Presented at BERA 1999 as part of the symposium, What does it mean to understand maths as social?
In addressing the question, what does it mean to understand maths as social?, I want to focus on the relationships between school mathematics and everyday, domestic practices, particularly shopping. These relationships are certainly involved in the concept of numeracy which, like literacy is frequently presented as a necessary condition for competent participation in a range of familiar activities, from shopping to critical citizenship. This is the idea that there are generalisable skills that can be transmitted within a pedagogic setting and that, once acquired, will facilitate optimum participation in everyday life. In this paper I want to introduce some elements of a sociological analysis of pedagogic and other practices that point to the fundamental importance of context in constituting a challenge to any simplistic interpretation of numeracy. Context is here being interpreted as a sociological category. That is to say it concerns the relations between social positions and the distribution of practices between these positions. I shall introduce the analysis primarily via a discussion of some examples that are predominantly extracts from mathematics textbooks, but I shall begin with an anecdote.
I recently visited a DIY store at a shopping estate. I noticed that the supermarket on the estate was fairly quiet, so I popped in to pick up some tomato puree; I was going to make a lasagne the next day and wasnt sure that I had enough. The store sold two sizes of their own brand of puree: 100g at 31p and 200g at 37p. I had intended to buy two 200g tubes. Now this was shortly before this particular supermarket chain announced the withdrawal of its own brand of tomato puree because of the difficulty in ensuring that it was free of GM tomatoes; they had sold out of the larger size and had, presumably, not restocked.
I have given a substantial amount of detail here in order to be able to suggest the complexity of the situation. This was not a conventional best buy situation. Nevertheless, I had a range of options that were open to me. It occurred to me that, had the 200g tubes been available, then buying one of these would have been the equivalent of getting the second of two of the smaller tubes for 6p. Well, a considerable saving in percentage terms, but the absolute amount was small; 50p difference for 400g; half the price of a copy of The Big Issue. On the other hand, I could always check out my local supermarkets (which Id probably have to visit anyway) after Id checked whether I did, in fact, need the puree. Or, again, I could buy another brand (I didnt check the price).
I think, as a general rule, I tend to value my time above moneyat least, small amounts like thisso I suspect that my usual response would have been to pick up the four small tubes and forget about the 50p. However, on this occasion, I remember being particularly incensed by the magnitude of the price difference: two at 31p each as against one for 31p and a second for 6p. I went for what was almost certainly the most expensive option, in terms of time, and which was uncertain in terms of outcome. I decided to find a manager and suggest that he let me have four of the smaller tubes for the price of two of the larger ones. The strategy worked. I got my puree, saved my 50p, satisfied my moral outrage (or was it a bit of machismo in not wanting to be ripped-off?) but was indeed rather later getting home that I had wanted to be.
This anecdote is, of course, a post hoc objectification of an event. Following Bourdieus persuasive argument at the beginning of Outline of a Theory of Practice (1977), I want to propose that the objectification of practices of necessity delocates them from their local conditions of elaboration transforming them, as it were, into something else. In this particular case, I have provided a kind of analysis of options that that may or may not have been consciously available to me at the time in a more of less modified form. I have certainly deployed more mathematical knowledge in relaying the event than I would have needed or may have used in negotiating the supermarket situation. As Jean Lave (1988) has argued and illustrated, decisions of this sort are made in situe and in real time and on the basis of criteria and employing strategies that are often highly localised. This certainly describes my actions in this instance. The relationship of these criteria and strategies to those constructed anecdotally, post hoc, or to those constituted by the generalising strategies of school and other discourses is, to say the least, highly uncertain.
In order to define the limits of the relations between domestic economic practices such as supermarket shopping and school discourses, I need to describe them in compatible terms. To do this I shall draw on elements of my language of sociological description that was originally developed in the context of the analysis of school mathematics texts (see Dowling, 1998, in press).
Firstly, I shall describe the supermarket as a text. By this I mean that it is a bounded instance of a broader social activity. An activity is understood as the contextualising basis of practice. This is to say, in general terms, it defines who can say or do what. It comprises, then, an organisation of social positions (who) and a distribution of practices (what). Activities are produced and reproduced(re)producedin texts and in human subjects; I shall not be theorising human subjectivity here.(1) Now I propose that the supermarket activity is dominated by a mode that I refer to as exchange activity (Dowling, 1999, in press). This kind of activity consists of a transmitter (here, the supplier side), one or more acquirer positions (the client side) and a privileged content (here, the particular range of commodities and services supplied). The nature of the relationship between transmitter, acquirer and content is such that the principles of evaluation of content reside with the acquirer.(2)
This description of the activity is clearly consistent with my earlier description of the text, which is to say my anecdotal account of my supermarket experience. The extent to which the supermarket management can or attempts to delimit the criteria or strategies that I deploy in decision-making is clearly very limited. As I have illustrated, these criteria and strategies are highly localised and may indeed be quite perverse when viewed from other perspectives.
I now want to look at a rather different text that is also concerned with the relationship between quantity and cost.
When a motorist buys petrol, the cost of the petrol is directly proportional to the quantity.
Doubling the quantity doubles the cost. Trebling the quantity trebles the cost and so on.
The symbol for is proportional to is µ. So we can write
Cost of petrol µ quantity
The graph of (quantity,cost) is a straight line going through (0, 0).
(SMP11-16 Book Y4, p. 62 ) (3)
|An illustration of the graph is included in the text.
Now what is going on here is obviously quite different from the supermarket situation. Briefly, the authorial voice of this text is acting transformatively on the practices of an exchange activity, delocating them from any local context of their elaboration (a particular filling station at a particular time and a particular motorist in particular circumstances and so forth). The result is a rather specialised form of commentary upon the practice of buying petrol. The specific nature of the commentary is achieved in and by the privileging of a mathematical discourse. I want to refer to this mathematical discourse as the esoteric domain of school mathematics and the specialised commentary that it generates as contributing to the public domain of school mathematics.
Now in this example, the public domain is presented as a way of entering mathematical discourse rather than as a way of conducting the purchase of petrol. The trajectory of the text is from public domain to esoteric domain. Furthermore, the reader of the text is constituted as having a career within the esoteric domain. It is the esoteric domain that is developed systematically in the chapter within which this extract appears. Thus the final piece of text in the chapter is the following task.
|A sketch of the relevant apparatus is included alongside the verbal text.
The public domain settings for a number of exposition and tasks within this chapter is, as in this case, school physics.(4) However, as is the case here, the text is constituted throughout such that the precise nature of the setting is irrelevant to the completion of the task. In the above case, all that is necessary in order to complete the task is the table and the question that follows it.(5) Furthermore, there would seem to be no central organising principle that selects the settings in the chapter. Even the physics settings cover quite a range of physical topics.
Thus in this school text there is a degree of arbitrariness of the public domain setting in terms of general topic and detail. The esoteric domain principles relating to the graphical and algebraic expression of forms of proportionality are clearly privileged both by this arbitrariness of the public domain and by the textual trajectories as illustrated by both of these extracts.
That this text operates in this way marks its referent activityschool mathematicsas distinct from the supermarket activity. In the school text the relation between the authorial and reader voices is such that the principles of evaluation of performances resides with the author, which is to say with the transmitter side of the activity. This is a pedagogic text that (re)produces a pedagogic rather than an exchange activity. To define this second ideal type formally: a pedagogic activity involves a transmitter, one or more acquirer positions, and a privileged content under conditions whereby the principles of evaluation of content reside with the transmitter. This is the logical inverse of the exchange activity in respect of the location of principles of evaluation so that the ideal-typical space is, in respect of this dimension, exhausted.
Now, I want to make two claims about the school that entail a disparity between school activity/texts, on the one hand, and supermarket activity/texts, on the other. My first claim identifies a structural disparity by describing the school as predominantly a pedagogic institution in the terms in which I have defined it above. Insofar as the supermarket is dominated by exchange, school activity/texts are likely to be incompatible with texts relating to the supermarket.
The second claim refers to a strategic disparity. Essentially, the school privileges a form of practice that I describe as exhibiting high discursive saturation (DS+) (see Dowling, 1994, 1998, in press). Such practices are dominated by specialising strategies that render their principles available within discourse so that texts are relatively context independent. This mode is distinguished from practices exhibiting low discursive saturation (DS-). Here, the practice is dominated by localising strategies that render texts context dependent. Examples of these ideal types are mathematics (DS+) and Japanese mingei folk pottery (DS-) (see Singleton, 1989; Dowling, 1998). In these terms, supermarket shopping is, I suggest, an example of a DS- activity.
The structural and strategic differences between school and supermarket entails a necessary disparity between supermarket texts and school texts constituting the supermarket as a public domain setting. I shall illustrate this disparity by reference to some more texts.
|Here are two packets of washing powder. The small size contains 930g of powder. It costs 84p.
The large size contains 3.1 kg of powder. it costs £2.56.
(a) How many grams do you get for 1p in the small size?
(b) How many grams do you get for 1p in the large size? (Remember you must work in grams and pence.)
(c) Which size gives more for your money?
(SMP 11-16 Book G7; p. 2)
|This task is accompanied by a photograph showing a large and a small box of Persil Automatic washing powder.
The first point to be made here is that the approach to best buy decisions that is implied in this text is not one that is commonly used. A more general approach is illustrated in the following extract.
|A shopper considered two rolls of paper towels, one costing 82 cents, the other 79 cents. The shopper noted the number of sheets in each roll, 119 versus 104, and proceeded to reformulate the problem, saying, That would be three cents more and you get 11 more, 15 more sheets. She concluded that the larger roll was probably a better
buy. The shopper decision is, precisely, whether to spend an additional three cents for 15 sheets. that is, she must judge whether the marginal value of the additional quantity is worth the marginal cost, a different and more relevant question than whether the larger or smaller size has the lower unit price.
(Murtaugh, quoted in Lave, 1988)
|The formulation by Murtaughs shopper shares similarities with my formulation of the tomato puree situation, but is quite different from the Persil question.
A second point also emerges from this example. From the perspective of school mathematics, it is not possible to address the shoppers questionis it worth spending 3 cents for 15 extra sheetsin mathematical terms. The criterion that determines the answer is the shoppers perception of the relative worth of 3 cents and 15 paper towels. Quite clearly, this cannot be codified mathematically.
As a pedagogic institution, the school privileges a specific body of knowledge that constitutes principles against which students performances are to be assessed. Now this is a very different situation from the supermarket. The latter certainly privileges a particular body of goods and services. It does not, however, evaluate the performances of its clients against these goods and services. We could, of course, reformulate the Persil question as which would you buy? However, whereas the supermarket leaves this as an open question, the school must foreclose on a privileged set of answers that maps onto its curriculum. In general, a mathematics curriculum must entail the privileging of mathematical principles. This relates to the structural difference between pedagogic and exchange activities.
In fact, the textbook that contains the Persil question also includes tasks that do not have fixed solutions. Generally, these tasks are bounded from the main text by a border and the title discussion point. Here is an example.
Look at each of the tins and packets in question A5. Look at the packets that give you less for your money. Some people might prefer a packet that gives you less for your money. Who might prefer a packet like this?
(SMP 11-16 Book G7; p. 3)
|The Teachers guide to the first textbook book in this series includes a general introduction to the materials. In this introduction, discussion is marked out as being particularly important.|
|Discussion between pupils, and between pupil and teacher, is perhaps the most useful mathematical activity possible; talking through with the teacher may be the only way to make the work relevant. Discussion should always precede a teacher-led lesson, and discussion can often follow a class game or investigation. Pupils may be asked to explain how they solved a particular problem, and the different methods used by pupils can then be compared. Often for these pupils, there is no single correct way of doing things. Rather there is one method which suits a particular pupil best for a particular problem.
Discussion of how to solve problems will be almost as valuable as actually solving them. A discussion of how to avoid congestion in the school corridors, which would be the best local school to amalgamate with, where to go to buy a bike cheaplyall these represent the sort of problem whose solution is mathematically valuable. They are, of course, absolutely specific to the pupils own environs and interests. Problems may arise topically from a newspaper of TV of a local incident. Valuable discussion can come out of unpromising territory.
(SMP 11-16 Teachers guide to Book G1; pp. 8-9)
|Now one point to note, here, concerns a statement in the second of the above paragraphs,
all these represent the sort of problem whose solution is mathematically valuable. On the face of it, this seems to propose the same kind of trajectory that characterised the petrol example, that is, from public to esoteric domains. This interpretation, however, is denied by my analysis of the series as a whole. Less than 10% of the page space of the series is occupied by esoteric domain text (Dowling, 1998). Nevertheless, there is a clear privileging of mathematical principles, hereValuable discussion can come out of unpromising territoryand there is again evidence of a retention of the principles of evaluation of performances by the teacher who is to supervise the comparison of student methods.
In the discussion point example quoted above, however, there is no obvious privileging of mathematical principles. It seems to refer more closely to the generation of relevance and to the principle that the method should suit the situation. On the face of it, this might be interpreted as an attempt to get at precisely the kind of context dependency that is illustrated by the tomato puree example. However, such an interpretation misses two points. Firstly, it ignores the transformative work of objectification that is illustrated by Bourdieus (1977) commentary upon Lévi-Strausss objectification of gift exchange. The discussion of necessity delocates decision making from the immediate context of its elaboration. To stay with Bourdieu (1991), it represents a move from practical logic to theoretical logic. In my language, the introduction of discussion in this way stands as a rudimentary generalising strategy that shifts the practice in a direction away from DS- and towards DS+. As I have argued above, the school is a form of institution that is characterised by its strategic privileging of DS+ practices. It generates generalised forms of commentary even where it appears to be engaging in highly localised practices. (6)
In the context of a mathematics lesson or textbook, we would expect much of the generalising to give way to specialising in the form of mathematical discourse.(7) In my analysis of the SMP 11-16 textbook scheme, however, I have found a distribution of the extent to which mathematical discourse is made available to the student reader. Essentially, the availability of esoteric domain mathematics is a function of ability which, in turn, is recognised in terms of an interpretation of social class (Dowling, 1998). The petrol example is taken from a text targeted at high ability students, so that the public domain to esoteric domain trajectory is consistent with this distribution. The shopping examples, however, are from a text directed at low ability students. Here, the partial substitution of mathematical discourse by general discussion on shopping is again consistent with the distribution. Essentially, high ability students get mathematics whilst low ability students get ad hoc principles that lay claim to but which clearly do not constitute official school knowledge. This effect is (re)productive of the strategic disparity between school mathematics and the supermarket.
There is, however, a second point to be made concerning these shopping/discussion texts. Here I am referring to the structural distinction that I have made between the pedagogic and exchange activities. Insofar as the school is dominated by the pedagogic mode, then apparently open tasks are likely to be transformed into closed tasks by students if not by teachers. This is illustrated in, for example, fieldwork by Doyle & Carter (1986). In this work, attempts by an English teacher to introduce open tasks triggered student strategies that tended to minimise risk and ambiguity and so encourage the teacher to close down the task by offering explicit principles of evaluation. Where she failed to do this, the lesson generally disintegrated.
In the context of text analysis, we may be able to see the manner in which a text can tend to foreclose on an apparently open task by, for example, the way in which such tasks are cued. The following extract is the final text from the shopping chapter.
|B6 200 g of Heinz tomato ketchup costs 29p. 198 g of Rip-of [sic] red sauce costs 15p.
(a) Which make gives you more for your money?
(b) Which do you think would be better value for money?
Value for money means more than cheapest. Quality counts as well! Does quality sometimes count more than cost when you buy things? What other things count, as well as quality?
(SMP 11-16 Book G7; p. 6)
|Task B6 is accompanied by a photograph of two bottles of sauce. Rip-of sauce is the only example in the chapter of what appears to be a fictional brand name. Rip-of very obviously indexes rip-off and low quality as opposed to the higher quality of the recognisable brand name. The ordering of the two products in terms of value for money is thus made possible. This introduction of ordinalisation is carried over into the discussion point: quality counts
; does quality sometimes count more
; what other things count
To reprise, the claim that I am making here is that the school and the supermarket are institutions that can be understood as being dominated by logically opposite modes of social activity and that deploy strategies that privilege opposite modes of practice. The supermarket is an exchange activity that locates the principles of evaluation of practice with the acquirerthe customer. In general, the supermarket privileges localised and context dependent, which is to say DS- practices. The school, by contrast, is dominated by the pedagogic mode which locates the principles of evaluation of practices with the transmitterteacher/textbook. Official school discourse deploys strategies that privilege context independent, DS+ practices. These structural and strategic disparities between school and supermarket pose a challenge to the contention that the school is or can be a site for the transmission of skills that are generalisable in any simplistic way to sites such as the supermarket.
This is not to claim that school mathematics can be of no use to shoppers. The evidence of Lave and colleagues (Lave et al 1984) is that shoppers make use of resources that are available to them in the context of the supermarket. This may generalise to discursive resources which may include school mathematicsalthough this is, of course, an empirical question. The point is rather to challenge the claim that mathematics occupies a privileged position in respect of the optimising of DS- exchange activities. Furthermore, to the extent that mathematics is to become available as a resource, it would seem to be incumbent upon teachers and the authors of textbooks to concentrate upon providing a more rectangular distribution of access to the esoteric domain of mathematics rather than on the public domain as mythologised shopping.
I am now in a position to summarise my response to the question, what does it mean to understand maths as social? in more general terms.
|1. Institutions that may be considered to implicate mathematics differ in respect of the structural and strategic principles of the activities that dominate them.
2. Resourcesmathematical or otherwisewhich are deployed in actions and texts are always subordinated to the structural and strategic principles of the activity in which the actions or texts participate. Thus structure and strategy motivate resources rather than the other way around.
3. The school is structurally dominated by pedagogic activity and strategically dominated by DS+ practices; the supermarket is structurally dominated by exchange activity and by DS- practices.
4. Within school mathematics, the supermarket is recruited as a public domain setting, which is to say, as a resource; similarly, school mathematics may or may not be recruited in the supermarket, where it is, it is constituted as a resource.
5. Thus a mathematics pedagogy that is dominated by public domain settings is ineffective in respect of both the extramathematical activities that are recontextualised in these settings and mathematics per se.
6. An effective pedagogy must focus on the transmission of knowledge and facilities that carry their own esoteric domain integrity, whether this be mathematics or juggling. The principles of their recruitment in other institutional activities must be located with the acquirer.
7. In other words, it you want to teach someone mathematics, then teach them mathematics; if you want to teach them shopping, then take them to the supermarket.
1. I want to introduce two riders. Firstly, the relationship between activity and text is interpreted as dialectical and not determinant. Secondly, both activity and text are analytic categories that are constituted within my general methodological approach which I refer to as constructive description (see Dowling, 1998, 1999; Brown & Dowling, 1998).
2. This is clearly an ideal type. Any substantive instance is likely to be most appropriately described as a combination of more than one ideal type, which is why I refer to the supermarket as dominated by the exchange mode. This description is adequate to the present discussion. I shall introduce a second ideal type below.
3. The School Mathematics Project (SMP) 11-16 textbooks that I shall refer to in this paper are: SMP. (1987). SMP Book Y4. Cambridge: Cambridge University Press; SMP. (1987). SMP Book G7. Cambridge: Cambridge University Press; and SMP (1985). SMP 11-16 Teachers guide to Book G1. Cambridge: Cambridge University Press.
4. It is an empirical question as to whether these settings are appropriately incorporated within the esoteric domain of school mathematics or whether they constitute public domain settings on a basis that is equivalent to the petrol setting. Ultimately, the decision would rest on the extent to which the precise nature of the selected settings is rendered arbitrary by the texts in which they appear. On the basis of my analysis of secondary school textbooks, I would suggest that physics settings are appropriately described as public domain in the 11-16 age range, but that they are incorporated into the esoteric domain as what used to be referred to as applied mathematics at post 16 school mathematics.
5. It does not, of course, follow that an empirical reader of the text will necessarily come to the same conclusion. Furthermore, a very long tradition of research would seem to suggest that setting-independent and setting-dependent readings are likely to be distributed on a sociocultural basis (for example, see Luria, 1976; Bernstein, 1977; Cooper & Dunne, 1998).
6. It is important to emphasise that I am referring to that which is formally privileged in and by the school. Clearly, in the day-to-day practices of the school there will be many instances of context dependent decision making. Where such decisions enter into formal school practice, however, they are likely to be rationalised or at least rationalisable (DS+) to a greater or lesser extent.
7. The distinction between generalising and specialising is that the latter extends whilst the former delimits the range of the discourse (Dowling, in press).
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