A MOMENT IN THE ZOOM OF A LENS: TOWARDS A DISCURSIVE PSYCHOLOGY OF MATHEMATICS TEACHING AND LEARNING
Stephen Lerman
Centre for Mathematics Education
South Bank University
London, UK
Abstract Psychology has undergone considerable changes in the last decade or so, in line with developments in cultural studies, feminist research, postmodernism etc. In mathematics education research one can see the evidence of such developments, albeit as yet in a limited way. In publications over the last few years I have argued the case for a cultural psychology and for a move away from constructivism. In this paper I attempt to outline a research programme for mathematics education which is firmly based in cultural, discursive psychology. I discuss theoretical issues, refer to some of the important literature, and present some research in aspects of the programme. (1)
Mathematics education as a theoretical field has its roots in mathematics classrooms, from nursery age to those in University. It draws on a range of theoretical resources in developing its own body of theory to account for what goes on in those settings in relation to teaching and learning. The concern of workers in the field is to find a language with which to describe the process of the acquisition of mathematics, and through which to draw inferences for what teachers might do to bring about that acquisition by as many students as possible. That language has to account for a great deal, much of it tacit in teaching, such as: a notion of development as being towards something; the goals of education in general and of mathematics education in particular; the relationship between teaching and learning; and the particularities of the nature of mathematics. That language has to be informed by empirical studies, and has therefore to incorporate a process for its own continuing elaboration. At the same time it needs to take account of relations of power, of voice and of silence of any theory, of any account of teaching and learning, of any set of goals for education, and for any notion, usually implicit, of development.

Traditional psychology, for all that its field of study is human behaviour, has offered little that can help to improve society.

modern psychology has been incapable of making serious contributions to Third World (sic) development... it is important to point out that mainstream psychology has also failed to make significant contributions to national development and the lives of the poorest sectors in Western societies. (Harré, 1995, p. 54)

In the process of individualizing its view of students, it (mathematics education) has lost any serious sense of the social structures and the race, gender and class relations that form these individuals. Furthermore, it is then unable to situate areas such as mathematics education in a wider, social context that includes larger programs for democratic education and a more democratic society. (Apple, 1995, p. 331)

 I believe they are right to say that psychology cannot provide such a language, at least psychology as understood to be the study of learning as the individual's cognitive reorganisation, albeit caused by social, physical or textual factors (von Glasersfeld, 1994, p. 6) through equilibration. I want to suggest, though, that psychology can be seen as 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. This is an adaptation of Rogoff's planes of analysis, into a dynamic metaphor in which one might envisage a researcher choosing what to focus on in research through zooming in and out in a classroom, as with a video or still camera, and selecting a place to stop. Rather than seeing social factors as causative of learning, they can be seen as constitutive (Smith, 1993). As such, I will argue that psychology from this perspective can respond to Apple's critique. A discursive, cultural psychology locates its interpretation of the individual at the intersection of overlapping language games in which the person has developed and thus is necessarily rooted in the study of cultures and histories. 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.

In this paper I address the issue of what is the terrain of such a psychology, that is, what counts as an appropriate language, for mathematics education. I will describe the move in psychology over the last decade or so (Cole, 1996; Harré, 1995) to one which is fully cultural and focused on the way in which consciousness is constituted through discourse. That move can be seen as part of the reaction in sociology and philosophy to the nineteenth century challenge by Durkheim and Marx to the image of the individual as the source of sense-making and as the autonomous builder of her/his own subjectivity. It is also a response to Wittgenstein's later work on language, and to the anti-essentialism of poststructuralism. I will attempt to engage with the elements of a theory for mathematics teaching and learning which I mentioned above: what might be understood as teleology/development; the connections between teaching and learning; the process of acquisition; the particularities of mathematics; the inevitable coercion and denial of voice of any theory; and the role of empirical study. I am choosing to undertake an impossibly large agenda for one talk as I wish to present as comprehensive a map as possible. It is more of a recipe for a life's work, and in that it is a recounting of developments in my own learning and research and a setting out of a programme upon which I am engaged with colleagues and with some of my students, it is self-consciously broad and over-ambitious. I will, however, endeavour to sketch the main outlines of this programme and give some illustrations of relevant research, my own or that of others, where appropriate. These illustrations will inevitably be briefly described, due to limitations on time and space.

Vygotsky, whose work has become better known in the mathematics education community in recent years (e.g. Bartolini Bussi, 1991; Lerman, 1992; Boero et al, 1995) is a major figure in the development of cultural and discursive psychology (Cole, 1996, p. 107; Harré & Gillett, 1994, p. viii). Feminist research, in particular, has invited us, researchers and writers, to be forthcoming about our biographies and to own up to where we are located in our work. I happily confess that I became fascinated and excited by Vygotsky's ideas when I first came across them some eight years ago and immediately found a strong resonance with the way in which I perceive myself to be culturally and socially situated (Lerman, forthcoming, a).

The theoretical resources to which I referred above come from outside mathematics education, and I will endeavour to signal from where I draw inspiration. I will also attempt to do justice to those ideas in spite of the limitations of time and space, an essential task for researchers. Whilst recontextualisation is inevitable (Bernstein, 1996) it is incumbent on researchers to take theories seriously. Elsewhere (Lerman, 1996) I have argued that the consequences of superficial readings can result in incompatible theories being conflated.

Development and Teaching/Learning

The question of what is consciousness and how it develops was the subject of Vygotsky's (1924/1979) first major public paper. In his subsequent writing he argued for development to be seen, from the first moments, as brought about by communication:

instruction and development do not meet for the first time at school age; rather, they are in fact connected with each other from the very first day of a child's life. (Vygotsky, 1956, in Wertsch, 1985, p. 71) Bruner's studies of babies illustrate vividly Vygotsky's descriptions of voluntary attention as learned, and of meaning as initially intersubjective, mediated by the adult (Harré (personal communication)): Bruner was interested in the very early moment of the development of intentionality and causality. Here's an attractive object. And an infant, a very small infant, reaches for it. Causal relation. Object is there...infant reaches out like that. What happens? Mother gives it to him. At that point the relationship between the reaching and the grasping changes...the mother has introduced into this game, baby wants, baby intends. Baby just acted causally...didn't get, so mother says baby wants. Next step, so she's completing the action, she is giving him the interpretation of his action. Next step, baby reaches forward, and compresses the air in his little chest and goes, 'Ah.' So, mother gives it to him. We've now turned the intentionality into a babble, the first primitive verbal utterance. Causality turning into intentionality. And the mediating role is the caretaker, who has done the defining, and done the supplementing. (2) This account of how the baby becomes conscious exemplifies Leont'ev's (1981, p. 57) description of how the mental plane is constituted in the process of internalisation. In this sense, learning is inseparable from 'teaching'. That teaching may not be deliberate and intentional, as it is in school or in parental instruction. In everyday situations of the child's life s/he learns how to be, in gendered, ethnic, class and other historical, socio-cultural identities. Learning how to be, or to become, is motivated by desire, goals and needs, to be accepted, to emulate a desired person, or to join a group. Lave and Wenger's (1991) account of learning in workplace situations presents that theory of learning as becoming, and Lave (1996) and Winbourne and Watson (1998) discuss the notion in relation to the classroom.

Concomitant with this view of learning and development is an interpretation of concepts and knowledge which is neither of the two choices rejected by Piaget, empiricism and innatism (1970/1972), nor the individualistic, constructivist world which he proposed in their place. It is one where meanings historically precede the individual, which the individual internalises, and through which the individual perceives the world.

How do I know that this colour is red? - It would be an answer to say: 'I have learnt English'. (Wittgenstein, 1958, 381)

I did not get my picture of the world by satisfying myself of its correctness; nor do I have it because I am satisfied of its correctness. No: it is the inherited background against which I distinguish between true and false. (Wittgenstein, 1969, section 94)

 Those meanings are not static and singular. As they are experienced and internalised from a range of discursive situations they may well carry the meanings, integrally imbued with affect, of those situations (Evans, 1993).

Vygotsky emphasised the presentation of scientific concepts to students and opposed the idea that they need to rediscover the development of mankind for themselves (1988). This formulation is taken to be very close to a transmission style of teaching by some. However, Vygotsky was opposed to merely telling learners. He was centrally concerned with the mediation of cultural tools and of metacognitive tools.

scientific concepts… just start their development, rather than finish it, at a moment when the child learns the new term or word-meaning denoting the new concept (Vygotsky, 1988, p. 159). Two aspects of a discursive psychological approach to development will be reported on here, first that of Vygotsky's argument for teaching from the general to the particular and, second, cultural mediation in learning, through Lave's work on learning as social practice.

i Vygotsky drew on Marx's notion of ascent from the abstract to the concrete in his theory of the acquisition of scientific concepts, and one development of this perspective has been towards the teaching of general principles to students, with particular questions being seen as instances in which the general principles need to be identified and applied (Galperin, 1969; Talyzina, 1981). This runs contrary to the usual tendency to work inductively from a range of everyday examples to general principles. We have been working on utilising this approach in mathematics classrooms. In a study of students in three inner city schools in the UK, across all levels of achievement according to national tests, principles for calculations of rates of processes were taught (Day, forthcoming). Students were shown a generalised model for the conception of the values involved and the relations between them, in the form of a visual structure for combining given data, such as time and rate, and using it to calculate the quantity. The teaching was orientated towards success in using the model for a range of problems of increasing complexity, both set by the teacher and invented by the students. That success was measured by a dynamic assessment procedure based on the amount and type of assistance they required. Analysis of quantitative data and of videos is currently taking place. Interim results show a level of achievement and change in attitude, across the ability range, which has surprised the classroom teachers and ourselves, and we have certainly found that the results support the argument for a 'theoretical learning approach' (Karpov & Haywood, 1998).

ii The metaphor of students as passive recipients of a body of knowledge is terribly limited: so too is the metaphor of students as all-powerful constructors of their own knowledge, and indeed of their own identities. Lave's (e.g. 1996) focus on the shaping of identity in social practice, extended by an analysis which takes account of the differences between schooling and the practices which she has studied (Lerman, forthcoming, c), emphasises the centrality of the social relationships constituted and negotiated during classroom learning. Lave talks of learning as "an aspect of participation in socially situated practices" (ibid., p. 150). Provided we do not expect those practices to be those of the mathematics teacher, or of the mathematician but, instead, of the practices of the classroom culture, her description of learning can be very fruitful, as is shown by Winbourne (1997), for example. In that study he described the demonstration of creativity and expertise in the use of graphical calculators amongst a class of 13 year-old girls, and the subsequent display of mastery and learning through participation in mathematical activities, which he has since described in terms of local communities of practice (Winbourne & Watson, 1998).

To summarise, Vygotsky outlined a method for accounting for development which is rooted in an historical, sociocultural notion of mind. This method brings together teaching and learning. In terms of a telos, or direction for development, Wertsch (in Cole & Wertsch, 1996) argues that Vygotsky offered, although not explicitly, a somewhat confused account of a telos of abstract rationality, an enlightenment principle, and one of a 'harmony of imagination', a kind of mythical thinking. The former is evidenced in his and Luria's studies in Uzbekistan (Luria, 1976), and the latter in Vygotsky's The Psychology of Art. These two teloi co-exist in dialectic with each other, much like thinking and speech. Lave offers an interpretation which is, inevitably, more recent and partial, and hence more appropriate to discursive psychology than Vygotsky's, although it has clearly grown from his ideas, that of the desire of the individual to 'become'. Again I mean here such desires as: to please parents; to emulate a sibling; to be a member of a desired group; to fulfil goals, etc.

The Process of Acquisition - The ZPD

For psychologists of education, as distinct from sociologists, cultural theorists and others who also study the situations in which meanings are manifest, the concern is with the process of acquisition of meanings. Vygotsky introduced the zone of proximal development in a lecture given in March 1933 (Van de Veer & Valsiner, 1991, p. 329), although he pointed out that the idea was not originally his own. He died only fifteen months later and clearly had not been able fully to elaborate his thoughts on the zpd. Along with Newman & Holzman (1993) I take it to be the explanatory framework for learning as a whole, both in intentional settings, such as schooling, and in informal settings; in other words all socio-cultural milieus. It recognises the fundamental asymmetry of the teacher-student(s) relationship, an asymmetry often denied or underplayed by more individualistic approaches. It provides the framework, in the form of a symbolic space (Meira & Lerman, forthcoming), for the realisation of Vygotsky's central principle of development:

In our conception, the true direction of the development of thinking is not from the individual to the socialised, but from the social to the individual. (Vygotsky, 1988, p. 32) Such a definition opens a space for a unit of analysis of consciousness that incorporates affect and cognition: When we approach the problem of the interrelation between thought and language and other aspects of mind, the first question that arises is that of intellect and affect. Their separation as subjects of study is a major weakness of traditional psychology... (Vygotsky, 1988, p. 10) The interpretation of Vygotsky's zpd upon which we draw (Meira & Lerman, forthcoming) is closer to that of Davydov (1988, 1990) and is less 'internalist' than Vygotsky's own version might appear. He wrote that the instruction process creates the zpd, but he also wrote that instruction is "fruitless if it occurs outside, below, or above the zpd" (1987, p. 213). He probably intended to emphasise jointly the roles of the learning activity and the learning potential of the child. The zpd is often described as a kind of force field which the child carries around, whose dimensions must be determined by the teacher so that activities offered are within the child's range. According to Davydov and followers, on the contrary the zpd is created in the learning activity, which is a product of the task, the texts, the previous networks of experiences of the participants, the power relationships in the classroom, etc. They speak of the ideas offered by one student potentially pulling other students into their zpd (Lerman, 1994a). The zpd is the classroom's, not the child's. In another sense the zpd is the researcher's: it is the tool for analysis of the learning interactions in the classroom (and elsewhere).

Teachers use a variety of strategies to try to create a zpd, including reminders of past lessons, events and language (Edwards & Mercer, 1987), references to outside school objects or meanings, etc. Much can be learned by the researcher, focusing on these strategies of the teacher. Similarly, pairs of students can create their own zone of proximal development if they are motivated, taught how to share ways of working, have an appropriate personal relationship, and/or other factors. Students can be, and very often are, pulled into their zpd by imposition. For reasons of desire to become like another person, or to please another person, to be accepted into a group, or achieve other such goals people will copy/emulate another, and subsequently that behaviour may become part of that person. However, it is certainly not the case that learning always takes place. In both teacher-student interactions and student-student interactions the participants may not engage together in the activity. They may act separately or one and/or the other may not act at all. In Meira & Lerman (forthcoming) we give some instances of a zone of proximal development being created, in which a nursery teacher almost 'grasps' a child's attention and orientates it towards what she wants the child to learn. In one scenario the interaction is initiated by the child's questioning gesture and as observers we were unable to ascertain whether the child was pointing at the objects on which the teacher chose to focus. This did not matter, though; the child responded to the teacher and became involved in the activity. We also offer instances where, despite the teacher's best efforts, a zone of proximal development is not created in the activity, the teacher missing the child's experience immediately preceding her intervention and offering something not relevant. In another study (Lerman, forthcoming, b) I argued that as much may be learned from incidents where a zone of proximal development is not created as when it is. Two thirteen-year-old students were engaged, ostensibly together, on a task to simplify a ratio expression, ab:ab. A close analysis revealed that they chose different methods and although they spoke, one might say, at each other about their methods, they did not pay attention to each other and so did not move forward. As we have described it elsewhere, students may not catch each other's ideas (Vile & Lerman, 1996) and hence not create a zpd. Creating a zpd is more about mutual orientation of goals and desires than about the intended content of the interaction. In that study I was looking at the interactions between the subjects of the video, trying to identify when they were communicating their ideas and reasoning to each other. Acts of communication, as objects of study, are the signs of sociogenesis, the social origins of psychogenesis and internalisation. I was looking for clues from all the elements of the data set, videos, transcripts, and interviews, to help in drawing inferences about the nature of that communication. For instance in that study I indicated that the behaviour of the teacher and the students with regard to their mathematical activity is framed by a discourse of ability. This constituted one of the students as more able and therefore more powerful in the interactions between them. Some time after the interaction used in the study the teacher looked at and read the video and the interactions of the students as confirming her evaluation in terms of ability.

This is the relevant extract from the transcript.
 
1.  M: What? Equals ab? [pause, D looks on M's page] Equals ab?
2.  D: Yeah.
3.  M: No, it equals 1.
4.  D: Wait a second...
5.  M: 'Cause 1, [punching calculator buttons] 12 times tw... no. One, look, look, look. One times 2, divide 1 times 2...it shouldn't equal 4. [M appears to be substituting the values 1 and 2 for a and b]
6.  D: [laughs]
7.  M: Um, yeah, it's, 'cause I'm doing [punching buttons] 1 times 2, divide 1 times 2, equals 1.
8.  D: So that's cancelled. The two b's are cancelled out.
9.  M: Equals 1.
10.  D: Right? The two b's are cancelled out.
11.  M: Hey, where'd my pen go? No come on, look, look, look, look. You've got to do BODMAS. Watch, watch, watch, watch. [punching buttons] One times 2, divide 1...come on, 1 times 2. That's stuffed up. [with emphasis] One.
The outcome was that the 'more able' student M ignored the ideas of the 'less able' D and firmly tried to impress his method on D. D seemed to lose heart quickly and did not press his method. In order to give an account which is adequately framed in a theory, and which also offers a description of the objects about which one, as researcher, is making statements, one has to delimit one's text. A Piagetian model would argue that power, in the form of authority, inhibits equilibration. Vygotsky's zpd, working from a sociogenetic perspective, assumes imbalances in social relations as part of being human and communicating, and therefore these were identified as an element of my account.

Following an account of teaching and learning and development from the perspective of discursive psychology, I have argued that the zpd offers a sociogenetic mechanism for interpreting learning particularly suited to microgenetic studies. In mathematics education research activity theory has been used in this way to take account of goals and needs as they change over time, by Crawford & Deer (1993), Bartolini Bussi (1996) and Lerman (1997).

Mathematical Meaning-Making

In mathematics education we are confronted with powerful models of the process of mathematics learning, based on Piaget's constructivism. According to Steffe and Wiegel (1996) a model of mathematics learning consists, at least, of a meaning for operations and for representation. The former "are part of a system of operations that is goal directed" (p. 486) and the authors draw distinctions between Piaget's notion of actions and activity and the Soviet notion in activity theory. The latter are of greatest concern to me here.

Many accounts of knowledge representation are misleading because they are based on the assumption that concepts are things - mental objects - "out there" to be represented… we regard mathematical concepts as mental acts or operations, and it is these operations that are represented. (ibid., p. 487) Piaget's familiar ontological choices, between reality imposing itself on a person empirically or platonically, and the person constructing her/his own internal individual world, are evident here. There is another option, that mathematical concepts are social acts and tools. Consciousness is constituted in historical, socio-cultural settings and cultural tools are internalised in the strong sense that the mental plane is formed in that process. Thus, cultural tools (analogous with Marx's thesis concerning physical tools) both transform the person and the world for that person, and these cultural tools precede the individual. Words and symbols are mediators of thought, "It is the world of words which creates the world of things" (Lacan, 1966, p. 155) and objects, including concepts, have meanings only within relations of signification (Walkerdine, 1988).

From the perspective of teaching and learning mathematics the research programme would therefore be to study empirically the semiotic mediation of those objects. The language of semiotic mediation, whereby the person and the world for that person are transformed by the acquisition or appropriation of cultural tools, is a theoretical resource that engages with the fact of signification as well as the specificity of relations of signification. It rejects the notion of decontextualised, abstract concepts. It offers a medium through which one can account for cultural specificity, such as the mathematical meanings of Aboriginal students (Klein 1997) and of multiple subjectivities as a result of the overlapping social practices of gender, ethnicity, class, family relations etc., in which people develop. The richness of the social and cultural implications of a discursive approach to psychology and the range of theoretical resources (e.g. sociolinguistics, cultural theory, semiotics, postmodernism) on which it draws is enough to support its adoption.

In Vile (1996) and Vile and Lerman (1996) students' work in linear equations and co-ordinates were examined as case studies for the elaboration of a developmental semiotics, taking together the science of signs and the functioning of the process of mediation in learning, which is interpreted as making meaning which allows appropriate use in relevant contexts. In the study of co-ordinates (Vile, 1996) students were asked a series of questions in clinical interviews concerning the distance between two points, in two, three and four dimensions. The intention was to examine the meanings which students gave when working in the different dimensions and to try to identify which meanings corresponded with successful transfer to four dimensions.

Students who retained a concrete, measuring meaning for the distance between two points were unable to cope with the distance between 4 dimensional co-ordinates… Those students who did make the appropriate symbolic meaning but only as a process… were unable to make recourse to a more iconic representation if they were unsure… Students who were able to make the more generalised sign-sign foregrounded meaning early in their development are able to transfer to the more generalised dimension with relative ease. (p. 176-177) This study pointed to the kinds of meanings that can mediate the generalisations needed in mathematical thinking, and hints at the classroom activities that might encourage their appropriation. The transformation in thinking and acting, when students learn mathematics using such software as Cabri Géomètre or technological tools such as the graphics calculator, is accessible with the notion of semiotic mediation, which also opens the possibility for analyses of social transformations of the classroom (Winbourne, 1997).

In another study (Finlow-Bates, 1997) we examined undergraduate students' mathematics proof activities through an analysis of proof as a process of social negotiation of meaning, rather than an 'understanding' or its lack.

Wertsch (1991) has argued that focusing on mediation offers a unity for analysis which neither the individual nor mathematical knowledge can offer, with their implied separation of subject and object. Drawing on meaning as the mediation of cultural tools enables the study of other aspects of the positioning brought about in learning, through the social and political associations of concepts, or knowledge as power. For example, recent sociological studies (Cooper & Dunne, 1998) offer insights into how contexts mediate differently for students of different social groups. In the first stage of their research on mathematics assessment items set in realistic (everyday) settings and esoteric ('pure' mathematical) settings, they found that working class children "failed to demonstrate competences they have" (p. 115), through mis-readings of the realistic settings. Cole (1996) has argued that a focus on the mediation of cultural tools does not take account of action on the world, in the sense of tool use that Marx described. I have suggested above that in the zone of proximal development one can study the mediation of cultural tools but that activity theory is more fruitful for longer-term studies, taking account of goals and needs. There is a dialectical unity in these two methodologies in that, whilst both are rooted in the cultural psychology of Vygotsky, mediation is a generalising principle, looking for similarities, whilst activity theory is a specialising one.

I gave this section the title 'Mathematical Meaning-Making', rather than mathematical understanding, with the intention of writing this paper without using the term. Indeed I have scarcely used it, apart from in this paragraph. The term is part of the 'regime of truth' which locates power in the hands of teachers who can say when a child understands or doesn't, independently of what s/he produces, verbally or in writing (Watson, 1995). Its entirely internal nature makes it a rather useless notion (Lerman, 1994b) whilst its association with closure places it in a positivist paradigm. Much of Wittgenstein's later work can be seen as a deconstruction of attempts to find essences behind social meanings. His well-known argument (e.g. 1974, p. 64) that to understand a concept is to know its use is to locate meanings in grammar and in rule-following.

Voice

Confrey (1995) argues that constructivism offers a space for individuality of interpretation, or voice, that Vygotsky's emphasis on scientific concepts replacing spontaneous concepts appears to deny. This aspect of his theory has often been interpreted, wrongly in my view, to recall the possibility of learning through transmission. First, spontaneous concepts do not disappear under scientific ones, which might be seen to lead to a uniformity which denies the possibility of individual voice. In general they coexist with spontaneous concepts, through a splitting of subjectivities, the child having learned in which situations the differing meanings are appropriate. As a rather simplistic example, a child might know to use "My half is bigger than your half" in the playground but not in the mathematics classroom. This offers a discursive interpretation of intuitions in mathematics (Fischbein, 1987). Second, as I discuss below, the notion of the zpd requires from the teacher, (desired) peers and texts, the particular experiences of individuals.

In my view the method which Vygotsky's work offers is also often misunderstood, in large part because of the time and forms in which it was used. Vygotsky died in 1934, at the age of 38. The theoretical discourses available at that time, and especially the particular circumstances of the Soviet revolution, limited the perspective for the theorising and therefore for the choice of research programmes of Vygotsky and his colleagues. This is inevitable and is actually an application of Vygotsky's own theory that concepts are related to their time and place. Thus Luria's work in Uzbekistan (1976) presents a strong image of the valuing of a particular interpretation of advanced societies as against primitive ones, and of progress. But Vygotsky's method is, through his argument for the priority of the intersubjective, to enable the study of consciousness as the internalisation of sociocultural meanings, the appropriation of cultural tools, and the transformation that this effects for the individual and for her/his world. The origins of individual meanings being located in socio-cultural tools roots 'voice' in its proper framework. It is not the individualism of private world views which has dominated the debate around subjectivity and voice in recent decades. In cultural, discursive psychology individuality is the uniqueness of each person's collection of multiple subjectivities, through the many overlapping and separate identities of gender, ethnicity, class, size, age, etc., to say nothing of the 'unknowable' elements of the unconscious.

Discourses which dominate in the classroom, and everywhere else for that matter, distribute powerlessness and powerfulness through positioning subjects (Evans, 1993). Walkerdine's (1989, p. 143) report of a classroom incident in which the emergence of a sexist discourse bestows power on five year-old boys, over their experienced teacher, dramatically illustrates the significance of a focus on discourse, not on individuals. In some research on children's interpretations of bigger and smaller (Redmond, 1992) we found some similar evidence of meanings being located in practices.

These two were happy to compare two objects put in front of them and tell me why they had chosen the one they had. However when I allocated the multilinks to them (the girl had 8 the boy had 5) to make a tower . . . and I asked them who had the taller one, the girl answered correctly but the boy insisted that he did. Up to this point the boy had been putting the objects together and comparing them. He would not do so on this occasion and when I asked him how we could find out whose tower was the taller he became very angry. I asked him why he thought that his tower was taller and he just replied "Because IT IS. " He would go no further than this and seemed to be almost on the verge of tears. (p. 24) Many teachers struggle to find ways to enable individual expression in the classroom, including expressing mathematical ideas, confronting the paradox of teachers giving emancipation to students from their authoritative position. But this can fruitfully be seen as a dialectic, whereby all participants in an activity manifest powerfulness and powerlessness at different times, including the teacher. When those articulations are given expression, and not denied as in some interpretations of critical pedagogy (Lerman, 1998), shifts in relations between participants, and crucially between participants and learning, can occur (Ellsworth, 1989; Walcott, 1994). Learning is predicated on one person learning from another, more knowledgeable, or desired, person.

In the classroom Davydov's learning activity structure of a lesson encourages, and actually requires 'voice', the expression of individual life experience and perspective. When a teacher offers an activity in a classroom, say to share 2 oranges between 3 children, the different answers offered by the children arise from their previous experiences, what has been called the zone of actual development, and potentially pull the others, including perhaps the teacher, into their zones of proximal development. Similarly, powerful technologies can offer possibilities for novel ideas by children which create zones of proximal development for other participants and change the social relations in the classroom.

The account of a discursive psychology for mathematics education which I have attempted to develop in this paper incorporates action, goals, affect, power and its lack, based on sociocultural origins. A psychology focused on the individual making her/his own sense of the world does not engage with social and cultural life: other theoretical discourses, such as approaches to sociology which merely describe, are not adequate for mathematics education either. I go along with Harré (1997) when he writes, referring to discursive psychology: "Psychology is the study of the skills necessary to live as a human being with others" (p. 189). It should be clear that such a definition, particularly when related to education, is open to contestation concerning what is valued as development and what constitutes cultural capital. A cultural, discursive psychology places that contestation at the heart of what constitutes consciousness, meaning-making and, in this paper, mathematics teaching and learning.

Theoretical and Empirical Fields

Theories need to account for their on-going development in relation to their empirical work. Brown and Dowling (1998) propose that "the research process itself is properly conceived of as the construction of the theoretical and empirical as increasingly coherent and systematically organised and related conceptual spaces" (p. 11). Since Kuhn (1970) researchers have been forced to recognise that they create the objects of their research, they are not entities existing independently of the research discourse or the researcher. This is not to prioritise theory but to recognise the dialectic between the two fields, the empirical and the theoretical, and it distinguishes between mathematics education as a set of practices and mathematics education as a field of knowledge (Patricio Herbst, personal communication). I began this paper by pointing out that there is an overlap, since all mathematics education has its roots in the classroom whether its aim is to say something about practice or about how one might think and speak about mathematics education. In the main, though, this paper has been about the latter. My intention has been to map out the field, from the point of view of a discursive psychology. I have also tried to indicate its implications for mathematics education as a set of practices, through the examples of research and other classroom illustrations.

Steffe & Wiegel (1996) challenge researchers to provide an account of the self-reflexivity of their theories, although why this should be a sign of a good theory is not spelt out, except as a counter to naiveté. They argue that, according to radical constructivism, theories of learning can be seen as making what they call second order models of students' understandings, which are understood as first order models constructed by students to order their experiences. This symmetry is very appealing.

In that the objects of research, the products of research, the theories drawn upon, the methodologies used, etc. are all cultural products, texts, the theoretical programme outlined here is, in its entirety, reflexive. Language precedes phenomena, which precede experience - this the sense I take from the sentence which heads this paper. To refer again to Kuhn, however, researchers are forced to admit their allegiances to their theories. In one direction, empirical research leads to elaboration of theory, as our work on, for example, the zpd, on developmental semiotics and on teaching general principles demonstrates. In the other direction, theory, as outlined in this paper, provides the resource for interpretation, and for methodology and its justification. Rarely does one's theory as a whole change (although see Lerman, 1989, in comparison with this paper!).

The metaphor of the zoom lens is part of my theory: to sustain the metaphor a little further, it has framed my writing here and thus offered me both possibilities and limitations. It is a rhetorical tool for expressing the need to take into account all of the social and cultural life of the classroom, but it cannot quite capture the histories of the participants, or the classroom, and perhaps it is too linear. However, if it is the zoom lens of a video camera it can capture development and change. How we read the tapes remains the challenge for research.

Notes

(1) My thanks to Peter Winbourne and Ros Sutherland for comments on an earlier draft.

(2) This quote comes from the transcription by Anne Watson of a tape recording of a seminar with Rom Harré held in Oxford University on December 6th 1997.

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