anyone for tennis? social class differences in children’s responses to national curriculum...

Anyone for tennis? Social class differences in children’s responses tonational curriculum mathematicstesting1

Barry Cooper and Máiréad Dunne

Abstract

Mathematics is a central part of the school curriculum. Alongside studiesin the dominant language of a society, success and failure in the disci-pline play an important role in the distribution of opportunities to chil-dren and young people. Until fairly recently, in England and elsewhere,success in primary school mathematics was achieved by demonstrating acapacity to memorise, reproduce and use relatively simple algorithms.However, in recent years, there has been considerable change in primaryschool mathematics with an increasing stress being laid, at least rhetori-cally, on understanding, investigation and the application of mathematicsin ‘realistic’ settings. It seems likely that such changes, in so far as theyaffect the form and content of National Curriculum assessment, will pro-duce changes in who succeeds and who fails, ie in selective processeswithin schooling.

The paper draws on preliminary results from an ESRC project which isexamining National Curriculum assessment in mathematics for 10–11and 13–14 year-old children in relation to class, gender and ‘ability’. Thepaper examines the ways in which children from different socioculturalbackgrounds approach assessment items which embed mathematics insupposedly ‘realistic’ contexts. Early data from the Key Stage 2 sample of10–11 year olds will be presented which shows that there does seem be asocial class effect in the response of children to ‘realistic’ items – onewhich leads to some working class children failing to demonstrate com-petences they have. The paper uses quantitative and qualitative methods,relating its findings to Basil Bernstein’s account of sociocultural codes –in particular his theorising of the social distribution of recognition andrealisation rules for reading educational contexts – and to Bourdieu’s the-orising of habitus.

© The Editorial Board of The Sociological Review 1998. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

Introduction

This paper reports some aspects of work funded by the ESRC whichhas as its focus children’s interpretations of and performance on thenational tests of mathematics for 11 and 14 year-olds in Englishschools.2 The project has collected data via tests and interviewsfrom children in three primary and three secondary schools. Thispaper will draw on the data from primary schools to demonstratewhy certain ‘realistic’ mathematics test items might be associatedwith the under-estimation of children’s actually existing capacitiesin mathematics and, furthermore, that the degree of such under-estimation varies by social class. The work of Bernstein andBourdieu is drawn upon to offer an explanation for the tendency ofthe competence/performance gap to vary by class. Although thefocus of the paper is on class differences, some data will be providedon associated gender differences. For an account which focuses onboth class and gender, and presents statistical analyses drawing onmore than 100 test items, see Cooper, Dunne and Rodgers (1997).

The policy background

There is no space here to provide more than a sketch of the policybackground within which this work has proceeded (for fulleraccounts, see Ball, 1990 and 1994; Brown, 1992 and 1993; Cooper,1994a and 1994b). However, there are several key points to note. The1988 Education Act introduced both a national curriculum andassociated national assessment of children at the end of Key Stages,including Key Stage 2 at age 11. There was considerable debate andconflict around this time and subsequently about the nature of boththe curriculum and the form of its assessment. This debate resultedin the initial proposals for assessment mainly by teachers in theirclassrooms being replaced by a stress on testing via group paper andpencil tests (DES, 1988; Brown, 1992). While there has been arestatement more recently on the importance of teacher assessment(Dearing, 1993), England now has an institutionalised pattern ofannual national testing of children at ages 7, 11, 14, and 16. The firstnational league tables for 11 year olds were published in early 1997.

There is one other key point which must be made. Mathematics –though it clearly has a central core concerning number, space, mea-sure, etc.3 – is, like other school subjects, not fixed and unchanging.As a result of periodic renegotiations of what counts as school


116 © The Editorial Board of The Sociological Review 1998

mathematics the cognitive demands made on children change overtime (Cooper, 1983, 1985a and b; 1994a). In England in recent yearssuch a renegotiation has led to an apparent weakening of theboundary between everyday knowledge and esoteric mathematicalknowledge both in the curriculum and in its assessment (Cockcroft,1982; Cooper, 1994a). While in the 1960s and early 1970s the pre-ferred version of school mathematics tended to favour ‘abstract’algebraic approaches (though, at the same time, dominant versionsof school mathematics also incorporated newer applications ofmathematics), the dominant orthodoxy since the time of theCockcroft report of 1982 has favoured the teaching and learning ofmathematics within ‘realistic’ settings (Cockcroft, 1982; Dowling,1991; Boaler, 1993). This preference within the world of mathemat-ics educators was reflected in the early versions of the national tests,as we shall see below. It has been argued that test items contextual-ising mathematical operations within ‘realistic’ settings might beexpected to cause particular problems of interpretation for workingclass students4 (Cooper, 1992, 1994b, 1996, 1998). It is this possibil-ity that we intend to explore here.

The project

The project is combining quantitative and qualitative approaches. Akey feature of the work is the use of sociological ideas concerningthe cultural nature of cognition to make theoretical sense of chil-dren’s responses to assessment contexts and items (Bernstein, 1996;Lave, 1988; Wertsch et al., 1995). The project is collecting a varietyof forms of data from three primary schools (10–11 year-olds) andthree secondary schools (13–14 year-olds).5 In early 1996 tests wereconstructed, using previous National Curriculum tests as an itembank. The tests were designed to include a variety of item types,contextualisations and topic areas. All of the primary students tookthe same test while the secondary school students were allocatedtests tiered by ‘ability’ as judged by their teachers.6 These tests wereadministered to some 140 primary and some 450 secondary schoolstudents in February 1996. Subsequent to this, all of the primaryschool children and some 100 of the secondary school childrenworked through the test for a second time in the context of an indi-vidual interview with one or other of the researchers. In May 1996all 590 children participated in the statutory National Tests inmathematics and we have also collected the marked scripts from this

Social class differences in children’s responses to maths testing

© The Editorial Board of The Sociological Review 1998 117

exercise (two papers per child). In addition we have collected otherdata from these schools: background information on the students(parental occupations, measured ‘ability’, gender) and teachers’assessments of children’s level of mathematical understanding. Thispaper will use the data from primary students (10–11 years of age).

Class, culture, cognition and testing

The idea that children (and adults) experience difficulty in under-standing and meeting the demands of formal educational tasks –and that these difficulties are not randomly distributed across thesocial structure – is hardly a recent one (Mercer and Edwards, 1987;Holland, 1981). Neither is it an idea particular to sociologists (egCole et al., 1971; Cole, 1996). however, it is sociologists who havebeen mainly associated with particular versions of the claim in thepost-war period (Bernstein, 1990 and 1996; Bourdieu, 1977 and1986). Early versions of Bernstein’s thesis of a cultural discontinuitybetween the (lower working class) home and the school attractedconsiderable critical debate, with the claim being made by variouscritics that his thesis was flawed by an attachment to a ‘deficit’model of the working class child and family. Bernstein has, ofcourse, employed considerable energy in attempting to answer thesecritics (Bernstein, 1990, 1996) and, perhaps more importantly, hasrevised his ideas substantially over a period of almost forty years.The resulting theoretical account (Bernstein, 1996) seems primarilyto be a relational account of cultural differences and their conse-quences within organised educational systems akin to that Bourdieuhas developed over a similar period. While there are important dif-ferences between the two (especially with respect to their use of theconcept of rules: Bourdieu, 1990a and b), both are arguing that sub-ordinate groups within society lack access, relatively speaking, tothe cultural resources which schools demand of children and which,in their nature, reflect the ways of life of dominant social groups.Bernstein captures this via his concept of code; Bourdieu via hisconcept of habitus. While, notwithstanding their expressed inten-tions, it may in both cases be possible to detect a residual tendencyfor ‘academic’, ‘abstract’ forms of thought to be favoured over alter-natives (eg Bourdieu and Wacquant, 1992; 83–89) this is hardly thecentral thrust of their relational sociologies of education and cul-ture. To the extent such a ‘preference’ remains visible in their workit probably reflects the partial dependence of both thinkers on



Durkheim’s analyses of the relation of cognition to the form of thedivision of labour in society and an associated realist claim that‘minds’ and mental processes do differ across social orders charac-terised by qualitatively different degrees of complexity of the divi-sion of labour. The debate on the latter claim continues, closelyconnected with debates about the effects of literacy and methods oftesting human capacities (eg Spradley, 1972; Keddie, 1973; Gould,1984; Street, 1984; Lave, 1988; Nunes et al., 1993; De Abreu, 1995;Cole, 1996).

We will show here that working and intermediate class childrenseem to be more predisposed than service class children, at age 11,to employ initially their everyday knowledge in answering mathe-matics test items and that this can lead to the under-estimation oftheir actual capacities with respect to the demands of the school dis-cipline of mathematics as it is currently defined. We will therefore beusing the idea of cultural difference in a critical examination of thedifferential validity of these tests.

Before presenting our account, a few comments on social classare in order. There has long been, and still is a vigorous debateabout whether and how to use this concept in social and educa-tional research (eg Crompton, 1993; Apple, 1995). We are currentlyemploying the class scheme developed by Goldthorpe and others(eg Erikson and Goldthorpe, 1993) and, in particular, are codingchildren’s social backgrounds in terms of a ‘dominance’ model asset out by these authors (Erikson and Goldthorpe, 1993: 238). Wehave data, inevitably of variable quality, on the current or mostrecent occupation of the parents/guardians of the children in oursample. We suspect that a proportion of our classifications will con-tain error. It is also the case that there is, at best, a simplification ofsocial reality involved in attempting to characterise a child’s familialcontext of socialisation by a measure taken at a point in time. AsFeatherman et al. (1988) have shown, there is considerable occupa-tional movement amongst parents of young children and much of itis across the class boundaries described by Goldthorpe. It is alsolikely that, to some extent, the ‘effects’ of class which we discuss inthis paper are ‘effects’ of parents’ educational careers, linked totheir own origin and destination class, ie of the distribution, inBourdieu’s (1994) terms, of educational capital (‘capital scolaire’).Partly as a result of Featherman’s findings and partly as a result ofour sample size, we have chosen to employ in the body of the papera three category version of the class scheme, with the original 11categories being collapsed into groupings previously described by



Goldthorpe and his co-workers as the service class, the intermediateclass and the working class (see appendix 1 for descriptions of thegroupings).

Some illustrations

It is often argued that what characterises formal educational know-ledge above all else is its disconnectedness from everyday life andconcerns (Neisser, 1976; Mercer and Edwards, 1987). Sociologicallyspeaking, this description might be improved by a reference towhose concerns the knowledge is disconnected from (Young, 1971).Since it is often argued in educational circles that there has been amove since the 1960s – at least until recently – to introduce applica-tions of a ‘relevant’ nature into curriculum areas like mathematics,it might be useful to begin with a brief illustration of the ways inwhich ‘realistic’ test items still demand a ‘disconnected’ responsefrom children in spite of their surface appearance. Consider the itemin Figure 1, drawn from the 1992 tests for 14 year-olds.

This item is one of a type much discussed in mathematical educa-tional circles. The ‘correct’ answer is 20 times. The key point is thatthe child’s answer must not be fractional. The lift can not go up(and down) 19.2 times. The child is required therefore to introduce a‘realistic’ consideration into his or her response. In fact the childmust manage much more than this. He or she must introduce only asmall dose of realism – ‘just about enough’. The child must notreflect that the lift might not always be full; or that some peoplemight get fed up and use the stairs; or that some people requiremore than the average space – eg for a wheelchair. Such considera-tions – ‘too much realism’ – will lead to a problem without a singleanswer, and no mark will be gained (see Cooper, 1992 and 1994b fora fuller discussion). Children’s and schools’ interests now hinge onmanaging these complexities in a legitimate manner.

Various writers have employed the notion of educational groundrules to capture what is demanded of children in these cases (Mercerand Edwards, 1987). There is clearly some affinity between this con-cept and those of recognition and realisation rules as recentlyemployed by Bernstein (Bernstein, 1996; Cooper, 1996; 1998).However, it can be seen that it would be quite difficult – if notimpossible – to write a set of rules which would enable the child torespond as required to the lift question. Certainly, the rule – in thesense of a mandated instruction – to employ ‘realistic’ considera-



tions would not do, since ‘how much’ realism is required remains adiscretionary issue. It is this problem that has led to a range ofattacks on the use of rules to model human activities (Taylor, 1993)and, in particular, has led Bourdieu to reject a rule-based account ofcultural competence. His concept of habitus aims to capture the idea of a durable socialised predisposition without reducing



Figure 1 The Lift Item (SEAC, 1992)

behaviour to strict rule-following (Bourdieu, 1990a and 1990b).Bourdieu sometimes describes what habitus captures as ‘a feel forthe game’ and we can see that this describes fairly well what isrequired by the lift problem and others like it (for a qualitative com-parison across a range of test items of two children who differmarkedly in their ‘feel for the game’ see Cooper, 1996, 1998).

Before moving to a more in-depth discussion of two items, draw-ing on the responses of more than 100 children, we want to give onemore illustration of what the ‘feel for the game’ can require of chil-dren taking the national tests. Consider Figure 2, which bases math-ematics on shopping – a fairly common ‘relevant’ topic for children.

This problem, for those socialised into ‘esoteric’ mathematics,reduces to the solution of the pair of simultaneous linear equationsset out below (in pence).

C + P = 902C + P = 145

One way of solving the problem – legitimately from the point ofview of the marking scheme – would be to see this embedded struc-ture and apply whatever technique has been taught for solving suchequations. Alternatively, the embedded equations might be solved ina more ‘intuitive’ manner. However, what the child must not do(note the ‘rule’-like formulation!) is to read the item as an invitationto draw on what they know about and from shopping contexts.Here is an example of what can happen if the child makes this ‘mis-take’:

Example 1Working class girl (written response from national test in May1996):I said to myself that in a sweetshop a can of coke is normally 40p soI thought of a number and the number was 50p so I add 40p and50p and it equalled 90p.

This response, instead of recognising the hidden structure of theproblem – the pair of simultaneous equations, uses just the first‘equation’ plus knowledge of the price of cola gained from shoppingin everyday life to generate a solution. In some respects this solu-tion, which includes correctly worked arithmetic, is efficient – if oneis rushing through a timed test – but it is, nevertheless, ‘incorrect’given the total information available and the context of a school



mathematics test. We are left not knowing whether the child has the‘competence’ required to solve the pair of equations taken together.What is it that the child has ‘failed’ do do? Has she failed to readthe whole question? Or to solve the equations? Or to bracket outher everyday knowledge? It may be that, in Bourdieu’s terms, this



Figure 2 The Shopping Item (SCAA, 1996)

child lacks the required ‘feel for the game’. (To answer this wewould need to examine her responses across a range of items.)

Notwithstanding the problems of ‘rule’ on the side of the child –the acquirer of educational knowledge – Bernstein’s use of ‘rule’does seem to capture a key feature of the power relations this childfinds herself in. Those defining the test items, and in particular,what counts as acceptable answers are indeed setting out a series of‘rules’ at least in the sense of regulations. Bernstein’s use of the con-cepts ‘recognition and realisation rules’ brings this out. His usage ofthese concepts can only be grasped within the overall structure ofhis account of educational codes, based in his account of classifica-tion and framing. A brief summary follows.

In his constantly developing general theory of pedagogic codesBernstein, on the one hand, describes the organisation of educa-tional knowledge on the side of the pedagogic transmitter in termsof particular values of classification and framing (translating powerand control relations) and, on the other hand, describes the learner,ie the acquirer, in terms of access to recognition and realisation rules(Bernstein, 1996). Very briefly, and crudely, classification refers tothe degree of insulation between categories – discourses, practices,agents, etc. Framing refers to the balance of control over pedagogiccommunication in local interactional contexts. ‘Classification refersto what, framing refers to how meanings are to be put together, theforms by which they are to be made public, and the nature of thesocial relationships that go with it’ (Bernstein, 1996: 27).Recognition rules, ‘at the level of the acquirer’, are the means by‘which individuals are able to recognise the speciality of the contextthat they are in’ (31). Realisation rules allow the production of‘legitimate text’. ‘The recognition rule, essentially, enables appropri-ate realisations to be put together. The realisation rule determineshow we put meanings together and how we make them public’ (32).These rules are potentially independent of one another, eg ‘manychildren of the marginal classes may indeed have a recognition rule,that is, they can recognise the power relations in which they areinvolved, and their position in them, but they may not possess therealisation rule. If they do not possess the realisation rule, they can-not then speak the expected legitimate text (32)’.

It is also the case that a child may have access to the realisationrule, but not the recognition rule which allows it to become opera-tive, as we shall see later in the discussion of the tennis item (Figure4). In the case of the child’s response to the shopping item above shemay have in fact been able to undertake the required arithmetic/



algebra. We don’t know, since she was not, in the test situation,given a second chance to respond to the item as a pair of simultane-ous equations modelling an entirely imaginary situation.7

What we will do in the remainder of the paper is describe in somedetail the patterns of response to two ‘realistic’ test items by socialclass, showing that there does seem to be a stronger predispositionamong working and intermediate class than service class 11 year-olds to draw ‘inappropriately’ on their everyday knowledge in thecontext of mathematics testing. In the second case we will demon-strate that this leads to an under-estimation of what children areactually able to do – and hence raises potential equity issues withrespect to these tests. The first item concerns an imaginary trafficsurvey (Figure 3); the second an imagined tennis competition(Figure 4).

The traffic item: the production of false positives?

This item (Figure 3) can be criticised severely on ‘mathematical’grounds (Cooper, 1997). Here, however, our concern is with chil-dren’s responses to it. The marking scheme asymmetrically alloweda mark for ‘likely’ or ‘very likely’ in the case of the car, and for‘unlikely’ in the case of the lorry.8 In the test of which it was a part,there was little apparent sign of any strong social class patterning tothe marks awarded (see Table 1).

What the test results cannot tell us, however, is anything aboutthe children’s response strategy and, in particular, whether theyutilised the given data or drew on their everyday knowledge (presumably an ‘incorrect’ response from the perspective of an



Table 1 Marks obtained on the traffic item in the February 1996test by class and sex

(2 marks available) Female Male Total—————— —————– —————–Mean Count Mean Count Mean Count

Service class 1.08 26 1.09 34 1.08 60Intermediate class .85 13 1.12 17 1.00 30Working class 1.15 13 .90 20 1.00 33Total 1.04 52 1.04 71 1.04 123

examiner, though method was not addressed in the markingscheme). In the context of the interview we can address this issue.Children were asked, after they had circled two choices, why theyhad chosen these. Two examples follow:



Figure 3 The Traffic Item (SEAC, 1993)

Transcript 1

A working class girl – response ‘esoteric’.M: I don’t like this one.BC: Don’t you? Why don’t you like it?M: I don’t know, I just don’t like it – too hard for me.(She laughs.)(She circles likely for lorry – no mark, very likely for car – onemark.)BC: Okay, how did you decide on those two?M: That one, you look at that, and there was only thirteen thatwent past in an hour, so it was not pretty like- (sic), it might havehappened but it wasn’t that likely it would happen, and the car,because there’s seventy five went past in an hour, it’s very likely butyou can’t be certain.BC: Right, why can’t you be certain?M: Just in – don’t know – you just can’t be certain.BC: You can’t be certain. OK. I thought you – why didn’t you likethat one then? You seemed to do it quite quickly.M: Yeah, but, I don’t like trying to figure which one’s which.BC: Well it’s fine. OK, number fourteen next.

Transcript 2

A working class boy – response ‘realistic’.(He circles unlikely for lorry – one mark, very likely for car – onemark.)BC: Now how did you decide on those two?Ra: Cos, because the lorry, there’s not as many lorries around asthere is cars.BC: What were you thinking of, whereabouts?Ra: Outside of school, more parents would come to like collect achild in a car than they would in a lorry.BC; That’s true, right, OK, did you look at these numbers at allhere? Did you read that part?Ra: No.BC: OK so you did the question without looking at that part?Ra: Yep.BC: Right, why do you think you didn’t bother to read that then,because you knew already?Ra: Yep.



We have coded each of the children’s responses to our request toexplain why they choose what they chose for lorry and car (takentogether) according to whether they employed the given data ortheir everyday knowledge of vehicles and roads. Table 2 shows thesocial class distribution of responses on a three category scale. Itcan be seen that working class children are almost twice as likely asservice class children to refer only to their everyday knowledge inanswering our enquiry (and, we might assume, the original questionitself). The parallel Table 3, on the other hand shows less sign of anysimple gender effect, with similar percentages of boys and girlsusing only everyday knowledge – though the fact that girls are twice



Table 2 Distribution of response strategies by class (traffic item –interview context)

Uses given Uses everyday Uses everyday Totalsdata alone knowledge knowledge

and given alonedata

Service class 38 10 11 59Percentage 64.4 16.9 18.6

Intermediate class 16 10 4 30Percentage 53.3 33.3 13.3

Working class 16 6 10 32Percentage 50.0 18.8 31.3

Totals 70 26 25 121Percentage 57.9 21.5 20.7

Table 3 Distribution of response strategies by sex (traffic item –interview context)

Uses given Uses everyday Uses everyday Totalsdata alone knowledge knowledge

and given alonedata

Girls 26 16 10 52Percentage 50.0 30.8 19.2

Boys 44 11 16 71Percentage 62.0 15.5 22.5


as likely to use both given and everyday data may deserve furtherstudy.

Table 4 draws on the interview data to show the relationshipbetween class, response strategy and mark achieved for the questionin the interview context. It is not easy to see any strong relation herebetween class and success9 – through there was a relation betweenclass and strategy (Table 2). Why? Possibly because the design ofthis particular question allowed an ‘inappropriate’ ‘everyday’response – one which ignored the given data in some cases com-pletely – to gain marks. Had the item employed given data that con-flicted with what children have experienced in their everyday livesrather than paralleled it, then we might have seen a clear relationwith class. It is worth stressing this point – the associations reportedhere result from, amongst other unexamined factors, the relationbetween what children bring to the context as a result of their acad-emic and general socialisation (in and out of school) and the partic-ular nature of the items. The lack of an association between class,response style and mark here can be seen as a possible example ofhow a really existing predisposition of a person may or may not,depending on context, lead to ‘predicted’ associations in a data set.The distribution of the required ‘recognition rule’, ie, seems to berelated to social class leading to working class children being morelikely to ‘misrecognise’ the demands of the problem background(Table 2). However, coupled with the choice of given data by the testdesigners, this results in an item which does not measure validlywhether children can produce the required probabilistic reasoningabout the given data.

In the case of the traffic item therefore, we have a clear case of aquestion which seems to generate ‘false positives’ in assessment



Table 4 Mean marks obtained on the traffic item (interview) byclass and response strategy

(2 marks available) Uses given Uses everyday Uses everyday Totalsdata alone knowledge knowledge

and given alone(n in brackets) data

Service class 1.21 (38) 1.20 (10) .91 (11) 1.15 (59)Intermediate class .88 (16) 1.40 (10) 1.50 (4) 1.13 (30)Working class 1.25 (16) .83 (6) 1.10 (10) 1.13 (32)Totals 1.14 (70) 1.19 (26) 1.08 (25) 1.14 (121)

terms (Wood and Power, 1987). Children are being awarded marksfor ‘non-mathematical’ behaviour – as judged by ‘esoteric’ criteria.However, in the light of the findings discussed in the next sectionconcerning the tennis problem (Figure 4), it will be seen that thechildren who used their everyday experience to decide on likeli-hoods perhaps could have used the given knowledge to the sameend – had they been asked to do so more explicitly.

The tennis item: the production of false negatives?

The tennis item (Figure 4) is intended to be difficult, being judgedby the test designers to be suitable only for higher attaining chil-dren.10 Why, however, should it be difficult? In an earlier paper,without reference to any data, it was argued:

This item seems to be based on the assumption that the childrenwill not be misled by acting in a common-sense way – by imagin-ing, for example, that they might be acting physically on the‘names’ on the cards in the context of an imagined ‘realistic’competition. On the contrary, in order to be successful, they areexpected, having abstracted the mathematical problem from itspictorial setting, to approach this in a Piagetian ‘formal opera-tional’ mode. The problem is that, even where children arecapable of undertaking the abstracted combinatorial act, theymight not demonstrate it in this case. After all, objects put insuch bags are normally there to be taken out (as, for example, intelevised draws for the FA Cup).11 If children were to operate onthis assumption they would have, in their imagination, to put the‘names’ back in order to generate all the possibilities. (In anempirical case, it could, in principle, given a run of bad luck, takea very long time to generate the nine possible pairs by thismeans.) It seems clear, therefore, that to achieve level 6, the childmust treat this as a mental exercise in combining names and mustbe able to avoid being side-tracked by any element of the deviceused to test the Statement of Attainment. The bags must not betaken to signify the request for the ‘empirical’ three pairs thatmight seem implicit in the physical act of removing names withoutreplacement (as in the case of real draws for knock-out competi-tions), when a step into mathematical discourse with replacementcan allow the production of nine. (Cooper, 1994b)



In the terms used by Bernstein, this item seems likely to be a candi-date for confusion about what is relevant knowledge to bring tobear. What might be ‘appropriate’ in the context of real sports com-petitions is not ‘appropriate’ or ‘legitimate’ in the context of amathematics test.12 There is a potentially confusing boundary tonegotiate.

That children find this a difficult boundary can be illustrated bythe fact that several children took the apparent national origins ofthe competitors into account in making their decisions (see tran-scripts 3–5). Two of these children initially produced three pairs,and one did so after some elaboration by the interviewer of the testitem’s demands.

Transcript 3: girl, service class

MD: Have you finished? OK, explain to me how you worked thatout then.Child: Well, Rob and Katy, they’re like normal names, and they’llbe OK together, and they sound like different country names.MD: What Rashid and Gita?Child: Yeah, and so, they would be quite happy as a pair becausethey come from a different country and they’ve got those namesand just put them together.

Transcript 4: boy, intermediate class

Child: Done.MD: OK, so how did you work that out?Child: Well, Rob and Ann’s sound like not really, um, (indecipher-able phrase, possibly ‘brother and sister’), so I’ll come back to them.Rashid and Gita are like, sort of the same names from a differentcountry.MD: Mm.Child: So it might be them two pairs, and Rob and Ann.MD: David and Ann do you mean?Child: I mean David and Ann.MD: OK, so you worked it out because of where the country, isthat what you’re saying?Child: Yeah.



Transcript 5: boy, working class

MD: OK, what have you done then – Rashid and Ann.Child: Rashid and Gita, sound like different country names so thatit wouldn’t exactly be fair if Rashid and Gita got together, becauseyou’ve got to give them a chance to meet other people.



Figure 4 The Tennis Item (SEAC, 1993)

Clearly, from the perspective of esoteric mathematics the children’snational origin is not a relevant consideration, though it might wellbe in the everyday life of school children. What does the full data setsuggest about children’s reading of this question? In fact, in theFebruary 1996 test, 28 of the 125 children produced three pairs astheir answer rather than the required nine, suggesting that for thesechildren everyday concerns may have dominated esoteric ones. Howdid success on this item relate to class and gender?

Tables 6 and 7 show that, in both the test and the interview theworking class children do considerably less well on this item, forwhich one mark was available, than the service class children. Theboys do slightly less well as a group than the girls in the test but thedifference disappears in the interview. We will concentrate on classin this rest of this section.

In this case, unlike the traffic item, since we do have relevant writ-ten text available to us,we have coded the written text produced inthe interview setting13 – as opposed to the children’s verbal



Table 5 Mean marks obtained on the traffic item (interview) by sexand response strategy

(2 marks available) Uses given Uses everyday Uses everyday Totalsdata alone knowledge knowledge

and given alone(n in brackets) data

Girls .96 (26) 1.38 (16) 1.00 (10) 1.10 (52)Boys 1.25 (44) 1.00 (11) 1.06 (16) 1.17 (71)Totals 1.14 (70) 1.22 (27) 1.04 (26) 1.14 (123)

Table 6 Mean marks achieved on tennis item in the February 1996test context

(1 mark available) Female Male Total—————— —————– —————–Mean Count Mean Count Mean Count

Service class .62 26 .56 34 .58 60Intermediate class .54 13 .41 17 .47 30Working class .38 13 .40 20 .39 33Total .54 52 .48 71 .50 123

responses to our questions. The marking scheme allows a mark onlyfor nine distinct pairs. Children therefore respond ‘appropriately’ tothis item by setting out nine pairs. If they do so, it is possible tocode the nine pairs as ‘esoteric’ or ‘realistic’ by reference to the waythey have been grouped. Here is what we would term an ‘esoteric’set:

Rob & KatyRob & AnnRob & Gita

Rashid & KatyRashid & AnnRashid & Gita

David & KatyDavid & AnnDavid & Gita

Here, by contrast, is a ‘realistic’ set:

Rob & KatyDavid & GitaRashid & Ann

David & AnnRashid & KatyRob & Gita

David & KatyRashid & GitaRob & Ann



Table 7 Marks achieved on tennis item in the interview context –initial response

(1 mark available) Female Male Total—————— —————– —————–Mean Count Mean Count Mean Count

Service class .81 26 .85 34 .83 60Intermediate class .77 13 .82 17 .80 30Working class .69 13 .60 20 .64 33Total .77 52 .77 71 .77 123

In the second case the pairs, taken in groups of three at a time,might engage in games of tennis – hence our coding of the set as‘realistic’. In the first case, each group of three could not, given therepeated use of one child’s name. In most cases where children pro-duced three pairs, in both the test and the interview, they were a‘realistic’ three, using each child just once. Now it might be arguedthat the second nine above is a ‘better’ response than the first – asfar as tennis is concerned – but it is arguably not from the point ofview of ‘esoteric’ mathematics. The first set of pairs, from the latterpoint of view, is more ‘abstract’, more obviously ‘mathematical’ inits ‘systematicity’. What about response style in this case? Table 8shows response style is strongly related to class.

Table 9 breaks down the marks achieved in the interview by socialclass and response style. Given the apparent relation between



Table 8 Response strategy on the tennis item (interview) by class

‘Esoteric’ Other ‘Realistic’ Totalspairings (typically pairings

mixed)

Service class 47 4 8 59Percentage 79.7 6.8 13.6

Intermediate class 20 2 8 30Percentage 66.7 6.7 26.7

Working class 14 5 12 31Percentage 45.2 16.1 38.7


Table 9 Mean mark obtained on the tennis item (interview) by classand response strategy (counts as above table)

(1 mark available) ‘Esoteric’ Other ‘Realistic’ Totalspairings (typically pairings

mixed)

Service class 1.00 .75 .00 .85Intermediate class 1.00 1.00 .25 .80Working class 1.00 .80 .25 .68Totals 1.00 .82 .18 .79

response style and mark,14 it becomes particularly important tolook in more detail at what happened when, in the interview, thechildren were given a ‘second chance’ to try to find all the possiblepairs. Given the apparent tendency for working class children to‘choose’ an ‘inappropriate’ self-defeating strategy in their initialresponse it becomes a critical question whether they might haveproduced the nine pairs had they not ‘chosen’ this initial responsestyle. In other words, is there some sense in which they might beable, in some cases, to do the mathematics ‘pure and simple’, given aminor cue that they have ‘misrecognised’ the context? Now, in fact,the cue was merely the request, once they had clearly finished writ-ing some number of pairs fewer than nine, to consider whether theyhad obtained all the pairs. One example of what happened in anumber of cases follows. The child here is a girl from the intermedi-ate class grouping.

Transcript 6: girl, intermediate class

She writes the three pairs thus:

Rob and KatyRashid and GitaDavid and Ann

MD; Done that one?E: Yeah.MD: OK, so tell me how you worked that one out.E: I put those two names and – so I did those two there can and Idid those.MD: David and Ann, Rashid and Gita, OK.E: Mm.MD: OK, see where it says there find all the possible ways that girlsand boys can be paired, do you think you’ve found all the possibleways?E: No.MD: You could find some more?E: Yeah.MD: OK, let me just do that, so I’ll know where you stopped forthe beginning.15 OK, go on then.(She works at the problem, silently.)(She adds six pairs to give:)



Rob and Katy Katy and DavidRashid and Gita Ann and RobDavid and Ann Ann and Rashid

Gita and DavidGita and RobKaty and Rashid

MD: OK, so have you finished that one now?E: Mm.MD: And you think you’ve got all of them?E: Yeah.MD: OK, do you know? – when you first did it you stopped, afterthree, why did you stop after three?E: I don’t know.MD: You don’t know, but why didn’t you continue?E: I didn’t think that you were supposed to.MD: OK, that’s a good reason, but why didn’t you think you weresupposed to?(The interview continues with the child not being able to give a rea-son.)

She initially produced three pairs but then obtained nine pairs whenencourage to reflect on whether this was enough.16 She did notthink she was ‘supposed to’ do nine – a rather rule-like formulation.How common was this type of response? Table 10 sets out thedetails of the 12 cases where an initial response of 3 pairs was fol-lowed, after the ‘cue’, by a subsequent response resulting in ninepairs.17 Some 10% of the sample under discussion here fell into thistrap of initially producing a ‘realistic’ three pairs and yet recoveredfully from this ‘inappropriate’ response when offered a chance toreconsider.18 Furthermore, children from intermediate and workingsocial class backgrounds were over-represented amongst this groupand children from service class backgrounds under-represented, asTable 11 shows. Notwithstanding the small numbers involved, thisresult seems to suggest that differentiated responses to items of thisparticular type may have some important consequences in produc-ing class-related patterns of success and failure in national testing.There is also a slight tendency for girls to be over-representedamongst this group (Table 12) and for the children to have lowerthan average ‘ability’ scores.

We can see that in the case of this question a child’s apparentresponse style can lead to his or her ‘mathematical’ competence



Barry C

ooper and Máiréad D

unne

138©

The E

ditorial Board of T

he Sociological Review

1998

Table 10 Details of the 12 cases where 3 pairs was followed by 9 in the interview situation – Transcript 6 child in bolditalics

Initial Initial Pairs after Overall Social class Sex Verbal Quantitative Non-verbalpairs response ‘second response ‘ability’ ‘ability’ ‘ability’

strategy chance’ strategy

3 ‘realistic’ 9 ‘realistic’ service male 75 70 793 ‘realistic’ 9 ‘realistic’ intermediate male 88 92 1003 ‘realistic’ 9 ‘realistic’ working female 101 94 1083 ‘realistic’ 9 ‘realistic’ intermediate female 89 83 873 ‘realistic’ 9 ‘realistic’ intermediate male 92 94 1063 ‘realistic’ 9 ‘realistic’ working male 89 89 913 ‘realistic’ 9 mixed intermediate female 95 88 1003 ‘realistic’ 9 ‘realistic’ intermediate female 84 94 893 ‘realistic’ 9 ‘realistic’ working female 91 96 943 ‘realistic’ 9 ‘realistic’ service female 99 108 104

‘realistic’ 9 mixed working male 110 116 1143 ‘realistic’ 9 ‘realistic’ service female 125 129 123

mean (of 94.83 96.08 99.58the 12)

mean (of 102.47 104.17 105.36the 125)

being underestimated because of the way in which he or she ‘chose’to read the question. These 12 cases, some 10% of the total, arearguably all ‘false negatives’ on first response – given that the chil-dren could do what was required when cued to reconsider what theyhad written.19 Their performance has not reflected their mathemati-cal competence – at least in the sense of the capacity to produce ninepairs from two groups of three items – and this seems to be relatedto the children’s capacity to recognise correctly or not the demandsof the context.20 In the light of Holland’s (1981) work with Bern-stein on children’s preferred classifying strategies it seems that theoverrepresentation of working class children amongst these falsenegatives is something we might have expected.

The traffic and tennis items compared

What can we learn from a comparison of the children’s responses tothese two items, one multiple choice and one open-ended? Thethreats to valid and ‘fair’ assessment of ‘competence’ inherent inboth types of question can be noted. In the case of the traffic itemthe children, because of its multiple choice nature, could obtainmarks by ‘inappropriately’ employing their everyday knowledge.



Table 11 Class distribution of 3 followed by 9 children compared tosample in general

Service Intermediate Working n (withclass (%) class (%) class (%) class data)

The 12 cases 25.0 41.7 33.3 12In whole sample 48.8 24.4 26.8 123

Table 12 Sex distribution of 3 followed by 9 children compared tosample in general

Girls (%) Boys (%) n

The 12 cases 58.3 41.7 12In whole sample 43.2 56.8 125

The nature of the response mode hides this from the assessor. Thisvalidity problem could have been avoided by making the given dataless typical of everyday settings known to the children, ie by intro-ducing ‘realistic’/‘esoteric’ dissonance. However, had this been done,we perhaps would have found the second problem coming to the fore– the equity problem that we found in the case of the tennis item. Onsuch a revised traffic item, working class children might have tendedto have lost marks not because they could not reason probabilisti-cally but because they referred more to everyday knowledge thantheir service class companions. We saw, in the case of the tennis item,that children who had the ‘mathematical competence’ did not alwaysdemonstrate it, without the elaboration of their response induced bythe interviewer, and that this behaviour was apparently related to thesocial class distribution of recognition rules.

Conclusion

This paper has concentrated on just two test items. In a parallelrecent paper, we have reported a statistical analysis, by class and gen-der, of children’s degrees of success on two broad classes of test items,termed by us ‘esoteric’ and ‘realistic’. The ‘realistic’ items embedmathematical operations within contexts containing people and/ornon-mathematical everyday objects while the ‘esoteric’ do not. All ofthe items discussed here would be coded as ‘realistic’. For two exam-ples of ‘esoteric’ items see Figures 5 and 6 in Appendix 2. We havefound a greater difference between service and working class meanperformances for the category of ‘realistic’ than for the category of‘esoteric’ items (Cooper et al., 1997). The current paper shows thatpart of an explanation of this general finding might be found in socialclass differences in the interpretation of the demands of ‘realistic’questions, with working class children being more likely to draw‘inappropriately’ on their everyday knowledge when responding toitems. This relative failure (and it is relative, not absolute) to recog-nise the strongly classified nature of school mathematics in the face ofsurface appearances which suggest the relevance of everyday know-ledge may be an aspect of the overall sociocultural predispositionsdiscussed by Bourdieu and Bernstein. The realised meaning of theitems seems to vary with social class, with the resulting negativeeffects on performance leading to the underestimation of the actualcompetence of more working than service class children.

What Bourdieu has argued concerning ‘popular’ classes’



responses to ‘art’ works seems to capture something of the featuresof some children’s responses to the two items discussed here. He haswritten, in Distinction:

When faced with legitimate works of art, people most lacking thespecific competence apply to them the perceptual schemes of theirown ethos, the very ones which structure their everyday percep-tion of everyday existence. These schemes, giving rise to productsof an unwilled, unselfconscious systematicity, are opposed to themore or less fully stated principles of an aesthetic. The result is asystematic ‘reduction’ of the things of art to the things of life, abracketing of form in favour of ‘human’ content, which isbarbarism par excellence from the standpoint of the pure aes-thetic. Everything takes place as if the emphasis on form couldonly be achieved by means of neutralisation of any kind ofaffective or ethical interest in the object of representation whichaccompanies . . . mastery of the means of grasping the distinctiveproperties which this particular form takes on in its relations withother forms (ie, through reference to the universe of works of artand its history). (Bourdieu, 1986: 44)

Furthermore, he clearly sees the explanation for the differences inresponse as grounded in different material conditions of life.

The aesthetic disposition which tends to bracket off the nature andfunction of the object represented and to exclude any ‘naive’reaction – horror at the horrible, desire for the desirable, piousreverence for the sacred – along with all purely ethical responses, inorder to concentrate solely upon the mode of representation, thestyle, perceived and appreciated by comparison with other styles, isone dimension of a total relation to the world and to others, a life-style, in which the effects of particular conditions of existence areexpressed in a ‘misrecognizable’ form. These conditions of exis-tence, which are the precondition for all learning of legitimateculture, whether implicit and diffuse, as domestic cultural traininggenerally is, or explicit and specific, as in scholastic training, arecharacterised by the suspension and removal of economic necessityand by objective and subjective distance from practical urgencies,which is the basis of objective and subjective distance from groupssubjected to those determinisms. (Bourdieu, 1986: 54)

These remarks, especially in their references to the concern with abstract form as opposed to human content which he sees as



characteristic of dominant class responses, seem to capture equallysomething of the differences between children who do and do notemploy the resource of their everyday knowledge in their testresponses. These patterns of response do seem to be related in ourdata to social class background. However, we have not presentedhere an analysis which would justify the claim that the patterns ofresponse result from predispositions to act in similar ways across awide range of test items (though the findings in Cooper et al., 1997,would lend some support to such a claim). The full examination ofthat claim will require further work.

There are a number of reasons why such work is important. Firstof all, it is clearly a contribution to the general study of the relationsbetween sociocultural background and cognitive processes andproducts. Secondly, it has important policy implications for those ineducation, especially for those concerned with equity issues in test-ing. Mathematics is one of the key areas of study within formal edu-cational institutions. Because of this children’s success or failure inmathematics is a key factor in the determination of their subsequentlife chances. In particular, as the climate of opinion seems to movein favour of greater selection at the end of primary schooling (see,eg, Hutton, 1995: 311), the Key Stage 2 tests may well become oneelement of a process of selection for secondary school. More gener-ally, researchers in the field of assessment in both the UK and theUSA are increasingly concerned about the implications for equalityof opportunity of recent changes in forms of assessment (Gipps andMurphy, 1994, Darling-Hammond, 1994, Baker and O’Neil, 1994).What the evidence in this paper suggests, especially if taken togetherwith the results reported in Cooper et al. (1997), is that the currentattempt by mathematics educators to maintain a concern with ‘rele-vance’ in the context of the national curriculum and its testing mayhave unintended – and perverse – consequences for another set ofconcerns they may well share – those concerning equal opportuni-ties issues (eg Secada et al., 1995).

Appendix 1: Occupational groupings

(combined from Goldthorpe & Heath, 1992 and Erikson &Goldthorpe, 1993)

1. Service class, higher grade: higher grade professionals, adminis-trators and officials; managers in large industrial establish-ments; large proprietors.



2. Service class, lower grade: lower grade professionals, administra-tors and officials; higher grade technicians; managers in smallindustrial establishments; supervisors of non-manual employees.

3. Routine non-manual employees.4. Personal service workers.5. Small proprietors with employees.6. Small proprietors without employees.7. Farmers and smallholders.8. Foremen and technicians.

9. Skilled manual workers.10. Semi- and unskilled manual workers.11. Agricultural workers.

We have collapsed 1 and 2 into a service class, 3–8 into an interme-diate class, and 9–11 into a working class.

Appendix 2: Two illustrative esoteric items



Figure 5 The Algebra Item (SCAA, 1996)

Figure 6 The Row of Numbers Item (SCAA, 1995)

University of Sussex Received 30 July 1997Accepted 22 September 1997

Acknowledgements

We would like to thank all of the teachers and children in the six schools for puttingup with our constant demands over most of a year; Beryl Clough, Hayley Kirby andJulia Martin-Woodbridge or their work in transcribing interviews so patiently, andthe Economic and Social Research Council for funding this work. We would also liketo thank Nicola Rodgers for her part in data collection, and various of our col-leagues at Sussex and Stephanie Cant for their comments on drafts of papers.

Notes

1 An earlier version of this paper was presented at Annual Conference of theBritish Sociological Association, York, April 1997.

2 Economic and Social Research Council Project (R000235863): MathematicsAssessment at Key Stages Two and Three: Pupils’ Interpretation andPerformance (1995–97). Nicola Rodgers has also worked on the project as aResearch Assistant for part of its life.

3 Without some way of describing activities as ‘mathematical’ it is difficult to retaina critical distinction between what children can ‘really’ do ‘mathematically’ andwhat they can apparently do when their activities are ‘measured’ by particularassessment procedures which embed ‘mathematical’ operations within differentcontextual settings.

4 There has been related work in respect of science problem-solving (see Morais etal., 1992).

5 The schools have been selected to cover the range of social class positions of stu-dents in the English state school sector.

6 We have done this, in the case of secondary schools, in order to replicate the offi-cial practice.

7 For a previous attempt, using data collected in a pilot exercise for this project, todevelop a diagrammatic model of the processes involved in the child’s interpretingthe meaning of test items, see Cooper, 1996 and 1998.

8 The Statement of Attainment being assessed is (supposedly) Use appropriate lan-guage to justify decisions when placing events in order of ‘likelihood’.

9 An analysis of variance supports this face reading. Neither class nor responsestrategy are statistically significant predictors of the mark achieved here. The sexdata is in Table 5.

10 The Statement of Attainment for this item is: Identify all the outcomes of combin-ing two independent events. The (very!) esoteric mathematical ‘equivalent’ of thetennis item is: Find the Cartesian product of the sets {a, b, c) and {d, e, f}.

11 The FA Cup is the annual knockout soccer competition organised by theFootball Association in England and Wales.

12 In Bourdieu’s terms the item requires a ‘scholastic’ rather than a ‘practical’response (Bourdieu & Wacquant, 1992).



13 We are using the interview data because we wish to explore below how childrenresponded to a cue to reconsider their initial response.

14 This procedure may seem to have a little tautology built into it. However, thereare children who produce nine pairs using a ‘mixed’ style (see, eg, Transcript 6).

15 The interviewer adds a mark at this point to indicate the first response (for latercoding).

16 Her account of why she stopped at three pairs points to the sort of subconsciousbehaviour which Bourdieu (1990b) describes via his concept of habitus.

17 We have coded each second response by the child in terms of whether we ‘com-promised’ their response via leading questions, etc. None of these 12 cases were‘compromised’ on this coding. Of the other seven children who had produced 3pairs initially, two stuck at three on their second attempt, two produced sevenpairs, one produced eight, one a very idiosyncratic set of ten pairs, and one weseem to have failed to offer the second chance to.

18 In Holland’s (1981) work with Bernstein on children’s classifying strategies, theworking class children tended to reproduce their initial non-esoteric classifica-tions a second time.This was also the case with the tennis item. 10 of these 12children ended up with a set of nine pairs which we coded as ‘realistic’ and twowith a ‘mixed’ set. To some extent, of course, this result was constrained by theiradding to their first ‘realistic’ set of three.

19 From an assessment point of view, when the context was manipulated to allowthem to produce their ‘best’ response.

20 This ‘capacity to recognise’ might, of course, be described as a cultural compe-tence at another level of analysis. Bourdieu’s (1986) use of the phrase ‘culturalcompetence’ would seem to be at this level.

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