Keywords

12.1 Introduction

This study presents a panorama of the research on philosophy of mathematics education in Brazil, based on the studies approved to be presented and discussed in Working Group 11 (WG11) , ‘Philosophy of Mathematics Education’ during the VI International Seminar of Research on Mathematics Education (Seminário Internacional de Pesquisa em Educação Matemática, SIPEM VI) in 2015. With that in mind, a hermeneutic analysis of each of the studies was carried out. In light of the representativeness of WG11 in the Brazilian Society of Mathematics Education (Sociedade Brasileira de Educação Matemática, SBEM) and the importance of SIPEM as a vehicle to present Brazilian research to society, in what has become a fruitful environment for discussion and exchange of ideas, we understand that the studies presented to the group of philosophy of mathematics during the 2015 edition of SIPEM reflect important aspects of the current scenario of research on mathematics education. This is especially true for the studies addressing philosophical thought on mathematics education, or the philosophical approaches through which these studies are articulated.

The philosophy of mathematics education turns its eyes to mathematics education in the quest to understand its own way of doing and proceeding. As an interface between philosophy, philosophy of education , and philosophy of mathematics , it is from these fields of inquiry that it acquires its mode of proceeding, always inquisitive and attentive so as not to naturalize statements, concepts, and objectives. We understand that the object of philosophy of mathematics education is the critical and reflexive analysis of proposals and actions in mathematics education, in the various contexts where it takes place: in public institutions, families, the street, and the media. These analyses resort to the works of several authors renowned for their importance in these specific themes, understanding and interpreting their cases as a means to communicate with these proposals and actions and thus articulate a new discourse for mathematics education itself. The inquiries undertaken by philosophy of mathematics education also address the theory/practice articulation in the very same reality where it becomes effective or put to action—which is the reality of schools and of the mathematics classroom as much as of any other environment where mathematics education comes to be—where teacher, student, teaching contents, and mathematics themes are in action. These actions, which are outlined in their own context, stand on a perceptional, and therefore spatial, temporal, and historical ground, where educational postures, the teaching proposal, and the conception of mathematical object and knowledge unfold.

The Philosophy of Mathematics Group is a working group (WG11) that, from a philosophical perspective, investigates embryonic topics in the scope of mathematics education, and may establish a communication with other perspectives. The objective of WG11 is to collect investigations, studies, experiences, discussions, themes, and debates about mathematics as well as mathematics teaching and educational processes from different perspectives, such as the epistemological, ontological, and axiological viewpoints. Some of the topics investigated by the group are ‘What is there?’ (ontology), ‘How do you know what is there?’ (epistemology), and ‘How much is it worth and why?’ (axiology) as embedded in the theme mathematics education. The studies submitted, reviewed, and approved for presentation to this WG are the fruit of a research that derives support from the practices of debate established concerning these questions.

In addition to representatives of the numerous institutions located in several regions of Brazil (southeast, south, northeast, and midwest), the members of WG11 during SIPEM or the National Meetings on Mathematics Education (Encontro Nacional de Educação Matemática, ENEM) also included teachers, researchers, and educators of various specialization levels, including elementary and high schools, as well as universities and graduation courses. Importantly, it becomes clear that the number of participants of WG11 increased gradually, from 10 in the first edition of SIPEM in 2003 to 25 in 2015.

This diversity of representativeness also manifests as a wide variety of research lines of the studies presented in WG11, among which we highlight Phenomenology and Mathematics Education, Philosophical-scientific Thoughts, Teaching and Learning of Mathematics and the Interfaces with Other Disciplines, Studies on Mathematics History, Philosophy, and Education, Conceptions of Mathematical Knowledge and of Teaching and Learning of Mathematics, Conceptions and Perspectives in the Formation of Mathematics Teachers, Trends in Mathematics Education: a Critical Analysis of Conceptions Addressed, Philosophical Conceptions of Mathematics Environments for Online Learning , Philosophy of Language , Philosophy Fundaments in Ethnomathematics, Philosophy and Technology .

Such list of themes addressed in WG11 shows that it establishes a dialogue with topics discussed in other WGs of the Brazilian Society of Mathematics Education (SBEM) , but always from the perspective of the philosophy of mathematics education, according to the way this group understands the discipline and describes it hereafter.

In addition to this variety of subjects, these studies also adopt different philosophical perspectives, which are frequently focused on more comprehensive research being conducted by the authors, covering the specific theme of a given article presented. Besides, the studies presented throughout the years translate a growing concern amidst members of the group with regard to resuming discussions held in previous meetings, both at SIPEM and ENEM , conducting the activities based on the presentation of studies that contribute more comprehensive and insightful notions in terms of what has been probed and requires further investigation by the WG. The effort to proceed with studies about issues understood as not fully discussed and that demand more engagement has become a constant purpose of this group.

Another characteristic observed in the activities of WG11 is the relationship that the studies under discussion establish with research previously presented and endorsed, like doctoral theses or master’s degree dissertations. It is through this relationship that subthemes of philosophy of mathematics education are rearticulated, promoting the development of articles. Also, the group discusses articles whose authors declare that these studies were carried out as part of more comprehensive research projects.

12.2 Investigation Procedures

The present investigation was carried out using the hermeneutic-phenomenological approach , according to which the contents of a text being considered have to be understood without being naturalized. In this sense, the 13 articles published in the annals of the VI SIPEM by WG 11 were analysed under strict interpretation standards, based on the following questions: ‘What are the article’s purposes/objectives/questions/problems/interrogations?’, ‘What were the investigation procedures?’, ‘Who are the authors of the reference works cited?’, and ‘What does the article say, considering its proposal?’

In this chapter, each article evaluated was assigned a number. For instance, The Ontological Conception of Mathematical Objects in the Theory of Objectification was named ‘article [1]’. Throughout this discussion, we will introduce the number assigned by us at the first mention of each article. The articles reviewed were distributed to the four authoresses of this chapter so that each of us analysed articles that were written by some other researchers, and reviewed the analyses made by other authoresses of the group. So, an article was reviewed by one authoress, while the review produced was analysed by another.

Phenomenological research includes successive reductions as a means to understand the comprehensive core ideas in the articulation of a text’s discourse. In phenomenology , rather than summarizing or simplifying the description under analysis, the term reduction is the articulation of perceived meanings that interleave as a network of meanings converging to a core. This core is more comprehensive than the individual parts that form it, which are the perceived meanings, the different perspectives, and the phenomena analysed. Also, through successive articulations that represent the journey of thought with what reveals itself in the world-life where one is with the other, this core develops ideas that indicate perceived meanings, transcending these and being presented as an articulation of meanings. Therefore, in the first part of this analysis, we underlined the passages of a text we considered significant, based on the criterion that they should cover the questions germane to the article. These underlined passages were called Meaning Units , since they make sense to the investigator, who constantly dialogues with the question proposed, with what is said by the author of the article and its respective historical-cultural context and, then, proceeds with the interpretation. Therefore, the first stage of the present chapter was called Ideographic Analyses , when the First Reduction was carried out, that is, the articles were read as many times as necessary so as to detect their Meaning Units . Next, we attentively analysed these Meaning Units looking into what was contained in the text, transliterating those units the way we considered the most appropriate, and renamed them Signification Units . These units were given this name because they carry our interpretations of what is understood from each article based on the dialogue we established with it. After, we revealed our comprehensions in our own sentences and attentive reading and, then, through articulation, combined these sentences according to the meanings that we made of them. The union of meanings of the Signification Units generated a new sentence, called Second Reduction Articulated Propositions , since these express the articulation of the understanding and the interpretations we had and made, and were thus named to explicit this understanding. This nomenclature, which summarizes a new comprehension made clear in the text, was transferred to a table and given a code.

In the next stage, called Third Reduction , all Signification Units of all articles were collected in four tables named in accordance with the questions that directed the study of the articles, namely Question 1, Question 2, Question 3, and Question 4. These tables were printed and had a blank sheet of paper attached, where the Articulated Propositions were listed as we interpreted these tables, now working as a group. These articulated propositions were identified and written down on the said blank page.

Next, we constructed a Table of Convergences to support the analysis carried out, though it is not shown here due to space limitations. The first column listed the Signification Units that generated the Third Reduction Articulated Propositions , presented in the second column of the table. The third column contained the Articulated Core Ideas , which were expressed according to the articulated consideration of the data given in the second. After the attentive inspection of the Articulated Core Ideas, we proceeded to the articulation of the Comprehensive Core Ideas .

Each question was analysed following the steps described. For the first question, ‘What is the purpose of the article, its objectives/questions/problems/interrogations?’ for example, three convergent core ideas were articulated, including the various objectives of the articles reviewed: (1) The cyberspace environment in mathematics education , (2) The philosophical research on mathematics education carried out in elementary school and higher education institutions, in undergraduate courses , and (3) Constitutive themes of the philosophy of mathematics education .

These Convergent Core Ideas were interpreted as a dialogue between the authors of the articles reviewed, the authors of the reference works cited in each article, and us. The interpretation effort, which is the core of this chapter, is described in the following section, called Interpreting the analyses carried out . This interpretation is given as an analytical, argumentative, and articulated text, emphasizing what the various studies reviewed propose to do, the investigation procedures that were proved to be important, the authors that were most often cited as reference, or, in other words, what the texts say in terms of their proposals.

12.3 Interpreting the Analyses

In this section, we present our interpretations of the Articulated Core Ideas concerning each of the four questions described above. Next, we name each of the topics and discuss our interpretations thereof.

12.3.1 Articulated Core Ideas When Investigating the Article’s Purposes/Objectives/Questions/Problems/Interrogations

We examined each article looking into what it proposed, trying to understand its aim, or question, or problem, or even its verbalized interrogation. From this analysis, we articulated three Convergent Core Ideas that cover the various aims of these articles: the cyberspace environment in mathematics education; the philosophical research on mathematics education carried out in elementary school and higher education institutions, in undergraduate courses; on the reality and nature of mathematical objects.

The cyberspace environment in mathematics education , introduced for discussion in philosophy of mathematics education by WG11 in 2015, was based on studies published by researchers and professors dedicated to clarify what to do and how to proceed in mathematics teaching and learning scenarios that address the reality of cyberspace, using its resources and the respective informational screen that, in turn, provides a list of possible activities. These include the activities developed based on information and communication technologies (ICTs) . Researchers investigate this reality, underscoring the ontological characteristics that those studies have made clear. They also examine the way communication is established between people, considering software logics, understanding that this communication mode also affords epistemological and anthropological comprehensions. The studies presented emphasized the dialogue enabled by situations in which the subject is with others near logic and near the possibilities created by the software. Such is the case of Article [5], Communication in cyberspace : dialogues about mathematics (Paulo & Ferreira, 2015), whose leading question was ‘How does the dialogue about mathematics content become possible and take place in communities in social media like Facebook and Orkut?’. Article [13], A study on the mathematics demonstration by/with the computer (Batistela, Barbariz, & Lazari, 2015) inquiries over the feasibility of an effective demonstration by a computer, from which the limitations or questions about computability vs. mathematics production are outlined. The article frames the following questions, ‘How is the computer being used by mathematicians in the activity of proving?’ and ‘Is mathematical proof carried out by/with a computer today?’ This question addresses the comprehension of the way through which mathematical deductive proofs are currently conducted in the life-world where technologies are present. The phrase ‘life-world’, as translated from the German word Lebenswelt, or world of life such as rendered by most authors whose first languages derive from Latin, is understood as the spatiality (modes of being in space) and temporality (modes of being in time) in which we live with other human beings, other living creatures and nature. Life-world is also seen to include all explanations of scientific, religious nature, or any other areas of human activity and recognition. The world is not a container, an object, but a space that extends itself as actions are carried out and whose horizon of comprehensibility widens as meaning becomes to each of us and to the community we live in (Bicudo, 2010, p. 23). Article [10], The moving and the formal (Figueiredo, 2015), declares that computer science is a special field for the exploration of the connections between the capacity to perceive changes, even movement, and the cognitive relationships with formal sciences .

The philosophical research on mathematics education carried out in elementary school and higher education institutions, in undergraduate courses, as we presented and investigated, underlines the study of the works of philosophers from a position of constant, critical, and inquisitorial dialogue. As they were being understood, we analysed these studies in light of the work scenarios in the school environment, seen as the epicentre of questions. Therefore, it is not about the application of philosophical theories to mathematics education defining what has to be done and why it should be done this way though these works reveal a philosophical stance by proposing, describing, analysing, and interpreting activities and taking the reality of schools as what is out there to be understood. Article [12], Teacher, who invented mathematics? The course of a question that becomes a problem and a problem that defines curricula (Clareto, 2015) frames one question already in its title. The question is alarming, and as it unfolds, it begins to be seen as a problem, that is, it is problematized in every school that mobilizes itself to face what is being asked. Article [11], Concerns and trends in research on ethnomathematics (Miarka, 2015), discusses the move towards the theorization of ethnomathematics and the theory that supports this area of research, including the ways through which it operates. Article [9], Teachers-to-be and their conceptions about supervised trainee programs in mathematics (Meneghetti & Oliveira, 2015), analyses the notions student teachers have of supervised trainee programmes considering the investigative character of the statements obtained and the issues surrounding mathematical knowledge and contents addressed in mathematics pre-service teacher education.

The core ideas collected on the reality and nature of mathematical objects address questions about the reality of mathematical objects, the way they are constituted and investigated by other authors, and background questions about mathematics education as observed in philosophical works. Several articles were presented in WG11 in this scope of discussions. Article [2], Hermeneutics in mathematics education : comprehensions and possibilities (Mondini, Mocrosky, & Bicudo, 2015), resumes the discussions held by WG11 in 2012, placing them on a widening horizon without judging lines of research, and much less deciding over any hermeneutics that may be seen as opposing the Heideggerian, Gadamerian, Habermesian, and other phenomenologies. The article attempts to reintroduce the possibilities of carrying out a hermeneutic study for debate, revealing comprehensions that aim to deepen the studies conducted by WG11, un-veiling the possible contributions for investigation and pedagogical practice in mathematics education. Article [3], The language of Gadamer: its image in research on mathematics education (Kluth, 2015), addresses language from Gadamer’s perspective and, based on the studies cited as reference, frames the question, ‘How does thought reveal itself in the movement of knowledge construction of algebra structures ?’ aiming to clarify the hermeneutical investigation about this question. Article [10] declares that we are creatures of change, that is, beings whose world opens itself full of meanings of change, and frames the question, ‘How can we understand the phenomenon of formal sciences ?’ The article highlights, as a doubt of philosophical nature, the human capacity of revealing meanings of change in the mathematical doing, for instance. In that article, ontological discussions stand out, emphasizing the formal and the moving. Yet, another article that poses questions about the ontology of mathematical objects is article [1], which investigates how the various theories conceive the ontology of mathematical objects, since not every modern theory of teaching in mathematics education discusses the ontological concept of its object of knowledge, more specifically, the object of the ontology of mathematical beings. Article [4], That mathematics: is it a problem? (Rotondo and Azevedo, 2015), also addresses mathematical contents of distributing and measuring, asking, ‘How to operate and produce meanings with the concept of division?’ Article [6], Curricular contents of mathematics in elementary schools in Brazil: a philosophical analysis (Baier & Bicudo, 2015), is an attempt towards philosophical investigation, analysing, criticizing, and reflecting over the curriculum adopted in elementary education, stressing science and mathematics issues in light of their sociocultural historicity in the Western world. Again, we mention article [12] that, in its own core question, focused on the mathematical object. Article [13] investigates mathematical proof, considered one of the pillars of this science such as it exists in the western world. Article [7], Directions towards a geometric philosophy of transformations (Detoni and Pinheiro, 2015), looks into the dynamic geometry of movement, mathematics, and flow aiming to understand it from the historical and epistemological perspectives that cover a variety of conceptions of geometries.

12.3.2 Articulated Core Ideas in the Analyses of Studies About Understanding the Investigative Procedures Adopted by Authors

When trying to understand the investigative procedures adopted by the authors of the different studies reviewed, we articulated three comprehensive core ideas that were called theoretical essay supported by bibliographic review, qualitative research that used experiences in everyday life at school as data, and qualitative research that used data obtained in the cyberspace.

Theoretical essay supported by bibliographic review articulates methods described in studies by authors who are relevant to the theme under discussion and that focus on key concepts of different theories to reveal these concepts and develop argumentations highlighting the narrowing or increasing distances between authors. Such is the case of article [1], a literature review based on the theory of objectification, on a discussion about the ontological principle of this theory, and radical constructivism. The article makes, in the authors’ words, some considerations on the ontology of mathematical objects while fundamental principle to be taken into account (or not) by modern theories of learning in mathematics education. These core ideas also include theoretical essays whose notions originate from literature reviews, as articles [2] and [3], as well as article [8], titled Ethnomathematics and the ideas in language games (Silva, 2015). Article [2] is a text that presents an initial literature review about hermeneutics, supported on a phenomenological perspective, with the aim of revealing comprehension as human existence and contributing with research about mathematics education focused on comprehension of teachers, students, texts, concepts, meanings, and signification in the scope of this science, while article [3] declares that this theoretical essay intends to clarify language from a Gadamerian perspective. Article [8] is also presented as a literature review, intending to analyse the possibilities of family resemblance between mathematics, ethnomathematics, and the philosophical thought of Ludwig Wittgenstein in his book called ‘ Philosophical Investigations ’.

Qualitative research that uses experiences in everyday life in schools as data is a discussion about Comprehensive Core Ideas that examines articles developed at two moments, namely in schools (with teachers, students, and staff), and in cyberculture environments. Regarding the research in and with the teaching environment, article [12] is an investigation carried out in a school based on a question framed by a student during a mathematics lesson. The ways through which the school faces the question are reported and discussed, revealing the process in which the curriculum is being developed. Article [4] is an investigation conducted in a workshop of in-service teachers taking a continued education course, when a problematization of the formation of teachers evolves. The operator division was used as a guiding cue that, during the movement of the formation action, problematizes itself, that is, becomes a problem that is more complex that initially conceived aiming for the formation of these teachers. Using a semi-structured questionnaire answered before and after the training period, article [9] introduces an analysis of the conception students of teacher education programmes have about supervised trainee programmes. It was included in these comprehensive core ideas because it addressed undergraduate teacher education students directly who were taking the supervised trainee programme in elementary schools .

In Qualitative research that uses data from the cyberspace environment , article [5] follows members of Facebook and Orkut groups and analyses the ways they express themselves. The article also highlights the qualitative aspects of the mode of being in cyberspace, such as the dread felt when communicating with subjects in cyberspace. Article [7] evaluates the convenience of the traditional geometry practice as described in curricula in terms of graphics software environment. It reveals the articulation between theoretical studies about this theme and the dynamic meaning of graphics software.

12.3.3 Articulated Core Ideas in the Analysis Covering the Authors Studied

In our first investigative attempt on the subjects addressed by WG11 in SIPEM VI 2015 , we also addressed the references cited in the articles, since these reference works reveal views of the world and knowledge present in the discussions held and in the production of knowledge of the researchers participating in this group.

As the common ground on which philosophical and mathematical thought flows across the participants of WG11, we introduce authors who, as we understand them, are relevant thinkers about these themes. Of the articles that address themes about mathematics and philosophy of mathematics, we highlight Bachelard (1968), Bourbaki (1950), Dedekind (1931), Euclid (2009), Galileu (1999), Hawking (1997), and Wussing (1969, 1989). Concerning philosophy authors, we mention Deleuze (1997, 2006), Deleuze and Guatari (2012), Gadamer (1999), Husserl (1997, 2008), Merleau-Ponty (1994, 2002), and Wittgenstein (1999).

Among the relevant authors cited in articles about mathematics education, we include D’Ambrosio (1979, 1984, 1998, 1999), Knijnik (2001), Knijnik and Wanderer (2008), and documents of the Brazilian legislation (Brasil, 1998, 2002). Also, one authoress is quoted as reference in several articles on various themes, especially concerning qualitative research procedures and questions about cyberspace: Bicudo (1993, 1998, 1999, 2000, 2010, 2011) .

12.3.4 Articulated Core Ideas in the Analysis Showing What the Articles Say Considering Its Proposal

The successive reductions carried out led our articulations to four Comprehensive Core Ideas about what the texts under analysis say concerning the answers and the understandings made explicit by the authors when concluding their articles. These reductions were: Constitutive themes of mathematics education , Constitution of mathematics objects , Philosophy of mathematics education focused on the elementary and higher education spaces , The cyberspace environment in mathematics education , and Themes of research presented in WG11 in the previous SIPEM. As we understand it, the Comprehensive Core Ideas articulated in the reductions carried out while we tried to understand the object of our first question framed about the articles reviewed, that is, ‘What are the article’s purposes/objectives/questions/problems/interrogations?’, are coherent with the Comprehensive Core Ideas articulated in our analysis of the fourth question proposed to the articles, ‘What does the article say, considering its proposal?’

The Comprehensive Core Ideas discussed herein clarify what WG11 considers important in philosophy of mathematics education in SIPEM VI . An examination of international literature panorama on the theme reveals that some items coincide with contributions made by various authors, like Paul Ernest, Merylin Frankenstein, Jean Paul van Bendengen, and Ole Skovsmose, among others who write about philosophy of mathematics education, as it is the case here considered in terms of Constitutive themes of mathematics education . However, the references cited by the members of the WG Philosophy of Mathematics Education in SIPEM 2015 were not restricted to the works quoted in studies developed in countries other than Brazil, though it reveals, in several aspects, the existence of a different way to understand reality and the constitution of mathematical objects. The theme philosophy of mathematics education focused on the elementary and higher education spaces , in undergraduate courses, seemed to innovate in this field of study. We understand that, considering the studies presented in this WG, we address the questions about philosophy of mathematics education and investigate them in the reality of school temporality and spatiality. Another relevant theme that surfaced in the studies reviewed and, as it seems to us, is important in the reality of the life-world as lived by us is the cyberspace environment in mathematics education . The authors writing on this theme investigate ontological, logical, and epistemological questions in this environment. Articles investigating themes not fully clarified in SIPEM V were also presented, requiring further examination. These articles were collected under the item Themes proposed for research in WG11of the previous edition of SIPEM .

Constitutive themes of the philosophy of mathematics education collects ideas about the conception of mathematical objects, the mathematical doing, ethnomathematics, the constitution of mathematical objects, the conception of knowledge, and knowledge as a technique. One of the concerns highlighted is the way through which these mathematical objects are constituted, which agrees with the idea that articulates the clarification of the conception of mathematical objects and that of doing mathematics.

One of the ways to understand this conception includes looking at mathematical objects as fixed standards generated during historical-cultural development through the action-reflection process promoted by social practice (Gomes & Morey, 2015). Another way to understand mathematical concepts is based on a given operation, as discussed in article [4], which considers division as distribution in equal parts, treating the operation as a mathematical concept that carries a need, that is, the division into identical parts though with no concern for the experience of the people who conduct this distribution in sociocultural concepts . Article [8] also reveals the existence of an extra-linguistic mathematical reality , which lends meaning to its propositions, going against the notion of mathematical objects that exist or pre-exist in the mental sphere, at empirical level, or even in social subjectivity. This is mathematical doing, that is, the procedures adopted in the construction of mathematical knowledge are addressed in article [7], which declares that transformations are not mere new clothing to Euclidean geometry . Nevertheless, the article claims that the dynamic transformations previously admitted and tacitly and axiomatically declared now become the epistemological medium of these transformations, which are now seen as a group so that the notion of group revolutionizes geometry doing, since the properties of geometric objects, rather than being embedded in them, are, or are not, a group with other objects. Article [13] discusses the question that doing mathematics is a method, not a procedure that inflexibly indicates the ways to go; on the contrary, it is a process of doing that involves judgements and choices of what is significant or useful.

In this collection of studies, discussions about ethnomathematics also are carried out to understand mathematics and address the question of mathematical objects. Article [8] discusses the comprehensions about what it calls mathematics associated with different life forms understood as sets of language games that share similarities and that may, according to the authoress, intersect the comprehensions of ethnomathematics . The author of article [11] discusses the ways through which ethnomathematics has been conceived: as a study of innate mathematical thought and of practices that emerge from various cultures, as the possibility of criticism and deconstruction of the universality of mathematics when decoding cultural practices and testing pedagogical possibilities that work with these decoded concepts. The article also emphasizes that these ways of conceiving ethnomathematics reveal a naturalized view (i.e. one that has not been problematized concerning the presupposed understandings) and the notion that ethnomathematics adapts itself to use.

WG11 has held constant discussions about the constitution of mathematical objects , since they have not been understood by the group’s members as not existing outside the human world, which, put simply, is the world where we live with one another in cultural environments that are socially organized and in contact with the natural world as seen through the possible interpretations. For example, the authors of article [1] resort to the theory of objectification as an explanatory theory that affords to assume that mathematical objects derive from social practice. Together with Nietzsche and Deleuze, the authoress of article [12], in emphasizing the question: ‘Who invented mathematics?’ points to the forces and desires that devise a mathematics. The authoress inquires over the invention of mathematics and the invention with mathematics. The authoress of article [4] discusses the invention that takes place in the composition with remainders, an invention of relationships, properties, and concepts that form the scope of what is called mathematics. This effort towards searching for the modes mathematics constitutes itself in the world is made by the participants of WG11, even though with distinct views and possibilities.

The discussion about the usefulness of mathematical objects is also addressed. In article [1], the usefulness of these objects lies in their role of models, standards, and essential matrices in the development of any human activity requiring mathematical knowledge. Article [11] discusses the decolonizing potential of ethnomathematics, revealing its political bias, and the decolonization of the educational system. Both political and educational views address the question: ‘What is ethnomathematics useful for?’, similarly to the question about the usefulness of mathematics.

Underlying the ways WG11 articulates the ideas presented above is the understanding that knowledge is something that constitutes us but is not to be owned by us, as discussed in article [2]. The same article presents argumentations about positivist science, revealing that, when rationality prevails, knowledge begins to be constructed through techniques that validate truth, blurring the search for questions whose answers cannot be visualized, as a consequence of the premises of theory considered from an objectified standpoint.

The Comprehensive Core Ideas The cyberspace environment in mathematics education addresses the comprehensions about the conception of computer, covering questions about formal language and programming language, and mathematical proving carried out by mathematicians who use the computer. These background questions help understand the way of doing mathematics when one uses the computer, creating possibilities to understand the communication spaces constituted in the cyberspace where mathematics is discussed, in a dialogue. Therefore, a space was created for the ways to understand mathematics when talking about the discipline in social media groups.

One of the ways to see the computer is presented in article [13]. According to the authors, the computer goes beyond the descriptions of sequences of operations carried out in three main elements, namely the data input unit, the data processing unit, and the output unit. On the contrary, the computer represents new opportunities to the mathematician, improving efficiency of calculations, generation and editing of graphic elements, which turns it into a more responsive communication medium. In article [10], the author highlights an important principle of computer science: whatever is static in computation does not exist to the machine. The underlying comprehension of this principle is the perception of perception of change. It has been argued that this perception takes on a special mode, to which computers connect directly: the perception of the transformations taken place as a result of the stimuli we send, which, when shown on the screen, are considered as perceived reaction. Also, the notions of interaction and interactivity, when associated with computers, are already embedded as possibilities in these transformations .

Concerning the work of mathematicians next to the computers, the discussions make clear, at the current state of investigations and debates, that, certainly, the computer has proved to be important in the work with mathematics and mathematics education in several aspects, like the marking of tests, for example. Besides, another aspect that was emphatically highlighted was the creative experience of the mathematician, that action that synthetizes reasoning, intuitions, perceptions, and that comes out as the solution for the problem. In this experience, intuition plays a central role in the work of the mathematician and, as we understand it, it is this experience that differentiates the work of the computer from the work of the mathematician in mathematics production. Article [13] alludes to Turing (quoted from Copeland & Shagrir, 2013), who describes intuition as a spark that shines for an instant and affords a glimpse of a possibility of adjustment. Creativity is responsible for the approach taken to this adjustment.

The space opened to communication by the computational structure affords modes of expression between people who communicate verbally, in writing, imagery, and mathematics. Article [5] presupposes that thought and its expression constitute themselves simultaneously in the life experience of the lived body, according to the concept developed by Merleau-Ponty (1994, 2002). By focusing on the reality of cyberspace where one may be with the other establishing dialogues, the article understands that, in the dialogue about mathematics thought may advance, call the other in, and thus share what is perceived, understood, and interpreted. In terms of the situations in which comprehension of this science stands out, some studies cited in this chapter reveal that discourse, understood as an articulation of intelligibility, may express itself in social media groups .

The philosophical research on mathematics education carried out in the teaching environment of elementary and higher education is a set of comprehensive core ideas that, as we see them, contribute distinctiveness to the research on philosophy of mathematics education that we have been carrying out. This difference lies in the work conducted in the school environment based on philosophical studies that improve the clarification of ontological questions (modes of being of mathematical objects), epistemological questions (modes of knowing and producing knowledge of mathematical objects), and axiological questions (modes of valuing actions and attributing values, as it is the case, for instance, of evaluation and ethical posture), in addition to the planning and actualization of investigation activities. These include the didactic-pedagogical activities developed by the researcher with professionals in training, elementary school students, and undergraduates.

In a study carried out in the classroom environment , the authoresses of article [4] use an item of teaching content and problematize it. The study considers division as distribution: a mathematical concept that, as said above, carries the need for equal parts and clarifies the operation as a mathematical object, explaining the way it operates. The study also shows that the operation takes place far away from several lives, even though it produces lives by imposing ways to operate that are often memorized and begin to dominate the ways a person proceeds beyond the school walls. The inquiry expressed in the article and that maintains itself as an inquiring dialogue in its clarification analyses how to turn mathematics into a problem. Therefore, the intention expressed by the authoresses of the article is not to consider mathematics objectively as given, but to work it in teaching and learning activities, problematizing mathematics itself. Also, the authoresses understand that a trace of occupation in this mode of proceeding is mathematics as being instituted, which fills it with objects of a world that is distant from the students who are there in a scenario of producing (themselves) with mathematical knowledge. This generates tension, and when the authoresses carry out the activities always in an inquiring way, the subjects with whom they are in a situation of teaching and learning mathematics start the process of producing ways of operating invented amidst of disquietude. So, they do not resort to the reproduction of instituted modes. We understand that, in this process, the researchers who authored the article take the conceptions with which they work in the mathematics classroom as a means to understand the activity of conducting the generation of knowledge and to understand the other with whom they are, listening to them in their disquietude.

The understanding of this WG11 is that mathematics is actualized in the school, that is, it realizes itself through the assembly of several forces, not only by what is determined by established science or by a previously conceived and inflexible curriculum in light of what is happening. These forces generate from the enrolment of bodies with materiality and historically constructed modes of knowing that intertwine with modes of being with teachers with students with textbooks with … with … with (Clareto, 2015). The vigour and diversity of these forces are so intense that it becomes impossible to name where they come from. However, we understand that these forces are perceived and that it is necessary, in the school environment, to work with them. We understand that the curriculum of activities to be carried out in the school has to consider all the historicity of the construction of science in the western world, from the modern age and the current times. Nevertheless, as proposed in article [6], one cannot avoid accepting the ideas surrounding the elementary concepts of the theory of fractals, emphasizing its connections with the theory of chaos so as to include uncertainty and the instabilities that prevail in the world we live in, where chaos and order coexist in a close relationship.

Also, the theme curriculum was one of the subjects highlighted by WG11 and addressed in one of the studies presented, which focus on the work with geometry at school. Article [7] introduces a debate about The New Scientific Spirit , in which Bachelard (1968) positions the dialectics of the realism/rationalism in geometric philosophy, in what becomes a tool for the epistemological comprehension of new geometries, that is, the understanding of the notion of group. Since a group is a set of invariants that manifest due to a transformation, epistemologically speaking a significant turn is observed towards the definition of what geometry is (geometries are) when it is the physical invariants that would lend rational value to the principles of permanence. Bachelard (1968) also declares that the western geometry tradition is Euclidean, and that, as ancient and widely practised as it is, this geometry may circumscribe itself in completed notions. Each object found its own concept and may be placed as an object among objects. We understand that our work at school, based on what has been discussed so far, has to make this tradition relative. This may take place when, with the advent of non-Euclidean geometric thoughts, we accept the task of reconstructing concepts. It is important to underscore the epistemological question framed by Bachelard, since it implies discussing how one geometry—or a certain way to make geometry—realizes itself de-realizing another, exposing it to criticism. When taken to the pedagogical field, this question exposes the differences between more traditional teaching practices as we follow one geometry and one practice that will propose geometrizing itself in a given possible way.

Another topic addressed in article [9] discusses the conception and importance of the trainee period in teacher education programmes based on a questionnaire prepared by the researchers and that the students answered. For the authoresses, students understand the trainee period as an opportunity for them to gain experience about the school environment. In addition, the respondents understood that this is one of the few situations during the course when they can get near schools and know different realities. The article places emphasis on this experience in the formation of teachers, suggesting that the training period should take place already in the beginning of the teachers’ education course, not at the end. It is also understood that a review of the activities proposed by supervisor professors and/or those allowed by the schools during training periods should be conducted, since these activities sometimes restrict the role of the student to a few moments during classes, when this student is mostly considered only an observer of the work conducted by the professor during the training period.

Finally, we discuss the Themes of research presented in WG11 in the previous SIPEM, which concerned ethnomathematics and hermeneutics. The questions that remained unanswered addressed the conceptions of ethnomathematics and the respective approaching or distancing from mathematics and the difference between the hermeneutics as presented by Gadamer and depth hermeneutics. In the article presented in that edition of SIPEM, the latter was mentioned as referring to the hermeneutics defined by Thompson.

Articles [8] and [11], which investigated ethnomathematics , clarified the subject and included themes covered in Constitutive themes of mathematics education , the reason why they will not be discussed again here.

Hermeneutics was discussed in two articles, [2] and [3]. The concerns expressed by the authoresses included clarifying the conception of hermeneutics and, for that, investigated the Heideggerian, Gadamerian, and hermeneutical views. The first article elucidated depth hermeneutics. Hermeneutics was understood from two standpoints: in its exegetic meaning, when it is methodical and normative, as it was seen between ancient times and the nineteenth century, and the philosophical hermeneutics, which was studied by Heidegger, Gadamer, and Ricoeur. In this case, hermeneutics is understood as a window to the world and, therefore, to the historic and the cultural, inasmuch as the act of understanding is seen as the constitutive of human beings who, by existing, understand themselves and their cultural works. This comprehension, which carries an interpretation and comprehension, supersedes the definition of hermeneutics as a method of interpretation, as some authors call it. The hermeneutic experience is that of linguisticity, in which language meets its mode of being in conversation and in mutual understanding. It is a movement that always brings the subject, the other, and the world into an interpretative hermeneutic circle, widening horizons and other possible comprehensions. It is in this frame that the appraisal made by Habermas is placed (as quoted by Negru, 2014 and Batista, 2012), an author who systematically investigates the distortions of communication, that is, of the linguistic world. For Habermas, communication is distorted and distortions should be elucidated by depth hermeneutics. This way of understanding hermeneutics is adopted by Thompson (2009), who, in the last chapter of his book, introduces a methodology to adopt with depth hermeneutics. For WG11, as observed during the discussions, any attempt to develop one interpretative method has to face the impossibility of comprehension since it limits its own method. The circle closes in the logicity of the method and does not open itself to the comprehension of the world and of ourselves.

In this sense, hermeneutics was understood as a possibility for research on mathematics education, as a means to understand the phenomenon of human language and its various cultural manifestations and expressions. Using the Gadamerian hermeneutics, article [3] investigates comprehension of algebraic structures.

12.4 A Comprehension of Philosophy of Mathematics Education

Philosophy of mathematics education, as the philosophy of any activity, turns itself to this very activity, aiming to understand it from the standpoint of its objectives and its logic. In this sense, it focuses on the investigation of the activities of this area, like teaching and learning mathematics, the importance of teaching mathematics to all elementary school students. Questions like ‘What is mathematics?’ ‘How does mathematics relate with society?’ ‘What is the nature of learning (mathematics)?’ ‘What is the status of mathematics education as a field of knowledge?’ were framed by Paul Ernst and by the group of organizers of the Topic Studies Group on Philosophy of Mathematics Education of the International Congress of Mathematics Education, 13th edition, in 2016.

The works presented and discussed in WG11 in SIPEM 2015 addressed several of these questions and strengthened the mode of proceeding mentioned during problematizations, analyses, and reflections about mathematical objects, the conceptions of mathematics, the relationships between mathematics and the school environment, the schoolarization of mathematics, the formation of the mathematics teacher, the different cultural mediators in mathematics teaching, with special emphasis on computational environments and ICTs, and ethnomathematics.

The analysis of what was carried out aiming at the essence of each article discussed by the group brings to light a uniqueness concerning the philosophy of mathematics education as it has been investigated across the world. This uniqueness manifests itself through the view about knowledge and the reality presented by several Brazilian authors that emerges from consistent studies and reflections about the works of authors like Deleuze, Gadamer, Husserl, Merleau-Ponty, and Wittgenstein. This view underlies the innovative investigations on research on philosophy of mathematics education as being realized in the very environment of elementary and higher education, besides investigations conducted hermeneutically. Concerning the latter, a quote extracted from the back face of The Philosophy of Mathematics Education (Ernest et al., 2016), with which we end this article:

[...] A case study is provided of an emergent research tradition in one country. This is the Hermeneutic strand of research in the philosophy of mathematics education in Brazil. This illustrates one orientation towards research inquiry in the philosophy of mathematics education. It is part of a boarder practice of ‘philosophical archeology’: the uncovering of hidden assumptions and buried ideologies within the concepts and methods of research and practice in mathematics education […]. (Ernest et al., 2016, back cover.)

Based on the analysis of the works presented in the group of Philosophy of Mathematics Education (WG11) in the latest edition of SIPEM, it was possible to draw an updated panorama of research on philosophy of mathematics and of the dialogue that these studies establish with works from both international and Brazilian authors as well as other lines of research on mathematics education, contributing a philosophical reflection to develop the ideas shaping this field of knowledge.

At last but not least, in being a group on philosophy of mathematics education, WG11 works philosophically, which means that the studies conducted by its members focus on several themes on the mathematics education investigated in the other working groups of SIPEM, since the attribute of philosophy is to reflect on what is taking place or what has been carried out. We can mention some of these themes as being: learning and teaching mathematics, mathematics education, informatics and distance education, the school reality where mathematics education takes place, and mathematics education carried out in the teaching environment of elementary and higher education.