This paper examines relationship between the sociology of mathematics and mathematics education. If in one hand we learn from the sociology of mathematics justifications that mathematics is bounded by the society that produces it, on the other, from mathematics education literature, we know that learning is strongly influenced by social based processes. Being so there are a number of questions that this paper points to as areas for investigation and research.
1. Social bases for mathematical curricula
When looking at recent history of education we find that social issues have been taking a more prominent role in the justification for the inclusion of mathematics in school curricula. By the end of late nineteenth century, mathematics became central in the curriculum of elementary and secondary education. The reasons invoked to determine such a curricular centrality were based on the efficacy of mathematics to exercise the mind, especially reasoning and there was no special place attributed to social aspects. With the increasing of schools' population and with the consummation of industrialization, such a criterion for mathematics curricular inclusion became old fashioned, and, by establishing as more compatible and adequate criteria for the social necessities of that time the cultural transmission and children's accurate preparation for future social performance, started a problematization of the role of mathematics within school curricula, as well as the search for a more utilitarian vision of mathematics education.
Despite all the social changes and the educational debate, the recognition of the need for mathematics for youths and children based upon mentalist criteria remained, i.e., educators continued to believe the virtues of mathematics as a means to effectively develop reasoning and logical power (Stanic, 1987). During the 60s, and in a context of a political race between two blocks, the Modern Mathematics movement sought to increase scientific background of students. Mathematics was then seen as a privileged mean to achieve excellency in a scientific competition. Up to this point the curricular mathematical content was compatible with an absolutist perspective (Ernest, 1994) of mathematics, which views mathematical objects as either independent of human action or as mere symbols.
More recently, the bases for the inclusion of mathematics in the curricula continued to embrace it as a means of cultural transmission and professional performance. The reasons for mathematical curricular inclusion are now more explicitly related to economic and technological dimensions. However, new goals are now added: mathematics incorporates an individual component fostering the possibilities for youth self-fulfillment and the definition of personal goals, and mathematics is perceived as a fundamental tool to understand its role in important social decisions by ensuring a more complete performance of citizenship (NCTM, 1990; Secada, 1990; Skovsmose, 1992).
Another aspect of the social influence over the justification for the inclusion of mathematics in the curricula is possible to be highlighted. On the one hand, there is an imperative for a respect for cultural choice and beliefs. On the other, there is a presence of a global cultural dimension. These two contradictory requirements of modernity, have been generating a new type of problematic that go beyond a local vision of the teaching and learning process of mathematics. There are claims of curricular diversification, made to preserve cultural and individual dissimilarities (NCTM, 1990; Souta, 1996), and we can testify a globalization of Western mathematics (Bishop, 1995). The social dimensions of mathematics education are now visible on the issue of cultural representativity in the mathematical curricula not only in each country and each classroom, but also in the underlying problem of the social status that Western mathematics has been acquiring as a symbol of promotion of the movement toward globalization.
Thus, what is emerging is that the requirements for mathematical curricular inclusion and extension increasingly incorporate a social ground of argumentation, accompanying a shift from the locus of mathematics education’s traditional problematic, not anymore simply located in the "mind", or the classroom, or the materials and techniques, or teaching methods to include a broader comprehension of world curricular similarities, as well as curricular diversity to include cultural representativity and the possibility of individual choice.
As mathematics educators, these challenges in curricular perspective prompt us to enlarge our vision of mathematics. This shift is questioning the very nature of mathematics. An absolutist vision has to be replaced by another that incorporates a new account for mathematical activity and a new perspective on the nature of mathematical objects. Claims for diversification and the valorization of students’ cultural and personal backgrounds are not compatible with an authoritarian view of the source of mathematical knowledge. It is the purpose of this paper to prospect ideas taken from the sociology of mathematics, as represented mainly by the work of David Bloor and Sal Restivo, looking for a comprehension of mathematics and mathematics education connected with their socially immersion and embodiment.
2. Mathematics as an empirical and social science
We turned first to the issue of the nature of mathematics. Under the denomination of Naturalistic Mathematics, David Bloor (1991) proposes an account of the nature of mathematical knowledge that incorporates contributions both from J. Stuart Mill and Gotlob Frege. Mill proposes that mathematical knowledge comes from experience. We know facts that apply to a wide range of things, namely ways of ordering, sorting, rearranging, etc. These patterns and groupings of physical things provide models for our thought processes, and when we do mathematics, we are tacitly using these models. Mathematics is a set of beliefs about the physical world that arise out of our experience of that world.
Frege criticizes Mill by rejecting the idea that the nature of number has a subjective, mental or psychological component. Moreover, he argues that number is not a property of external things. For him numbers are objects of Reason, or Concepts, which have the important property of objectivity. Frege views objectivity (Manno, n/d) as denoting something that is independent from our sensations, intuitions and imaginations but is not independent from reason. We can call objective to which is subject to laws, what can be conceived and judged, what can be expressed with words.
Bloor argues that, as Frege points out, experience alone does not provide an adequate background for mathematical knowledge. "The characteristic patterns" of objects, as Mill puts it, are not on the objects themselves. These patterns are social, rather than individual, entities and they are at the very root of the objective objects of Reason proposed by Frege. "Mill’s theory only concerns itself with the merely physical aspects of situations. It does not succeed in grasping what it is about a situation that is characteristically mathematical" (Bloor, 1991, p. 100), i.e., it does not do justice to the objectivity of mathematical knowledge, to the obligatory nature of its steps, or to the necessity of its conclusions. This missing component are social norms that single out specific patterns, endowing them with the kind of objectivity that comes from social acceptance. Bloor explains that "the psychological component provide[s] the content of mathematical ideas, the sociological component deal[s] with the selection of the physical models and accounted for their aura of authority" (p. 105).
The connection between the empirical and social aspects of mathematics is also highlighted by researchers in mathematics education. Although mathematics education research has been deeply influenced by psychology in its emphasis on the empirical bases of mathematical understanding, and the role of sensory experience has been at the center of research in mathematics education, specially in the area of mathematics learning, the focus on the external, material aspects of learning has been gradually shifting to encompass other perspectives, as it happened in the shifting of curricular perspectives we pointed out previously.
The perspective that there is an objective student "out there" that can be known has been criticized by researchers from various fields. On the one hand, attention has been drawn to the ways in which our cognition has strong ties to features as basic as our own body (Johnson, 1987). On the other hand, cognition has been shown to rely heavily on cognitive models that have strong social aspects (Lakoff, 1987). In mathematics, in particular, uses of terms like "height", "basis", "ten", "increase" denote an individual cognitive base relating to corporeal features, together with their social acceptance. This blend of individual and social aspects provide the base for the construction of those mathematical concepts as metaphors (Matos, 1992; Pimm, 1990; Presmeg, 1991).
The empirical nature of mathematical knowledge suggests that on the one hand formal drill with written symbols should be discarded in favor of more relevant didactical strategies, and on the other, learning of mathematical ideas should be created out of experience, namely from those very models that underlie mathematical knowledge. The social nature of mathematical knowledge proposes that special attention should be given to the creation of a classroom culture fostering the development of intersubjective means of establishing mathematical truth.
By extending Mill’s theory sociologically and by interpreting sociologically Frege’s notion of objectivity Bloor opens the door to what he calls "alternative mathematics". Alternative mathematics would look as error to our mathematics. These errors should be "systematic, stubborn or basic" (p. 108) and they should be "engrained in the life of a culture" (p. 109). Bloor presents four types of variations in mathematical thought which can be related to social causes.
Two lines of research in mathematics education have been converging in legitimizing alternative mathematics. Researchers with a constructivist stance have been pointing out that students’ own constructions of mathematics may lead them in a natural way to the development of mathematical significances divergent from what counts as school mathematics (Steffe, Cobb, & von Glasersfeld, 1988). Researchers working within the ethnomathematics framework have been pointing out the necessity for school mathematics to take into account the rich mathematical backgrounds that students bring to school (Gerdes, 1996; Powell & Frankenstein, 1997).
3. Mathematics as a social activity
Sociology of mathematics helps us to enlarge our perspective of mathematics activity as a social endeavor. The study of teaching and learning mathematics as a social activity has been a concern of several researchers in mathematics education: social construction of mathematics in classrooms (Walkerdine, 1988), the study of micro-cultures and interactional processes (Bauersfeld, 1980; Voigt, 1992), among others. Analogies between the mathematics scientific culture and the classroom mathematics culture have been drawn. As Cobb, Wood, Yackel and McNeal (1992) point out, both are created by a community and both influence individuals’ construction of mathematical knowledge by constraining what counts as a problem, a solution, an explanation, and a justification. To incorporate the social aspects of the creation of mathematical knowledge as a subject with its own specificity, and to articulate its location within a broader social problematic have been a challenge to these studies.
Restivo may provide a way to think about the production of the mathematical knowledge in a more comprehensive view. He proposes (1990) that mathematical ideas, beings, models, etc. are not only social products but also socially constructed, and as such they embody social interests and practices that are inserted and belong to a more vast form of society. As Restivo shows, the organization of the community of mathematicians, what they produce, and context of creation, are all intertwined and are not independent of the larger social organization and interests. Namely, what is generally categorized in our society as mathematical pure ideas are a result of a set of procedures carry out by the community of mathematicians that started at the XIX century strongly related with the process of scientific specialization and professionalization.
In addition, Restivo highlights, that the circumstances that gave rise to the necessity of an ideology of purity are not only linked with the constraints imposed by specialisation and professionalisation of science but also with the "imperatives of social conflict" (1990, p. 136), thus, to all this "movement toward ‘purity’" (Restivo, 1991, p. 163) is not strange the fact that we live in a society whose social model is built upon a dominant class, and, to the promotion of mathematics in our society are not strange its success yearned by the problem solving of material and ideological question posed by the powerful as for example those linked with technology.
A main challenge to mathematics educators can be taken from Restivo (1991). Mathematical literacy and mathematical education in general can be improved by focusing attention in "revisions, reforms, and revolutions in mathematics always with an awareness of the web of role, institutions, interests, and values mathematics is imbedded in and embodies" (p. 172). Mathematics educators have been paying attention to this area (Frankenstein & Powell, 1994; Mellin-Olson, 1987; Powell & Frankenstein, 1997). For them, mathematics may be used as a means to achieve social change. Special attention has been given to mathematics rooted in specific cultures. However, Western mathematics can also be used as a subject of change and critic since findings from the sociology of mathematics show it as a subject embodied in a scenario of social struggle.
Since mathematics education plays an active role in the design of contents that promotes what counts as mathematics to a larger public, we cannot put anymore the social construction of mathematics out of mathematics education.
4. Mathematics education influences in the social construction of mathematics
Another contribution of the sociology of mathematics to our understanding of mathematics education is related to the role mathematics education plays helping to shape mathematics itself. Researchers in mathematics education have been denominating the process of transformation of mathematics as a science to mathematics as a didactical discipline as "didactical transposition" (Brousseau, 1986). The sociology of mathematics shows that there is also a converse process and mathematics education influences mathematics.
Several authors have put forward the idea that mathematical development, namely that higher levels of abstraction, has been possible because of the special focus put in the transmission of mathematical knowledge to the next generations. For example: "abstraction depends on realising opportunities for producing, publishing, and disseminating ideas in a specialised community of teachers and students" (Restivo, 1990, p. 136).
The capacity that mathematicians showed to grant a generational continuity of ideas, processes and their own mainstream culture (supported, or not, by social agreement) underlain the social changes that appeared in the beginning of last century in their community, and is highly responsible for the cohesion of the field as well as for themselves to emerge as a new specialised group.
In addition to the idea that the teaching of mathematics was important to the development of mathematics itself there are also references in the literature of mathematics to suggest that the teaching organisation and the contents organisation have influences of their own. The creation of schools for mathematicians at the beginning of the XIXth century in France and Germany showed, for example, how these multiple aspects relate to each other to sustain social modification of the organisational forms of the community of mathematicians (Restivo, 1990). As Struik points (1986)
In conclusion, by the late XIX century, the picture of mathematics reveals that the imperatives of mathematics professionalization and especialization lead to an all set of changes in the community of mathematicians that put on teaching an enormous responsibility on its social and internal growing and recognition that conduces to the creation of teaching materials that ultimately configures a particular vision of mathematics. Thus, among the causes that influenced today's vision of mathematics is the way its teaching was organised and promoted: teaching materials of mathematics shaped the body of contemporary mathematical knowledge. An idea of mathematics as a social structured activity has brought out, and by and large implemented in school curriculum, privileging a vision of mathematics as an idealised science.
Mathematics education is ultimately responsible for the enhancement mathematical literacy of a generation. Mathematical knowledge of one generation is also a reflection of what has been taught to them and entails a particular vision of mathematics. Therefore, we may also try to understand additional ways in which we may see what is the social construction of mathematics. Since mathematics education plays an active role in the design of contents that promotes what counts as mathematics to a larger public we cannot put it anymore out of the social construction of mathematics. If previously we argue the importance of the social construction of mathematics for the field of mathematics education, we are now point out a converse conclusion.
5. Final considerations
This paper tries to put forward the idea that work by sociologists of mathematics may help mathematics educators to understand the activity of mathematics, its teaching and learning as a social process embedded in larger social problematics. We began by outlining social requests that are affecting mathematics curricula leading to tensions between a uniformity in mathematics content worldwide and a diversity accounting for distinct cultural backgrounds in students. This issue drove us to the sociology of mathematics, looking for views of mathematics incorporating strong social explanations.
We learned that mathematics is seen more and more as a social dependent field. Although mathematics may have a basis on empirical grounds, social exchanges are vital to the establishment of the "authority" of mathematical objects. Learning of mathematics tends to be broadly understood in a social conjuncture enlarging individualistic or mentalistic perspectives. The understanding of mathematics activity as a social effort was our next concern. Although mathematics educators have been investigating the nature of social exchanges in classrooms, the sociology of mathematics may allow us to deepen this perspective. The point is not that social processes occur in the classroom. They do, and the ways in which mathematics is created in schools influence mathematics learning. But also the very nature of mathematical knowledge embodies a social component. Sociology of mathematics also points to the fact that mathematics is created within a broader social context. It is not possible to account for the complexity of mathematics within the process of its teaching and learning without reference to the complex forces that drive it in the society. A final idea was supported by the sociology of mathematics. Research suggests that the way teaching and content are organized have their own role in the development of mathematics itself.
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