place weekly, at night, and lasted three hours.

They were recorded in audiovisual form and were later transcribed to become the universe of data. Along this path, and according to ^{Talizina (2009}, p. 29), in order to be successful in using this methodology, it is necessary that “the researcher knows the objective structure of the activity that he will form”. Therefore, next, in Table 1, there is the organizational structure of the actions developed with teachers in continuing education.

Source: Prepared Structure of the training experiment

Table 1 by the authors (2024)⁵; ⁶; ⁷  

Chart 1 exposes the organizational structure of the formative experiment, a scope of how it happened and an attempt to present the reader with a way to understand it in its entirety. Such peculiarities will be highlighted in the analysis composed of units, episodes and flashes in order to better understand the contributions of the movement that took place at the Mathematical Club.

5 The path in which the phenomenon reveals itself: analysis

In order to understand the evidence of the contribution of this experiment, an analytical structure is needed so that the phenomenon under study can be apprehended in its entirety, that is, it is sought to apprehend the subjects not only “as an existence, but as a convenience, as he should be as a result of his practical activity” (^{Kopnin, 1978}, p. 62).

To this end, a structure was designed based on Vygotski (²⁰⁰⁷) who, by replacing product analysis with process analysis, highlights that the investigation must have as its basic objective becoming a reconstruction of the development of the phenomenon. To achieve this, Vygotsky elaborates the method of analysis by units. Anchored in the same perspective, ^{Moura (1996}) develops a complementary structure to the units of analysis and establishes within them the need to compose them into episodes.

In line with the path indicated by ^{Vigotski (2007}) and ^{Moura (1996}), ^{Silva (2018}) points to a complementarity in the analysis structure proposed by these authors. In the analytical framework composed of units and episodes, Silva (²⁰¹⁸) inserts flashes, here understood as signs of a process of transformation of the subject's thinking, that is, “the observable signs that would prove the existence of the process of composition of the subject's meaning” (^{Silva, 2018}, p. 149).

In this movement, analysis is consolidated as the instant of apprehension of objective reality that allows the researcher to point out the interface between what has already been stated and new determinations about the research object in question. In this process, Table 2 shows the development and understanding of the phenomenon based on the following analysis structure:

Source: Prepared by the authors (2024)

Table 2 Structure of the analysis 

In this unit of analysis and episodes, it is sought to understand from the teachers in training that the acceptance of the theoretical-methodological structure of the Mathematics Club entails the understanding of this context as a space, in which the teacher's intentionality to carry out teaching is a key point, starting for establishing the action plan that will define the mediating instruments of these actions, in a process of “reflection, analysis and synthesis on the part of the teacher [...], which could give new quality to their general way of organizing teaching” (^{Moura; Sforni; Lopes, 2017}, p. 72).

Next, the data analysis unfolds.

6 Final considerations

The research carried out reveals advances in understanding the continued training of teachers who teach Mathematics in the final years of Elementary School I, with the particularity that this process took place in a peculiar training space: the Mathematics Club. Therefore, the actions experienced regarding the organization of teaching mathematical concepts were based on the TGA proposal and what were the formative contributions for these teachers to stand out.

Among these contributions, it was noticeable the apprehension of the historical-logical movement of the concept as a possibility for them to detach themselves from the traditional proposals for teaching Mathematics, which they experienced during the period they were undergraduate students and also when planning their classes at school. The training proposal offered to them enabled them to overcome the didactic approach that is based on the formalism of the concept, with a subsequent symbolic registration, devoid of meaning and meaning for the student and the teacher.

The training of teachers who teach mathematics and the learning of mathematics are processes that, as they are seen in educational reality, devalue the logical-historical character of mathematical concepts, covering up and ignoring the contradictions that constitute the reality pertaining to their emergence and development. Carried out in this way, they predominantly reduce the teaching training process and student learning to an adaptive perspective, in which the option given to these subjects is to comply, to accommodate themselves to what is proposed to them, without many questions.

However, in the training perspective offered by the Mathematics Club, a break in this devaluation was noted by presenting the relevance of understanding the process of emergence and development of mathematical concepts, but also how such process can be incorporated into situations that trigger learning to starting from the study of the logical-historical movement of these concepts. In this way, conditions were provided to promote the assimilation of content to the teacher and student (^{Davidov, 1988}), which goes beyond immediacy, towards meeting the perception of the need to appropriate mathematical knowledge as a promoter of their development.