Discipline and Mathematise: the myth of relevance in education1

Paul Dowling

Culture Communication & Societies
Institute of Education, University of London

{1995, Perspectives in Education, 16, 2, pp 209-226}

p. 209

In a stimulating and well-argued paper, Johan Muller and Nick Taylor2 offer an important caveat for 'hybridising' curriculum developers. Specifically, they point to the dangers of too readily assuming a continuity between academic and everyday or commonsense knowledge. Here, I shall attempt to move towards a sociology of what I perceive as a fundamental epistemological distinction. I shall draw on my own theoretical and empirical research in the sociological analysis of mathematics texts3 and on some recent developments by Basil Bernstein4. The proposition that I wish to develop is that the division of labour constitutes two distinctive modes of social relations. These modes generate, respectively, academic and everyday practices and knowledges which are, thereby, mutually incommensurable. Thus, the widely held belief that school knowledge can be made relevant to the everyday and to working practices in any direct sense is, I shall argue, mythical. I shall examine the nature of the incommensurability of these practices and, in the final section of the paper, I shall consider how it might be utilised in productive dialogue.

An important qualification must be stated at the outset. The expression "division of labour" may be interpreted as oppressive, in the Marxist sense, involving alienation and exploitation. Alternatively, it may invoke the interdependency of Durkheim's5 "organic solidarity". However, organic solidarity itself may entail the exclusivity of specialisms. My use of the term assumes the possibility of the division of labour penetrating the individual. In arguing the value of academic

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practices, therefore, I am not seeking to maintain an elite, but rather to generalise access. The incommensurability of the academic and the everyday does not imply that we must be restricted to one or the other.

I shall begin with a brief consideration of some points made in the African National Congress (ANC) discussion document, A Policy for Education and Training6, which has established the crucial arena for current debate on the curriculum in South Africa.

introduction: discipline and knowledge
The national core curriculum for the [General Education Certificate and Further Education Certificate] will:
(vii) Foster self-discipline7

One of the problems-or, perhaps, one of the advantages-associated with any document that is intended for wide circulation is the necessary dependency upon assumptions of commonsense understandings. In everyday conversation, "self-discipline" is likely to be an uncontested and unproblematised term referring to a positive personal quality. We might expect it to be glossed over without a second thought. In the context of such an important issue, however, it makes sense to enquire about the origins of the discipline.

"Self-discipline" would seem to refer to some internalising of the regulative structures and values of society. Foucault8 suggests a metaphor for the disciplinary society in Bentham's panopticon. The prisoner cannot see whether or not s/he is being watched from inside the dark central tower. S/he, therefore, takes on the responsibility for her/his own surveillance in what is precisely self-discipline. An apparently pessimistic image. Durkheim9 associates a strong common consciousness with mechanical solidarity. This strong disciplining of consciousness is substantially replaced by interdependence within the complex division of labour of organic solidarity. This release from the tight bonds of the collective consciousness allows and is the result of an increase in the scope for individual expression.

It is, perhaps, the integration of the individual into a system of interdependence that the ANC document is getting at by the use of the term "self-discipline". Certainly the document opposes self-discipline to the imposed, authoritarian discipline of the apartheid state and of an examination-driven curriculum. The core curriculum will "[b]e learner-centred and non-authoritarian and encourage the active participation of students in the learning process" and "[p]roblematise knowledge as provisional and contested"10. Yet there appears to be a certain tension between this emphasis on learner autonomy and the document's apparent reluctance to problematise its own knowledge and authority. The policy framework proposes

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central control over certification and over the evaluation of textbooks. Its only mention of educational research is presented in an authoritarian way:
The gradual introduction of the language of wider communication as a language of learning is based on the research evidence which strongly suggests that the conceptual development of children is facilitated by initial learning in their home language.11

"The research" is not identified for our scrutiny, but is presented in a way that subordinates the reader to the greater authority of the text.

In a similar vein, the document repeatedly announces a commitment to the integration of education and training, of the academic and the vocational. The existing curricular provision is claimed to lack relevance12 and science and mathematics education is too abstract and theoretical:
... science and mathematics education and training, both school-based and work-based, must be transformed from a focus on abstract theories and principles to a focus on the concrete application of theory to practice. It must ensure that students and workers engage with technology through linking the teaching of science and mathematics to the life experiences of the individual and the community.13

It would be disingenuous to claim that the ANC document is deliberately dogmatic and authoritarian. This is very far from being the case. Nevertheless, and its status as a discussion document notwithstanding, there are strong suggestions of the non-negotiable. Amongst those with which I wish to take issue are the apparent devaluing of the academic as appropriate school knowledge and the emphasis on student-centred-which I take to imply constructivist-pedagogy. John Volmink has described a number of features of the latter, of which one is an extension of the mathematics curriculum to incorporate "situated problems":
the bulk of the curriculum should be based on authentic problems which will force students to engage with mathematical ideas and construct their own modes of expressing these explanations. The role of the teacher would be to negotiate with students as to how these forms of representations can be refined and expressed more accurately, succinctly and elegantly.14

Clearly, a shift is proposed in the balance of authority in relation to a more traditional curriculum. This is the case both in terms of the teacher's authority-who must now 'negotiate' with students-and the authority of mathematics as a discipline. Authenticity is to be related to relevance, foregrounding, perhaps, engagement with what Volmink has described as "situated problems":
Situated problems arise outside formal mathematics, but give rise to mathematical actions, structures on [sic] insights. Their solution directly affect decisions in our own lives or about something we really care about. In [one example], Ian made a mathematical construct (histogram) to help him to negotiate with his parents. It led to a decision that affected his own life as well as that of his sister and brother.15

A curricular programme which places subjectivity within a Rousseauian expressive and constructive student and which weaves knowledge as a seamless robe

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romanticises both the subject and the everyday. I do not want to deprive Ian of his histogram, nor to challenge its value as a possible resource in respect of the everyday. I do, however, intend to argue that the mathematics classroom is not the place to get him to solve his problems with his parents. I shall initiate the argument with a discussion of an aspect of Muller and Taylor's paper.

From boundaries to rearticulations

In their paper, Muller and Taylor introduce the following polarising of the current debate on curriculum:
Insulation stresses the interdictory and impenetrable quality of cultural boundaries, of textual classification, of disciplinary autonomy. It highlights the integral difference between systems of knowledge, and the difference between the forms and standards of judgement proper to them. It stresses the virtues of purity and the dangers of transgression. Hybridity, by contrast, stresses the essential identity and continuity of forms and kinds of knowledge, the permeability of classificatory boundaries, and the promiscuity of cultural meanings and domains.16

In Bernstein's terms17 we might label "insulation" as an epistemology of strong classification (C+) and "hybridity" as one of weak classification (C-). Muller and Taylor are particularly interested in the strength of classification between formal, or academic knowledge and everyday knowledge. Their position is that, whilst curriculum theory should be concerned with how this boundary is traversed, this may not be best achieved by wishing it away:
The strategy of the constructivists is to reason that exclusion occurs because of an unwarranted disparity between curricular content and the sensuous content of the everyday lives of the children who must learn this foreign and hostile knowledge. This disparity constitutes an unnecessary barrier to the learners: it is a barrier arbitrarily constructed by agents of the status quo, and it must therefore be removed in the interests of empowerment and emancipation. This strategy is, we argue, extrapolated from a strong social constructivist epistemology which can be read to say that, since barriers of this sort are socially or discursively constructed, they can just as easily be dismantled by the same means.18

Without necessarily adopting the contrary position themselves, Muller and Taylor wish to explore the possibilities of considering commonsense and esoteric knowledge as distinct forms. Following Shotter, for example:
First, the subject matter of common sense is determined by sensuous events and is thus wholly contingent on circumstance, whereas that of academic discourse is predetermined by the arbitrary systematics of the canon. Secondly, because of its systematic objectivisation, the subject matter of academic discourse can, as Wittgenstein has said, be 'surveyed' in rational contemplation, that of common sense not.19

From this perspective, academic discourses constitute their own objects. They are, Muller and Taylor argue, arbitrary. Commonsense, on the other hand, is not. The

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two forms of knowledge are, in this respect, incommensurable and ignorance of this situation entails danger:
Common sense cannot accede to arbitrariness; esoteric discourse cannot proceed without such accedence. To insist on the radical equality of worlds of discourse is, paradoxically, not to threaten the "standards" of academia (pace the conservatives), but [...] to threaten common sense, by the wholly inadvertent imposition on common sense of an arbitrary academicised way of constructing objects.20

"Hybridising", then, involves a moralising of the everyday by arbitrary discourses.
My own position coincides with that described (if not necessarily adopted) by Muller and Taylor in respect of the incommensurability of these two forms of knowledge. However, I disagree with the reason that they put forward. As a sociologist, I must focus my attention on the distinctive nature of the social relations which are constitutive of the two "worlds of discourse". Before doing so, however, I shall discuss two key differences that I have with the position represented by Muller and Taylor.

Firstly, the description of commonsense as non-arbitrary appears to be predicated on an assumption that the everyday is driven by something which is pre-social. This might, for example, refer to the "natural" world. However, this would appear to underplay the human facility to constitute material practices in terms of the modality of social relations. This is a facility that is clearly understood by Marx. Human societies constitute their everyday material practices in diverse ways. These practices are, in other words, just as arbitrary as are academic practices and to deny this is to deny the everyday its creativity.

In my own work, I have made a distinction in terms of the degree of "discursive saturation" (DS) which characterises practices. An academic discipline entails a highly complex and articulated regulation within language: a "discursive formation"21. This regulation will comprise a combination of explicit principles and canonical texts. The utterances of such a discipline are highly generalised and relatively independent of context. Such activities are described as exhibiting high discursive saturation (DS+). Everyday activities, on the other hand, do not, generally, exhibit regulation within language in the same way or to the same degree. The principles of such activities are, in general, tacit rather than explicit and there are no canonical texts. The utterances of such practices are highly local and context-dependent. These practices exhibit low discursive saturation (DS-). There is, however, no distinction between DS+ and DS- activities in terms of arbitrariness: they are all arbitrary. The social, in other words, is characterised by heteropraxis as well as heteroglossia22.

My second point of difference with Muller & Taylor's representation concerns the notion of "boundary". Essentially, the bounding of knowledge suggests a notion of knowledge as contents. Indeed, this is the way in which Bernstein originally referred to classification:

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Classification, here, does not refer to what is classified, but to the relationships between contents. Classification refers to the nature of the differentiation between contents. Where classification is strong, contents are well insulated from each other by strong boundaries. Where classification is weak, there is reduced insulation between contents, for the boundaries between contents are weak or blurred. Classification thus refers to the degree of boundary maintenance between contents.23

Whilst the metaphor is entirely appropriate for Bernstein's ongoing project (and may, indeed, be appropriate for that of Muller & Taylor), I cannot conceptualise 'boundary' within my general methodology, which is, essentially (post)structuralist24. I start with a heuristic notion of a Global Semantic Universe25 of signs. In my head, I have an image of one of those molecular models one finds in (well-equipped) chemistry laboratories. Any specific practice/knowledge-I use the term "activity"-will constitute and will be constituted by a particular state or articulation of that Universe. The value of signs within the system will be given by the specific form of relations with other signs established by the articulation as a specific activity26. The possibility of a boundary between activities thus evaporates. Crudely, a sign may quite easily be carried between activities, but its signification will, of necessity, be transformed, because it will participate, relationally, in distinctive systems.

The original state, the Global Semantic Universe, is also erased, as all heuristic devices-like construction lines in a scale drawing-must be. There is, in my conception, no positivist or realist universe which lies beneath all of its representations. We move, not between truth and representation, but from representation to representation. Uncertainty does not arise out of the gap between a yet-to-be-grasped truth and the misrepresentation of that truth within a given activity. Rather, the unthinkable may be construed as the inevitable incompleteness of articulation and the ultimate ineffableness of the non-discursive with respect to the discursive.

Boundary metaphors make it all too easy to invent militaristic strategies involving the breaching of barriers and the invasion of territories. Alternatively, perhaps, they invite the softer, hermeneutic strategy of Gadamer27 in fusing horizons. The possibility of unifying the cultural domain, whether by the sword or the pen, becomes possible, even if only as a vision. The sheer enormity of the task of rearticulating the semantic universe into a single, substantive system is clearly beyond curriculum developers28. We may, however, realistically postulate the existence of such a unique articulation: prior to the advent of human life on this planet-and after its demise.

Then what, precisely is the nature of the relationship between activities? In order to address this question sociologically, I shall have to move beyond the cultural level to which the discussion has thus far been confined. Before doing so, however, I shall introduce some examples which may provide further heuristic orientation.

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Mythologising the everyday

The first example is from one of the curriculum developers described by Muller and Taylor as "hybridisers", Paulus Gerdes. In considering the non-industrialised production practices of Mozambicans Gerdes asserts that:
There exists "hidden" or "frozen" mathematics. The artisan who imitates a known production technique is-generally-not doing mathematics. But the artisan(s) who discovered the techniques, did mathematics, developed mathematics, was (were) thinking mathematically..29

Elsewhere, Gerdes illustrates the intended political impact of his "recognition" techniques in the classroom:
"Had Pythagoras not ... we would have discovered it". The debate starts. "Could our ancestors have discovered the 'Theorem of Pythagoras'?" "Did they?" ... "Why don't we know it?" ... "Slavery, colonialism ...". By "defrosting frozen mathematical thinking" one stimulates a reflection on the impact of colonialism, on the historical and political dimensions of mathematics (education).30

Colonial power has artificially created boundaries where, in fact, there are none. The mathematics entailed in building huts and weaving buttons, in Mozambique, and in making sand drawings in Angola31 is essentially the same knowledge as that which is celebrated as the product of European civilisation. Luckily, another European is at hand to reveal the real value of their culture to those whom his ancestors have oppressed.

Of course, the oppression is real. But Gerdes' intervention is an extension of it and not its overthrow. Gerdes' valuing of Mozambican and Angolan culture is being made in precisely European-mathematical-terms. And he goes further, as I have argued elsewhere32. "The artisan who imitates a known production technique is-generally-not doing mathematics": Gerdes is denying creativity to the Mozambican practitioner by projecting onto the practices of the latter the structure of the "European" production line for which creativity is equated with resistance. "But the artisan(s) who discovered the techniques, did mathematics, developed mathematics, was (were) thinking mathematically": and this is precisely the projection of a European "great man" approach to history that sees each technological development as a reason to celebrate in the individual the intellectual power of its culture. The hut builders and sand drawers may or may not be ignorant of European mathematics. Gerdes is apparently ignorant of his own Eurocentrism.

I do not want to challenge Gerdes" political intentions. However, his writing is exhibiting the mythologising power of the European academic gaze when cast onto non-European practices. The gaze can recognize only exotic forms of itself. The European constructs the other as the public domain of its own expression. This public domain is constructed as a mythical plane on which African homunculi

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participate in their everyday practices according to principles which European culture can divine, even though they may go unrecognised by the participants themselves. Gerdes' mathematical anthropology is a less sophisticated version of the objectivising structuralist gaze criticised by Pierre Bourdieu33.

This mythologising gaze is a more general property of academic discourses. The following extract, from the UK secondary school mathematics scheme SMP 11-1634, mythologises shopping.
Soap powder is sold in "Euro-sizes".
These are Euro-sizes E5, E10 and E20.
In the McGee family there are 2 adults and 3 children.
The McGees use six E10 packets of soap powder each year.
C5 What weight of soap powder do the McGees use in a year?
C6 The E10 packet costs £2.59.
How much does soap powder cost the McGees each year?
C7 The E20 packet contains twice as much powder as an E10 packet. It costs £3.89.
(a) How many E20's [sic] would the McGees use in a year?
(b) How much would powder cost them if they used E20's?
(a) How many E5 packets would the McGees
use in 1 year.
(b) The E5 packets are on special offer. At the moment, an E5 packet costs 95p. How much would the McGees soap powder cost in a year if the used E5 packets?35
Anthropological work by Jean Lave raises serious questions about the validity of constructing continuities across different activity/settings such as supermarket shopping and school mathematics36. However, recourse to such research is hardly necessary to reveal some of the violence that has been done to the setting in the above extract. None of the questions really make much sense within the context of domestic practice: whoever would want to know what weight of soap powder they would use in a year? Furthermore, the long timescale of the narrative ought, perhaps, to make allowances for inflation37, and the answer to C8 (b) given in the Teacher's Guide (£11.40) suggests that either the "special offer" lasted all year (rendering it rather "ordinary") or the McGees bought a year's supply all at once. This, despite the family's apparent storage or money problems indicated by their customary use of the smaller and cheaper middle sized packet rather than the "better value" E20 size. The conditions have to be fixed in order to allow the exercising of the mathematical intention of the chapter, which is "to provide pupils with

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experience of ratio problems"38. This recontextualising is, however, apparently belied by the realism of the task formulation. The reader is provided with some "factual" information about shopping (Euro-sizes) and is introduced to a family. The task, itself, apparently concerns the optimising of the McGees' domestic routine. The setting, in other words, is highly localised in terms of both task detail and the community space of the family. The narrative is comparatively closed in terms of the level of detail given: this might be the reader's family, either now or in the future; or it might connote a "soap" family with which the reader may identify. Despite the distorting effects of the gaze, the task is being presented as if it is really about shopping. School mathematics has constructed a public domain which is a mythical plane of domestic activity.

The task is mathematically structured, but the text is silent on the matter of how it is to be read and on how the task is to be carried out. There is, deliberately, no explicit pedagogising of method in this chapter, as is common in the G series:
There are many methods of solving problems where ratio is involved. We have deliberately not set out a "standard method". After pupils have done a few questions, we hope that discussion with them will bring out these various methods, and thus help them tackle the next batch of problems.39
There is an assumed transparency in the task text with regard to its reading and a repertoire of "strategies" within the classroom. The repertoire is activated by the "realistic" task and made explicit in "discussion". Thus, pedagogic practice, here, is a celebration of essential competences through official recognition and sharing in discussion. At the end of the chapter there is a bordered set of "discussion points", the first two of which are, "Why are things cheaper when you buy them 'in bulk'?" and "Sometimes it is not sensible to buy in bulk. When is it silly to buy huge packets?"; the Teacher's Guide gives the following answers to these discussion points:

Cheaper in bulk: easier to handle, less packaging
costs, less transportation costs, take up less
space etc.
Large packs not best when you have little
money, you can't store large packs, food might
go off before you can eat it, prices might go
down etc.40
However, the inclusion of such discussion only serves to accentuate the domestic setting which has apparently been incorporated by school mathematics. In other words, we might ask why shopping should appear in a mathematics textbook. One reason might be to initiate a route into what I refer to as the 'esoteric domain' of mathematics-the domain of specialised, unambiguously mathematical knowledge. Alternatively, the performance of the shopping tasks might be constructed as mathematically therapeutic. However, the text does not leave the public domain and, furthermore, there is very little (if any) explicit articulation of these shopping tasks with other tasks and exposition in the textbook series. Thus the tasks seem to be constituted for the benefit of optimising shopping practices themselves.

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Mathematics, even if this is signified only by the subject or book title, is being constructed as a prior condition for optimum participation in the mythical plane of the public domain. This is a "myth of participation".
Two additional points may be made. Firstly, as has been indicated, there is no pedagogising of method. Rather, methods are constructed as residing within the reader or within the setting: the reader's competence in participation is constructed as the prior condition for mathematics, whilst the latter is constructed as the prior condition for the reader's competent participation in the setting. The myth thereby deconstructs itself41. Secondly, the setting, itself, is a recontextualising of shopping. The mathematical principles regulating the mythical plane of domestic activity are, in fact, not substitutable for the predominantly tacit regulation of domesticity. The criteria for successful completion of the tasks within the mathematics context are, therefore, different from and so not available within shopping practices. There is, in other words, no basis for the reader's competence. The myth of participation is precisely a shibboleth.

The shopping task is taken from a book in a series-the "G" series-intended for "lower ability" students in year ten of compulsory schooling. The SMP 11-16 scheme includes other series of textbooks aimed at different levels within the "ability" spectrum. The "Y" series, for example, is intended for students in the "top twenty percent" of the "ability range". This series incorporates a different mode of mythologising, as is illustrated by the following examples from Book Y1 (year 9 students):

Shopkeeper A sells dates for 85p per kilogram. B sells them
at 1.2 kg for £1.
(a) Which shop is cheaper?
(b) What is the difference between the prices charged by the two
shopkeepers for 15 kg of dates?42

Britannia best British flour cost £0.71 for 12.5 kg.
Uncle Sam's best American flour cost $1.30 for 3.5 lb.
If 1kg = 2.2lb and £1 = $1.85, which brand of flour
was cheaper, and by how much per kilogram?43
Here, there is a clear semantic distance between the reader and the shopping setting and the narratives have become far more open. The first example uses a recontextualised algebraic resource, generalising through the use of letters which depersonalise the shopkeepers. This example also refers to a slightly exotic commodity and to a highly unrealistic quantity. The second case presents a transatlantic comparison which clearly cannot be relevant to most shoppers who would not be expected to board Concorde in order to get a cheaper pack of flour. In other words, the tasks include an exaggerated element of strangeness which ensures their dislocation from the domestic setting to which they now only indirectly relate. The reader is being invited to participate, not in domestic activity, but in the

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mythologising itself. This is descriptive mythologising. Whilst the "G" reader is projected into the mythical plane of domesticity, the "Y" reader is to be drawn up into academic mathematics.

The tendency of the two textbook series to specialise their modes of mythologising constitutes a distributing strategy. "Higher ability" students are apprenticed into descriptive mythologising; "lower ability" students are provided with participative mythologising. Elsewhere44 I have illustrated ways in which such distributing strategies are accompanied, in this textbook scheme, by positioning strategies which associate low and high "ability" with the working and middle classes, respectively. As far as textual strategies are concerned, then, "relevance", in terms of optimising participation in the everyday, is a fundamental criterion only to the "lower ability"/working class students. However, the relevance is, as I have illustrated, only apparent, because of the mythologising work of the gaze. Furthermore, participative mythologising tends to render invisible the principles of this mythologising. "Lower ability"/working class students are, thus, provided with "relevance" at the expense of either mathematical or everyday use-value.

Descriptive mythologising, on the other hand, tends to reveal these principles, because it always constitutes a move away from the public domain towards the esoteric. The "higher ability"/middle class texts also contain far more esoteric domain text45. Furthermore, the esoteric domain text in these texts tends also to realise the systematic complexity of DS+ practices, whereas such esoteric domain text as there is in the "lower ability"/working class series is procedural and highly local in nature, corresponding more to the DS- practices of the everyday. "High ability"/middle class students are thus to be apprenticed into academic mathematics and into the principles of the descriptive gaze.

Thus far, I have offered a description of the current situation as represented by the writings of one of the hybridisers and by the SMP textbooks. The next and fundamental problem is to consider the extent to which things might be otherwise. Is it possible, in other words, to constitute a participative and relevant curriculum which concerns the everyday, but which is not mythologising?
Horizontal and vertical practices: the everyday and the academic46

Shopping is an example of what I have referred to as DS- practices. This concept has been introduced as referring to practices which do not entail a highly complex and articulated regulation within language. Utterances within the context of such practices are always highly local and context-dependent. I want to suggest that the social relations which characterise this kind of practice can appropriately be described as "horizontal". This is a concept which has been introduced by Basil Bernstein47 to describe practices which are constituted as a reservoir of strategies.

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The individual will have immediate access to a repertoire, which is a selection from the reservoir. The acquisition and extension of an individual repertoire is achieved via the sharing of strategies within the context of weak framing. That is, there is no institutionalising of sequencing and pacing in pedagogic action. Furthermore, there is no institutionalising of pedagogic action in the form of an explicit grammar, or esoteric domain. Nor is an esoteric domain established in terms of canonical texts. On the contrary, texts and their grammars are always dependent upon the immediate material context. Under these circumstances, there can be no general evaluative principles. The evaluation of strategies can only be achieved in local and pragmatic terms, that is, "is it working?"48. The division of labour regulating such practices is simple and closer to mechanical than organic solidarity.

Schooling is constituted within a complex articulation of the state, the family and the economic. The state regulates schooling in the normalising of a period of childhood and youth in terms of a hierarchical educational and training structure. Boards of governors and school inspectors are appointed, control is maintained over teacher supply and training, and the state frequently reserves the right to validate textbooks and certification agencies. The ANC document, discussed earlier, represents a strong programme of state regulation of schooling, as does the Education Reform Act in the UK49. The use of textbooks and other commercially produced materials and the role of certification for selection purposes establishes a direct link between schooling and the economic sphere. The achievement of this complex is the institutionalising of pedagogic relations and of their cultural realisations in academic knowledge. Schooling is thus characterised by explicit evaluative principles (which may or may not be made visible to the student). These evaluative principles are predicated upon institutionalised esoteric domains of knowledge which comprise a combination of more or less explicit and systematic grammars and canonical texts. The esoteric domains are organised in more or less strong framing in terms of the selection, sequencing and pacing of their transmission. The organisation of the academic, in these terms, exemplifies what Bernstein describes as "vertical" practices. The division of labour in respect of schooling is comparatively complex, in terms of disciplinary specialisms, and hierarchical, in terms of teacher-student and student-student relationships. This division of labour is closer to organic than mechanical solidarity50.

The implications of this analysis for the pedagogising of the everyday are clear and are illustrated in the earlier discussion of the mythologising gaze of school mathematics. The institutionalising of an esoteric domain, of strong framing and of explicit evaluation principles constitutes school knowledges as DS+ activities. The incorporation of DS- practices by and within school knowledges must, therefore, constitute a recontextualising, a mythologising. The pedagogising of shopping in the "McGees" example, discussed earlier, exhibited certain similarities with the informal pedagogic action of horizontal practices. That is, strategies were not imposed, but "brought out" in discussion. The students are to be encouraged to

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share their repertoires of strategies. However, rather than grasping the lived reality of the horizontal domestic practices, the mathematising of domesticity constitutes a mythical plane which occupies a space outside of both mathematics and the quotidian. The students are objectified by the mathematical gaze and recontextualised as homunculi which inhabit not the everyday world, but the mythical plane. Mythologised shopping is now subject to the generalisable evaluative principles of the vertical practice of school mathematics, but, at least in the "G" books, these principles are not made available to the students. This submission of horizontal practices to the mythologising gaze of vertical discourses is precisely the moralising of the everyday that Muller and Taylor signal.

There is a further difficulty arising out of the question of selection and sequencing. The gaze effects a selection of metonyms on the basis of esoteric domain elements. In other words, the everyday is reduced to sets of its features which are to be mapped onto specific mathematical structures. This entails that the sequencing of the public domain content is determined by the sequencing of the esoteric domain. This week we are doing "trigonometry", so we can project triangles onto the world and incorporate the result as mythologised "reality". Next week we will be doing "percentages", so we can search around for instances of this particular signifier and constitute a unity of what we find as a public domain region. There is, of course, no "everyday" motivation for putting together discounts in sales, mortgages, and the steepness of hills (measured, on continental European roadsigns, as percentages). The mythologising power of the discipline can be displayed by the extensiveness of the range of the public domain. This is further facilitated by the possibility of open, or highly partially described settings and extremely brief narratives: the everyday setting is, after all, no more than a token. The esoteric domain can always be prioritised. This is entirely consistent with the pedagogy of the descriptive myth, which invites the student to participate in the vertical discourse itself.

The pedagogy of the participative myth, however, is more problematic. Clearly, there is again a selection of metonyms which is achieved on the basis of esoteric domain elements. However, the exigencies of relevance dictate that the selection must be presented in terms of the importance of the everyday. This criterion must also inform the sequencing, which may, for example, be related to age-related interests rather than any pedagogic sequencing of the esoteric domain. The public domain sequencing of the "G" books, for example, constructs a trajectory out of the childish settings of the classroom and play to the adult world of the supermarket. A student-led curriculum in which students bring their own problems to the classroom opens up a space for an even more intrusive moralising and does nothing to render more visible the principles of the mythologising gaze.

The notion that schooling can have direct relevance to the everyday, which is to say, non-academic world is thus revealed as itself mythical. What, then, are the

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possibilities for a more productive relationship between the horizontal and the vertical?
Conclusion: towards a productive dialogue
I have attempted, in this paper, to problematise the emphasis on non-academic relevance which characterises the ANC discussion document and much other current writing on curriculum development. My argument has culminated in an introduction to a sociology of the academic and of the everyday which establishes these as incommensurable practices. This does not, however, lead inevitably to the reactionary position that appears to be exhibited in the SMP texts which I have described and which has, as I understand the situation, characterised much South African schooling. That is, the recognition of a fundamental sociological distinction between the academic and the everyday does not entail a necessary division of labour built on racial or on class lines.

Furthermore, it is certainly the case, as has been pointed out by Johan Muller51, that the kind of distinction that I have made draws on a Durkheimian interpretation of the division of labour and that Durkheim's model was evolutionary. However, there is no essential evolutionism in my analysis. My argument is that everyday, or horizontal, practices constitute us all and that this is unavoidable and inevitable. Academic, or vertical, practices have been systematically distributed on class and racial lines, however. This has entailed the effective exclusion of the majority of the populations of both South Africa and Europe from the academic. This is variously achieved via the non-existence or inadequacy of schooling provision or, more subtly, by the insistence of the inclusion of the everyday and the relevant in terms of participative mythologising.

It might be noted that, in the ANC document, the only curriculum to remain untouched is that of the university in which academic freedom is to remain protected52. The concern, then, is that those who ultimately produce the curriculum will have access to the principles of the esoteric domains which will dictate its form, but that these principles will remain inaccessible to all those who do not aspire to or are materially prevented from university education. How often do the vanguard prescribe something other than their own culture for those whom they would lead to freedom!

Vertical, DS+ practices, in the form of academic disciplines, are to be valued precisely because they facilitate the scrutinising and interrogation of the everyday. The horizontal context-dependency of the everyday ensures that utterances are always localised. Whilst the recognition of oppression and resistance against it is always possible within everyday practices, the scope for self-interrogation is limited by this localising. The principles of democracy inspiringly celebrated in the ANC document dictate, however, that those who do the moralising are those who are to be moralised: the emphasis must be on self-discipline and not oppressive regulation

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by a state which guards the secrecy of the principles upon which the rest of us are to be judged. The democratic principles dictate, then, that the schools open up the availability of academic discourses to all53. The acquisition of such vertical practices necessarily involves subjugation to the evaluative principles of these discourses. Subjectivity of necessity entails subjugation. Academic subjectivity entails subjugation to a discipline. Only in this way can the DS+ practices be made available as structured resources for the interrogation of everyday practices by the practitioners themselves: discipline and then and only then mathematise.

Academic discourses, of course, cannot be taken to be timeless. This knowledge, too, must be problematised as provisional and contested as advocated by the ANC document. Firstly, their introspection is potentially interrupted by the intrusion of the empirical which is grounded in the everyday. Secondly, a dialogue between academic discourses-between sociology and mathematics, for example-may enable the focusing of attention on their own conditions of existence and the structures of power that are relayed through them. Both forms of dialogue, however, are predicated upon prior apprenticeship into the disciplines of the academic. Dialogue occurse between subjects who at once inhabit a position and recognise the possibility of the unthinkable within it.

The opening up of the academic, in these ways, can only be productive in terms of its own development. This is, of course, particularly important within the context of a society which is beginning to loose the shackles of European oppression. Both the academic and the everyday can benefit from the extension of "dialogue". If dialogue is to have social implications, however, then enfranchisement must proceed on the basis of genuine access to sociologically distinctive forms of knowledge. Self-discipline is nothing without the provision of both the material and discursive resources which enable it to structure itself. Oppression is precisely a state of power whereby technologies of the distribution of vertical and horizontal practices and discourses operate selectively on categories of individuals and groups which they thereby define and delimit. Self-discipline is predicated upon the free acquisition of both horizontal and vertical practices. This is the condition for democratic dialogue within a sociocultural domain which is characterised by both heteroglossia and heteropraxis.
I am grateful for productive dialogue with Parin Bahl, Basil Bernstein, Andrew Brown, Paula Ensor and Johan Muller in the generation of this paper. I am also grateful for the critical comments made by the referees.
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Notes and References

1 An earlier version of this paper was presented as a public lecture at the University of Cape Town School of Education in April 1994.
2 J. Muller and N. Taylor, "Schooling and Everyday Life: knowledges sacred and profane", forthcoming.
3 P.C. Dowling, "Textual Production and Social Activity: a language of description", Collected Original Resources in Education, 16, 1 (1992); "A Language for the Sociological Description of Pedagogic Texts with Particular Reference to the Secondary School Mathematics Scheme SMP 11-16" (PhD thesis, Institute of Education, University of London, 1993); "Theoretical 'Totems': a sociological language for educational practice", in C. Julie et al (eds), Political Dimensions in Mathematics Education: curriculum reconstruction for society in transition (Cape Town: Maskew Miller Longman, 1993); "Discursive Saturation and School Mathematics Texts: a strand from a language of description", in P. Ernest (ed.), Mathematics, Education and Philosophy: an international perspective (London: Falmer, 1994); A.J. Brown & P.C. Dowling, "The Bearing of School Mathematics on Domestic Space", in R. Merttens et al (eds), Ruling the Margins: problematising parental involvement (London: IMPACT, University of North London, 1993).
4 B.Bernstein, Class, Codes and Control volume 5, (London: Falmer, in press).
5 É. Durkheim, The Division of Labour in Society, (Basingstoke: MacMillan, 1984).
6 African National Congress, A Policy for Education and Training, (Johannesburg: ANC, 1994).
7 Ibid., 69.
8 M. Foucault, Discipline and Punish: the birth of the prison, (London: Penguin, 1977).
9. Op cit.
10 ANC: 69.
11 Ibid., 64.
12 Ibid., 67.
13 Ibid., 84.
14 J. Volmink, "When We Say Curriculum Change, How Far are We Prepared to Go as a Mathematics Community", in C. Julie et al (eds), Political Dimensions in Mathematics Education: curriculum reconstruction for society in transition, (Cape Town: Maskew Miller Longman, 1993): 127.
15 Ibid., 126.
16 Muller and Taylor: ms. p. 1, my emphasis.
17 See B. Bernstein, Class, Codes and Control Volume 3: towards a theory of educational transmissions, second edition, (London: RKP, 1977); Class, Codes and Control, volume 4, (London: RKP, 1990).
18 Muller and Taylor, ms. p. 12.
19 Ibid., ms. p. 7.
20 Ibid., ms.p.8.
21 See M. Foucault, The Archaeology of Knowledge, (London: Tavistock, 1972).
22 Cf. M. M. Bakhtin, Speech Genres and Other Late Essays, ed. C. Emerson and M. Holquist, (Austin: University of Texas Press, 1986); V.N. Volosinov, Marxism and the Philosophy of Language, (Cambridge, Mass.: Harvard University Press, 1973). Bakhtin, however, does not need to make the distinction, since, for him, the sign is always grounded in concrete practice: heteroglossia entails heteropraxis.
23 Bernstein, 1977: 8.

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24 The parenthesising of "post" intends that (post)structuralism is not non-structuralism. Rather, it denies unity or fixity in langue (or its analogue).
25 The term is Umberto Eco's; I am using it for heuristic purposes. See U. Eco, A Theory of Semiotics, (Bloomington: Indiana University Press, 1976).
26 Cf. F. de Saussure, Course in General Linguistics, (London: Duckworth, 1983).
27 H-G. Gadamer, Philosophical Hermeneutics, (Berkeley: University of California Press, 1976).
28 This is not, of course, to deny that attempts are constantly made to establish such a stable hegemony, nor that such developments are transformative (see, for example, E. Laclau & C. Mouffe, Hegemony and Socialist Strategy: towards a a radical democratic politics, (London: Verso, 1985)). An analogy might productively be made, here, between the tensions and constant reorganisings of a sociocultural domain which is energised by hegemonic and counter-hegemonic strategies, and the dynamic of cognitive development entailed in Piaget's hierarchised concept "equilibration" (see J. Piaget, "Opening the Debate", in M. Piattelli-Palmarini (ed.), Language and Learning: the debate between Jean Piaget and Noam Chomsky, (London: RKP, 1980)). I would, however, want to assert a greater degree of stability in the sociocultural chronotope than in that of the cognitive. Even if it is only by virtue of the sheer number of human beings alive at any one time, the transformation of the social is not to be grasped within a lifetime.
29 P. Gerdes, "How to Recognize Hidden Geometrical Thinking: a contribution to the development of anthropological mathematics", For the Learning of Mathematics, 6, 2, pp. 10-12: 12.
30 P. Gerdes, "On Culture, Geometrical Thinking and Mathematics Education", Educational Studies in Mathematics, 19, 2, pp. 137-62:152.
31 P. Gerdes, "On Possible Uses of Traditional Angolan Sand Drawings in the Mathematics Classroom", Educational Studies in Mathematics, 19, 1, pp. 3-22.
32 P.C. Dowling, "Theoretical 'Totems'".
33 P. Bourdieu, Outline of a Theory of Practice , (Cambridge: CUP, 1977); The Logic of Practice, (Cambridge: Polity, 1990).
34 SMP 11-16 is by far the most popular secondary mathematics scheme currently in use in the UK, being used by between 40 and 50% of secondary schools in England and Wales. It is published by Cambridge University Press.
35 SMP 11-16 Book G4, p. 18, drawing omitted; the vulgar use of the apostrophe to indicate a plural is representative of a common inelegance in language.
36 See J. Lave, Cognition in Practice: mind, mathematics and culture in everyday life, (Cambridge: CUP, 1988); J. Lave, M. Murtaugh and O. de la Rocha, "The Dialectic of Arithmetic in Grocery Shopping" in B. Rogoff and J. Lave (eds), Everyday Cognition: its development in social context, (Cambridge, Mass.: Harvard University Press, 1984).
37 The possibility of prices going down is mentioned in the "discussion points" in the Teacher's Guide, as noted below.
38 Teacher's Guide to Book G4: 14.
39 Ibid.
40 Ibid., 16.
41 See J. Derrida, "Semiology and Grammatology", in Positions, (London: Athlone Press, 1981); J. Sturrock, Structuralism and Since: from Lévi-Strauss to Derrida, (Oxford: OUP, 1979).
42 SMP 11-16 Book Y1: 55.
43 Ibid., 56.
44 P.C. Dowling, "A Touch of Class: ability, social class and intertext in SMP 11-16", in D. Pimm and E. Love (eds), Teaching and Learning School Mathematics, (London: Hodder & Stoughton,

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1991); "Gender, Class and Subjectivity in Mathematics: a critique of Humpty Dumpty", For the Learning of Mathematics, 11, 1, pp. 2-8; "Textual Production and Social Activity"; "A Language for the Sociological Description of Pedagogic Texts".
45 In "A Language for the Sociological Description of Pedagogic Texts", I established an estimate of more than 40% of the "Y" series as comprising esoteric domain text. The corresponding figure for the "G" series was less than 10%.
46 The analysis which is introduced here is developed somewhat in P.C. Dowling, "Against Utility in Mathematics and Research: a voice from the Twilight Zone" (plenary address to the annual conference of the Southern African Association for Research in Mathematics and Science Education, University of Cape Town, January 1995) and in "Spectres of Schooling and Utopia", Arena, (in press, Jan/Feb 1995).
47 Class, Codes and Control, volume 5.
48 Such evaluation is often tacit. That is, a negative evaluation is raised in consciousness only at a point of breakdown in recognition of parapraxis (see Freud, Introductory Lectures on Psychoanalysis, (London: Penguin, 1973)). In making breakfast, one morning, I was suddenly thrust into the light of objectification when I became aware that I was pouring cornflakes into the teapot.
49 See P.C. Dowling and R. Noss (eds), Mathematics versus the National Curriculum, (London: Falmer, 1990); M. Flude and M. Hammer (eds), The Education Reform Act 1988: its origins and implications, (London: Falmer, 1990).
50 A Freudian interpretation of the resonance between horizontal/vertical, prostrate/erect, feminine/masculine is recognised, but not elaborated here. Walkerdine's description of mathematics as a masculine phantasy is, however, not to be ignored any more than the fact that her own theorisings are also rationalising myths, as are mine (V. Walkerdine, The Mastery of Reason, (London: Routledge, 1988). Philosophical deconstruction, however, does not relieve us of the need to act with as well as on what we have.
51 Private Communication.
52 ANC: 116.
53 That schools may be perceived as both having attempted and persistently failed to achieve this availability may have something to do with their misconstrual of the notion of discipline. Apprenticeship properly entails the construction of subjects, not the automation of objects (see Dowling, 1995, in press).