Resumo:
O artigo discute como o discurso da Matemática específica para ensinar tem proposto a condução da conduta de professores(as) de Matemática. Tratase de um ensaio teórico decorrente de uma problematização foucaultiana que incidiu sobre estudos que tratam do Conhecimento Matemático para o Ensino, do Conhecimento Especializado do Professor de Matemática e da Matemática para o Ensino. Esse discurso põe em exercício diferentes tipos de poder, mobilizando estratégias e táticas a fim de acionar a tecnologia nomeada Tecnologia da Especificidade Matemática. Os resultados sugerem que as condutas disponibilizadas por esse discurso transitam entre a lógica individual e disciplinar à lógica governamental.
Palavraschave:
Educação Matemática; Discurso; Poder; Governamentalidade
Abstract:
The article discusses how the Mathematics discourse specific to teaching has proposed the conduct of mathematics teachers. It is a theoretical essay resulting from a Foucaultian problematization that focused on studies dealing with Mathematical Knowledge for Teaching, the Specialized Knowledge of the Math Teacher and Mathematics for Teaching. This discourse puts into practice different types of power, mobilizing strategies and tactics in order to activate the technology named Technology of Mathematical Specificity. The results suggest that the behaviors made available by this discourse move between individual and disciplinary logic to government logic.
Keywords:
Math Education; Discourse; Power; Governmentality
Introduction
In the area of Mathematics Education, different speeches have been circulated in order to argue that, to teach Mathematics, a teacher should mobilize specific mathematical knowledge for teaching (Ball; Bass, 2003BALL, Deborah Loewenberg; BASS, Hyman. Making Mathematics Reasonable in School. In: KILPATRICK, Jeremy; MARTIN, Gary; SCHIFTER, Deborah (Org.). A Research Companion to Principles and Standards for School Mathematics. Reston: National Council of Teachers of Mathematics, 2003. P. 2744.; Ball; Thames; Phelps, 2008; Barwell, 2013BARWELL, Richard. Discursive Psychology as an Alternative Perspective on Mathematics Teacher Knowledge. ZDM Mathematics Education, v. 45, Issue 4, p. 595606, jun. 2013.; Carrillo; Climent; Contreras; MuñozCatalán, 2013CARRILLO, José; CLIMENT, Nuria; CONTRERAS, Luis Carlos; MUNÕZCATALÁN, María. Determining Specialized Knowledge for Mathematics Teaching. In: CONGRESS OF THE EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION. v. 8, 2013, Ankara. Anais… Ankara, Turkey: M.E.T. University, 2013. P. 29852994.; Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014.). The term discourse is used in accordance with the Foucauldian theorization, which extrapolates the notion of a simple combination of words or phrases referring to a set of practices that designate the things^{7} 1 For Foucault (2014a) these things are men in their relations with everything around them: resources, customs, ways of acting or thinking, misfortunes like hunger, epidemics, death, etc. they speak about. Thus, a discourse is not reduced to an act of speech, it not only represents the thing of which one speaks, but constitutes it (Foucault, 2014aFOUCAULT, Michel. A Ordem do Discurso: aula inaugural do Collège de France, pronunciada em 2 de dezembro de 1970. 24 ed. São Paulo: Edições Loyola, 2014a.; 2016).
We identified the specific Mathematics discourse to teach circulating in studies of the area about different denominations, among which we highlight: Mathematics Knowledge for Teaching (MKT), Mathematics Teacher’s Specialized Knowledge (MTSK), and Mathematics for Teaching (MfT), given their widespread dissemination (Hoover; Mosvold; Ball; Lai, 2016HOOVER, Mark; MOSVOLD, Reidar; BALL, Deborah; Loewenberg; LAI, Yvonne. Making Progress on Mathematical Knowledge for Teaching. Mathematics Enthusiast, v. 13, n. 12, p. 334, apr. 2016.).
In a previous study (Grilo; Barbosa; Maknamara, 2020GRILO, Jaqueline de Souza Pereira; BARBOSA, Jonei Cerqueira; MAKNAMARA, Marlécio. Discurso da Matemática Específica para Ensinar e a Produção do Sujeito ‘Professor(a)deMatemática’. Ciência & Educação, Bauru, v. 26, e20040, 2020.), we suggested that the discourses circulating in these studies can be organized in two groups: i) cognitiverepresentational discourses, for which the teacher is able to decompress, connect, anticipate, articulate, understand, and prove mathematical ideas in a way associated with the specific demands of teaching. Therefore, he or she chooses activities appropriate to the demands of teaching, is critical of didactic and curricular materials, is efficient, regent of tasks related to teaching, and attentive to inequalities; and, ii) sociodiscursive discourses^{8} 2 From a Foucaultian perspective the two groups are of a discursive order. The register of the word discursive aims to demarcate that, when operating from this perspective, there is a recognition that there is not a single a priori mathematics to be taught, but that it emerges from the discursive interactions of a social practice. , for which the teacher is regulated by the principles of pedagogical practice, flexible, explorer of opportunities, and formulator of mathematical concepts according to the context in which he/she participates. Because it is an integral part of a social practice, it is collectively constituted, therefore, it is evolutive, participative and engaged and its practice results from a collective and unstable repertoire.
This way of organizing them is intended to point out that there are differences between the epistemological affiliations that underlie them theoretically, without disregarding the possibility that they influence each other, regardless of the terminologies adopted. In this sense, we have adopted the notion of specific mathematics to teach in an attempt to capture the variability of CME, CEPM and MpE discourses in a single expression. Therefore, when we use the phrase specific mathematical discourse to teach we are referring to a discourse formation that brings together the discourses of WEC, CEPM and MpE. A discursive formation is assured by a set of relationships that is established between instances and objects of discourse, considering their historical conditions, their regularities and dispersions (FoucaulT, 2016).
These discourses have been widely disseminated among mathematical educators in different countries, through research that relies on items that advocate for themselves the task of reflecting on real situations faced by teachers when teaching Mathematics. By way of example, we cite one of the projects of the group of researchers led by Deborah Loewenberg Ball^{9} 3 This researcher was awarded in 2017 by the International Commission on Mathematical Instruction (ICMI) with the Felix Klein medal, the highest award in academic recognition in the Mathematical Education community, in recognition of her leadership and contribution to improving the practice of teaching mathematics and teacher training, with emphasis on the development of the theory of mathematical knowledge for teaching (CME) (Source: <https://deborahloewenbergball.com/>). , the Learning Mathematics for Teaching Project (LMT), which provides more than 100 publications^{10} 4 The list with the publications can be found at: <http://www.umich.edu/~lmtweb/research.html>. that address this issue: measures the group uses to study (Schilling; Blunk; Hill, 2007SCHILLING, Stephen; BLUNK, Merrie; HILL, Heather. Test Validation and the MKT Measures: Generalizations and Conclusions. Measurement  Interdisciplinary Research and Perspectives, v. 5, n. 23, p. 118127, 2007. ); what they believe to be the structure of the teacher’s mathematical knowledge (Hill; Schilling; Ball, 2004HILL, Heather; BALL, Deborah. Learning Mathematics for Teaching: results from California’s mathematics professional development institutes. Journal of Research in Mathematics Education, n. 35, p. 330351, 2004. ); how teachers learn mathematical knowledge for teaching (Hill; Ball, 2004HILL, Heather; SCHILLING, Stephen; BALL, Deborah. Developing Measures of Teachers’ Mathematics Knowledge for Teaching. Elementary School Journal, v. 105, p. 1130, 2004.); of how teachers’ knowledge relates to students’ mathematical performance (Hill; Rowan; Ball, 2005); of the relationships between teachers’ knowledge, curriculum and teaching quality (Charalambous; Hill, 2012CHARALAMBOUS, Charalambos; HILL, Heather. Teacher Knowledge, Curriculum Materials, and Quality of Instruction: unpacking a complex relationship. Journal of Curriculum Studies, v. 44, n. 4, p. 443466, 2012.), among others.
The LMT is pointed out as one of the pillars of another project entitled Teacher Education and Development Study in Mathematics (TEDSM). This project, conducted under the aegis of the International Association for the Evaluation of Educational Achievement (IEA), was designed to provide information that could be used in the development of policies and practices for training mathematics teachers (Tatto, 2013). According to Tatto (2013), TEDSM involved seventeen countries^{11} 5 Botswana, Canada (four provinces), Chile, Chinese Taipei, Georgia, Germany, Malaysia, Norway, Oman, Philippines, Poland, Russian Federation, Singapore, Spain, Switzerland, Thailand and USA. and was the first transnational study developed to provide data on the knowledge that future teachers acquire during initial training, focusing on knowledge of mathematical content and pedagogical knowledge of mathematical content. The measures elaborated and used by these projects are the result of the refinement made by Ball, Thames and Phelps (2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008.) of Shulman’s (1987SHULMAN, Lee. Knowledge and Teaching: Foundations of the New Reforms. Harvard Educational Review, v. 57, n. 1, feb. 1987.) proposal to describe the professional knowledge of teachers.
This refinement is the result of an attempt by researchers to develop the notion of Mathematical Knowledge for Teaching (MKT) which aims to emphasize the importance of mathematical knowledge which is specific to teaching, a mathematical knowledge which, according to Hoover, Mosvold, Ball and Lai (2016HOOVER, Mark; MOSVOLD, Reidar; BALL, Deborah; Loewenberg; LAI, Yvonne. Making Progress on Mathematical Knowledge for Teaching. Mathematics Enthusiast, v. 13, n. 12, p. 334, apr. 2016.), is different from the mathematics typically taught at school (although it includes knowing the mathematics taught to students) and the mathematics needed by other professionals who are not teachers. In Ball, Thames and Phelps (2008), Knowledge of Content is subdivided into: Common Knowledge of Content, Specialized Knowledge of Content and Knowledge of Content on the Horizon. Likewise, the Pedagogical Knowledge of the Content, is subdivided in: Knowledge of the Content and Students, Knowledge of the Content and Teaching and Knowledge of the Content and Curriculum.
In this scenario, other discourses appear in dispute, either to propose an emphasis on the Specialized Knowledge of the Content, such that the specialized nature would define all knowledge regarding the teaching of mathematics (Carrillo; Climent; Contreras; MuñozCatalán, 2013CARRILLO, José; CLIMENT, Nuria; CONTRERAS, Luis Carlos; MUNÕZCATALÁN, María. Determining Specialized Knowledge for Mathematics Teaching. In: CONGRESS OF THE EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION. v. 8, 2013, Ankara. Anais… Ankara, Turkey: M.E.T. University, 2013. P. 29852994.), or to propose the nondichotomization between, for example, the knowledge of mathematical content and the pedagogical knowledge of mathematical content (Huillet, 2009HUILLET, Danielle. Mathematics for Teaching: an anthropological approach and its use in teacher training. For the Learning of Mathematics, v. 29, n. 3, p. 410, 2009.; Pournara et al, 2015POURNARA, Craig. Can Improving Teachers’ Knowledge of Mathematics Lead to Gains in Learners’ Attainment in Mathematics? South African Journal of Education, v. 35, n. 3, 2015.; Tatto; Burn; Menter; Mutton; Thompson, 2018TATTO, Maria; BURN, Katharine; MENTER, Ian; MUTTON, Trevor; THOMPSON, Ian. Learning to Teach in England and the United States: the evolution of policy and practice. London: Routledge, Taylor & Francis Group, 2018.). There are also discourses that challenge Ball, Thames and Phelps (2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008.) for considering it extremely focused on the individual and choose to present mathematical knowledge to teach from a social, collective, emerging perspective (Adler; Hulliet, 2008ADLER, Jill; HUILLET, Danielle. The Social Production of Mathematics for Teaching. In: SULLIVAN, Peter; WOOD, Terry (Org.). International Handbook of Mathematics Teacher Education: v. 1. Knowledge and Beliefs in Mathematics Teaching and Learning Development. Rotterdam: Sense Publishers, 2008. P. 195222.; Barwell, 2013BARWELL, Richard. Discursive Psychology as an Alternative Perspective on Mathematics Teacher Knowledge. ZDM Mathematics Education, v. 45, Issue 4, p. 595606, jun. 2013.; Davis; Rennert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014.).
Even not integrating TEDSM, countries like Saudi Arabia, South Africa, South Korea, Brazil, among others, did not remain on the sidelines of these discussions. Among these countries, it is possible to identify investigations that use the items proposed by LMT (Haroun; Ng; Abdelfattah; Alsalouli, 2016HAROUN, Ramzi; NG, Dicky; ABDELFATTAH, Faisal; ALSALOULI, Misfer. Gender Difference in Teachers’ Mathematical Knowledge for Teaching in the Context of SingleSex Classrooms. Int J of Sci and Math Educ, v. 14 (Suppl 2), p. S383S396, 2016.; Kwon; Thames; Pang, 2012KWON, Minsung; THAMES, Mark; PANG, Jeongsuk. To Change or not to Change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea. ZDM Mathematics Education, v. 44, p. 371385, 2012.), as well as there are those that adopt a more social perspective for research on mathematical knowledge to teach (Pournara et al., 2015POURNARA, Craig. Can Improving Teachers’ Knowledge of Mathematics Lead to Gains in Learners’ Attainment in Mathematics? South African Journal of Education, v. 35, n. 3, 2015.; Davis; Rennert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014.; Santos; Barbosa, 2016SANTOS, Graça Luzia Dominguez; BARBOSA, Jonei Cerqueira. Um Modelo Teórico de Matemática para o Ensino do Conceito de Função a Partir de um Estudo com Professores. Revista Iberoamericana de Educación Matemática, n. 48, p. 143167, diciembre 2016.).
Faced with the scope and how these discourses have influenced Mathematics Education, we consider it opportune, inspired by Michel Foucault, to problematize how these discourses have proposed the conduct of mathematics teachers. We understand that the conduct is the practices of control over the subjects’ lives, whether they are in the scope of individual or collective life, which aim to establish a possible field of action. In this sense, we have developed a theoretical essay which, according to Meneghetti (2011MENEGHETTI, Francis. O que é um ensaioteórico? Revista de Administração Contemporânea, v. 15, n. 2, p. 320332, abr. 2011. ), requires deep and detailed reflections for the understanding of things. Our reflections focused on studies in which CME, CEPM and MpE discourses circulate and which are widely disseminated in Math Education journals, as described above.
The Power is Shown in Exercise
As we sought to investigate the conduct of mathematics teachers, our gaze was focused on the effects of these discourses on their possible actions. This led us to mobilize the government theme (Foucault, 1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989.; 1999a; 2008) as a great umbrella, in the sense that it absorbs the rain of power exercises that are put into operation by specific Mathematics discourse to teach.
Power shows itself in action when it is exercised, therefore, it is not possessed, it is not won and it is not lost either. If power does not emanate from a center, it is taken “[…] as a productive network that runs through the entire social body much more than a negative instance that has the function of repressing” (Foucault, 1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989., p. 8). Thus, power does not apply to an individual or is applied by him, power crosses them in order to direct behavior.
When Michel Foucault set out to investigate the subject of government, he turned to identifying and describing the power technologies aimed at conducting himself and others. From some of his works (Foucault, 1999a; 2008), it is possible to understand that the term technology was used by Michel Foucault “[…] as a system of practices invested with strategic rationality” (Villadsen, 2014VILLADSEN, Kaspar. Tecnologia Versus Ação: uma falsa oposição atribuída a Foucault nos estudos organizacionais. Revista O&S, Salvador, v. 21, n. 71, p. 643660, out./dez. 2014., p. 3).
For Castro (2016CASTRO, Edgardo. Vocabulário de Foucault: um percurso pelos seus temas, conceitos e autores. 2 ed. Belo Horizonte: Autêntica, 2016., p. 412), “[…] studying practices as techniques or technology consists in placing them in a field defined by the relationship between means (tactics) and ends (strategy). Therefore, to describe a technology consists in situating it in terms of strategies and tactics. Strategy is the means used to make work or to maintain a technology of power, it aims at the routine training that shapes individuals from previously planned objectives and tactics is what puts the strategy into operation. According to Maknamara and Paraíso (2013MAKNAMARA, Marlécio; PARAÍSO, Marluce. Pesquisas PósCríticas em Educação: notas metodológicas para investigações com currículos de gosto duvidoso. Revista da FAEEBA  Educação e Contemporaneidade, v. 22, n. 40, p. 4153, 2013., p. 49), “[…] while strategy is meticulously architected [...], tactics are opportunistically activated”. Foucault (2014bFOUCAULT, Michel. Vigiar e Punir. Petrópolis: Vozes, 2014b.) illustrates these concepts considering the military scope for which strategy can be understood as a way to conduct war and tactics would be the existence of the army as a principle to maintain the absence of war in society.
The use of the word government should not be confused with the use that is currently given to the word as government of a state; bodies, population are governed. The population is a political character that appears in the 18th century (Foucault, 2008FOUCAULT, Michel. Segurança, Território, População: curso dado no College de France (19771978). São Paulo: Martíns Fontes, 2008.), as a new collective subject, “[…] a new body: multiple body, body with countless heads, if not infinite, at least necessarily numerable” (Foucault, 1999a, p. 292).
In the present study, we consider the set of mathematics teachers as the population for which the mathematics discourse specific to teaching structures a possible field of actions. This discourse has provided information that subsidizes both public educational policies, whether at the level of basic schooling (Hill, 2007HILL, Heather. Mathematical Knowledge of Middle School Teachers: implications for the No Child Left Behind Policy initiative. Educational Evaluation and Policy Analysis, n. 29, p. 95114, 2007.) or teacher training (Hill; Ball, 2004HILL, Heather; BALL, Deborah. Learning Mathematics for Teaching: results from California’s mathematics professional development institutes. Journal of Research in Mathematics Education, n. 35, p. 330351, 2004. ), and the pedagogical relationships that occur in the classroom (Charalambous; Hill, 2012CHARALAMBOUS, Charalambos; HILL, Heather. Teacher Knowledge, Curriculum Materials, and Quality of Instruction: unpacking a complex relationship. Journal of Curriculum Studies, v. 44, n. 4, p. 443466, 2012.). Given their breadth, both in terms of what they subsidize or may subsidize and in terms of territorial scope, they put into operation government practices that aim to structure the possible field of action for all Mathematics teachers, moving from individual and disciplinary logic to government logic.
In trying to make explicit the transition from individualizing power practices to massifying practices, Michel Foucault coined the concept of gouvernementalité which, according to Fimyar (2009FIMYAR, Olena. Governamentalidade como Ferramenta Conceitual na Pesquisa de Políticas Educacionais. Educação & Realidade, v. 34, n. 2, p. 3556, maioago., 2009.), translates the effort to create governable subjects through control techniques, normalization and molding of people’s behaviors. For VeigaNeto (2002)VEIGANETO, Alfredo. Coisas do Governo... In: RAGO, Margareth; ORLANDI, Luiz; VEIGANETO, Alfredo (Org.). Imagens de Foucault e Deleuze: ressonâncias nietzschianas. Rio de Janeiro: DP&A, 2002. P. 1334., the word governance would be the most appropriate translation for gouvernementalité. Thus, governance is government practices that have their object in the population and that intend to structure the possible field of actions of themselves and others (Foucault, 2008).
In taking the concept of governance, Michel Foucault does not disregard that these powers continue to act on individual bodies with the pretension of conducting conduct. Thus, in order to unveil how the Mathematics discourse specific to teaching has proposed the conduct of mathematics teachers, we analyze different types of power put into operation by this discourse. To do so, we rely on the genealogy on the subject of government carried out by Michel Foucault, who denies the possibility of a center of power, generally represented by the State in classic theories on government, and turns to show how power is diluted, crossing the whole social structure, in the defense that “[…] power is everywhere; not because it encompasses everything, but because it comes from everywhere” (Foucault, 1999bFOUCAULT, Michel. História da Sexualidade I: a vontade de saber. 13 ed. Rio de Janeiro: Edições Graal, 1999b., p. 89).
The Exercise of Power: from disciplinary to governmental logic
In investigating the theme of government, Foucault (1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989.; 1999a; 2008) identified and named different types of power, of which we highlight: sovereign power, pastoral power, disciplinary power and bio power. These powers were identified in different historical times, but still today they can coexist by acting mutually on bodies and populations.
We will begin our digression into the sovereign power that is based on the existence of the figure of a sovereign and his subjects. According to Foucault (1999aFOUCAULT, Michel. Em Defesa da Sociedade: curso dado no College de France (19751976). São Paulo: Martíns Fontes, 1999a.), the sovereign, in order to defend his territory or himself, holds the right over the life and death of his subjects. This allows us to ask, for example, why the study developed by Shulman (1987SHULMAN, Lee. Knowledge and Teaching: Foundations of the New Reforms. Harvard Educational Review, v. 57, n. 1, feb. 1987.), which intended to revolutionize the way the professor’s knowledge had been researched until then, was not enough to account for the knowledge required of the mathematics professor? How sovereign is the Mathematics that leads Ball, Thames and Phelps (2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008.) to establish specific domains in the conceptual map of teachers’ knowledge elaborated by Lee Shulman, even recognizing the importance of his ideas on content knowledge and pedagogical knowledge of content in teaching? The authors themselves outline a possible answer to these questions:
Our hypothesis is that teachers’ opportunities to learn mathematics for teaching could be better adjusted if we could identify these types [of knowledge] more clearly. If the mathematical knowledge needed for teaching is indeed multidimensional, then professional education could be organized to help teachers learn the range of knowledge and skills they need in focused ways. If, however, the mathematical knowledge needed for teaching is basically the same as general mathematical ability, then it would be unnecessary to discriminate against professional learning opportunities. Based on our analysis of the mathematical demands of teaching, we assume that Shulman’s knowledge of content could be subdivided into CCK [common knowledge of content] and specialized knowledge of content and his pedagogical knowledge of content could be divided into knowledge of content and students and knowledge of content and teaching (Ball; Thames; Phelps, 2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008., p. 399).
Thus, this Mathematics that is specific to teaching would have a prominent place, exercising a sovereign power that crosses its subjects, the teachers of Mathematics. Therefore, the territory to be governed is that which involves teaching practices, whether in initial or continuing education courses or the pedagogical practices established in the school spaces. The subjects’ sacrifice is identified in the sense of making die a teaching practice based on a Mathematics that is not specific to teaching, allowing letting live a teaching practice that is aware of such specificities.
The sovereign power exercised by mathematics, as a body of scientific knowledge that needs to be scrutinized before being moved for teaching purposes, runs through all those who deal with it. Thus, as Foucault (1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989.) suggests to us, we look at this power not as located in a center; but, as a network, from its ramifications when crossing this multiple body that are the teachers of Mathematics.
The categorization proposed by Ball, Thames and Phelps (2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008.) (Figure 1) shows that Mathematical Knowledge for Teaching requires, among other domains, a Specialized Knowledge of Content that is different from the general mathematical ability that would be expressed by a Common Knowledge of Content.
These domains are described in most of the studies where cognitiverepresentational discourses circulate on the specific mathematical knowledge to teach (Barwell, 2013BARWELL, Richard. Discursive Psychology as an Alternative Perspective on Mathematics Teacher Knowledge. ZDM Mathematics Education, v. 45, Issue 4, p. 595606, jun. 2013.). For Ball, Thames and Phelps (2008BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content Knowledge for Teaching: what makes it special? Journal of Teacher Education, v. 59, n. 5, p. 389407, nov./dec. 2008.), Mathematical Knowledge for Teaching is multidimensional and once the types of knowledge required to teach are identified, teacher training courses could be organized to help teachers learn the range of knowledge and skills they need to teach.
Other studies propose that teachers’ knowledge is specialized (Carrillo; Climent; Contreras; MuñozCatalán, 2013CARRILLO, José; CLIMENT, Nuria; CONTRERAS, Luis Carlos; MUNÕZCATALÁN, María. Determining Specialized Knowledge for Mathematics Teaching. In: CONGRESS OF THE EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION. v. 8, 2013, Ankara. Anais… Ankara, Turkey: M.E.T. University, 2013. P. 29852994.; Moriel Junior; Wielewski; 2017MORIEL JUNIOR, Jeferson; WIELEWSKI, Gladys. Base de Conhecimento de Professores de Matemática: do genérico ao especializado. Revista de Ensino, Educação e Ciências Humanas, v. 18, n. 2, p. 126133, 2017.) and configure the notion of Specialized Knowledge of Mathematics Teachers. As can be seen in Figure 2, CEPM also organizes itself into domains and subdomains that would focus on demands related to the teaching of Mathematics in particular. Thus, in these discourses it is not enough to recognize that to teach Mathematics the teacher needs to have a Specialized Knowledge of the Content: it is necessary to conduct it in such a way that it moves between one type of knowledge and another.
Even supported by distinct epistemological perspectives, in a similar way, the social discourses of mathematics specific to teaching also turn to the modes of displacement, no longer of the teacher as an individual body, but of the population of teachers. The interest lies in observing as teachers, while a body with countless heads moves between the emphases (Figure 3) that would constitute the ways to communicate a given mathematical concept.
Backed by the theory of complexity, Davis and Revert g (2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014.) understand that the emphases (achievements, panoramas, linkages, and combinations) would be more focused on the real mathematical content of teaching. Still according to the authors, these emphases do not occur in a linear manner or as stages; they would be coimplicated, emerging and evolving, which would require participative, collective and ongoing commitments from teachers.
The strategy used to exercise sovereign power is to make a Mathematics that is specific to teaching live in order to let a Mathematics that is not specific to teaching die, in the defense that there is a mathematical knowledge that is specific to the teaching of Mathematics. Together with this strategy, the tactic of differentiation is activated, which intends to strip away categories of professional knowledge proposed by Shulman (1987SHULMAN, Lee. Knowledge and Teaching: Foundations of the New Reforms. Harvard Educational Review, v. 57, n. 1, feb. 1987.) in order to make explicit the difference between Mathematics to be made available to the teacher and Mathematics required by other professionals. Both discourses, cognitiverepresentational and sociodiscursive, operate in the sense of making die pedagogical practices based on a Mathematics that is not specific to teaching, in order to let live practices that are established through a specific Mathematics to teach.
We observe, however, that other forms of power are put into operation by these discourses moving from the expression of an individualizing and disciplinary power to the massifying logic that makes teachers live by their rules. The disciplinary power turns to control multiplicity, to organize it in such a way as to use it to the maximum; “[…] it defines how one can have dominion over the body of others, not simply so that they do what one wants, but so that they operate as one wants” (Foucault, 2014bFOUCAULT, Michel. Vigiar e Punir. Petrópolis: Vozes, 2014b., p. 135). The disciplinary power “[…] takes individuals both as objects and as instruments of their exercise” (Foucault, 2014b, p. 167).
In search of this training of the bodies, the disciplinary power resorts to discipline. The discipline is revealed in “[…] methods that allow for the thorough control of the body’s operations, that carry out the constant subjection of its forces and impose on them a relationship of docilityutility” (Foucault, 2014bFOUCAULT, Michel. Vigiar e Punir. Petrópolis: Vozes, 2014b., p. 135). The discipline acts on the bodies; it is subtle, and its success owes to the use of simple instruments: the hierarchical gaze and the normalizing sanction, in addition to the examination that results from the combination of the former. Let us return our gaze on the exam.
The exam “[…] is a normalizing control, a vigilance that allows qualifying, classifying and punishing. It establishes a visibility over individuals through which they are differentiated and sanctioned” (Foucault, 2014bFOUCAULT, Michel. Vigiar e Punir. Petrópolis: Vozes, 2014b., p. 181). The examination reverses the logic of the visibility of power, making it increasingly invisible as it is more insidious; it provides a documentary field of records of a series of individual traits that make it possible to classify, categorize and set standards that become homogenizing.
By turning to cognitiverepresentational discourses, we identify the use of the exam as an instrument for the exercise of disciplinary power. Projects such as the Learning Mathematics for Teaching Project (LMT) and the Teacher Education and Development Study in Mathematics (TEDSM) (Tatoo, 2013TATOO, Maria (Org.). Teacher Education and Development Study in Mathematics (TEDSM): policy, practice, and readiness to teach primary and secondary mathematics in 17 countries: Technical Report. Amsterdan: IEA, 2013.) use the exam technique through the elaboration/application/refinement of instruments that are applied on a large scale to teachers and students with a view to thorough control of their bodies. These instruments are used to examine the structure of the mathematical knowledge of the teacher (Ní Ríordáin; Paolucci; O’ Dwyer, 2017NÍ RÍORDÁIN, Máire, PAOLUCCI, Catherine; O’ DWYER, Laura. An Examination of the Professional Development Needs of OutofField Mathematics Teachers’. Teaching and Teacher Education, v. 64, p. 162174, 2017.); how these teachers learn this knowledge for teaching purposes (Mosvold; Fauskanger, 2013MOSVOLD, Reidar; FAUSKANGER, Janne. Teachers’ Beliefs about Mathematical Knowledge for Teaching Definitions. International Electronic Journal of Mathematics Education  IΣJMΣ, v. 8, n. 23, p. 4361, 2013.); how they can relate them to the mathematical performance of students (Delaney, 2012DELANEY, Seán. A Validation Study of the Use of Mathematical Knowledge for Teaching Measures in Ireland. ZDM Mathematics Education, v. 44, p. 427441, 2012.; Tchoshanov, 2011TCHOSHANOV, Mourat. Relationship Between Teacher Knowledge of Concepts and Connections, Teaching Practice, and Student Achievement in Middle Grades Mathematics. Educational Studies in Mathematics, v. 76, p. 141164, 2011.); refining the evaluation instruments (Phelps; Kelcy; Jones; Liu, 2016PHELPS, Geoffrey; KELCEY, Benjamin; LIU, Shuangshuang; JONES, Nathan. Informing Estimates of Program Effects for Studies of Mathematics Professional Development Using Teacher Content Knowledge Outcomes. Evaluation Review, v. 40, p. 383409, 2016.), among other purposes.
To exemplify how the use of the exam, as an instrument of disciplinary power, is put into practice by cognitiverepresentational discourses, we present below the summary of the article by Kwon, Thames and Pang (2012KWON, Minsung; THAMES, Mark; PANG, Jeongsuk. To Change or not to Change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea. ZDM Mathematics Education, v. 44, p. 371385, 2012.).
This article examines the challenges in adapting the measures of mathematical knowledge for teaching (MKT) developed in the United States for use in Korea. After an initial analysis of the candidates’ questions regarding the ‘adjustment’ of the items to the Korean contextif the items were known, authentic, and realistic, [...]  we adapted and administered an instrument developed by the Learning Mathematics for Teaching project with 93 Korean teachers and conducted followup interviews with nine teachers. Based on the analysis of this data, we conducted a second round of review and then administered the revised test to 101 Korean teachers. The results showed that small modifications that were made to increase the adjustment often increased the teachers’ performance on the items, as expected, but the impact of the changes was sometimes difficult to interpret. For several items, the modifications introduced unexpected validity problems. The article discusses the dynamics that arise when making changes to MKT items  in particular, the strain of modifying items to increase adjustment to specific educational contexts while maintaining validity (Kwon; Thames; Pang, 2012KWON, Minsung; THAMES, Mark; PANG, Jeongsuk. To Change or not to Change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea. ZDM Mathematics Education, v. 44, p. 371385, 2012.).
From what has been explained in this study, we see how the exam is used for body control when it takes teachers as an object of study (when responding to items) and, at the same time, as an instrument of power exercise (when providing information that allows the adjustment of such items). In the logic of the exam, teachers are increasingly visible as the disciplinary power is more insidious by providing records on specific mathematics to teach. These records are first of all individual, but they allow classifying, categorizing, and setting norms that become homogenizing.
Sociodiscursive discourses are contrary to the way the exam is used by cognitiverepresentational discourses as observed in the study of Davis and Renert:
The emphasis of contemporary research on identifying and measuring what each teacher can articulate explicitly is, in our view, simply inadequate  both as tools to assess what teachers really know and as means to support the development of the M4T’s vibrant body of knowledge (Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014., p. 116).
However, this does not mean that they do not exercise disciplinary power by using the exam as an instrument of teacher control. What these discourses propose is the use of
[...] more refined analyses than largescale evaluations, largely because many of the most important aspects of teachers’ knowledge are simply not available for explicit and immediate evaluation. They are tacit and can only emerge through participation in collective explorations, such as concept studies.
That said, we engage in some pragmatic strategies to increase our population base. For example, the University of Calgary’s teacher training program has been restructured so that all teacher candidates  at the elementary and secondary levels  declare a specialization. For those who choose a specialization in mathematics, one of the main components of their twoyear (foursemester) experience is the study of the concept. In addition, the Faculty of Education offers a twoyear master of education class [...] and a fouryear postgraduate certificate that emphasizes the study of the concept.
These efforts, of course, focus on individuals. Another strategy is to focus on collective levels, such as schools and school districts. A project of this kind is just beginning, involving most teachers who deal with mathematics at a primary and secondary school in Calgary. Again organized around the study of the concept, this fiveyear project is analyzing the possible impact on school culture (Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014., p. 124).
The intention to have even more refined analyses about the variability of accomplishments of a certain concept by teachers’ points to the use of the Concept Study as an instrument of control that would allow the thorough examination of what teachers effectively do or can come to do when teaching Mathematics classes. Once again, teachers are given full visibility as an object and instrument of disciplinary power. Teachers are taken as an object when it is intended to record the different forms they use to communicate a mathematical concept and, consequently, they are also instruments of power, for it is they who provide the data for these records.
As we have observed, the disciplinary power comes into play triggered by the strategy that aims to identify what and how teachers teach Mathematics. Two tactics are activated in order to achieve this strategy. One of them, associated with the cognitiverepresentational discourses, is the evaluation tactic that, through the exam, performs a scrutiny of the pedagogical practices established in the school spaces in order to qualify and classify both teachers and students. Through the tactic of evaluation, one can see that the government of men no longer gives itself through obedience, but through the manifestation of what it is (Foucault, 1997FOUCAULT, Michel. Do Governo dos Vivos. In: FOUCAULT, Michel. Resumo dos Cursos do Collège de France (19701982). Rio de Janeiro: Jorge Zahar, 1997 P. 99106.). Sociodiscursive discourses activate the tactic of collaboration in which teachers are invited to participate in collaborative groups so that their souls may be explore, so that they reveal the ways in which they carry out a certain mathematical concept and are conducted between different emphases, in order to broaden their repertoire on the concept under study.
The participation of teachers in collective exploration activities, such as the Study of the Concept or the standardized testing of individual measures with homogenizing purposes, trigger other forms of expression of power: pastoral power and biopower. Michel Foucault discusses pastoral power by referring to the figure of the shepherd of sheep in the JudeoChristian tradition. For the author, “[…] the power of the shepherd is essentially exercised over a multiplicity in movement” (Foucault, 2008, p. 168). Thus this exercise of power does not occur over a specific place, but over a flock, the teachers, more specifically, over the flock in its displacement, in the movement that makes it go from one point to another. That is, pastoral power is put into practice by these discourses when they lead teachers between one type of knowledge and another, in the case of cognitiverepresentational discourses; or between one emphasis and another, when they deal with sociodiscursive discourses.
Although the mathematical discourse specific to teaching predicts the participation of teachers in groups, what is expected is that in the end each of them will achieve their individual salvation. In other words, it is hoped to offer each of them different ways of dealing with mathematics for teaching purposes, either by moving between one type of knowledge and another or between one emphasis and another. That is why it is so important to integrate the group, so that each one allows the other to know what he knows, directing them, leading them to each other.
Regarding pastoral power, the strategy used is confession, so that the teachers, when participating in formation groups, confess what they know and how they teach a certain concept. Linked to this strategy is the tactic of movement in which teachers are led to move between one knowledge and another or between one emphasis and another of a concept, depending on the epistemological affiliations to which the discourses are submitted, in order to achieve the promised goal: to learn a specific mathematics for teaching.
It is with the emergence of the population, as a new social body, that the bio power comes into action. Bio power, different from sovereign power and disciplinary power (but not separate from them), is not exercised over one body, but over a multiple body, the population, through regulations that seek the government of life (Foucault, 1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989.).
When we consider the specific mathematics discourse to teach, we see this kind of power in action when they subsidize, regardless of their epistemological affiliations, public policies, and programs to train mathematics teachers, for example. To illustrate how this power is shown in exercise in cognitiverepresentational discourses, below are some extracts from studies that suggest how they seek to regulate the government of teachers’ lives.
This discovery provides support for policy initiatives aimed at improving students’ mathematical achievement by improving teachers’ mathematical knowledge (Hill; Rowan; Ball, 2005HILL, Heather; ROWAN, Brian; BALL, Deborah. Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal, v. 42, n. 2, p. 371406, 2005.).
Curricular implementations in mathematics are unlikely to provide the expected benefits to students if written guidance to teachers is interpreted and promulgated differently than policy makers and curriculum planners intend (Foster; Inglis, 2017FOSTER, Colin; INGLIS, Matthew. Teachers’ Appraisals of Adjectives Relating to Mathematics Tasks. Educational Studies in Mathematics, v. 95, n. 3, p. 283301, 2017.).
The implications include limiting the minimum requirement to a fouryear university degree and requiring teachers to teach various levels of education over a period of time (NG, 2011NG, Dicky. Indonesian Primary Teachers’ Mathematical Knowledge for Teaching Geometry: implications for educational policy and teacher preparation programs. AsiaPacific Journal of Teacher Education, v. 39, n. 2, p. 151164, 2011.).
Its regulations aim to determine from minimum requirements to be a Mathematics teacher to expected proficiency levels of these teachers. It is intended to control the entire population of mathematics teachers, including determining that they must follow the guidelines developed by others, or they will be blamed for the poor performance of their students. They believe that by regulating minimum requirements and quality measures it is possible to improve students’ performance in Mathematics.
On the other hand, social discourses ponder the difficulty of grasping the complex, emerging, adaptive and evolutionary nature of mathematics for teaching. However, they argue that
[...] the [scientific] community has an obligation to work together to explore, test and create new possibilities [...] [for] much could be achieved if the teaching of mathematics and teacher training included emphasis similar to concept studies (Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014., p. 120).
Recognizing the difficulty in governing teachers through policies based on largescale evaluation results, social discourse proposes that this government be based on monitoring the relationship between, for example, students’ performance in standardized evaluations and their interest in pursuing careers that require mathematical knowledge.
[...] Alberta province, Canada, has comparatively good performance in national and international math achievement tests. Even so, enrollment in mathrelated university programs has been in steady decline for decades. [...] As important as personal achievement is, if the impacts of education are not being registered at the social and cultural levels, we question its effectiveness (Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014., p. 120).
In addition, they stress that it is important to note that there is a strong relationship between initial and continuing teacher training (Santos; Barbosa, 2016SANTOS, Graça Luzia Dominguez; BARBOSA, Jonei Cerqueira. Um Modelo Teórico de Matemática para o Ensino do Conceito de Função a Partir de um Estudo com Professores. Revista Iberoamericana de Educación Matemática, n. 48, p. 143167, diciembre 2016.; Davis; Renert, 2014DAVIS, Brent; RENERT, Moshe. The Math Teachers Know: profound understanding of emergent mathematics. NY: Routledge, 2014.) and argue for projects that combine the articulation of these trainings in the context of partner schools in which the Concept Study can be developed (Davis; Renert, 2014).
The bio power is exercised having as strategy to constitute educational public policies and teacher training programs. Two tactics are used, respecting the epistemological affiliations of their discourses: the tactic of generalization triggered by cognitiverepresentational discourses and the tactic of evolution triggered by sociodiscursive discourses. The generalization tactic, by means of regulations (laws, guidelines, resolutions, training programs), establishes general criteria to be adopted for the conduct of teachers. They establish through levels of mathematical proficiency homogenizing norms. On the other hand, the evolutionary tactic dispenses with the use of general criteria and is mobilized when it proposes articulation between the initial and continuous training of teachers because it believes that this would achieve mutual evolution, an improvement in the population, in performance, to the point of constituting a group of teachers with a cohesive professional profile and consistent with the specific demands related to the teaching of Mathematics.
We observe that the discourse of Mathematics specific to teaching puts into practice different types of power that move between individual and disciplinary logic (by mobilizing strategies and tactics affectionate to sovereign and disciplinary powers) to governmental logic (by mobilizing strategies and tactics affectionate to pastoral power and bio power). These powers in exercise aim to lead the conduct of mathematics teachers in the sense of: making a specific mathematics for teaching live; identifying what and how teachers teach mathematics; confessing their knowledge and teaching practices; and constituting public policies and teacher training programs.
Effects of Power or What Teaching Conducts?
We showed in the previous section that the conduct of mathematics teachers is only achieved after passing through powers affectionate to sovereignty and discipline, using “[…] more tactics than laws” (Foucault, 1989FOUCAULT, Michel. Microfísica do Poder. 8 ed. Rio de Janeiro: Graal, 1989., p. 166). When we turn our gaze to the types of power put into operation by these discourses we identify and describe strategies and tactics that constitute a power technology that has been addressed to mathematics teachers, which we name Technology of Mathematics Specificity.
This technology wants to account for a set of practices, equipped with a strategic rationality that has as its object the teaching of Mathematics, which allows us to show how power and knowledge are articulated to constitute governable Mathematics teachers. The Technology of Mathematical Specificity is driven by discourses that suggest the existence of a Mathematics that is specific to teaching and, therefore, differs from the Mathematics required by other professionals.
These discourses disclose required behaviors to Mathematics teachers guided by the understanding that to teach it one must know a specific Mathematics, which would be achieved in the interweaving of different strategies and tactics. Initially, the teacher is led to the recognition of the sovereignty of Mathematics in relation to other fields of knowledge, in view of the need to make available to the population of teacher’s specific domains of knowledge. Another way to lead teachers is to expose them to examination, materialized in standardized evaluations, producing as an effect an accountability on the performance of themselves and others, because when a country, a state or a school does not reach the desired educational indexes, the responsibility falls on all teachers.
It is also expected that teachers will keep moving, that is, that they will master the different knowledge required to teach Mathematics and that they will know how to move between them as they are required in the task of teaching. In addition, teachers are expected to integrate groups so that they can be collectively led to maintain a specific Mathematics for teaching. Finally, teachers are condemned to live under public policies and/or training courses that aim to subsidize minimum requirements to be a Mathematics teacher.
In this sense, we infer that the teaching behaviors desired by the specific Mathematics discourse to teach go through the constitution of a subject teacher (Grilo; Barbosa; Maknamara, 2020GRILO, Jaqueline de Souza Pereira; BARBOSA, Jonei Cerqueira; MAKNAMARA, Marlécio. Discurso da Matemática Específica para Ensinar e a Produção do Sujeito ‘Professor(a)deMatemática’. Ciência & Educação, Bauru, v. 26, e20040, 2020., p. 3). A subject equipped with capabilities that would meet specific demands related to the teaching of Mathematics, which go through the recognition of the existence of different types of knowledge or conceptual emphases and which can be acquired through: differentiation of Mathematics in relation to other areas; evaluation and collaboration of what is known; movement in search of learning new knowledge, generalization and evolution of teaching practices.
Final Considerations
To achieve the objective proposed in this essay, we analyze how power is put into operation by the discourse of mathematics specific to teaching. Our analysis has shown that these discourses put into practice different types of power. In presenting sovereign power, we discuss how mathematics needs to be scrutinized before it is moved for teaching purposes. In dealing with disciplinary power we show how the exam is used as an instrument that scrutinizes the life of each teacher by providing thorough control of their bodies. Moving from individual and disciplinary logic to government logic, we saw that pastoral power is shown when discourses intend to lead teachers collectively between different types of knowledge or between different emphases of a mathematical concept and, when analyzing bio power, we saw that the leadership of teachers is established through regulations that fall upon the entire category of mathematical teachers.
These powers in action, often in a mutual way, mobilize different strategies that are triggered by different tactics all at the service of a power technology that we have named as Mathematical Specificity Technology. It was through the analysis of this technology that we were able to show the expected behaviors for Mathematics teachers made available by the specific Mathematics discourse to teach.
The government aimed at Mathematics teachers aims to lead them to the understanding that there would be a specific Mathematics for teaching and that to achieve it the teacher would need to make a Mathematics that is specific for teaching and let a Mathematics that is not specific for teaching die. In addition, these teachers would be subjected to a thorough examination of their practices to provide information that would be able to improve the pathways to be followed in order to improve/evolve Mathematics specific to teaching and subsidize general regulations on teacher training.
Other studies are needed to reflect on the multiple forms of resistance operationalized by the power relations that are mobilized by the specific Mathematics discourse to teach. To this end, it is necessary to question the conditions of the existence of these relations, to direct one’s gaze toward the order of strategy and the struggle against the forms of subjection of the subject to himself and to others that reject, among other things, the scientific inquisition that seeks to determine who we are.

^{1}
For Foucault (2014a) these things are men in their relations with everything around them: resources, customs, ways of acting or thinking, misfortunes like hunger, epidemics, death, etc.

^{2}
From a Foucaultian perspective the two groups are of a discursive order. The register of the word discursive aims to demarcate that, when operating from this perspective, there is a recognition that there is not a single a priori mathematics to be taught, but that it emerges from the discursive interactions of a social practice.

^{3}
This researcher was awarded in 2017 by the International Commission on Mathematical Instruction (ICMI) with the Felix Klein medal, the highest award in academic recognition in the Mathematical Education community, in recognition of her leadership and contribution to improving the practice of teaching mathematics and teacher training, with emphasis on the development of the theory of mathematical knowledge for teaching (CME) (Source: <https://deborahloewenbergball.com/>).

^{4}
The list with the publications can be found at: <http://www.umich.edu/~lmtweb/research.html>.

^{5}
Botswana, Canada (four provinces), Chile, Chinese Taipei, Georgia, Germany, Malaysia, Norway, Oman, Philippines, Poland, Russian Federation, Singapore, Spain, Switzerland, Thailand and USA.
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Publication Dates

Publication in this collection
12 Apr 2021 
Date of issue
2021
History

Received
19 Nov 2019 
Accepted
27 Nov 2020