1 Introduction

Investigating the training of Mathematics teachers is a constant provocation. So, how is it possible to move forward and find answers to the challenges that involve the training of these teachers? How do training contexts deal with these issues? Are there other spaces capable of training this teacher from another perspective than the currently dominant one?

The dominant conceptions for the training of these teachers have disconnected the idea that mathematical concepts are responses to human needs throughout the history of man on the planet. Thus, they do not take into account the fact that these concepts are products of historical-social problems, which man has confronted (^{Radford, 2011}; ^{Moretti, Radford, 2015}; ^{Sousa, 2018}).

Therefore, this should not be discussed in the classroom in an arbitrary way and disconnected from their history, as ^{Borasi (1992}) points out, because, if this is done, the understanding of the social decision-making process that the human communities had to do for such concepts to emerge will not be considered (^{Arcavi, 1991}).

On the path of searching for answers and finding possibilities, this field of investigation accumulates extensive theoretical production. This is due to the growing social demand for Mathematics teachers. Although this fact contrasts with the terrible objective conditions of their training and work.

Anchored in such needs, the question is: What are the formative contributions that the organization of teaching mathematical concepts, offered by the Mathematics Club (MathClub), based on the logical-historical movement, can bring to Mathematics teachers in initial training?

MathClub is a training space that aims to conduct teacher training so that teachers can find a general way of organizing their main activity: teaching.

This formation takes place in a way that highlights the importance of the interaction between history and society, which produced the concepts to be studied. This is a prominent element within MathClub, so that teachers in training can be led to “understand conceptual developments”, placing “the knowledgeable subject and the entire mathematical activity under study within their cultural conception” (^{Radford, 2011}, p.82).

In this way, the research that supports this article took as its object the initial training of Mathematics teachers in a special space: Mathclub.

In order for the reader to understand the entire process developed, the text is organized in such a way that, firstly, it will be clarified what the logical- historical movement of concepts is and how it interrelates with the proposal of the Learning Triggering Situation (LTS). Next, the Mathematics Club is presented and how this space embodies the logical-historical movement in its formative actions.

Next, the research methodology is presented, by didactically characterizing the formative experiment developed in its four moments.

From this perspective, sequentially, there is data analysis, in which the development of the phenomenon can be monitored based on the analytical structure that is made up of units, episodes and flashes. Finally, there are some considerations about the research carried out.

2 Logical-historical movement (LHM) of concepts: didactic proposal present in LTS

Understanding the LHM is an issue that is intertwined with the incessant search for didactic proposals for the training of Mathematics teachers, which allow for an organization of the teaching of mathematical concepts that allows the understanding of the need to "overcome the conception that it is enough to understand the context in which the concepts were developed to be able to teach them” (^{Leandro; Sousa; Andrade, 2020}, p. 396).

This perspective ensures the defense that teacher training, which takes it as a premise, must be based on the idea that the teacher, in the process of learning how to teach, must experience situations which place him in the presence of the “movement of thought in the context in which such concepts were conceived and developed” (^{Leandro; Sousa; Andrade, 2020}, p. 396).

In this line of thought, what can the LHM allow for the training of Mathematics teachers?

It can favor transformations in the meanings in which subjects, in teacher training, attribute to the process of emergence and development of mathematical concepts, a fact that directly interferes with the constitution of new practices in the classroom.

In this way, these practices will have theoretical-didactic conditions to establish themselves as defining the understanding of the LHM's potential in providing resources for an organization of teaching mathematical concepts that reveals the importance of social practices, linked to activity, socially, human and, historically, built.

In the search for a theoretical and methodological organization structure, which aimed to achieve the LHM, it resorted to the concept of Teaching Guiding Activity, proposed by ^{Moura (2007}). According to this author, it would be a teaching activity guided, intentionally, by those who teach and, because of this, it

has an objective, instruments and modes of action for its achievement; considers the learning possibilities of the subjects who participate in the activity and the complexity that involves the logical-historical formation of the concepts that are constitutive of the activity (^{Moura; Sforni; Lopes, 2017}, p. 87).

In order to involve the logical-historical dimension of concepts in the Teaching Guidance Activity, it is concerned, in all its stages, with meeting its constitution. The first of these is the historical synthesis, which seeks, in the History of Mathematics, the path of emergence and development of a referred concept. Subsequently, in the second stage, the LTS (which can be offered as a game, virtual story (VH) or emerging everyday situation) seeks to materialize this logical-historical path of concepts, through the presence of triggering problems.

The triggering problems have, at their core, the need that induced humanity to construct the concept. Therefore, it is paid attention to the fact that the LTS, regarding the concept of number, which was materialized as an HV in the format of a Comic Book (CB), brought, in its plot, human primordialities, which led the man to create this concept: quantification and its consequent control of variation in quantities, the ability to measure and establish units of measurement, among others (^{Silva, 2022}).

^{Ifrah (2005}) highlights that, in this process of controlling the variation of quantities (represented in ‘The Agnuns⁴), it appears that, initially, the physical objects remained closely linked to those that were intended to be counted. However, little by little, they moved away from the material indicative and represented abstract symbols, which can be linked to any object.

^{Caraça (1989}) corroborates the conception of ^{Ifrah (2005}) when says that such distance is based on the fact of the interface existing between mathematical concepts and the needs that determined them, and "[...] mathematical concepts arise, once problems of capital, practical or theoretical interest are posed” (Caraça, 1989, p. 125).

In this sense, ^{Caraça (1989}) emphasizes that the “natural number, arising from the need for counting”, has a close connection with physical objects, however “the rational number” is born linked to the needs “of measurement”, while “the number real" is born "to ensure the logical compatibility of different acquisitions” (^{CARAÇA, 1989}, p. 125). In this way, rational and real numbers move away from material references and the need to be linked to physical objects, as they are the product of theoretical-practical essentialities.

Thus, covering the essence of the needs that drove humanity, in the search for solutions that enable the social and historical construction of concepts up to the present moment, is a component of the movement of apprehension of the concept itself, when it materializes in the logical. Therefore, the teacher in training must deal with mathematical concepts as syntheses of the history of human work, which is embodied in activities (^{Leontiev, 1978}).

In this movement, the historical aspect is integrated with the logic of a particular object of study, and it is only in this dialectical unity that the knowledge of that object is admissible. According to ^{Kopnin (1978}, p. 186), the “historical aspect implies not only the history of the object, its production and development, but also the history of how humanity appropriated this object, that is, the history of its knowledge”. Therefore, “the logical-historical pair is the criterion for the systematization of knowledge to be appropriated by the student. The teacher is responsible for making this pair visible [...]” (^{Moura; Sforni; Lopes, 2017}, p. 91).

Thus, the subjects, by appropriating these once historical aspects, understand the path taken by human thought so that such concepts are currently in a certain form, which constitutes the logical aspect. In such a way, the logical is “the reproduction of the essence of the object and the history of its development in the system of abstractions” (^{Kopnin, 1978}, p. 183), constituting itself as the appropriation of history by man's thought. And, “for this reason, the logical is the historical freed from the casualties that disturb it” (^{Kopnin, 1978}, p. 184), aiming at the way in which human thought carries out the task of apprehending the historical (^{Kopnin, 1978}).

Such problems do not need to have the factual history present in the History of Mathematics. But rather the one that is saturated in the concept when considering that it embodies a historically allocated human need, “That is, a problem that brings the essence of the need that led humanity to create the concept to be taught, the core of the concept to be appropriate” (^{Moura; Sforni; Lopes, 2017}, p. 91).

Training designed in this way takes into account that such relationships change, as teachers assimilate what has historical, social and cultural value, consequently, interpreting their social worlds in different ways. Therefore, they act on them in different ways, which, in turn, impact the dynamic relationship between the subjects' development and the historical-social situations that they experience.

Among the possible spaces to allow and encourage such interrelations, MathClub stands out, being an environment capable of maintaining the interface between the University and the School, while the teacher, in the process of learning how to teach, learns the teaching process. This phenomenon is a process of making sense and meaning, according to the involvement with the world to which it is part (^{Moura, 2021}). The next topic provides an understanding of how MathClub aims at this teacher training proposal, based on the LHM.

3 Mathematics Club: space for teacher training based on the logical- historical movement

MathClub is a training space that emerged in 1999, as a supervised internship project, linked to the Faculty of Education of the University of São Paulo. The concepts defended within this space were disseminated throughout the Brazilian territory⁵ em diversas Instituições de Ensino Superior.

At this juncture, specifically, in 2017, MathClub was created within the scope of the State University of Goiás, Câmpus Sudoeste - Headquarters - Quirinópolis⁶, which continues until today.

In this training space, emphasis is placed on the importance of understanding LHM concepts. This is an element of relevance within the Club: highlight the process in which teachers in training can “understand conceptual developments”, placing “the knowledgeable subject and the entire mathematical activity under study within their cultural conception” (^{Radford, 2011}, p. 82).

From this conception, and based on ^{Sousa (2004}), ^{Kopnin (1978}) and ^{Saito and Dias (2013}), it is highlighted that MathClub, by proposing a learning that approach to teach Mathematics linked to an organization of teaching mathematical concepts, based on the logical-historical perspective, defends the need to “understand the immutability and mutability of things and concepts [...], realizing

that the logical is a reflection of the historical [...]” and that this unity “[...] presupposes the possibility of understanding the movement of thought to grasp the object [...]” (^{Leandro; Sousa; Andrade, 2020}, p. 395).

At MathClub, the importance of concepts being studied in their production process is highlighted based on the meanings intrinsic to the historical period that emerged and developed, as well as the interrelationship of this process with human activity.

According to ^{Radford (2006}, p. 112), this activity "[...] generates conceptual objects, which become the root of changes in the activities themselves”. Still according to Radford (2006), it is important to pay attention to the explanation of how the acquisition of concepts is carried out, which are arranged at the school as content, that is, a fundamental problem in the training of Mathematics teachers, in particular, in learning in general.

Therefore, MathClub is a space that allows the understanding that the teacher in training, in this process of dealing with change, does not do so as a simple response to his behavior, which was from the student and now becomes a teacher, but it involves a process full of contradictions, which takes on new forms as it develops and generates its relationships with the historical-social situations of its development.

Thus, within MathClub, emphasis is placed on the dynamic relationship that may exist between individuals and the historical-social situations of the emergence and development of concepts so that they are capable of promoting the ability to see them as a result of historical-cultural needs, experienced by humanity.

To this end, it is necessary to understand the process of emergence and historical development of mathematical concepts and how this proposal can be possible in a training process for Mathematics teachers, which is connected with the organization of teaching mathematical concepts in Basic education.

But what can the existence of a training space, which enables the dynamics of the logical-historical movement of concepts, mean for the training of Mathematics teachers?

To find answers to this question, another question will be needed: What profile of Mathematics teachers is necessary to train?

On the way to find answers to these demands, researchers such as ^{Cedro (2008}), ^{Radford (2011}), ^{Sousa (2018}), ^{Panossian, Moretti and Souza (2017}), ^{Moura (2017}), ^{Leandro, Sousa and Andrade (2020}), among others, raise questions in their studies about the relevance of LHM in the training of Mathematics teachers. To this defense is added the perception that such a proposal could lead to another organization of pedagogical activity. To achieve this, the existence of training contexts, such as MathClub, was essential.

In this approach chosen by CluMat, “[...] concepts are understood as living productions in direct relationship with the needs of the subjects and historical times that produced them [...]", which is why one of the objectives of this training space is allow the subjects, who are part of it, to appropriate “[...] a certain concept, understood as historical and cultural production [...]” (^{Moretti, 2014}, p. 34).

In this way, they can be understood based on the constitution of/in man's social practice, while life itself develops and becomes more complex, allowing subjects to acquire social and cultural conditions of thinking, in addition to theorizing about this social practice, from its objects and constitutive phenomena.

This condition highlights another postulate: the understanding and the use of such a historical movement of elaboration and subsequent logical increment of mathematical concepts that shows a potential possibility for the development of teacher training activity, in this special context, called MathClub.

Thus, this space aims to overcome the training of Mathematics teachers in training that has, in recent decades, focused exclusively on the contents of the curriculum and the assessment model, which is a copy of exhaustively repeated exercisese (Cochran-Smith et al., 2008).

Formed based on this model, the teacher will hardly understand the Vygotskinian premise that it is not important to teach a quantity of knowledge as it is to instill the ability to acquire such knowledge and make use of it (^{Vigotski, 1997}).

Contrary to this training, 'which trains the future teacher', at MathClub, man's search to understand and to explain the reality that surrounds him is valued, and in the meantime, he creates and acquires knowledge and, therefore, develops it.

To this end, thought follows from what already exists, but without being limited to it to arrive at something that did not exist (^{Kopnin, 1978}). Thus, the path requires creative activity from the subject, rules and laws that govern objective reality.

In this sense, man lacks new concepts that expand his possibilities of mastering the reality that surrounds him. Only in this way, by going beyond the limits of current schemes humanity creates concepts that allow it to make choices to solve its problems and, thus, thought will not be directed in a rigid manner, but with a certain freedom (^{Kopnin, 1978}).

Therefore, in the training proposal offered by MathClub, teaching activities that can capture the process of emergence and development of mathematical concepts are valued in a way that not only “[...] photographs the real historical process with all its casualties, zigzags and detours” (^{Kopnin, 1978}, p. 184), but which can reflect history in theoretical form and, thus, interpret the historical process of the genesis of this concept.

However, in order to understand didactically how such actions are carried out, a methodological path is necessary to anchor this proposal. In the next topic, it will be presented highlighting the process of how MathClub's actions take place.

4 Methodological path: the formative experiment as an option

The Mathematics teacher training model, based on the ideas defended by MathClub, highlights that learning is a social process, because the interaction between subjects plays a prominent role in its development.

In this environment, it was decided to develop a formative experiment, which according to ^{Silva (2018}), supported by Davidov and Markova (1987), is an investigative structure carried out in several stages, in which all processes occur simultaneously.

In this way, the experiment allows highlighting the laws of the domain of reality of the object being researched. Therefore, for ^{Freitas and Libâneo (2022}, p. 6), the formative experiment would be the “process of identifying, understanding and explaining the historical genesis of human psychic functions in concrete conditions, revealing the movement of their emergence and transformation in social relations” (Freitas; Libâneo, 2022, p. 6).

Didactically, the experiment was organized into planning, development and collective evaluation meetings, which took place weekly and lasted 5 hours (there were 32 meetings per year). The participants were approximately 30 (thirty) undergraduates (all were students of the Mathematics course hosted by MathClub, this is the only condition to be participants, there is no selection process), who became club members from 2017 to 2023. The research actions were developed at the University (during the planning and development of activities when Basic Education students attended the Club, or when the Club members went to schools).

The data collection instruments were audiovisual recordings of all moments (planning, development and evaluation), which, later, became possible data for analysis.

The Club's activities have a very peculiar organization, which are shown in Chart 1.

Source: Prepared by the authors (2024).

Chart 1 - Organizational structure of the training experiment 

Due to the impossibility of analyzing in this article all the activities⁷ carried out by MathClub throughout the period of time that it exists, the LTS was chosen regarding the concept of number.

See in the analysis presented later how this phenomenon unfolded among the subjects participating in this training space.

5 Data analysis: understanding the phenomenon

In a scientific investigation, the production of knowledge only makes sense as long as its function is to reveal reality in its contradiction, in counterpoint and in the separation of appearance and essence, of what is not important to show what is fundamental.

Therefore, only through this process can its internal connection and, with that, its particular character emerge. “In this process, the secondary is not left aside as unreal or less real, but reveals its phenomenal or secondary character through the demonstration of its truth in the essence of the thing” (^{Kosik, 1969}, p. 18).

In order to achieve this understanding for the analysis, a structure was used that starts from the concept of unity, proposed by Vygotski (¹⁹⁹⁸), who states that it is “the result of the analysis which, unlike the elements, enjoys all the fundamental properties characteristic of the whole and constitutes a living and indivisible part of the totality" (^{Vigotski, 1998}, p. 20).

Likewise, in the necessary search for singularities that make up the universality of the phenomenon investigated, it is looked for episodes, which would be, according to ^{Moura (2004}, p. 267), the moment, which “can reveal interdependence between the elements of a formative action”.

Continuing the explanation of the proposed data analysis structure, there are flashes, which, for ^{Silva (2018}, p. 150), would be the parts of the episodes that would configure “the signs of the conscious and internalized reflection of reality”, which “they implicitly embody the motives and needs, the meaning and meaning that are expressed in language, but, above all, they are not reduced to it” (^{Silva, 2018}, p. 150).

However, the analysis, through units, episodes and flashes, strives to understand, through the captured flashes, the movement that portrays reality as they see it. “This movement was captured here by the subjects’ speech, understood as a mediating instrument of thought and an element that supports the development of all higher functions” (^{Silva, 2018}, p. 154). In this way, the analysis is based on the understanding of human thought, based on verbal language. To this end, and in accordance with ^{Luria (1991}, p. 17), “[...] conditions to overcome the limits of immediate sensory perception of the external world are necessary”.

From this procedural movement, the following composition of the analysis presented in chart 2 arises.

Source: Prepared by the authors (2024).

Chart 2 - Structure of data analysis 

These episodes were selected among many others possible, because they are able to reflect, make new connections and relationships between the actions of the formative experiment, allowing conditions to form concepts and carry out new analyzes and syntheses about the formative process that took place with Mathematics teachers in initial training who participated in the Mathematics Club.

6 Considerações finais

This article was established with the aim of investigating how a LTS, based on the LHM theoretical proposition of concepts, in a particular training space, contributed to the training process of Mathematics teachers.

This training space is the MathClub, which considers that the LHM is able to cooperate in understanding the important role of teaching intentionality in organizing teaching, with this organization being the starting point for establishing “the action plan, through knowledge about the idealized object” and, to this end, he needs to make use of “theoretical assumptions, define actions supported by these assumptions, seeing them as mediating instruments of these actions and, when acting in a process of analysis and synthesis, he aims his activity” (^{Moura; Sforni; Lopes, 2017}, p. 84).

Thus, at the Club, the logical-historical conception adopts the function of a didactic training proposal “that takes history as a provider to the extent that it makes it possible to perceive the movement of thought in conception and development” (^{Leandro; Sousa; Andrade, 2020}, p. 395) of the concepts and the way they are presented in school: as content.

Thus, when planning the LTS - The Agnuns -, it was expected that it would be embodied in the very idea of the emergence of the concept of number. The mathematical concept is understood here as the synthesis of human knowledge up to the current historical moment. The VH, which was presented as a comic, served as a springboard for new syntheses of the subjects, involved in a perennial movement of analysis and synthesis about the laws that govern objective reality.

In this way, the proposed teaching organization, offered at MathClub, presented them with mathematical concepts as ideas, through which man creates a “subjective image of the objective world” (^{Kopnin, 1978}), unifying the subject and of the object (^{Moura, 1996}). Such an approach is rare in other training contexts.

In this theoretical line, the requested teaching learning in Mathematics allowed a conscientious organization of individuals' training processes, through the purposeful organization of teaching that allows subjects to appropriate knowledge and ways of thinking.