A - Perspective of teacher training based on technical rationality

In this category, based on the theoretical contributions presented, we understand that teachers whose practice is guided by this rationality act as a technician or executors of pre-established routines and behaviors, and their work simply consists of applying the knowledge produced by applied science. Thus, there is an emphasis on mastering the technique, as well as the belief that their learning, perceptions, and knowledge have no validity or prominence and should not be taken into account for classroom action. Therefore, pedagogical practice is based on the domain of technique and control over the practice resulting from theories thought by researchers and/or higher education professors.

In the context of this study, we realized that the teacher who is anchored in this perspective of formation, when participating in the FMat, socializes a work that is restricted only to demonstrating mastery and technical dexterity used in the application of theories and achievement of pre-established results, whether in research or the classroom. It does not dare, as it is stuck in the reproduction, implementation, or application of models recommended by educational theorists and/or researchers. One possibility is that this teacher is a repeater of practices, that is, it is not that technical rationality is important to him, it may be the only possible way to operate, it is normal, whether consciously or unconsciously.

A work exhibited at FMat based on this perspective can present a text tied to definitions, models, or prescriptions from others, coming from sources such as textbooks or their initial/continuing training when guided by technical rationality. It is an activity tied to teaching considered more traditional, guided by the exercise paradigm, and may bring some contextualizations, but these are generic or not made thinking in the context of their learning subjects. It does not bring reflections on what was developed or considerations of the political dimension of its educational practice.

Within our interpretation, we perceive that the writing of the report, in this perspective, tends to evidence the reproduction of practices alien to the teachers, denying their possibilities of criticality and judgment. It emphasizes contents and skills that can be measured or observed, which will highlight and reinforce the applicability of scientific knowledge that resolves, in a somewhat mechanical action, where the context is reproduced and disregarded.

Our perceptions stem from the fact that the training perspective based on technical rationality is epistemologically based on the positivist philosophy, which praises a teacher trained with a broad theoretical basis who will apply this theory to solve everyday problems. This was noticed, for example, in T13, when the authors sought with the work “to analyze the performance of the textile effluent treatment plant, quantifying in percentage the efficiency of removal of the polluting potential of the textile industry effluent treatment plant, which is composed of the biological and physical-chemical treatments” and as results, they expressed that “the annual efficiencies that most varied were the parameters: N Am, ST and SSed, and the ST showed efficiencies below 10% in all years". The writing reveals that this teacher makes use of a knowledge of the academy in his/her research practice, acting as a technician, someone who operates a knowledge arising from applied science and evidence technical skills. Not that it is not important. The problem lies in the fact that he/she does not dare to extrapolate the results obtained to perceive their implications in the context in which he/she is inserted, which such results indicate in the face of the context and problem analyzed. Nor do they suggest what could be done from these results. The results are only for the results, since, for him, the important thing, as expressed by ^{Schön (2000)}, is to apply known and controllable techniques to solve problems, in this case, his/her research.

We also highlight texts from works that bring intervention proposals to the classroom, derived from the application of scientific knowledge, as was the case of T24, when he/she sought to “present a proposal for a didactic sequence that promotes the study of the polynomial function of the high school and that can help identify and overcome obstacles that students bring with them, which can significantly hinder or make learning impossible”. Writing makes it clear that, for them, there is a world of theory, represented by the university and the knowledge that derives from it, and the world of practice, in school institutions that must implement proposals arising from the application of systematic and normative knowledge thought by the former. In this way, they are in line with the belief that this sphere is responsible for theorizing and proposing solutions for the educational field, leaving for the teacher the role of following a prescribed model. Text T24 suggests that its authors may be defending the idea that “the professional activity consists of the instrumental solution of a problem made by the rigorous application of a scientific theory or a technique” (^{SCHÖN, 2000}, p. 21) as if this proposal would be the adequate solution to the problems of contextualization, interest, and learning. The emphasis of the text is on “know-how”, characteristic of this instrumental perspective, with “primacy of technical knowledge over educational practice” (^{FONSÊCA, 2009}, p. 93). Like T13, it does not bring other aspects that may imply positive results for learning and may be included in a type of training based on technical rationality, described as a traditional academic model. As described by ^{Cochran-Smith and Lytle (1999}, p. 6), these teachers may emphatically consider that knowledge for practice is constructed “through various training experiences that give access to the knowledge base. To improve teaching, [...] they need to implement, translate, or put into practice what they learn from experts outside the classroom”.

In this same perspective, we cite T34, who tried to present a planned workshop that, although it brought technological resources to show geometrically how to find the roots of a quadratic equation, only performed the steps of the tasks without reflecting on the purposes, interpretation of what was accomplished or changes in the educational context compared to the developed context, which can characterize a process in which a manual that provides the step-by-step mechanically, evidencing mastery and control over the educational action is only followed. The teachers may be limited to extrapolating the rigid models, established mainly by the teaching materials, as they understand that they must be followed in full and that there is no space for authorship based on their perceptions or the reality in which they live.

Another perspective of texts that make up this first category is those that bring up practices developed in the classroom, such as T38. In it, the authors report a work that begins with a diagnosis and, later, a didactic sequence. However, for the elaboration of this, according to the text, the deficiencies of the students identified in the diagnosis were not taken into account, and we were surprised by the fact that it was only used with a protocol without pedagogical implications. The central part of T38's proposal was around the study of textbook definitions, related to the concepts of matrices, determinants, systems, and vectors, which may show a posture more aligned with technical rationality, given the emphasis on common definitions and procedures present in textbooks. The structure and description of the proposal expressed in the text led us to understand that the classes of these teachers could be developed based on the definition scheme, examples, and exercises, which can be seen when they express that “the contextualization took place after these classes, with the study of digital images. There are two types of digital images that we call matrix or vector ” (T38).

In the same text, the authors present a critique of textbooks right at the beginning when they express that: “Starting with the analysis of some textbooks, we realize that, in general, examples are contextualizing Algebra in everyday life and utilitarian aspects and operations with matrices are introduced mechanically, without concern for the geometric investigation of these transformations” (T38). However, throughout the text, they do not bring or leave little evidence of the new search source, since they present a proposal very close to the traditional one made up with an application. Texts like this can evidence the existence of a teacher who presents and executes something still very much related to the textbook or some material produced by the academy, usually with little or no adaptation and reflection. They almost faithfully reproduce something already elaborated, exempting from venturing or not perceiving themselves as subjects capable of being authors.

Regarding this perspective of training, we identified that 23.7% of the texts reveal that, in summary, the practice reported by the teacher is closer to the perspective of training based on technical rationality, which shows an excessive appreciation of the knowledge produced in the academy, in detriment of the knowledge acquired through practice, putting the teacher in a position of just applying methods, techniques, and strategies without questioning them. To this result, we attribute two interpretations: first, we can have a teacher who is getting to know new theories and possibilities and, as the first stage of this training, replicates prescribed models without great participation, that is, with little protagonism and authorship; second, we realize that these teachers are based on a training paradigm that puts them on the sidelines when it comes to proposing solutions for teaching-learning relationships. Both interpretations need to be problematized if we believe in a formative movement that moves towards a perspective more aligned with the precepts of FMat.

In line with ^{Floriani and Zermiani's (1985}, p. 12) emphasis that the “Mathematics Fair, if it wants to contribute to the transformation of teaching, must awaken people's creativity and lead them to innovation”, we indicate that perspectives of training aligned with a technical paradigm need to be highlighted and problematized so that the teacher can overcome this paradigm and advance towards perspectives that give space for authorship and protagonism, characteristics that, in our view, are fundamental for having a practice. innovative.

B - Training perspective closer to practical rationality

In this second category, supported by theoretical contributions, we understand that a teacher who has his/her epistemological assumptions based on practical rationality guides his/her pedagogical action by the experience acquired with the practice over the years or by guidelines passed on by more experienced professors and colleagues, does not being concerned with studying theoretical aspects to guide their choices or reflect on them. What defines teaching work is the practical and deliberative nature of its interpretation and understanding of concrete situations that develop in the context of classroom interactions, teaching and learning situations, as well as their meanings and senses. Self-reflection, even without theorizing, can lead to changes in their practice, as the emphasis is on knowledge in action.

The teacher who has this perspective of training, when participating in the FMat, socializes a work resulting from his/her teaching activities in which he/she sought alternatives to problems in the classroom or related to his/her profession. This work emphasizes the practical character and seeks to remove or extinguish any theory that can anchor, justify or guide its choices and results. He/she will not be a technician who performs an action guided by others, but someone who seeks to validate actions in his/her successful experiences through narratives about his/her practices with his/her peers.

The analysis indicated that 42.1% of the works seem closer to this training perspective. They present a text in the form of a report based on previous experiences and/or topics of interest to students, which, in most cases, consist of a sum of activities that do not always converge to a central objective. There are attempts to break with the practice of traditional teaching, with the use of games or instructional materials, and project work, but without the awareness of their reasoned use of them, in a somewhat empirical way, perhaps because they acquired that knowledge observing the practice of others and ignoring the theory and studies behind it. The texts make it clear that, for these authors, “teaching is, to a certain extent, an uncertain and spontaneous craft situated and constructed from the particularities of everyday life in schools and classrooms” (^{COCHRAN-SMITH; LYTLE, 1999}, p. 15).

Thus, writing will essentially be used to report or describe an authorial pedagogical action performed, a didactic material or game developed for a certain purpose, devoid of any theoretical support that guides, anchors or justifies the choices and the results. After all, tacit knowledge will prevail, as the emphasis is on professional autonomy. The text shows creation, a certain boldness in the teacher's actions, and the results found with practice, generally related to the learning of their students, indicate solutions for pedagogical action or that support or redesign new actions.

Texts included in this category bring experiences developed and centered on what we call “themes of interest”, didactic sequences, or didactic materials or games. These texts come from works motivated mainly by empirical and not theoretical justifications.

An example of “themes of interest” is T27, which deals with the “social function of mathematics”. In it, the authors sought to contemplate a sum of activities that do not always converge to a clear and defined central objective, that is, from the choice of a theme, they plan a series of activities enhanced by the subject, and not by the objectives they intended to achieve with their development. When such objectives are expressed, they are usually broad and difficult for the teacher to assess or follow up on. In this type of work, it is not the objectives that determine the choice of tasks, but the theme. This can be seen in T27 when the authors report in the text that they aimed to “analyze the social function of mathematics existing in early childhood education and its importance in deconstructing the view that mathematics education only happens to older children”, for this purpose developing a didactic sequence based on the “children's song 'Sítio do Seu Lobato', which allowed the study of mathematical concepts, through playful approaches such as children's stories, mathematical games, songs, and games”. The writing illustrates the results of practices planned and/or developed based on the theme “sítio do seu Lobato”. It denotes the authorship and protagonism of the teachers, who make use of their expertise in the elaboration of activities, most of the time originating from the children's desires or some aspect of the context, such as in T16, T31, and T35. There is a movement of creativity, in empirical study, but we do not know if the choice is conscious or random.

In T27, although they expressed the objective described above, the authors focus on the presentation of activities carried out with or by the children, but everything through the experience of practice, that is, what that set of activities would provide to the little ones. In addition, they do not bring any study that claims that there is a “view that mathematics education only happens to older children”, which may indicate a certain departure from the theoretical knowledge of the area and the existing literature on the subject.

When we direct our view to the conclusions that are made, such as, for example, “we conclude that there has been noticeable progress in children, the ability to reason and solve problems has become fast. The interaction with the parents was gradual, but satisfactory, achieving our goals. Proving that mathematics education is possible in this age group” (T27), we observed that it lacks a comprehensive or reflective essay that shows some change in the teachers' condition.

Also, the activities belonging to the practices are not guided, justified, or anchored in any theoretical support, after all, the focus is on the action and not on what can result from it because the knowledge results from it. The decisions made by teachers do not intend to apply scientific and pedagogical knowledge as in the model of technical rationality, but rather to bring to light a space with something derived from their autonomy, in which knowledge is constantly generated and modified, potentiated from the validation of their peers in the FMat socialization spaces.

In this same category, texts that bring didactic materials and games developed in and for the classroom, derived from their practice, as was the case of T8, also stand out. In this type of text, there is evidence of an effort by the teacher to work the contents less traditionally, making use of concrete materials, posters, everyday situations, and exits from the classroom. However, all this is done in a very empirical way, which characterizes and highlights practical knowledge for teaching actions and its direct relationship with classroom performance (^{COCHRAN-SMITH; LYTLE, 1999}). T1, T7, and T29 also fit into this same perspective. The texts probably express what teachers know and are part of an accumulation of experiences.

The reason for the lack of reflection in texts belonging to this category is still unknown. We do not know if the discussions are not included in the text due to lack of space (after all, there are about seven pages), because they believe that FMat is not the place to bring this up, or because they do not have conceptual appropriation and mastery of writing. Thus, we believe that practical rationality is closer to the tacit than to the exercise of “deliberate reflection on the experience” carried out (^{COCHRAN-SMITH; LYTLE, 1999}, p. 15).

Furthermore, it called our attention that, in most texts, the experiences are always successful, devoid of problems or difficulties felt or had in the course of the work. They only highlight positive results, as if the experiences were free of difficulties or obstacles that may have arisen, both from the students and the teachers. This can lead the reader to an illusion of a truly romanticized environment, in which everything occurs in perfect harmony, without obstacles and setbacks, which may indicate a naive conscience (^{FREIRE, 1979}) of the teachers in what they should expose in the report. The validation of the developed proposal is made by the teacher's judgments and without aspects that justify or anchor the statements, only in the experience of practice. There can be two pretensions: to become an expert or to win the approval of one of them.

Finally, we emphasize that similarly to our study, the proximity character of the training concept to practical rationality had already been identified in the study by Silva (2014), although it was carried out in a temporal ballast before ours and using another corpus of analysis. This shows that little progress may have been made in FMat in this regard.

C - Perspective of teacher training closer to a critical rationality

In this third and final category, we will present our understanding of the conception in which the teacher is seen as a professional capable of reflecting, questioning, and theorizing. Based on the theoretical contributions already explained, we can say that a teacher whose practice is guided by critical rationality seeks to base practice guided by his/her experience, his/her knowledge, and the knowledge of colleagues acquired through experience with the practice, as well as the theories that support them. such choices. After all, he/she feels the need to understand phenomena from the historical and social context, highlighting his/her intentions with the educational action. He/she constantly seeks to reflect on the practice based on theoretical references, dialoguing and socializing such experiences with his/her peers. It would be a movement of non-hierarchy between knowledge of practice and academic knowledge, understanding that both have relevance and importance for a pedagogical practice aligned with the precepts defended by Mathematics Education. That is, he/she is someone who does not separate formal knowledge from practical while valuing the interrelation of the two.

We realize that the teacher who has this perspective of training, when participating in the FMat, socializes a work in which a process resulting from praxis is evident (^{PIMENTA, 1994}), in which action and reflection are articulated. The teacher justifies the action taken with arguments that come from his/her experience, as well as from theoretical foundations that anchor his/her choices and his/her criticisms, considering the implications in the context in which they develop. He/she can create from experience, as well as reflect on practice based on theoretical references, realizing that these give him the confidence to dare in the classroom (^{GUÉRRIOS, 2002}). He/she does not stick to prescribed models, but he/she is aware of the importance of theory to contribute to his/her practice, assuming a position of permanent training. In this understanding, we see that search is the keyword. Participation in collaborative groups can be a great ally for the teacher who wants to work within this perspective (^{COCHRAN-SMITH; LYTLE, 1999}).

His/her writing was essentially used to report practices in which the teachers went beyond presenting or describing what the students did, also contemplating a conscious analysis of the actions developed, whether for learning, for practice, or the contribution to the area in question. The description thickens a critical reflection, showing that they can highlight the knowledge produced by doing and reflecting and/or questioning what was done, and why it was done. Therefore, the writing must include, in addition to the description of the action, an interpretation of what is being reported enriched through reflection and discussion of the results, since the implications of what has been developed is also a concern of the author. All of this is anchored in the theoretical assumptions that support the actions and reflections.

Regarding this perspective of training, we identified that 34.21% of the texts reveal that, in summary, the practice reported by the teacher is closer to the perspective of training based on critical rationality. As we can see in excerpts from T11, in which the teacher reports activities with games as follows: “When playing, the students recorded (writing or drawing) about the game, describing their learning, their doubts, their opinions and their impressions about the experienced practice”. He/she goes on to argue that “games do not appear in a sequence to be used from beginning to end. They were designed to offer different levels of complexity, for different grades/years, involving different concepts and mathematical procedures”. In this same text, the teacher explains details of the planning and execution of his/her practice and gives recommendations to his/her peers on how to make good use of this methodology, as can be seen when he/she expresses: “For the teacher to introduce games into the daily life of their institution, he/she needs to recognize in play and games space for investigation and construction of knowledge about different aspects of the social and cultural environment in which children live. For the child to have a good performance in mathematics through play, there must be an educational intention, which implies teacher planning, aiming to achieve predetermined goals. The role of the teacher is fundamental, intervening and mediating the games, respecting the rhythm of each child, their discoveries of stimuli and challenges”. That is, he/she exposes his/her practice as well as proposes/cause others to have a reflective practice, in which they perceive as subjects capable of discovering the knowledge produced by doing and reflecting on what was done. Reflective practice requires, in addition to reflection on action, knowledge in action (tacit knowledge) and reflection on reflection in action. Practical rationality and/or reflection in action are not enough if it is not critical and can demonstrate and value praxis more than practice (^{PIMENTA, 1994}). Moments of reflection were also highlighted in T6 and T18.

In the T28 report, the teacher moves between theory, practice, and reflection, based on the theoretical contribution of the Interdisciplinary Islands of Rationality by Gérard ^{Fourez (2008)}. Even having made use of a teaching methodology proposed by a theorist from the academy, we realized that this work departs from technical rationality because the teacher reports that there is no need to follow steps linearly. He/she cites other authors to state that the route must be open and flexible when putting the eight steps as a suggestion, as well as warning that they can be revisited and/or deleted if necessary. The teacher also reports that he/she brought to his/her room professionals from other areas who were of paramount importance for the clarification of doubts, as well as for guiding the future paths of the project. In doing so, he/she is in a position of also a learner along with the students. Another highlight was the statement that during the application of the project “many questions were answered, others were not”, which we interpret as a statement consistent with a classroom environment that is very dynamic and heterogeneous, which would make it impossible for questions about any topic were completely exhausted. Supported by ^{Cochran-Smith and Lytle (1999}, p. 28), we understand that “it is not that teacher research provides all the knowledge needed to improve practice, nor that the knowledge generated by university researchers is of no use to teachers”. For the authors, when one is working within the conception of knowledge of practice, there is a movement in which

specialist teachers and others who are studying them (collaboratively or not) generate a new type, or a supplementary type, of formal knowledge about competent teaching practices. But neither are they assumed to generate and codify a new body of practical knowledge based on epistemological standards different from but derived from those of formal knowledge. On the contrary, the implicit idea of knowledge of practice is that, through investigation, teachers throughout their professional lives - from novice to experienced - problematize their knowledge, as well as the knowledge and practice of others, placing themselves in a different relationship with knowledge. (^{COCHRAN-SMITH; LYTLE, 1999}, p. 28, emphasis added).

This attitude is evidenced in the teacher’s invitation for other specialists to enter his/her classroom, as well as in stating that “in each step we went through, we experienced the deepening of issues related to the understanding of the information contained in food labels, which was the theme of the project. These questions, in addition to involving the prior knowledge of students and their families, were also deepened with all the rigor of the scientific discipline”. In the field of problematization and constant reflection on his/her pedagogical work, we have in T6 the indication that “as a teacher of Early Childhood Education, I am constantly reflecting on my pedagogical work. I have been looking for countless alternatives in the classroom, so that children always learn more, in a more pleasant and meaningful way. Of these alternatives already applied, the game stands out as one of the most effective ways to involve the student in the activities carried out. The child learns better by playing and all content can be taught through play, in predominantly playful activities”. Most of the works perceived as closer to the critical perspective bring up moments of reflection, searching other sources, and looking at the student as someone who will bring him information that must be taken into account in the planning and execution of practices.

It called our attention that six of the thirteen works that were perceived as closer to this training perspective brought quotes from Professor Kátia Smole: T5, T11, T14, T17, T21, and T36. This leads us to believe that the author can be a reference for this group of teachers. Another highlight was that the first record of allusion to the National Common Curricular Base was made in T25, which was presented in 2016. We expected that authors such as Lev Vygotsky or Jean Piaget would be referenced, given that guiding documents of the curriculum of the Basic Education of state and municipal public education networks, such as the Santa Catarina Curriculum Proposal and the National Curricular Parameters, have an explicit alignment with the Historical-Cultural Theory and the constructivist perspective, respectively. We bring this up because the level of performance of the exhibiting teachers is located in EI and AI. However, few works brought up the considerations made by these two authors.

Following our analysis, the author of T17 reports that it was the students who “showed in the sequence everything that we would forward and what they would bring us as a source of research [...] At each stage, a learning process, a meaning, and the certainty that every attempt is built in a work articulated with the other areas, providing opinions, discoveries, formulations of hypotheses”. We agree with Cochran-Smith and Lytle (1999, p. 34) when they state that it is highly formative for teachers to challenge their assumptions, to perceive themselves as solution-generating agents, “studying their students, classrooms, and schools; building and rebuilding curriculum; and assuming leadership and activism roles in the pursuit of transforming classrooms, schools and societies”. When the author of T18 reports that “as a teacher of Early Childhood Education, I am constantly reflecting on my pedagogical practice” and that “during the application of interdisciplinary play, I worked a lot on thinking, instigated, provoked, led the student to think, to create hypotheses, asking a lot for their participation, mainly orally”, we envision a practice convergent to a critical perspective. Whether the teacher has this awareness or intentionality is something that was not possible to identify in this text, nor most of the others highlighted within this conception. However, we agree with ^{Diniz-Pereira (2014}, p. 40), when he/she states that “a community of professors-researchers, with students as co-investigators, establishes a democratic and student-centered process through which the curriculum is constructed “in the bottom to top” instead of “top to bottom”. Thus, even if the teachers have not declared themselves to be critical in their text, their attitudes demonstrate a posture aligned with this perspective.

In other texts, such as T4, T14, and T36, we see that the teachers make notes on the continuity of the work, which leads us to realize that the reported practice is not something stagnant or that it was thought only to expose in the FMat, but rather which is the exposition of something that happened and that will still happen in the classroom. This converges with one of the main objectives of FMat, signaled since 1985 by its creators, which would be to expose a practice that occurs naturally in the classroom. We cannot say that this convergence occurs due to knowledge of this information or the teachers' convictions.

In summary, we noticed in this collection of outstanding works that in the texts the authors dare to bring writing that includes authorship that goes beyond the report of practice by practice or mere reproduction of something already planned. We noticed that there is a theoretical foundation in the teacher's writing but in an implicit way. Although they are often not referenced, by the terms used we perceive assumptions that underlie the practice and/or the methodological choices. A point of convergence between the reported practices was to assume a posture of constant search for a more assertive practice that would meet the students' desires, as well as envisage providing conditions for future learning. As recommended by the critical stance, where there is a need for constant study, with a view to an ideal that is always in the future, in utopia. Perhaps because it is a report, there may be an understanding that the teacher should give more emphasis to what he/she did and how he/she did it, with both theoretical and practical justifications for the choices being in the background. Another limitation is certainly the number of pages, which prevents a longer discussion.

An overview of the analysis

After having discussed the training perspectives revealed in the analyzed texts, we consider it relevant to create an overview. The analysis identified that the texts present descriptions or narratives of practices, experiences, and reflections on the performance of teachers considering the most varied contexts and focus of study. In Table 2, we present the reference context of the authors' work or study and the training perspective evidenced in the analysis.

T11 refers to work focused on two different stages of education - EI and AI. In T4, T9, and T13, the focus is not directly related to teaching or teaching practice, but to applied research that generates knowledge for the technical and technological areas. What caught our attention was the fact that (future) teachers, undergraduate students in mathematics, direct their studies towards the production of pedagogical didactic material focused on AI (75%) and not on the area of (future) performance - AF and EM. Furthermore, in these, the technical perspective of training is evidenced.

We identified that most of the published texts come from EI teachers who come to FMat to socialize practices resulting from their teaching practice. These teachers tend to build their texts closer to the critical and practical perspectives and further away from the technical perspective. The context with the highest percentage close to practical rationality comes from works whose focus was on AI. Finally, we emphasize that the teachers who present work in the professor category at the FMat are those who work (and/or study) in the stages closest to the initial stages of Basic Education and who, for the most part, bring to the FMat the report of a work whose object is the student subject and not the teacher subject. The reason for this is not yet known.