Constructivism as a liberal bourgeois discourse

ROBYN ZEVENBERGEN

CONSTRUCTIVISM AS A LIBERAL BOURGEOIS DISCOURSE

ABSTRACT. Using the theoretical tools offered through the writings of Pierre Bourdieu, this paper develops a critique of constructivism. Constructivism has assumed a dominance with the field of mathematics education but, as an epistemology, it ignores the social implications of the construction of meaning. It is argued that constructivism is a liberal discourse which valorises the individual construction of meaning. In doing this, the social and political contexts in which mathematical knowledge is located is ignored and the marginalisation of many social and cultural groups is legitimated.

My goal in writing this paper is to highlight the processes of legitimation for socially recognised, valued and reified knowledge and the implications this has in the wider social and political context within which such knowledge is constructed. In writing this paper, I have been informed by the writings of Pierre Bourdieu whose theory of schools as a conservatory force (Bourdieu, 1982) and theorising of the notion of field has provided a framework within which to position this critique of constmctivism.

This essay is working from outside those dominant discourses within the field which address the processes of learning. These discourses, which include constructivism, behaviourism and Vygotskian perspectives are essentially discourses of description which can only provide limited critique. The critique developed from within a particular discourse does not touch the core of the theory. In contrast, this paper is political and argues that dominant research paradigms produce particular forms of knowledge which have implications for the outcomes of schooling. A critique of the knowledge which is produced within the constructivist research paradigm is proposed. It is argued that constructivism, in particular, the radical version, with its focus on the individual construction of meaning, denies the social and political context of mathematics and in doing so, legitimates the marginalisation of many social and cultural groups. The theoretical framework from which this essay is located can be conceived as critical and can be represented by Bourdieu's argument that

...the limits of human understanding must be defined, not in universal terms, but in terms of their social construction. At every moment, we are limited by social censorship (Bourdieu, 1992, p. 48).

To this end, my purpose in this essay is to raise concerns about the limits of constructivism; how it assumes such a dominance within the

Educational Studies in Mathematics 31:95-113, 1996. �9 1996 Kluwer Academic Publishers. Printed in the Netherlands.

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field; the implications for conceptualising the learning of mathematics; and its inability to theorise adequately the marginalisation of significant numbers of students.

What I propose will be useful in understanding the social censorship that guides and constrains the work within the field of mathematics education, a problem that I see as critical to the development of work within this field. The logic which is dominant within the field of mathematics constructs and constrains what is seen as legitimate and hence, will determine what are seen as truths within the field.

The Dominance o f Constructivism

Within education as a whole, psychology dominates what is seen as legitimate and hence provides the framework within which a significant amount of knowledge is generated, but also to provide a lens through which education is perceived. Walkerdine (1984) has defended such a position by arguing that Piagetian psychology provides a lens through which teachers' work is constructed and constrained. Further to this she argues that the concentration on cognitive psychology to understand performance in mathematics fails "to engage with certain central aspects of the way in which oppression is experienced" (Walkerdine, 1990: p. 51).

Taking a constructivist view to understanding the learning of mathematics is to deny the role that mathematics plays in the production and reproduction of social differences within our society. This is achieved through the individualism and mentalism embedded in constructivism which constrains what is seen as legitimate and does not offer an effective tool through which macro issues associated with the construction of social differences and marginalisation can be explored.

At this point in time, the constructivist epistemology has assumed a dominance with mathematics education. This is evidenced in the amount of journal articles which adopt a constructivist framework. Furthermore, specific editions of journals such as Educational Studies in Mathematics and Journal for Research in Mathematics Education, along with other non-mathematics education journal such as the Educational Researcher have devoted special issues to constructivism. Similarly, the Reforms in the U.S.A., the Profiles in Australia and the National Curriculum in the U.K. all use constructivist language to underpin the learning and teaching of mathematics. Within the field, mathematics educators, for example, Ellerton and Clements (1992) and Lerman (1993) have explicitly stated that constructivism has become a dominant paradigm. It is the dominance of this view that I want to challenge since it constrains and confines what is seen as legitimate within the field.

CONSTRUCTIVISM AS A LIBERAL BOURGEOIS DISCOURSE 97

My particular focus is with the constructivist writings in mathematics education and how this epistemology does not take account of the processes by which a significant proportion of students come to be marginalised as a consequence of their exposure to mathematics. Constructivism does not take issue with this process, but instead, with its focus on the individual construction of meaning, ignores the wider socio-political context within which learning occurs, and the implications of that learning beyond the formal school context. As such, the implications of constructivism need to be understood within the wider socio-political context within which it is being espoused and embodied. Much credit is given to it as representing a break from the essentialist epistemologies and methodologies of positivism and behaviourism which have been dominant within the field for some time. However, as an epistemological position, it has similar social consequences of these earlier positions. It is at this point that the limits of a constrnctivist epistemology become apparent and it is this which is the focus of my argument.

At best, constructivism can offer to "study mathematical learning, to develop appropriate teaching strategies, and to reflect on everyday prob- lems of schoolteaching" (Noddings, 1992: p. 18) since it concentrates on developing a theory of cognition and a methodological perspective as opposed to a theory of the construction of social and cultural differences. Yet as most mathematics educators, indeed most educators are critically aware, a significant portion of students who, upon entering the formal mathematics learning environment of schools and their consequent exposure to the content and pedagogies of these institutions, become positioned in ways which construct them as ineffective learners of mathematics.

My intention is to explore how constructivism has assumed a dominance in the field and the constraining implications that this has for current and future theoretical and methodological work. My concern is with the structuring influence of constructivism on mathematics education, and its failure to account for the wider social implications that the learning of mathematics has on those students exposed to institutionalised mathematics.

MATHEMATICS EDUCATION AS A FIELD

In this section, I intend to develop Bourdieu's notion of the field, where mathematics education is seen as a field, so as to deconstruct the practices which work to enable certain forms of knowledge to be seen as legitimate.

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Fields are at all times defined by a system of objective relations of power between social positions that correspond to a system of objective relations between symbolic points: works of art, artistic manifestos, political declarations, and so on. The structure of a field is defined at a given moment by the balance between these points and among the distributed capital (Mahar, Harker, and Wilkes, 1990: p. 8).

Mathematics Education can be seen to be functioning within its own logic and rules which define and constrain what is seen as valued and legitimate. Situated within the broader field of Education, there is a par- ticularised intellectual legitimacy which is produced and recognised and is dependent upon the conditions that operate at any given time. At any time participants have recourse to a number of strategies through which they can establish legitimacy. However, such strategies are often not at the level of awareness, but to establish credence within the field it is necessary to comply with the unspoken rules and logic, and through the amassing of "capital". For Bourdieu, social groups, in this case mathematics educators, are constantly pursuing and attempting to accumulate forms of capital which will increase their standing within the society.

In contemporary mathematics education, constructivism can be seen to be a form of symbolic capital whereby those participants who amass the forms of capital are able to convert this symbolic capital into other forms of capital to convey status, power and legitimacy within the field. Once participants have gained status within the field, certain forms of knowledge are socially legitimated, and as a consequence, also censored. Accordingly, within mathematics education, constructivism has come to assume a certain legitimacy which will confer power and status to those who embody it in the ways that are valued within the field. As such, constructivism has become a form of capital which can be exchanged for power and status within the field.

Fields and Capital

There are certain dominant practices which confer power and legitimacy to some practices and people while relegating others to marginal status. For example, in mathematics education certain discourses and theoretical positions do not have the same legitimacy as those which comply with the unspoken rules goveming the field. One only has to consider what knowledge is most often published in the major international journals or presented at conferences. More often this is embedded in the empirical sciences.

Bourdieu's notion of capital is central to understanding the construction of social space and power where "the kinds of capital, like the aces in a game of cards, are powers that define the chances of profit in a given field"


(Bourdieu, 1985: p. 724). Bourdieu (1983: p.43) argues for three forms of capital: economic capital which be directly related to the money and institutionalised as property rights; cultural capital, which can be related to money under certain conditions and is institutionalised in the form of educational qualifications; and social capital, which can also be related to economic capital but is most frequently seen as social "connections" and is institutionalised in the form of nobility titles. Of particular relevance to the position being developed here is cultural capital for it enables a theoretical position to be proposed where there is unequal achievement and success within an field. It also permits an understanding of the processes through which the field acts as a form of censorship since at any time a discourse is both an "enactment and effect of symbolic violence" (Moi, 1991: p. 1022). Rather than explaining the success of participants from different social groups as individual and natural attributes, Bourdieu saw the distribution of cultural capital as the linchpin to such success. Bourdieu (1983) argues that cultural capital exists in three forms:

[T]he 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.., which are the trace or realization of theories or critiques of these theories, problematics, etc.; and in the institutionalized state, a form of objectification which must be set apart because, as will be the case in educational qualifications, it confers entirely original properties on the cultural capital which it is presumed to guarantee. (p. 243)

Power, legitimacy and status within a field is determined by the distribution of specific forms of capital, and those who are invested with the right to speak on behalf of the field are those agents who have been recognised as the possessors of capital. Those who yield power are those who are able to yield symbolic power. According to Wacquant (1987) symbolic capital refers to

the prestige or 'social credit' conferred by socially accepted or socially concealed uses of other forms of capital . . . . It is, as Bourdieu phrases it, "capital denies, recognised as legitimate, that is, misrecognised as capital." (p. 67)

To achieve power and legitimacy within the field demands the amassing of the symbolic capital that is determined by the rules and logic of the field. For Bourdieu, symbolic capital is any form of capital which is represented in a relationship with knowledge (Bourdieu, 1983: p. 255). In most instances, the knowledge which is embedded in the empirical sciences confers more status and power to those who are aligned with this paradigm and its forms of knowledge. To work outside the empirical sciences effectively engenders exclusion, marginalisation and stronger forms of censorship. The practices which confer power and status within the field serve to legitimate those practices and discourses which support

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the existing status quo. For example, mathematics education occupies a significant amount of intellectual space within the school curriculum and resource allocation, and as a research body, much of the work within the field serves to preserve and protect its existing state of authority. This is achieved through and by those agents who are seen to be supporting and legitimating the existing status quo. By and large, practices in mathematics education support the importance of the field and the major concerns of researchers and educators is aimed at improving the teaching and learning of mathematics as opposed to developing critiques of the field. The former supports the purported importance and status of the learning of mathematics, while the latter calls into question the legitimacy of the status of the discourses of the field.

Within mathematics education certain rules and logic are dominant, so that to be constituted as an effective participant in the field, it is essential to assimilate the tacit rules which govern what is seen as legitimate and valued knowledge. Those who amass this symbolic capital are able to convert it into other forms of capital such as in the form of professorial positions (institutionalised capital) or higher salaries, research grants, or bonuses (economic capital). At this point in time, constructivism provides a vehicle by which many mathematics educators are able to amass symbolic capital to become positioned as legitimate and powerful stakeholders in the field. Those agents who are seen to holcl more of the desired capital are those who speak with legitimacy and gain control over the field, whereas those who are "lacking in the desired capital" are not given the right to speak with authority and relegated to marginal status. Discourses which are critical of these practices are at risk of exclusion and not being seen as legitimate. This becomes a form of censorship which serves to protect the status quo within the field.

Constructivism and Mathematics Education

A constructivist approach to understanding how students come to make meaning within the mathematics classroom has tended to focus predomi- nantly on the cognitive aspects of learning, although there is an increasing awareness of the role of social interaction in the learning process as witnessed by the work of Purdue Project. From the work conducted in this project, it is argued that

In the course of a mathematical communication, meanings are negotiated and particular mathematics practices are institutionalized and taken for granted beyond justification by members of the classroom community. Mathematics is therefore both an individual constructive activity and a human social activity (Cobb et al., 1992: p. 137).


However, whether the constructivism is of a social or radical persua- sion, there are many common shared assumptions, which include the active and subjective role of meaning making. The emphasis of constructivism on the active construction of meaning by an individual has represented a distinct break from the earlier behaviourist epistemologies where learning was organised in a hierarchy of complexity and that learning would occur through reinforcement (Leder and Forgasz, 1992). Constructivism has moved to a position in which learning for understanding and the active, individual construction of meaning has replaced models of rote learning and moved away from the assumptions of Skinnerian models of learning. This has required a radical reconceptualisation of the teaching/learning processes along with profound implications for classroom practice.

The great strength of constructivism is that it leads us to think critically and imaginatively about the teaching-learning process Believing the premises of constructivism, we no longer look for simple solutions, and we have a powerful set of criteria by which to judge our possible choices of teaching method (Noddings, 1992: p. 18).

The impact of constructivism has been considerable (Ellerton and Clements, 1992; Lerman, 1993) as can be evidenced by the burgeoning literature and research over the past decade.

Constructivism represents a distinct break from the passive reception of knowledge to more active process of meaning making as apparent in von Glasersfeld's (1991, p. xiv-xv) comment that "the results of our cognitive efforts have the purpose of helping us cope in the world of experience, rather than the traditional goal of furnishing an "objective" representation of a world that might "exist" apart from us and our experience." Con- structivism has been able to challenge a very dominant ideology within education that knowledge is transferable from one person to another. In its place, constructivism has espoused a view that knowledge is a pri- vate, individual construction. The more radical version of constructivism extends this position to include two main principles:

1. Knowledge is not passively received either through the senses or by way of communication. Knowledge is actively built up by the cognizing subject.

2. a. The function of cognition is adaptive, in the biological sens of the term, tending towards fit or viability

b. Cognition serves the subject's organization of the experiential world, not the discovery of an objective ontological reality (yon Glasersfeld, 1992: p. 23).

There have been critiques that this hypothesis engenders a solipsistic position since there is no knowable objective reality which are countered by von Glasersfeld (1991) as being a consequence of "superficial" and


"emotionally distracted" readings. My interest is not in this debate but rather in establishing how this hypothesis is able to gain so much credence and legitimacy in the field. Constructivism whether radical, social or oth- erwise, represents a break from the models that had dominated learning theories since the 60s. This break represented a significant change in the ways in which teaching and learning were theorised, and gave scope to understanding the misconceptions that students held about mathematical concepts and knowledge in its broad sense. This change represented a break with the knowledge that held legitimacy within in the field.

The resistance to the Radical position can be epitomised in Kilpatrick's (1987) address at the Eleventh Psychology of Mathematics Education Conference where he expressed concern about the denial of reality. In a community where a dominant view is one where mathematics is seen to represent an ontological reality (Ellerton and Clements, 1989), the loss of certainty can be seen to be a real concern to those who hold such a view. The Radical position challenged many of the assumptions that were dominant within the field but managed to gain legitimacy in spite of initial resistance by those who controlled and held power within the field. Some seven years since Kilpatrick's critique, there has been a growing acceptance and recognition of the value of some aspects of Radical constrnctivism. This change reflects Bourdieu's proposal that the field should not be seen as static and fixed but as dynamic and open to current changes. The growing acceptance of Radical constructivism within the field suggests that there are changes in the structuring structures which influence what is seen to be legitimate.

Arguably constructivism, radical and others, were able to gain recognition and legitimacy within the field since they did not represent a distinct break with what was seen as legitimate knowledge. With its roots in Piage- tian theory, constructivism resonated strongly with many of the discourses and practices that were already established as legitimate and valued. As an epistemology it does not challenge many of the structuring structures as the twin notions of assimilation and accommodation were familiar concepts. The objectivity of mathematics as a body of knowledge remained stable, it was not called into question, so that the ontological foundations of mathematics were not challenged. When challenges were made to it being a solipsistic position and challenges made to its communicability, von Glasersfeld offered counter positions such as:

If everyone had a different experiential world, they tend to argue, we could both agree on anything and, above all, we could not communicate. There is nothing much wrong with that argument, but the fact that we do agree on certain things and the we can communicate does not prove that what we experience had objective reality in itself. If two people or even a whole society of people look through distorting lenses and agree on what they see, this


does not make what they see any more r e a l - it merely means that on the basis of such agreements they can build up a consensus in certain ways of their subjective experiential worlds. Such relative agreement are called "consensual domains", and one of the oldest in the Western world is the consensual domain of numbers. The certainty of mathematical "facts" springs from mathematicians' observance of agreed-on ways of operating, not from the nature of an objective universe (von Glasersfeld, 1991: p. xv-xvi).

What was questioned was whether or not students were able to construct knowledge that resembled legitimate mathematical knowledge. As such, the objective structures of the field remained unchallenged and unquestioned. Within the Radical position, students would reconstruct their knowledge if their existing conceptual structures were no longer viable in the light of new experiences. The inability of students to construct what was seen as legitimate was seen as an individualistic construction. Students' construction of knowledge is compared to some pre-existing knowledge. It is noble to recognise the individual construction of knowledge, but when this is compared with other forms of knowledge that are seen as legitimate within the field, then proponents of Radical constructivism are failing to recognise the knowledge which structures the field. For example, a student may be able to construct some individual knowledge, which, based on her/his history may be quite viable, yet when compared with what is seen as legitimate in the field, is quite invalid. Constructivism does not acknowledge how the processes of what is seen as "legitimate" knowledge in the field is constructed. In this light, it is the teacher who is responsible for the construction of viable knowledge, for it is her/his responsible to organise the learning environment in such a way that would evoke certain forms of knowledge construction that are in line with what is socially recognised knowledge.

To be constructed as effective learners of mathematics, students must construct knowledge which resembles that which is seen as legitimate in the field. This is repeatedly witnessed in the formal examination procedures at the completion of secondary schooling when students begin to compete for entrance into university and other tertiary institutions. In this setting, students are not rewarded for the construction of their own meaning, but rather, for the construction and reproduction of knowledge that is seen as legitimate by the examining bodies. It is those students who are able to reproduce the socially recognised mathematical knowledge and as a consequence are able to gain entrance to the university courses which will convey status, power and economic rewards (Zevenbergen, 1994). Constructivism with its valorising of individual, subjective construction of meaning ignores this political aspect of meaning making.


MATHEMATICS LEARNING AS A BOURGEOIS ENTERPRISE

The denial of the socio-political dimension of meaning making is not sur- prising given that most of education is founded on the liberal discourses that espouse the value of the individual and the dominant discourses within the field are imbued with psychology's individualism. Examining the discourses that teachers use to frame their work, the power of psychology is blatantly clear (Zevenbergen, 1994). The valorising of the individual and cognition are paramount in teachers' explanations of students' learning of mathematics. As products of their own histories, which will include teacher education institutions where psychology occupies high status, teachers have been indoctrinated with an ideology that purports to value the individual and seeks for every individual to reach her/his potential. Constructivism does not violate this ideology in that the knowledge individuals construct will have some legitimacy within the classroom. By acknowledging that "mathematical insights are always constructed by individuals and their meaning lies within the framework of that individual's experience" (Con- frey, 1991: p. 113), constructrnvism valorises the individual construction of meaning, although, as Confrey (1991) suggests, it does recognise that the status of these individual constructions is negotiated within a certain mathematical cultural context.

The individualism associated with the constructivist position focus- es attention on the individual construction of meaning. This is a rather ambiguous phrase which needs some clarification since it can refer to meaning that is the construction of the individual, or that when a student has misconstructed some knowledge, it is the role of the individual teacher to organise the learning environment in such a way as to encourage the student to construct meaning in a way that is aligned with that knowledge that is seen as legitimate within the formal school context. What is important to this discussion is that at all times the focus remains at the level of the individual construction of meaning. This position is in line with the dominant psychological paradigm and liberal notions of schooling. This orientation engenders an approach to schooling where the individual is seen to be given access to knowledge, skills and processes and it is the "discretion" of the individual whether these options will be exercised. Ultimately, the outcomes are such that those students who "exercise" the options that have been made available to them will be constructed as effective learners of mathematics, and those students who don't choose to elect to use their options, will be constructed as ineffective learners. Constructivism, as part of this overall ideology, fosters the same outcomes whereby those students who do not construct the legitimate knowledge of institutionalised mathematics are constructed as having misconstrued their knowledge or having


some misconceptions. The terms used, while recognising differences, have similar assumptions. In these instances, the construction of knowledge is something which is individualistic, and hence the capacity to construct socially recognised knowledge is something which is a natural talent. Within this ideology, it is often to referred to as ability or intelligence, and having the capacity to construct knowledge is seen as something which is rewarded in the schooling system, and the wider society.

Constructivism cannot account for the disparity in performance across social groups other than the recognition that past experiences provide the basis upon which new knowledge is to be constructed. Within a constructivist paradigm, the individual construction of meaning recognises that students from social, cultural and gendered groups are likely to construct different meaning based on their past experiences and it is this which can be seen to responsible for the poor performance of many marginal social groups. In contrast, Bourdieu (1982) argues that schools are not organised to recognise what are seen as "natural" talents, but rather social talents so that students from certain cultural and social groups are more likely to be constructed as effective learners of mathematics because of their congruency with the social context of formal schooling. Students whose social/cultural background is similar to that which is recognised within the formal school context, are more likely to construct that knowledge which is seen as valid within this context. The success in constructing meanings which are congruent with that of the formal mathematics curriculum is not an so much an individual attribute, but rather a form of symbolic capital.

What constructivist writings refer to prior experiences, Bourdieu has a more encompassing concept, the habitus, which can be seen to be the embodiment of culture. Schools, and mathematics, recognise the habitus of the dominant group as a natural phenomenon which is the ideal for the whole society. Harker (1990) argues that

the dominant habitus is transformed into a form of cultural capital that the schools take for granted, and which acts as a most effective filter in the reproductive processes of a hierarchical society. (p. 87)

Accordingly, the different achievement of groups of students is not something due to the inherent cultural differences of these groups, but rather in the ways in which schools and mathematics operates. Bernstein (1990: p. 79) argues a similar point when he suggests that the organisa- tional and cultural assumptions of the school have an "implicit ideological basis to pedagogy itself which acts selectively for those who acquire what it relayed". His point is that schooling qua schooling contains social class assumptions, quite independent of the institutional content. In a Bernsteini- an sense, the relay sets limits to what can be input and the messages that

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can be relayed. These theses, by definition, propose that schooling and mathematics are socially and culturally biased. Kemmis (1992) proposes that the patterns of language, work and power are integral to understanding the ways in which schools are organised so as to recognise and reward the dominant habitus. Students need to be conversant with those recognised and valued within formal schooling. Bourdieu (1974) argues that students who come from the lower middle classes and the working classes can only succeed in schools with great effort since they must acquire the dispositions of cultivated groups which are seen as natural attributes. In the following sections the importance of these aspects of schooling will be explored.

Constructivism and Language

Constructivist writers suggest that meaning cannot be transferred from one person to another and that meaning will be negotiated through interaction as borne out in Richards' (1991) comment:

For mathematical inquiry to take place students and teachers must learn to carry on a mathematical discussion - they must learn to speak inquiry math....schools must provide means for students to become mathematically literate adults. They must give student the language and tools to continue the conversation for themselves (p. 17).

Richards goes on to argue the importance of discussion for the construction of meaning where meaning which is achieved through consensus as opposed to transmission. Constructivism recognises that language will facilitate or impede the construction of socially recognised meaning (Pirie and Kieren, 1992; Richards, 1991; van den Brink, 1991), but it fails to recognise the actual codes (Bernstein, 1982) which constitute mathematics. In a Bernsteinian sense, the linguistic code of mathematics can be seen to be distinct code and as such, to be constructed as a successful learner of mathematics requires students to be fluent in that code. Students who come from social and cultural groups whose language is dissimilar to that of formal schooling are going to be at a distinct disadvantage when it comes to reproducing the unique language of mathematics. The students who come from middle-class families use an elaborated code, while students from working-class families use a restricted code. Brice-Heath (1983) and Walkerdine and Lucey (1989) have shown the styles of questioning in working-class and middle-class families are quite different; the restricted, declarative statements of the working-class families and in sharp contrast to the pseudo-type questions of the middle-class families. It is this latter type which are common in the classroom so that middle-class students entering the school context will have a familiarity with the patterns of questioning found in schools, whereas their working-class peers will be at a distinct disadvantage. Malin (1990) has shown similar differences in the


quality and forms of interactions for Australian Aboriginal students when they enter the formal school context.

The actual language of mathematics is also a recognised obstacle to the learning of mathematics. Consider the student from a poor, working-class Catholic family who is exposed to the measurement component of the mathematics curriculum. Such a student is more likely to experience more difficulty constructing meaning when confronted with a unit on mass that is akin to that espoused by the teacher and schooling system than her/his middle-class Protestant peer. Not only is there a disparity in the codes that will be used in the school context, but also the past experiences could provide an incongruence between "mass" from the past experiences provided by Catholicism and the current experiences provided in the mathematics classroom.

Walkerdine (1988) has explored this aspect in greater detail and has provided an excellent analysis of the incongruence between the use of more/less, arguing strongly that the differences between the home/school, working-class/middle-class differences in language use are potentially mathematical differences. This language disadvantage becomes manifest- ed as mathematical disadvantage and reified as something lacking within the individual, usually described as something cognitive or a more innate ability rather than a linguistic difference. Constructivism does not call into question the actual linguistic code of mathematics.

Constructivism, Work and Power

The focus on individualism and mentalism denies the social context of schooling and the political processes associated with schooling. In the case of constructivism, the micro analysis of classroom interactions is mostly concerned with the individual construction of meaning and the social negotiation of meaning as it applies to the didactic learning common in most classrooms. The following comment from von Glasersfeld is representative of this position:

When teachers look at a student in their class, they see that student in a complex environment. There is the physical environment of the desk (or whatever piece of furniture the student occupies), the classroom, the land- or townscape around the school building, parts of which might be visible through the window; there is the social environment of the classmates, the teachers themselves, and the student's family, which may to some extent be known to the particular teachers, and there is also the didactic environment of the school books, the educational props, the curriculum, and more, important but less perceivable, the teachers' views and beliefs as to how education should be achieved (von Glasersfeld, 1991: p. xv).

This focus enables educators and researchers to examine the micro social context without consideration of the macro context within which


the construction of meaning is occurring. This macro context facilitates processes which favour certain social and cultural groups. The issue of language was discussed in the previous section, but as Kemmis (1992) points out, it is important to consider further the patterns of work and power that are integral to the social context. Within the mathematics classroom certain patterns of interaction and patterns of work are valued. Most of the practices of work have been informed by Piagetian notions of cognitive development. Often this is overtly stated as apparent within the Rigby: Moving into Maths series of mathematics resource materials, or covertly through the progression from preoperational thought in which free play and the interaction with materials through to formal operational thought and the introduction of algebra as students enter the post-primary years.

I recently had the experience of working with a group of teachers who were researching their own practice. One teacher raised a problem she was having with a Grade 4 student (about 9 years old) who had recently arrived from Papua New Guinea. The boy had no idea of how to use the manipulatives commonly found in Western classrooms. This raised a number of issues for the teacher, including confronting her own beliefs about the learning of mathematics and the patterns of work in the mathematics classroom. She was concerned that although the boy was able to perform mathematically when writing mathematics, he was not able to use these materials, For the teacher, this raised serious implications for she was not able to undertake the usual mathematical activities with this child, so that the valued patterns of work were disrupted. The boy had no idea of how to use MAB equipment or other manipulatives to represent or aid his learning. This raised serious concerns for the teacher since she had to either teach the child how to use this equipment even though he was quite pro- ficient in the subject, or to devise new methods of teaching concepts. The teacher was confronted by the limitations of her beliefs about how students "naturally develop" from concrete to abstract thinking. This set of beliefs provided the framework for her classroom practices. Further compounding the issue was the challenging of an embedded assumption about certain cultures. For this teacher, she had assumed that since the boy was from Papua New Guinea he would be a concrete thinker, since "it has a very concrete culture." The insertion of this student into a mainstream Western classroom challenged the valued patterns of work which are dominant in Western mathematics classrooms.

One of the most pervasive myths in mathematics education, which is thoroughly supported by the ideology of constructivism, is the development of thought whereby students "naturally" progress from concretisations to abstractions. In the above example, the PNG boy was violating this myth

CONSTRUCTIVISM AS A LIBERAL BOURGEOIS DISCOURSE I 0 9

and the teacher was very confused about how to teach him. The implications arising from such a situation are to either acculturate the student into the dominant Western ways of working and to effectively deny his cultural background. Alternatively, the teacher could to try to reconcile the incongruence between the beliefs about the progression of thought and the reality that this student did not need materials for constructing mathematical meaning. In the latter option, this would mean a revision of the patterns of work in the classroom but more importantly, a substantial change in the teacher perceptions of the "naturalness" of cognitive development.

KNOWLEDGE AND POWER

The Radical constructivist perspective has pushed the boundaries of Pla- tonism associated with Mathematics Education arguing instead, that there can be no objective reality, but rather a "consensual domain" where people agree to certain aspects of "their subjective experiential worlds" (yon Glasersfeld, 1991: p. xvi). This has had major implications for the field given the purported objectivity of mathematical knowledge and the relationship to the pedagogies that have been so dominant and taken-for-granted by those who participate within mathematics education regardless of their level of interaction. The Radical Constructivist perspective has challenged the assumption that knowledge can be transmitted from teacher to student, or vice versa. In its place, a position which acknowledges the social and cultural differences of participants as per their experiential background, recognises that students will construct different meanings because they have had different experiences upon which they are to accommodate or assimilate new information. This position made possible the recognition that students from different social and cultural groups would inevitably construct different meanings than their peers. For example, Aboriginal students from Central Australia would be expected to construct substantially different meanings of the same concept than from their peers in the wealthy inner suburbs of Sydney. As a consequence, Radical Con- strucfivism gave some glimmer of hope for understanding how and why students were marginalised as learners of mathematics in that it challenged notions of some innate ability, and gave credence to the social background of the student.

While this position may prove useful in looking at micro processes of learning, it fails to address the structural aspects to teaching and learning. In particular, what is not recognised are the forms of knowledge that are seen as legitimate. In spite of yon Glasersfeld's defence that Radi- cal Constructivism is not a solipsistic position, it does not recognise that


within the field of mathematics, certain forms of knowledge convey more power than others. For example, within the Australian context, there is a certain conceptualisation of time that is legitimised through institutionalised schooling. This is the Western construct of time which is linear and closely linked to the quantifiable and economic ideology that dominates Western thought. Harris' (1984) work on Australian Aboriginal concepts of time suggests that it is quite different from that of Western thought. In contrast to Western conceptions of time, it is the quality of time that is important, time is circular, it is fluid and is thought about in relation to seasonal events or in relation to phases of the moon. Accordingly, these Aboriginal students entering the formal school context are recognised as having disparate conceptualisation of time, and their construction of time as portrayed in Western schooling is expected to be quite different from that of Anglo-Australian students who will have had substantially different experiences with time. Within a Radical Constructivist perspective this incongruence would be expected and recognised, perhaps even celebrat- ed. The experiences that many Australian Aboriginal students will have had prior to the exposure to formal schooling will provide substantially different knowledge and meanings from that of non-Aboriginal students. This background will provide the fundamental knowledge upon which the new experiences of school must be incorporated. For students whose culture is different from that which is represented in formal schooling, the construction of meaning will be more demanding when it is expected that they should construct meanings similar to those represented in the formal school context.

What remains problematic and unresolved within the constructivist theories is that schools do not recognise nor reward the individual construction of meaning. The structuring practices of schooling recognise particular constructions of knowledge. Students are rewarded for constructing knowledge that is akin to that represented in and through the formal school system. As mentioned at the beginning of this paper, it is not so much the cultural differences p e r se that are responsible for the construction of educational difference, but rather the schooling system. The meritocra- cy associated with schools as institutions rewards particular knowledge constructions over others. While constructivisrn recognises that Australian Aboriginal students are less likely to construct concepts of time akin to that of Western schooling, it fails to acknowledge that it is the Western construct of time which conveys power, status and economic rewards. Stu- dents who are able to display the knowledge that is seen as legitimate are those who will be constructed as effective learners of mathematics and hence enjoy the social rewards associated with school success.


Radical constructivism in particular, and constructivism in general, fails to acknowledge or recognise that certain forms of knowledge are seen as legitimate and reified through the processes of schooling and university. The recognition and valorising of individual construction of meaning, ignores the political consequences of individual meaning. To become empowered, as evidenced by the accumulation of cultural capital, neces- sitates students displaying those aspects of the culture which are seen as valued and legitimate within the society.

Limits of Constructivism

The socially critical perspective of this paper is deliberately political. My intention has been to raise the implicit processes through which constructivism has gained power within the field of mathematics education and the implications this has for classroom practice. I have then offered a critique of these practices.

Constructivism recognises the individual construction of meaning which is a very worthy ideal yet it masks the hegemonic process of exclusion. Within society certain forms of knowledge and practices are seen as legitimate and valued. Mathematics is seen as high status knowledge in Western societies. To be constructed as an effective learner of mathematics and to have the associated social, and economic power associated with effective learning in mathematics being conveyed to the learner, it is necessary to be aligned with the legitimate knowledge of the field. Con- structivism has been able to theorise, and provide empirical evidence, that students do not necessarily construct the same meanings as their peers or that which the teacher had intended. This represented a significant advance on earlier theories of learning. However, it is limited by its failure to acknowledge that the schooling system recognises only particular constructions of meaning. Students who come from social and cultural groups whose culture is not that of the dominant culture, are at a distinct disadvantage when entering the school system. Constructivism can only partially address this issue.

Mathematical knowledge is socially recognised and validated through the objective structures of the field. It is those people who hold significant amounts of symbolic capital within the field who are able to influence what is seen as legitimate mathematical knowledge. The constructivist paradigm has assumed a dominance within the field and hence is able to define what is seen as legitimate and valued practice. Constructivism does not address this process. Its non-recognition of this process supports the status quo and in doing so fails to address the impact of constructivism on the construction


of socia l d i f ference th rough its celebrat ion o f the individual cons t ruc t ion

o f meaning .

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Constructivism as a liberal bourgeois discourse

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