Structural exclusion through school mathematics:using Bourdieu to understand mathematicsas a social practice

Robyn Jorgensen & Peter Gates & Vanessa Roper

# Springer Science+Business Media Dordrecht 2013

Abstract In this paper, we explore a sociological approach to mathematics educationand offer a theoretical lens through which we can come to understand mathematicseducation as part of a wider set of social practices. Many studies of children’sexperiences in school show that a child’s academic success is a product of manyfactors, some of which are beyond the control and, sometimes, the knowledge of theclassroom teacher. We draw on the sociological ideas of Pierre Bourdieu to frame ouranalysis of the environment in which the pupils learn and the ways in which thepractices help to create parallel worlds which are structured quite differently insideand outside the classroom. Specifically, we use Bourdieu’s notions of habitus, fieldand capital. Using two cases, we highlight the subtle and coercive ways in which thepractices of the field of mathematics education allow greater or lesser access to thehegemonic knowledge known as school mathematics depending on the cultural backgroundsand dispositions of the learners. We examine the children’s mathematical learning trajectoriesand reflect on how what they achieve in the future will, in all likelihood, be shaped by theirsocial background and how compatible this is with the current educational climate.

Keywords Bourdieu . Sociology . Equity . Habitus . Field . Capital

1 The marketization of education

For some years now, there has been a belief within government circles around the world thatthere are problems within national systems of education and that the route to improving

Educ Stud MathDOI 10.1007/s10649-013-9468-4

R. Jorgensen (*)Griffith University, Brisbane, Australiae-mail: r.jorgensen@griffith.edu.au

P. GatesThe University of Nottingham, Nottingham, UK

V. RoperNottinghamshire Local Authority, Nottingham, UK

standards lies in applying the commercial concept of a “free market”, in which schools willimprove by being in competition with each other for students. In such a free market system,schools are measured on various criteria, ranked into “league tables” and are encouraged tosee themselves in competition for scarce resources and urged to raise themselves up theseleague tables. In both the UK and Australia, national testing has become the measure thatdefines the success of schools. In Australia, all students in years 3, 5, 7 and 9 are tested inliteracy and numeracy. Scores for schools are then published on a national website(ACARA, 2010). At this stage, there are no formal rankings or interventions for under-performing schools which is in stark contrast to the UK where the effect of national testinghas become so powerful. In spite of some recent and limited retrenchment, the results oftesting in the UK can effectively move a school into a category requiring significantimprovement—sometimes through the removal (or “resignation”) of the head teacher—orit may ultimately be closed.

These “failing” schools are a product of the political system, and often, the reason they‘fail’ is due not to bad teaching or leadership, but to the structures of the education system;rarely are ‘failing schools’ located in middle-to-affluent suburbs. Most frequently, they arelocated in poor, working-class and/or multicultural areas (Bell, 2003; Lupton, 2004). That isnot to say that we can absolve leaders and teachers from responsibility for poor practices.Rather we seek to understand the systemic failure of disadvantaged students and communi-ties which becomes reified in curriculum and through testing and management processes. Byunderstanding how these practices are structured to marginalise particular social and culturalgroups in ways that are coercive and invisible, we will be better able to change thosepractices. When the normalised practices within education are not challenged and the statusquo is preserved, then the most disadvantaged groups suffer through symbolic violence(Bourdieu, 1972) whereby they take on board the value-laden processes of education andbecome victims of those approaches through which they are effectively excluded andmarginalised.

In writing this paper, we draw on our experiences in working-class and culturally diverseclassrooms to illustrate the ways in which social practices work to marginalise particularstudents in their study of school mathematics while preserving the hegemony of thedominant classes; the same sort of pupils tend to succeed, and the same sort of pupils fail.The structuring of the field of education is a result of strategies engaged in by pupils andteachers within the specific field of school mathematics.

The political rhetoric in favour of the current policy direction suggests that testing,accountability, league tables, etc. are strategies to “drive up standards”, and in this way,state education will be improved for all—with the high-profile “successful” schools “pullingup” those seen as “failing”. At the same time, however, there is also a policy discourse aboutthe need for social justice with a consequent drive to support learners from less affluentbackgrounds.

Research, however, does not support the claim that current policies are reducingsocial exclusion. Whitty argued these structural shifts are “policies that do nothing tochallenge deeper social and cultural inequalities” (Whitty, 1997, p. 58). In fact, theshift towards a “free market” in education seems to have made little change at all tothese inequalities (Power, Halpin & Fitz, 1994, p. 39). The education systemcontinues to favour those whom it has always favoured—those of higher socio-economic status and those who know how to work the system and have a “feelfor the game” (Bourdieu, 1990, p.9). We would argue that this privileging does nothappen through oversight or accident but is a result of some deliberate, thoughpossibly covert, strategies.

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2 Using Bourdieu (1): introducing the habitus

A socially critical stance sees that “more affluent parents are more likely to have the informalknowledge and skill… to be able to decode and use marketized forms to their own benefit”(Apple, 2000, p. 248). Such parents do what they can for their children through both explicit andimplicit practices, and informal and formal procedures by drawing on their own sets of disposi-tions, habits and preferences—their taken-for-granteds. Bourdieu defines these dispositions as the“habitus” (Bourdieu, 1984)—a collection of informal skills and knowledge which participantshave constructed over time. But Bourdieu argues these are rather more than mere preferences asthey are operationalised by class positions. Consequently, learners engage in an activity (such aslearning mathematics) with a habitus that has already been shaped by their early socialisationwithin the family, home and immediate environment, and this shapes the way they act in andinterpret their worlds. Although the habitus is to a large extent durable and stable, it can bereshaped or transposed (Bourdieu, 1979, p. vii) by practices and sets of rules and expectations.Consequently, if students are to be positioned as successful, it is a task of schools to align theindividual habitus with the field, bringing their dispositions in line with those of the ideal pupil.These dispositions then operate to construct the pupil as “like a fish in water” (Bourdieu &Wacquant, 1992, p. 127) making sure they fall in linewith expected practices. It has been observedthat the habitus of more affluent parents often coincides with that expected in schools since

… the match between the historically grounded habitus expected in school and in itsactors and those of more affluent parents, combined with the material resourcesavailable to more affluent parents, usually leads to a successful conversion of eco-nomic and social capital into cultural capital. (Apple, 2000, p. 250)

This means that the children equipped with the right habitus are able to gain an advantage inschool and can exchange their dispositions for other rewards—grades, certificates and so forth.Effectively, this means that the embodied culture, or as Bourdieu prefers—the “habitus”—nowbecomes a form of capital, or in his terms “cultural capital”, that can be exchanged for othergoods within the economy of the school. Effectively, this process aids success in schoolwhereby those who are most likely to benefit are those from the groups whose class habitusaligns with the practices of the school. These students are generally not those from the lowersocio-economic classes. This explains, in part, why the marketization of education has donelittle to change the traditional models in place. The class with the habitus to take advantage ofthe market consists of those whose values are actually best reflected by the current system, andtherefore, it does not require or even desire change. Conversely, those whose class habitus doesnot resonate or align with the practices of school mathematics are in need of a reconstruction ofthe familial habitus if they are to be successful in school.

For society more generally, as well as for many parents, the subject of mathematics istraditionally held as important—as a gatekeeper to travelling successfully through the educationalsystem and as an inherent marker of intellect. Mathematics has been described as a “badge ofeligibility for the privileges of society” (Atweh, Bleicher, &Cooper, 1998, p. 63) which itself begsthe question of how this privileging works. Mathematics acts as a marker of success in schools,and consequently, mathematics is a useful context in which to explore the inequality apparent inthe education system as a whole because it performs a role of social segregation.

Mathematics education fails too many children; it fails children on the margins ofsociety, it fails children from ethnic minorities, and it fails children from social andcultural backgrounds that are different from the majority of mathematics teachers.(Gates, 2001, p. 7)

Structural exclusion through school mathematics

With this in mind, the field of mathematics education can provide us with a critical lens toobserve the day-to-day processes that lead to social segregation, and it is this process weexplore in this paper.

3 Using Bourdieu (2): introducing the field

We have already outlined the significance of the habitus as a key construct in understandinglearning of mathematics. However, Bourdieu’s stance is that because our dispositions arehistorically and socially constructed and sustained, the habitus does not operate in isolation;it operates within a set of socially organised rules through which power and control aredissipated and legitimised. This set of rules defines what Bourdieu calls the field of power,and for a Bourdieuian analysis, the field is a key organiser. Social fields are described as thesystem or set of objective social relations of power between those holding different positionswithin the field, but who share a related set of dispositions. In a social field, such asmathematics education, various individuals interact, and these encounters produce theaccepted social practices that typify the field (Griller, 1996, p. 6). In our case, the field ofmathematics education is a particularly appropriate unifying field because it encompassesand defines a clear set of rules that hold the discipline together. For example, the mathe-matics curriculum is structured in a particular way that privileges certain forms of thinking(Walkerdine, 1988; Apple, 1979), pedagogy is structured to distinguish between differentlearners, expectations become organised around visions of different futures, behaviours areshaped around the image of the ideal pupil, relationships with parents place teachers in veryspecific positions of authority and so on. All of which, when taken together, define thepractices we see in classrooms and relations between the learner and teacher, the home andschool contexts and between government and schools. Mathematics holds a privileged placein the school curriculum, and ability in mathematics is highly prized and valued, defininglearners as “can do” or “can’t do”. The way in which the social practices then becomeorganised and reified is defined by the balance between these points and among thedistributed social capital that groups of individuals have (Mahar et al., 1990, p. 8).

We feel comfortable locating our analysis within what we argue is a field of mathematicseducation because the practices within mathematics education are often unique to this field.For example, mathematics has an almost unique place in the school curriculum of allcountries; it draws on specific language patterns, its teaching is heavily structured and itcan be represented as a set of hierarchically organised skills usually divorced from applica-bility. As another example, literacy educators often see the practice of reading a numeracyproblem as a literacy event. However, in the field of mathematics education, it is much more.The reader/student needs to have a very specific interpretation of the world which the fieldaccepts as legitimate (Cooper, 2001). Not only does the student need to read, interpret andrespond to the literacy event per se, but he/she also is required further cognitive work thatgoes beyond literacy. Unlike literacy, the numeracy aspect requires significant mathematicalinterpretation of the task in order to be able to solve it, which extends the literacy event inways that are beyond a typical reading. Why are we being asked the question? Who wants toknow the answer? What will you do with the answer? What assumptions are behind thegivens? The role of language in shaping mathematical understanding for diverse groups hasbeen extensively studied by Cooper (2001), Cooper and Dunne (2000) and Dowling (1998).

The field will have particular practices that value and convey status on particulardispositions and learning. The accepted and common practices within mathematics educa-tion will differentially acknowledge what is seen as valued within that field, and the field

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will organise itself by imposing an objective structuring upon pupils and teachers throughcurriculum, pedagogy and the organisation of learners. For example, the students who areable to enter the field with a habitus that has been shaped by familial interactions that arevalued within the field will have those displays of language sanctified within the field. Thepractices within the familial context, which include language, dispositions, knowledgesystems, become internalised by the child, who then enters the school with these attributesembodied as their cultural background—their habitus. Within the field of mathematicseducation, the students who enter formal schooling with a linguistic repertoire (or linguistichabitus) that has a middle-class register are more likely to experience success (Zevenbergen,2000, 2001). Consider the middle-class families who, as Brice-Heath (1983) showed, engagein the standard classroom triadic dialogue—initiation–response–feedback—commonly usedby teachers of mathematics. These students are better able to engage with the interactionalstyle found in school than their working-class peers whose familial habitus is one whereparents engage in declarative interactions.

4 Using Bourdieu (3): integrating habitus and field

The habitus becomes a form of culture that is now exchanged for capital which is reifiedthrough practices (such as assessments) which effectively impose structure upon thoseparticipating in the field. The field values particular dispositions over others, and thesebecome entrenched into part of the habitus that can be exchanged for forms of capital.However, Bourdieu sees that cultural capital exists in three forms:

the embodied state, i.e. in the form of long lasting dispositions of the mind and body;in the objectified state, in the form of cultural goods (pictures, books, dictionaries,instruments, machines, etc.) which are the trace or realization of theories or critiques ofthese theories, problematic, etc.; as in the institutionalised state, a form of objectifi-cation which must be set apart because, as will be seen in the case of educationalqualifications, it confers entirely original properties on the cultural capital which it ispresumed to guarantee (Bourdieu, 1983, p. 243.)

Using Bourdieu’s framework, we are in a better position to understand the systemicfailure of students from marginalised backgrounds rather than looking at the problem as aresult of individual deficiencies on the part of particular pupils and parents. By usingBourdieu’s framing, therefore, we are taking a move away from deficit and individualisedthinking to a more encompassing and systemic approach. By analysing practices withinschool mathematics, the reification of social disadvantage—social, cultural or linguistic—can be challenged because we can see it as an arbitrary set of responses by individuals withpower rather than a natural state of how things just are.

To explain how power is enacted in a field, Bourdieu relies on the games metaphor whichenables us to offer a theorisation of how the practices within the teaching of mathematicsgive some students greater access to mathematical knowledge while excluding others.Through this game analogy, Bourdieu is able to describe how capital is realised within afield because “the kinds of capital, like trumps in a game of cards, are powers which definethe chances of profit in a given field” (Bourdieu, 1991, p. 230). How well one succeeds inthe game (or field) is determined by the “overall volume of the capital… and the compo-sition of that capital” (ibid., p. 231).

Using this metaphor, we see that learners enter the field of mathematics education—someof whom will be very aware of how the game is played in that field, while others have little

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idea and must learn the game if they are to be constructed as successful learners. Consideryoung preschool children whose parents have walked down streets talking about thenumbers on houses. These experiences position them well when they encounter the math-ematics classroom. Not only do they have knowledge about numbers, but some sense ofwhat bigger numbers mean and perhaps an intuitive sense of place value, and a sense of oddand even numbers. Collectively, these experiences rest well with the curriculum they will goon to experience. Furthermore, the informal teaching games that the parents or caregiverhave used to immerse the student in these number experiences are likely to resonate with thepedagogical practices the teacher will use in the classrooms. Already these students come toschool with knowledge of number and teaching episodes. These experiences are likely tohave been embodied by the children as part of their culture or as part of their habitus. Thesechildren, when playing the game of schooling, are more likely to be able to participate in thepedagogical practices of school and to display knowledge valued by the teacher, thuspositioning them as somewhat more “knowing” than peers who have not had suchexperiences.

Bourdieu (1984) argued that those players with the most resources are able to exert themost force over how the game is played. One only has to consider the power of the teacher,in comparison with the young students working in a classroom, in defining how interactionsare played out, or what is seen as desired in activities or responses. Bourdieu also argued,“…it is often the state of relations of force between the players that defines the structure ofthe field” (Bourdieu & Wacquant, 1992, p. 99). In this case, the teachers and students areable to exert and display what is seen as valued mathematical knowledge and ways of beingin the classroom that ensure that the field, with its inherent practices, remains intact.

For students coming in from middle-class families, their exposure to particular languagepractices provides them with better/more trump cards than their working-class peers.Walkerdine and Lucey’s (1989) work has shown how the mother–child interactions ofworking-class and middle-class families resonate less or more with the discursive practicesof schooling. For example, they found that middle-class mothers were more likely to useboth “more” and “less” in their interactions with their children whereas working-classmothers were more likely to use only the signifier “more”. This discursive positioning enablesmiddle-class students to enter the field of mathematics education with a linguistic habitusthat is more closely aligned to the practices within the field than their working-class peers.

Considering early childhood settings, teachers rely considerably on the use of thesignifiers “more” and “less” to develop many mathematical concepts—which number is 2more than 3; which group has more; what number is less, 3 or 5; and so on. Where earlylearners have access to this discourse, they are able to engage with the substantive learningas well as perform on test items that reflect this language. The converse is the case forstudents who do not have this linguistic repertoire when they enter schooling. This exampleillustrates some of Bourdieu’s key concepts that are integral to analysing the systemicexclusion of some social and cultural groups in their learning of school mathematics.

5 Language and social class—social heritage and the linguistic habitus

In this section, we draw on these concepts to explore how significant practices are allowed tooperate—the ways in which language and the linguistic habitus are differentially acknowl-edged and rewarded in mathematics. In doing so, students are effectively included orexcluded on the basis of their backgrounds, and use of language is a key aspect of thatbackground. Language is not merely a form of communication, but is a relationship that

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determines certain behaviours between individuals and groups, and thus defines powerstructures that in turn define the shape of the field, the forms of the habitus and theacquisition of social capital.

Language acts in numerous ways to define the forms of practices acceptable. Forexample, one needs to know how to “read” mathematical questions as real or imaginarycontexts (Cooper, 2001; Cooper, & Dunne, 2000; Dowling, 1998), and by extension, oneneeds to learn to interpret examination questions to be successful at certification; “hence”,“show”, “prove”, “find”, etc. each carry specific nuanced calls for type of mathematicalthinking and behaviour. Yet, understanding what is expected in school mathematics is amuch larger issue than just understanding test questions—the problems of interpretinglanguage within test questions is symptomatic of a difficulty with understanding thelanguage used within the school mathematics. Zevenbergen argues

The rich language of middle-class parents prepares children for the language they willencounter in school mathematics. Conversely, working-class children encounter formsof language in the home environment different from that which they encounter in theschool. (Zevenbergen, 2001, p. 43)

Hence, it is not simply what is being said but the structure of how it is said that constituteslinguistic competence (Bourdieu & Wacquant, 1992). Such competence is a form of capitalthat can be exchanged for success in the classroom. Middle-class children find the structureof classroom interactions familiar—they already have a large amount of linguistic capitalfrom home. To working-class children, the structure is much more confusing.

Without substantial reconstruction of their familial habitus, effective participation inthe mathematics classroom is transitory and intangible, making access to mathematicsand success difficult to achieve. (Zevenbergen, 2000, p. 220)

The need to fully understand the broader meaning behind mathematics tasks has seriousrepercussions for working-class children who are placed in higher-ability sets. This has alsobeen theorised by Bernstein’s ideas of linguistic competence and restricted codes.

Forms of spoken language in the process of their learning initiate, generalize andreinforce special types of relationship with the environment and thus create for theindividual particular forms of significance. (Bernstein, 1971, p.76)

The idea of a restricted code is a less formal form of talk, structured on shorter phrasesoften with tags such as “you know” and “know what I mean”. Forms of language drawing onelaborated codes have a sentence structure that draws on more complex forms, often usingunusual words and phrases, but also in the orientations to meaning (Bernstein, 1966). Thisorientation to meaning is most obvious in the moves towards abstraction and generalisationwhich is in contrast to meanings that are bounded by context specificity.

Boaler found that the lessons taught to higher-ability children proceed at a fast pace(Boaler, 1997a), which means those who are struggling with familiarising themselves with anew linguistic structure may get lost and hence fall behind before they have had a chancefind their feet. Here, the children’s social and linguistic capital serves as a filter, and one seesthat the cultural capital a child possesses must impact heavily on their success within school.

Language is an integral part of the social heritage that is brought into school mathematicsand becomes reified through various objective structuring practices. Language, in very broadterms, not only conveys particular concepts but also provides a medium through which thoseconcepts are conveyed. It is therefore important to consider not only the concepts that arebeing considered but also the medium of instruction. The subsequent success (or failure) of

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students is most frequently interpreted as an innate ability that facilitates, or not, success incoming to learn the disciplinary knowledge within the field of school mathematics. Theways in which failure and success are organised become an entrenched and naïve beliefwithin the field and one that is exposed through Bourdieu’s framework.

To better understand the processes by which the home habitus of working-class studentsclashes with, and constrains their access to, school mathematics, the work of Bourdieu canbe useful to theorise the symbolic violations that occur when the clash between school andlearners is foregrounded. While his focus was on social class, Bourdieu explains thateducators need to understand the processes around the conversion of social and culturalbackgrounds into school success. He argued that:

To fully understand how students from different social backgrounds relate to the worldof culture, and more precisely, to the institution of schooling, we need to recapture thelogic through which the conversion of social heritage into scholastic heritage operatesin different class situations. (Bourdieu, Passeron, & de Saint Martin, 1994, p. 53)

The notion of social heritage thus becomes a central variable in coming to understand thedifferential successes in school mathematics. Using a Bourdieuian framework, the lack ofsuccess for some social groups becomes a non-random event; it is a product of institutional-ised practices of which participants may be totally ignorant.

School mathematics represents a particular and powerful example of how social heritageconverts to academic success. Language is an integral part of the social heritage that childrenbring into school mathematics and is part of their “virtual school bag” (Thomson, 2002).This becomes reified to be seen as an innate ability. We do not subscribe to this view ofinnate ability, but it is an entrenched and naïve belief within the field. The language, in verybroad terms, not only conveys particular concepts but also provides a medium throughwhich those concepts are conveyed. It is therefore important to consider not only theconcepts but also the medium of instruction. In many disadvantaged communities, the clashbetween the culture of school and the culture of learners contributes significantly to thefailure to experience success of many learners.

6 Social class and ability grouping as social filters

One particularly pertinent issue within mathematics education is that of the dominantpractice of ability grouping, and this is a significant part of the structuring capability ofschool mathematics. What we describe later in this paper are the detailed ways in which thisprocess works at a micro level and how it acts as one element of a larger set of socialpractices. Using Bourdieu’s framework to understand the implications of ability grouping(Zevenbergen, 2003, 2005) has shown how the objective and subjective structuring practicesof ability grouping make for stratified learning. In this paper, we extend this work toexplicitly address issues of social class looking across two national cultures.

7 Applying Bourdieu’s theory

In the remainder of the paper, we draw on two illustrative cases which are drawn from alarger study and are used here to exemplify the concepts within Bourdieu’s theory and how itcan be applied to mathematics education. The larger study looked in detail at a group oflearners in a school and in particular at how social demographic patterns are linked with

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achievement. As part of this larger, more comprehensive study of the students, a range ofdata collection methods were employed. Mathematics lessons were observed, and pupilswere followed in other subjects. Pupils were interviewed and talked to informally, andparents were contacted at parent consultation evenings and followed up through interviewsin the home. Demographic data were obtained on where pupils lived. Whilst this larger studyfocussed on how pupils were organised and distributed through the structure of teachinggroups, for this paper, we just focus on two pupils who are in starkly differing social andschool positions. We use some demographic data but largely draw on interview data withpupils and parents to enable the application of Bourdieu’s model to understand the posi-tioning of the students through the practices to which they were exposed as part of theirschool mathematics experiences. In our analysis, we looked into patterns of language, formsof representation, as well as understandings of self and others.

Ethically, the study was approved through the procedures of the university, participants wereinvited to take part and all names are pseudonyms. The third author was a teacher ofmathematics at the school and obtained approval from the head teacher and head of departmentto undertake the study. Particular care was taken over the power relationships in data collection.

The two pupils we focus on, a boy (Cory) and a girl (Caitlin), come from the same schoolin the UK. The school population is diverse, so the cases serve to illustrate the manifestationof practices within the field and how they can position students as learners of mathematics.We particularly draw on Bourdieu’s constructs to illustrate the application of his theory tobetter understand the marginalisation and reification of students in and through schoolmathematics. It is our intent to focus on the social demographics of the students as thesewere the basis for the selection of these illustrative cases. “Ability grouping” is standardpractice in UK schools, much more so than in Australian schools or indeed other countrieswhere the practice is outlawed. While we are opposed to the use of the term ability grouping,due to its alignment with the hegemonic discourses of innate ability, we use the term here asit is one that is adopted by teachers and educationalists. Our intent is to explicitly disrupt theuse of such a term that reifies social characteristics as being somehow innately related toachievement or “ability”. The correlations between ability grouping and social background,as is made possible through Bourdieuian lenses, enable a serious challenge to one of themost hegemonic discourses in school mathematics—that of ability. The analysis that we nowundertake highlights the power of a social theory to critically appraise a taken-for-grantedpractice in school mathematics.

Caitlin Cory

Gender Girl Boy

SES/class status Middle/affluent Working/disadvantaged

Maths group High-ability group Low-ability group

8 Case study 1: Caitlin

Caitlin was placed in the highest-ability group in year7—the first year of secondaryeducation in the UK. She was put into this group on the basis of achieving a 5a in her UKNational Standard Assessment Test taken in the final year of primary school (year6), thisbeing the highest possible level she could have achieved. This already gives Caitlin anadvantage through the “dividend” of being in a top set.

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The benefit of being in a top set rather than a low set was particularly strong inmathematics, where on average the top set dividend was just under one grade atGCSE. (Ireson, Hallam, & Hurley, 2005, p. 455)

Caitlin came over in class as an articulate and confident girl with a high degree ofmathematical skill and in general achieved well on tests and other markers within the schoolcontext. She was described by her teachers as a student of “high intelligence”. She isinvolved in many extra-curricular activities in school including playing for a netball teamand performing in school productions. Her year6 teacher described her as “precocious”, andher year7 form tutor described her as a “very prominent member of the form” who waselected by her peers as the class representative on the School Council. Caitlin is an only childwho lives with her mother. She is clearly very close to her mother and spends time with herout of school doing a range of activities—“shopping, swimming and visiting differentplaces”. She regularly visits her estranged father. The most recent such visit consisted ofgoing to “a small music festival” and seeing all her “friends and family from dad’s side”.

Caitlin’s mother has just completed a PhD and is taking up a teaching post inuniversity; Caitlin talks about this with a lot of pride. She explains that they have hadmoney troubles because her mother has been completing this but that it is worth it.Interestingly, she identifies not only future financial rewards but also the personalsatisfaction her mother has gained. Caitlin sees education as an important part of life.Caitlin’s mother identifies “happiness and her goals” as being what she hopes Caitlinachieves at school, and this seems to sum up Caitlin’s attitude towards school; shebelieves learning can provide personal fulfilment as well as professional goals butfeels no pressure to quantify these goals.

In spite of a relaxed attitude towards school, Caitlin does have a healthy desire to achievehighly. Having just completed the end-of-year mathematics examinations, she made thefollowing comments:

It is frustrating because I know I could have achieved a 6a. I needed two more marks.It was just because I didn’t pay attention to the units.

She was not worried by making mistakes but was keen to talk about her errors andexplained that she found working out why she made mistakes interesting and added, “Itmeans I can do better next time”.

Caitlin professes both to enjoy mathematics and to believe that it will be important in thefuture, although, beyond numeracy and computers, she cannot quite explain why it will beuseful and just expresses a belief that it will important for her to have an all-round education.Functionality and utilitarianism here do not matter; the key is to be successful and to trust inthe system to bring that success.

Both Caitlin’s mother and her father influence Caitlin’s positive attitude towards math-ematics. In spite of describing mathematics as “not my strong point”, Caitlin’s mother is verypleased that her daughter is achieving highly in mathematics. Echoing the importance Caitlinattached to mathematics, she describes mathematics as:

One of the most essential and basic requirements of education, particularly in ameritocracy.

Caitlin’s father is clearly keen on mathematics as a subject. Caitlin says,

My Dad tells me a lot of stuff. I have a head start in maths because he is very good at itand likes to talk to me about it. Sometimes he will mention something and I’m like‘I’ve never seen that before’ and he will explain it to me.

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Caitlin attended parents’ evening with her mother and contributed fully to the conversation.Caitlin clearly felt her opinion was valued and equally valued her mother’s and her teacher’sopinion. This linguistic competence and confidence is part of Caitlin’s habitus. She is clearlyused to conversing in this way at home; she has a respectful manner but participates as an equal.This style of conversation is most compatible with interactions that take place in her mathe-matics classroom and clearly illustrates a high level of agreement between home and schoolhabitus; she has a strategic “feel for the game” (Bourdieu, 1990, p. 9).

Having a feel for the game is important; it enables the player, in this case the student, tobe able to read the game, predict the expectations, anticipate actions and engage in activitiesin a meaningful way. Caitlin’s parents were creating opportunities for her to engage in thegame of school mathematics in ways that were enabling. In so doing, they were creating thepossibilities of a mathematical habitus whereby Caitlin could engage with and experiencethe game in productive and meaningful ways. Part of the habitus includes aspects oflanguage whereby not only the spoken and written language are valuable, but also the waysof interacting become important. Bourdieu refers to this as the linguistic habitus. A specificexample of the linguistic capital Caitlin has received from home came when Caitlindescribed how her father helps her with homework when she is staying with him.

Caitlin: If I ask him a question, he asks me to read it out and he goes ‘you know thisone what is it about?’ Then he makes me work it out for myself and just keeps askingquestions to help me. It’s frustrating. [She laughs]. No, I know it’s a good thing really.

This description could have been about a teacher helping a student. The use of question-ing to help scaffold a student’s thinking is common in most classrooms. Caitlin is alreadyfamiliar with that structure from home and even recognises it, albeit a little reluctantly, as aneffective technique. The linguistic capital is utilised in school. The transference of thehabitus she has developed through her interactions with her father has enabled her to engagewith the game played by the teacher. In so doing, this habitus is recognised and validated bythe practices within school mathematics, thus positioning Caitlin as a successful learner inthe context of school. This is evident when she talks about the experiences within her groupand her interactions with the tutor:

Caitlin: I just get along with everyone. They’re of the same ability and can understandwhat you say. If you work in a group in the tutor group sometimes you have to explainthings to them and they have to explain things to you.

Her comment indicates fluency and familiarity with the discursive practices adopted bythe school and was evident in her interactions with her father. The linguistic habitus she hasbeen able to create through her familial interactions are exchangeable within the economy ofthe school.

While it is not possible to argue here with any certainty, it would appear that herfather’s questioning techniques may also have created a habitus in which Caitlin has astrong sense of mathematical processes and questioning. Many of the practices inschool mathematics require individual work and problem solving. As is the case inboth the UK and Australia, students are expected to engage with problems that arenovel so they need to have a repertoire of skills that enable them to sustain interest inorder to solve the task. It would appear from our interactions with Caitlin’s familythat such attributes have been fostered. Caitlin enjoys working independently and,says “I quite like tests actually. I know people don’t but I like it because it’sindependent”. She displays a high level of self-motivation and has a series ofstrategies to employ in a test situation.

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A disposition for perseverance and engaging with a problem is a key characteristicfor success in many classrooms. Caitlin appears to have developed such dispositions.These can be seen in her dialogue when she talks about how she solves mathematicaltasks:

I: What do you think you need to do to be successful in maths tests?

C: You need to go through everything in your mind when you’re stuck on a question.Or work out a few different ways if you can’t remember how you did it before. Likethe question about the cake in the test, I didn’t understand it at first but ended upgetting it correct.

I: How do you think you did that?

C: First I tried to ask myself the question in different ways so I understood it better.Then I worked it out in a few different ways and got different answers and thendecided what was the most logical one.

I: Why do you think you’re able to work question out that way?

C: I don’t know… it just comes to me. When I don’t really understand the question,I’ll work the completely wrong answer out but then it’ll bring me closer to the realanswer.

What is clear from Caitlin’s case is that her parents have been inducting her intoparticular patterns of practice, enabling the construction of a particular mathematicalhabitus—one that aligns well with the practices within her high-ability group. She hasdeveloped a mathematical habitus that aligns with the recognised practices of themathematics classroom so she is able to exchange these dispositions for rewards, anda healthy concept of herself as a learner of school mathematics. However, in addition,to the school she becomes positioned as “able” and receives the privileges thatbecome associated with such a label.

9 Case study 2: Cory

Our second illustrative case is of a boy, the same age as Caitlin but who is placed inthe lowest-ability group. Like Caitlin, he lives with his mother and two siblings, andsees his estranged father on a regular basis. Cory is hoping to be a sports coach andhis mother “hoped” he would be able to achieve the grades to enable him to go tocollege. The language of “hope” here is indicative of the perceived lack of controland agency in this process. We see this language of “hope” as a recognition that thesystem is already positioning Cory and his family as having little control of theireducation. Unlike Caitlin’s family, Cory’s family had a very different position onschool and mathematics as evident in his mother’s comment:

I don’t think a lot of the maths taught are necessary as most of the kids will never usethe difficult stuff again. I think people like Cory should be taught and concentrated onpractical and everyday maths.

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Unlike Caitlin, he is uncertain about his father’s opinion and indicates that he rarelydiscusses school with his father. Cory’s mother has a strong opinion against homework:

I don’t like the idea of homework. I think it should only be sent home when absolutelynecessary and then with clear guidelines of how to do it, since Cory sometimes can’texplain what he needs to be able to do. Maybe a help leaflet for parents to jog theirmemory could be sent—it’s twenty years since I left school and I can’t remember howto do much of the stuff. Also, he needs longer to do some pieces because I’m notalways there to help him.

The familial habitus that is potentially created by Cory is one that is incongruous with thedemands and expectations of schooling, thus positioning him outside the practices ofschooling. Cory is less likely to be rewarded by the school and hence has less chance ofbeing seen as successful.

The ethos within Cory’s mathematics classroom is quite different. The group is small butchallenging so that not only is there an impoverished offering of mathematics, but theengagement by the group is not with mathematics. It is far more social. This creates a verydifferent scenario for Cory than for Caitlin in terms of possibilities for construction of amathematics habitus, and the recognition of dispositions that are validated within thestructuring practices of school mathematics.

In the maths class there are some people who I get distracted by. They make me laugh.

In contrast to Caitlin’s experience, where her family habitus had positioned her well forthe practices within her mathematics classroom, Cory’s habitus was not aligning with thedominant and powerful practices of school mathematics. Unlike Caitlin, Cory oftenappeared to understand part of the question but to lack the confidence or ability to inferthe full meaning from this. In fact, when referring to some questions he had left unanswered,he said that he “thought that’s what it meant” but that he “wasn’t sure”. This lack ofconfidence seems fuelled by his belief that to be good at mathematics you need to“remember a lot and work stuff out quickly”. Whilst Caitlin saw and utilised the interrelatedstructure of mathematics, Cory has a compartmentalised view of the subject as a series offacts and methods to be memorised—a viewpoint which makes it much more difficult tosucceed.

Within a Bourdieuian framework, we see one of the structuring practices of the field, inthis case ability grouping, has created an environment for the constitution of a particularmathematical habitus. Within this, the subjective structuring practices, in this case Corey’ssense of himself as a learner of mathematics, which have come about as a result of thestructuring practices of assessment, have created a learner who is not confident of himself asa learner of mathematics. This is seen when he describes the important values within thefield as being linked to being able to remember and work quickly. These dispositions conveypower in Corey’s mind, and so without them, he lacks what is seen as valued, thuspositioning him as a not-so-strong learner of mathematics.

Another problem for Cory was in responding to the numerous questions that requirestudents to “Explain your answer”. When interviewing Cory, it was clear that he didunderstand how and why he had given the answers he had. However, the explanations hegave were not always immediately clear, and he needed to be prompted to give furtherclarification. Again, in subsequent discussions with Cory, it became apparent that in lessons,he was usually given few, if any, mathematical prompts; it was more that he needed answersto be given some linguistic structure. Clearly, this would translate to problems when tryingto write down a written explanation.

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On further observation, the lessons taught to the lowest-ability group contained lesscomplex language; questions were usually closed and easily understandable, single step;explanations were accepted without pushing for greater clarity when there was a feeling ofhaving seen evidence of understanding. Consequently, Cory is not exposed to the richerlanguage common in Caitlin’s classroom. This means he is not given so many opportunitiesto develop his linguistic competence, which will continue to adversely affect his test results.It also affects his understanding of what it means to “do mathematics”. Perhaps a greateremphasis on explanations, with greater exploration of the “why”, might allow Cory to makemore connections between concepts and to develop understanding which would move himaway from seeing doing mathematics as memorising facts and methods.

10 Discussion

This examination of these two pupils’ differing learning trajectories in mathematics focuseson the influences of familial habitus and linguistic capital alongside the environment,namely the ability group, they are taught in. The ability grouping of both students in manyways is already showing how it is starting to determine their future attainment.

The consequences of setting and streaming decisions are great. Indeed, the set orstream that students are placed into, at a very young age, will almost certainly dictatethe opportunities they receive for the rest of their lives. (Boaler, 1997b, p. 594)

It is unlikely that either pupil will change ability group even at this early stage of theirsecondary education, as this grouping seems in many ways to be a product of much morethan mathematical knowledge. In fact, it seems to be the grouping that has an effect on thetype of mathematical knowledge the two are presented with and consequently how far theyare able to attain. Yet this form of organisation becomes natural and normal to the school andthe teachers, so much so that, in the UK at least, it is inconceivable to many/mostmathematics teachers to consider how it might be otherwise. Furthermore, there is evidencethat teachers extend this to their beliefs about pupils’ capabilities, The pupils such as Cory,who do not fit the ideal pupil mould, experience a mathematics education that is consider-ably structured, restricted and controlled, by teachers who assume such pupils are not(cap)able of higher-order thinking (Zohar, 1999; Zohar, Degani & Vaaknin, 2001; Zohar& Dori, 2003).

Caitlin shares many similarities with her classmates. She lives in a similar area, usessimilar language and has similar values. Her mother may not currently possess a hugeamount of economic capital, but this does not hold her back. She and Caitlin’s father haveendowed their daughter with much social, cultural and linguistic capital. As she enters theeducational field, she has distinct advantages; her capital earns her an enhanced reputation, acomfortable position in the highest-ability set and high attainment. As Grenfell summarises,

Capital attracts capital, but, as in the case of education, we do not enter fields withequal amounts, or identical configurations, of capital. Some have inherited wealth,cultural distinctions from up-bringing and family connections. Some individuals,therefore, already possess quantities of relevant capital bestowed on them in theprocess of habitus formation, which makes them better players than others in certainfield games. Conversely, some are disadvantaged. (Grenfell, 1998, p. 21)

Here we can see how Caitlin is systematically and structurally privileged andhow her progression through the schooling system has been thoroughly efficient.

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Her familial habitus is in close agreement with that of the school and has requiredlittle adjustment. The linguistic structure at home and school shares many common-alities. The attitudes towards education, largely derived from parental opinion andactions and experiences Caitlin has had, are compatible and help to create aproductive approach towards learning mathematics. Her experiences have effectivelycreated a habitus that aligns with the structuring practices of school mathematics,that is, the field. This enables Caitlin to exchange aspects of her habitus for therewards of the field so that her culture becomes a form of capital valued within thatfield.

Cory’s progression unfortunately has not been so smooth, and he has not achievedthe same degree of academic success as Caitlin. There is no extreme friction betweenCory and the school system, but they do not fit together as naturally as Caitlin andthe school seem to. As with Caitlin, Cory’s attitude towards education echoes hismother’s opinions and his impression of his father’s disinterest. His view of educationis narrower than Caitlin’s; he sees the purpose of schooling as purely functional andlooks at individual skills and knowledge as opposed to the value of an all-roundeducation. Although schools often emphasise the functionality of what they teach, it isthe case that they more fully embrace the middle-class notion of the importance ofbeing well educated. In Cory’s case, his familial habitus is not resonating stronglywith the practices of the field, and his culture has little value within the field, so hehas been positioned as a poor learner of mathematics—and he himself accepts this.The conventional is thus taken as natural by all parties—it is the way it is and has tobe. In this way, Cory becomes the willing victim of his own limitations. Here, we seeBourdieu’s idea of “symbolic violence”—which he sees operating though “The complicity ofthose who do not want to know they are subject to it or even that they themselves exercise it”(Bourdieu, 1991, 164).

Specifically, Bourdieu sees this as how the dominated come to accept their owndomination as legitimate (Bourdieu & Wacquant, 1992, p. 167) in exactly the wayCory does. In the field of mathematics education, symbolic violence has been enactedagainst Cory, but such violence works only when the participants willingly accept thepractices and outcomes—it is “particularly insidious due to the fact that it is exercisedwith the agents’ full complicity” (Nolan, 2012, p. 205). This is the case with Coryand his family. His position in the low setting or ability group and the reification ofhis lack of success act as a form of symbolic violence against the culture that hebrings to school. This process, whereby both sides of a social power divide (theorthodox and heterodox in Bourdieu’s terminology) adopt a tacit acceptance of thedominance within the field, is termed by Bourdieu as the “doxa” and is described as“The set of core values and discourses of a social practice field that have come to beviewed as normal natural and inherently necessary” (Nolan, 2012, p. 205) andrepresenting where “There is a correspondence between the objective order and thesubjective principles of organization the natural and social world appears as self-evident” (Bourdieu, 1972, p. 164).

A Bourdieuian analysis allows us to conjecture patterns in the practices adoptedby those occupying similar positions in the field. We can expect therefore theexperiences of Cory to be not too dissimilar from others who share his engagementin the field.

On a more practical note, Cory’s mother is less well equipped than Caitlin’s father to helpwith mathematics homework. In fact, the whole issue of homework in some ways discrim-inates against Cory. Whilst Caitlin receives additional knowledge and develops linguistic

Structural exclusion through school mathematics

capital through completing homework with her father’s input, Cory’s mother struggles tohelp him. In one family, it is an opportunity to share and develop the child’s education; in theother, it is more of a burden. As de Carvalho comments

Indeed, from the family point of view, homework may be seen either as alegitimate need and a desirable practice, or as a burden and an imposition,depending on variable material and symbolic conditions of diverse families.(De Carvalho, 2001, p. 116)

Language also plays a huge role in the disparity between Caitlin and Cory.

Within the mathematics classrooms, legitimate participation is acquired and achievedthrough a competence with written or spoken texts, or both. To be constructed as aneffective learner of mathematics, students must be able to display a competence withthese forms of texts. (Zevenbergen, 2000, p. 202)

Caitlin has a significant amount of linguistic capital supplied from home. Inaddition, the classroom environment she learns in is rich in language. Drawing onVygotsky’s idea of social learning, we see that Caitlin learns in an environment betterequipped to facilitate learning, which from this perspective is the internalisation ofsocial interactional processes, most commonly realised through language (Peterson,Janick, & Swing, 1981). Cory does not possess the linguistic capital that Caitlin doesas a result both of social background and classroom environment, as Bourdieuexplains that habitus is an evolving concept, and in that process, habitus and fieldact on each other (Wacquant, 1989). Cory enters the school field with less linguisticcapital and is placed in the lowest-ability group. Then the school field, in particularthe classroom setting, which is advantageous for Caitlin, fails Cory. Cory’s classmates,in all likelihood, possess similar linguistic competence, and the teacher, in a doublebind, has few options but to attempt to “pitch the work correctly” and so fails toenrich the language they are exposed to. This, in turn, has fostered a limited andunproductive view of mathematics, which contrasts with the more positive modelCaitlin has constructed.

One important effect of the disparity in linguistic capital is performance in tests. Caitlinhas developed a successful approach to tests; she has the ability to independently deciphertest questions and respond to them clearly. Cory, on the other hand, under-achieves in a testsituation as he fails to understand questions, applies an incorrect level of “appropriateness”(Cooper, 2001) and is unable to structure coherent explanations. As a result, Cory wasplaced and remains in the lowest-ability group, and through this, comes the creation offutures.

11 Conclusion

What we have tried to illustrate in this paper is the application of Bourdieu’s theory to twoillustrative cases. It is an example of how the dominant power relations in the field ofmathematics are operating to convey an expected orthodoxy (that set of behaviours andrelationships demonstrated by Caitlin) and marginalising any heterodoxy (that set of behav-iours and relationships demonstrated by Cory). So what is actually only one set of responses(Caitlin’s) is seen as taken for granted and natural. By offering this misrecognition of thearbitrary as the natural way of things, it forces actors to accept as legitimate that which ispossibly against their best interests. Cory and his family accept his lot; he is just a struggler

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forced to play a game and yet accepting of playing a game that is not in his best interests. AsBourdieu says:

The earlier a player enters the game and the less he is aware of the associated learning,the greater his ignorance of all that is tacitly granted through his investment in the fieldand his interest in its very existence and perpetuation and in everything that is playedfor it, and his unawareness of the unthought presuppositions the game produces andendlessly reproduces, thereby reproducing the conditions of its own existence. (Bourdieu,1990, p. 67)

Cory and Caitlin’s position within the mathematics setting system seems to have beendetermined by far more than “mathematical ability”. The differences originate in social statusand familial habitus. Research suggests this fact is not restricted to these individuals, andcertainly, the groupings in this one school demonstrate that working-class students are over-represented in the lower ability sets. A vicious circle has developed: working class students aredisadvantaged on entering the school field as they have a less compatible habitus; this manifestsitself in underachievement in tests and in less impressive contributions in the classroom, whichresults in placement in lower ability sets. Here, they are surrounded by pupils with similarhabitus and linguistic incompatibility with the school mathematics discourse, which results inslower progression and continued underachievement in assessments, thus widening the gapbetween these students and the, largely middle-class, pupils in the higher sets.

What these detailed case studies of Caitlin and Cory suggest is that there is somesubstance to the joke about how to be more successful at school—change your parents.For as our analysis has indicated, the parents as the primary socialisers have endowed theirchild with success or struggle (habitus) in an education system (field) that is providingadvantages for some and restrictions for others (capital).

We might ask how we can do something about this process of exclusion in the field ofmathematics education because it is exercised not only with the complicity of those whoultimately suffer (the Corys of this world) but with the implicit and explicit complicity ofteachers of mathematics. Nolan (2012) used a Bourdieuian analysis to look at the practices ofmathematics teacher education where she reminds us that the school is a “site of reproductionand regulation” (p. 213) where teachers have their own sets of dispositions shaped by their ownjourney through education and professional training. She argues there is a set of professionaldiscourses that orient the field to operate in such a way to create Caitlins and Corys. Nolansuggests (2012, pp. 206–211) (and we paraphrase only slightly) that mathematics teachers arepressed for time to deliver a curriculum devoid of creativity and innovation which draws largelyfrom traditional pedagogical paradigms where testing and competition are paramount. Thisprocess creates the Corys and the Caitlins who become self-fulfilling prophecies of the successand failure; Caitlin cannot be a success without the failure of Cory.

A problem here for mathematics education research is that it often (re)presents a fieldwhich has incongruencies within the field of mathematics education. By exposing conflictswithin current practices, we offer a heterodoxy to schools and classrooms which serve as theorthodoxy. A possible way forward would be to broaden teacher education courses toencourage new teachers to examine the nature of social conditions in schools and theorisethe lack of fit between some but not all pupils and the demands of mathematics education.Furthermore, though, we ought to be prepared to expose the lack of fit between schoolmathematics and mathematics teacher education (Adkins, 2004, p. 191; Nolan, 2012,p. 212). This process of self-reflection, however, is not assisted by the failure of muchmathematics education research to recognise the importance of social backgrounds inmathematics educational achievement.

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