Getting used to mathematics: alternative ways of speaking about becoming mathematical Abstract Mathematics learning has a long association with cognitive psychology, with its focus on deep and hidden processes called 'understanding'. In this paper I argue for a discursive, cultural view of learning mathematics in which the focus for teachers and researchers is on the process of the formation of particular identities. Von Neumann, the great 20 th Century mathematician, said that it's not a matter of understanding mathematics, but of getting used to it. Given that education in general and mathematics education in particular has been and still is obsessed with the notion of 'understanding', this is quite a radical thing to say. It brings us to Wittgensteinian ideas of a focus on social practices in the search for meanings and the manner of their acquisition. That in turn brings to issues of how social practices are constituted, maintained and changed; of the forms of participation and regulation within those practices, and of the kinds of identities are produced in them. In this paper I will look at these issues in relation to becoming mathematical in school. I have focused here on the school and not becoming mathematical in general because I take on board Bernstein's (1996) arguments about recontextualisation, that practices become other when they are relocated from their original site. Social theories in mathematics education In the field of production of mathematics education research what can be termed a social turn (Lerman, 2000a) has developed over the last 15 years or so. This is not to imply that previous theories, mathematical, Piagetian, radical constructivist or philosophical have ignored social factors (Steffe & Thompson, 2000; Lerman, 2000b). The social turn is intended to signal something different, namely the emergence into the mathematics education research community of theories that see meaning, thinking, and reasoning as products of social activity. This goes beyond the idea that social interactions provide a spark that generates or stimulates an individual's internal meaning-making activity.
The social turn has developed from three main intellectual resources: anthropology, as situated theories or communities of practice (Lave, 1988; Lave & Wenger, 1991; Wenger, 1998); sociology (Bernstein, 1996; Walkerdine, 1988); and cultural/discursive psychology (Cole, 1996; Harré & Gillett, 1994; Lerman, 1998), with its roots in Vygotsky s theories. All these bodies of work have a common origin in Marx and Durkheim. In the nineteenth century they challenged the image of the individual as the source of sense-making and as the autonomous builder of her or his own subjectivity. Consciousness was to be seen as the product of social relations; in particular, relations to the means of production. It is not the consciousness of men that determines their being but, on the contrary, their social being that determines their consciousness. (Marx, 1859, p. 328/9) I will briefly examine how this important common theme manifests itself in each of the intellectual resources to which I referred, in relation to mathematics education. I will also indicate where each of these resources has to be developed to focus on the notion of getting used to mathematics. Communities of Practice A community of practice is a set of relations among persons, activity, and the world, over time and in relation with other tangential and overlapping communities of practices (Lave & Wenger, 1991, p. 98). Lave s studies of the acquisition of mathematical competence within tailoring apprenticeships in West Africa led her to argue that knowledge is located in particular forms of situated experience, not simply in mental contents. Knowledge has to be understood relationally, between people and settings: it is about competence in life settings. People s identities are formed in participating in a practice; one can speak of learning as developing identities. Wenger (1998) elaborated this view of learning,
emphasising reification and participation as key processes through which persons become what can be described as the unit persons-in-practice (Wertsch, 1991). Now whilst these ideas are clearly recognisable in relation to workplace practices, classrooms are clearly a more difficult case since there is a collection of overlapping practices at play. Often more important to students than learning what the teacher has to offer are aspects of their peer interactions such as gender roles, ethnic stereotypes, body shape and size, abilities valued by peers, relationship to school life, and others. The ways in which individuals want to see themselves developing, perhaps as the classroom fool, perhaps as attractive to someone else in the classroom, perhaps as gaining praise and attention from the teacher or indirectly from their parents, leads to particular goals in the classroom and therefore particular ways of behaving and to different things being learned, certainly different from what the teacher may wish for the learners (Boaler, 2000). The teacher may perform the role of master for some students in relation to some aspects of what we might call the mathematical identities produced, most often specifically the mastery leading to further study of mathematics. But the teacher will not stand as the master for most of the students for most of the classroom social practices that are important for them. How, then, might one extend Lave and Wenger s notion? It may be fruitful to refer to multiple models of mastery offered in the complex of classroom practices. Expertise/mastery may be represented in a person or not, hence models, and those masters may be present in the classroom or not. In terms of what can be called role models, other students might perform many of the roles that students may desire to emulate. The teacher s personal style is often reported as having been a significant factor in people s identification with, or rejection of, aspects of schooling including mathematics. In relation to people absent from the classroom, parents stories of, for instance, their ability or lack in relation to mathematics can function as model for a student and a sibling or valued other similarly. So too images of who students want to become can act as models, including media personalities. This identifies the need for more complex studies of individual trajectories in the classroom, perhaps through narrative accounts (Winbourne, 1999; Santos & Matos, 1998; Burton, 1999), examining who are the models and what are the practices that are important to individual students.
Lave's analysis, whilst clearly a social theory that follows the programme of Marx and Durkheim, even if not explicitly, lacks, I believe, a sociological orientation in that the way people are socialised in practices may differ according to social class, gender, ethnicity or other groups (Walkerdine, 1997). Sociology Sociological studies in mathematics education include: Restivo (1992), who demonstrates how mathematical ideas, including those of abstraction and proof, are rooted in social and cultural practices; those based on Bernstein s work (e.g. Dowling, 1998; Brown, 1999; Ensor, 1999; Cooper & Dunne, 1999), and on poststructuralism (Walkerdine, 1988; Klein, 1999; Walshaw, 1999; Evans, 2000). In all of these studies, sociology offers a resource for studying the teaching and learning of mathematics that enables materialist analyses of how social forces are realised as positions in the classroom. What constitutes appropriate school mathematical practices is produced in different forms according to positioning in discourse (poststructuralism), or according to recognition and realisation rules (Bernstein). Lave and Wenger (1991) argue that " A community of practice is an intrinsic condition for the existence of knowledge, not least because it provides the interpretive support necessary for making sense of its heritage" (p. 98). Thus, meanings signify, and they signify within social practices. But significations matter, they are not neutral meanings: situating meanings in practices must also take into account how those significations matter differently to different people. Practices should be seen, therefore, as discursive formations within which what counts as valid knowledge is produced and within which what constitutes successful participation is also produced. Non-conformity is consequently not just a feature of the way that an individual might react as a consequence of her or his goals in a practice or previous network of experiences. The practice itself produces the insiders and outsiders. Individual trajectories in the development of identities in social practices arise as a consequence of our identities in the overlapping practices in which each of us functions but also emerge from the different positions in which practices constitute the participants.
We can capture the regulation of discursive practices by talking of the practice-in-person as the unit, as well as the person-in-practice. Becoming used to mathematics, then, has both its forms and its content open to historical and sociological analyses. Cultural/Discursive Psychology Vygotsky s psychology was an application of Marx s theories to learning, providing a framework whereby the socio-cultural roots of thought become internalised in the individual. Every function in the child's cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (interpsychological), and then inside (intrapsychological)... All the higher functions originate as actual relations between human individuals. (Vygotsky, 1978, p. 57) Vygotsky was not directly concerned with social practices. At the time of the Russian revolution the singular discourse of dialectical materialism and the drive for progress from a feudal society to communism did not allow for the availability of other theoretical resources. His early death in 1934, at the age of 38, precluded any engagement with more relativistic social theories. However, Vygotsky s psychology is a cultural psychology (Cole, 1996; Daniels, 1993), a method for interpreting how persons become social beings, and it opens up spaces for different analyses than those which appeared during Vygotsky s life. Vygotsky s work is generally taken to be about the individual learning in a social context, but his notion of the zone of proximal development (zpd) offers more than that. First, in that consciousness is a product of communication, which always takes place in a historically, culturally and geographically specific location, individuality has to be seen as emerging in social practice(s). Second, all learning is from other persons-in-practices, and as a consequence meanings signify, they describe the world as it is seen through the eyes of those socio-cultural practices. In
his discussion of inner speech Vygotsky makes it clear that it is the process of the development of internal controls, metacognition, that is, the internalisation of the adult. Again, these are mechanisms that are located in social contexts. Finally, the zpd is a product of the learning activity (Davydov, 1988), not a fixed field that the child brings with her or him to a learning situation. The zpd is therefore a product of the previous network of experiences of the individuals, including the teacher, the goals of teacher and learners, and the specificity of the learning itself. Individual trajectories are key elements in the emergence, or not, of zpds (Meira & Lerman, submitted). Vygotsky's work provides a mechanism for learning that is absent from sociological theories and that is required to complement Wenger's notion of participation. That all three theories can be seen to owe their origins to Marx and Durkheim makes the process of drawing these theories together into a coherent account that we can use to speak of becoming mathematical. Conclusion The task of researchers working with the notion of becoming mathematical is to make the links between structure and agency and between culture, history and power and students learning of mathematics. Individuality and agency emerge as the product of each person's prior network of social and cultural experiences, and their goals and needs, in relation to the social practices in which they function. I proposed the metaphor of a zoom lens for research, whereby what one chooses as the object of study becomes: A moment in socio-cultural studies, as a particular focusing of a lens, as a gaze which is as much aware of what is not being looked at, as of what is Draw back in the zoom, and the researcher looks at education in a particular society, at whole schools, or whole classrooms; zoom back in and one focuses on some children, or some interactions. The point is that research must find a way to take account of the other elements which come into focus throughout the zoom, wherever one chooses to stop." (Lerman, 1998, p. 67)
But the object of study itself needs to take account of all the dimensions of human life, not a fragment such as cognition, or emotion. Vygotsky searched for a unit of analysis that could unify culture, cognition, affect, goals, and needs (Zinchenko, 1985). According to Minick (1987): In 1933 and 1934 Vygotsky began to reemphasize the central function of word meaning as a means of communication, as a critical component of social practice (p. 26). Combining the two elements I have described above, we might talk of practice-in-person-in-practice as a unit of analysis; or, to use and extend Vygotsky's own expression, society-in-mind-in-society. The legacy of cognitive studies in mathematics education is of a natural process of conceptual development that normal active children achieve in the course of appropriate classroom activities. Understanding, seen as the formation of decontextualised schemata of mathematical concepts, is carried out by the normal child through reflection. I am arguing here that mathematical learning should be viewed as a highly unnatural process which can only take place through socialisation into what constitutes school mathematics. Children become mathematical by getting used to what counts as being mathematical, which is constituted in the social practices of the classroom. This may be a more fruitful way of speaking about learning, in which learning is about speaking. References Bernstein, B.: 1996, Pedagogy, Symbolic Control and Identity: Theory, Research, Critique London: Taylor and Francis. Boaler,J.: 2000, Mathematics from Another World: Traditional Communities & The Alienation of Learners. Journal of Mathematical Behavior 18(4), 1-19. Brown, A. J.: 1999, Parental participation, positioning and pedagogy: a sociological study of the IMPACT primary school mathematics project. Unpublished PhD Thesis, University of London Library. Burton, L.: 1999, The implicatons of a narrative approach to the learning of mathematics. In L. Burton (Ed.) Learning mathematics: From hierarchies to networks (pp. 21-35) London: Falmer Press.
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Stephen Lerman Professor of Mathematics Education Head of Educational Research Centre for Mathematics Education Faculty of Humanities and Social Science South Bank University 103 Borough Road London SE1 0AA, UK Tel: +44 (0)20 7815 7440 E-mail: lermans@sbu.ac.uk