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Paul Ernest

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Social constructivism is appearing in the work of some researchers in the psychology of mathematics education as an alternative perspective to radical constructivism. However there are widespread disparities in what is meant by social constructivism. To overcome this ambiguity, the roots of social constructivism in sociology, symbolic interactionism, philosophy and social psychology are traced, and two types of social constructivism are distinguished. One is based on a radical constructivist (Piagetian) theory of mind, and either bolts on aspects of the social, or adds it on as an alternative complementary perspective. The other type is based on Vygotskian theory of mind, and is more thoroughly social.

Introduction: the problem

The central problem for the psychology of mathematics education is to provide a theory of learning mathematics that facilitates interventions in the processes of its teaching and learning. Thus, for example, Piaget's Stage Theory inspired a substantial body of research on hierarchical theories of conceptual development in the learning of mathematics in the 1970s and 1980s (e.g. Hart, 1981). Piaget's constructivism also led to the currently fashionable radical constructivist theory of learning mathematics, which accounts for the individual idiosyncratic construction of meaning, and thus for systematic errors, misconceptions, and alternative conceptions in the learning of mathematics. It does this in terms of individual cognitive schemas, which it describes as growing and developing to give viable theories of experience by means of Piaget's twin processes of equilibration; assimilation and accommodation.

Although a number of different forms of constructivism exist, the radical version most strongly prioritises the individual aspects of learning. It thus regards other aspects, such as the social, to be merely a part of, or reducible to, the individual. A number of authors have criticised this approach for its neglect of the social (Ernest 1991b, 1993d; Goldin 1991; Lerman, 1992, 1994). Thus in claiming to solve one of the problems of the psychology of mathematics education, radical constructivism has raised another: how to account for the social aspects of learning mathematics? This is not a trivial problem, because the social domain includes linguistic factors, cultural factors, interpersonal interactions such as peer interaction, and teaching and the role of the teacher. Thus another of the fundamental problems faced by the psychology of mathematics education is: how to reconcile the private mathematical knowledge, skills, learning, and conceptual development of the individual with the social nature of school mathematics and its context, influences and teaching? In other words: how to reconcile the private and the public, the individual and the collective or social, the psychological and the sociological aspects of the learning (and teaching) of mathematics?

One approach to this problem (and there are of course others not discussed here) is to propose a social constructivist theory of learning mathematics. On the face of it, this is a theory which acknowledges that both social processes and individual sense making have central and essential parts to play in the learning of mathematics. Consequently, social constructivism is gaining in popularity. However a problem that needs to be addressed is that of specifying more precisely the nature of this perspective. A number of authors attribute different characteristics to what they term social constructivism. Others are developing theoretical perspectives under other names which might usefully be characterised as social constructivist. Thus there is a lack of consensus about what is meant by the term, and what are its underpinning theoretical bases and assumptions. The aim of this paper is to begin to clear up this confusion by clarifying the origins and nature of social constructivism, and indicating some of the major differences underlying the use of the name.

Background traditions

Although there are few explicit references to social construction in the work of symbolic interactionists and ethnomethodologists such as Mead, Blumer, Wright Mills, Goffman and Garfinkel, their work is centrally concerned with the social construction of persons and interpersonal relationships. They emphasise conversation and the types of interpersonal negotiation that underpin everyday roles and functionings, such as those of the teacher in the classroom. Indeed, Mead (1934) even offers a conversation based social theory of mind. Following on from this tradition, a milestone was reached when Berger and Luckmann (1966) published their seminal sociological text 'The social construction of reality'. Drawing on the work of Schütz, Mead, Goffman and others, this elaborated the theory that our knowledge and perceptions of reality are socially constructed, and that we are socialised in our upbringing to share aspects of that received view. They describe the socialisation of an individual as "an ongoing dialectical process composed of the three moments of externalization, objectivation and internalization... [and] the beginning point of this process is internalization." (Berger and Luckmann 1966: 149)

From the late 1960s or early 1970s, social constructivism became a term applied to the work of sociologists of science and sociologists of knowledge including Barnes, Bloor, Knorr-Cetina, Latour, Restivo, and others. This tradition drew upon the work of Durkheim, Mannheim, and others, and its primary object is to account for the social construction of scientific knowledge, including mathematics (Restivo 1988). Recently, there has been work in this tradition (e.g. by Restivo and Collins) in developing a social theory of mind (drawing on the work of Mead and Vygotsky).

Not long after the development of these sociological traditions, in the 1970s social constructionism became a recognised movement in social psychology through the work of Coulter, Gergen, Harré, Secord, Shotter, and others. These authors have been concerned with a broad range of social psychological issues such as the social construction of the self, personal identity, emotions, gender, and so on (Gergen, 1985). A shared starting point, elaborated by different researchers in different ways, is that of Vygotskian theory. Consequently, one of the special features of social constructionism in social psychology is the explicit central use of the metaphor of conversation for mind, as well as for interpersonal interaction.

Within psychology there are other inter-related traditions which build on the work of Vygotsky, and which propose more less well developed social theories of mind. These include both Soviet Activity Theorists (Vygotsky, Luria, Leont'ev, Gal'perin, Davydov), what might be termed 'dialogists', including Volosinov, Bakhtin, Lotman, Wertsch, and sociocultural theorists such as Lave, Wenger, Rogoff, Cole and Saxe.

The term 'social constructivism' was not applied in philosophy, to the best of my knowledge, until the late 1980s, when the growing interdisciplinarity of sociological and social psychological studies, and their terminology, spilled over into philosophy. However, a social constructivist tradition in philosophy can be identified, with its basis in the late work of Wittgenstein, although some scholars, such as Shotter, trace it back to Vico. There are strands in various branches of philosophy which might be termed social constructivist. This includes the tradition of ordinary language and speech act philosophy, following on from Wittgenstein and Ryle, including the work of Austin, Geach, Grice, Searle and others. In the philosophy of science, a mainstream social constructivist strand includes the work of Hanson, Kuhn, Feyerabend, Hesse and others. In continental European philosophy there is an older tradition including Enriques, Bachelard, Canguilhem, Foucault which has explored the formative relations between knowledge, especially scientific knowledge, and social structure. In social epistemology there is the work of Toulmin, Fuller and others. In the philosophy of mathematics there is a tradition including Wittgenstein, Lakatos, Bloor, Davis, Hersh, and Kitcher. Ernest (1991a, In press) surveys this tradition, and represents one of the few specifically philosophical approaches to mathematics to adopt the title of social constructivism.

In the early 1970s the social construction (of knowledge) of reality thesis became widespread in educational work based on sociological perspectives, such as that of Esland, Young, Bernstein, and others. By the 1980s theories of learning based on Vygotsky were also sometimes termed social constructivist, and although we might now wish to draw distinctions between their positions, researchers such as Andrew Pollard (1987) identified Bruner, Vygotsky, Edwards and Mercer, and Walkerdine as contributing to a social constructivist view of the child and learning .

To the best of my knowledge the term 'social constructivism' appeared in mathematics education from two sources. The first is the social constructivist sociology of mathematics of Restivo, which he explicitly relates to mathematics education in Restivo (1988). The second is the social constructivist theory of learning mathematics of Weinberg and Gavelek (1987). The latter is based on the theories of both Wittgenstein and Vygotsky, but also mentions the work of Saxe, Bauersfeld and Bishop as important contributions to the area, even though they might not have explicitly drawn called themselves social constructivist. Unfortunately, to my knowledge Weinberg and Gavelek never developed their ideas in print. Bishop (1985) made a more powerful impact with his paper on the social construction of meaning in mathematics education, but he did not develop an explicit theory of learning mathematics, and focused more on its social and cultural contexts. Social constructivism became a more widely recognised position following Ernest (1990, 1991a, 1991b), but a number of authors have used and continue to use the term in different ways, such as Bauersfeld (1992) and Bartolini-Bussi (1991). There are also a number of contributions to mathematics education which might be termed social constructivist, in one sense or other, even though they do not use this title (e.g. the socioconstructivism discussed below).

Thus it can be said that social constructivism originated in sociology and philosophy, with additional inputs from symbolic interactionism and Soviet psychology. Subsequently it influenced developments in social psychology and educational studies, before filtering through to mathematics education. However, because of the diverse routes of entry, and because of the varying paradigms and perspectives into which it has been assimilated in mathematics education, social constructivism is used to refer to widely divergent positions. What they share is the notion that the social domain impacts on the developing individual in some formative way, with the individual constructing (or appropriating) her meanings in response to experiences in social contexts. Nevertheless, this description is vague enough to accommodate a range of positions from a slightly socialised version of radical constructivism, through sociocultural and sociological perspectives, all the way to fully-fledged post-structuralist views of the subject and of learning.

The problematique of social constructivism for mathematics education may be characterised as twofold. It comprises, first, an attempt to answer the question: how to account for the nature of mathematical knowledge as socially constructed? Second, how to give a social constructivist account of the individual's learning and construction of mathematics? Answers to these questions need to accommodate both the personal reconstruction of knowledge, and personal contributions to 'objective' (i.e. socially accepted) mathematical knowledge. An important issue implicated in the second question is that of the centrality of language to knowing and thought.

Elsewhere I have focused on the first more overtly epistemological question, concerning mathematical knowledge (Ernest 1991a, 1993b, In-press). However, from the perspective of the psychology of learning mathematics, the second question is equally important. It is also the source of a major controversy in the mathematics education community. In simplified terms, the key distinction among social constructivist theories of learning mathematics is that between individualistic or cognitively based theories (e.g. Piagetian or radical constructivist theories) and socially based theories (e.g. Vygotskian theories of learning mathematics).

Although this is a significant distinction, there is an important feature shared by radical constructivism and the varieties of social constructivism discussed here. This is a commitment to a fallibilist view of knowledge in general, and of mathematical knowledge in particular. This is discussed further elsewhere (see e.g. Ernest 1991a, In press).

Social Constructivism with a Piagetian Theory of Mind

A number of authors have attempted to develop a form of social constructivism based on what might be termed a Piagetian or neo-Piagetian constructivist theory of mind. Two main strategies have been adopted. First, to start from a radical constructivist position and add on social aspects of classroom interaction. That is, to prioritise the individual aspects of knowledge construction, but to acknowledge the important if secondary place of social interaction. This is apparently the strategy of Yackel, Cobb and Wood (In press), who claim to be radical constructivist, but also lay a special emphasis on the social negotiation of classroom norms. Indeed these researchers adopted the term ‘socioconstructivist’ for their position, but have since reverted to the term ‘constructivist’. However it is possible that these researchers should be interpreted as having adopted a complementarist position (the second strategy, discussed below), for certainly in some publications (e.g. Cobb, 1989) they explicitly write of the adoption of multiple theoretical perspectives and of their complementarity. Overall, a number of developments in radical constructivism would seem to fall under this category, in all but name (e.g. Richards, 1991)

The second strategy is to adopt two complementary and interacting but disparate theoretical frameworks. One framework is intra-individual and concerns the individual construction of meanings and knowledge, following the radical constructivist model. The other is inter-personal, and concerns social interaction and negotiation between persons. It can also extend far enough to account for cultural items, such as mathematical knowledge. A number of researchers have adopted this complementarist version of social constructivism, including Driver (In press), who accommodates both personal and interpersonal construction of knowledge in science education. Likewise Murray (1992) and her colleagues argue that mathematical knowledge is both an individual and a social construction. Bauersfeld (1992: 467) explicitly espouses a social constructivist position based on "radical constructivist principles... and an integrated and compatible elaboration of the role of the social dimension in individual processes of construction as well as the processes of social interaction in the classroom". Most recently Bauersfeld (1994: 467) describes his social constructivist perspective as interactionist, sitting between individualist perspectives, such as cognitive psychology and collectivist perspectives, such as Activity Theory. Thus he explicitly relates it to the symbolic interactionist position mentioned above, but he retains a cognitive (radical constructivist) theory of mind complementing his interactionist theory of interpersonal relations.

In Ernest (1991a) I proposed a version of social constructivism, which although intended as a philosophy of mathematics, also included a detailed account of subjective knowledge construction. This combined a radical constructivist view of the construction of individual knowledge (with an added special emphasis on the acquisition and use of language) with Conventionalism; a fallibilist social theory of mathematics originating with Wittgenstein, Lakatos, Bloor and others.

The two key features of the account are as follows. First of all, there is the active construction of knowledge, typically concepts and hypotheses, on the basis of experiences and previous knowledge. These provide a basis for understanding and serve the purpose of guiding future actions. Secondly, there is the essential role played by experience and interaction with the physical and social worlds, in both the physical action and speech modes. This experience constitutes the intended use of the knowledge, but it provides the conflicts between intended and perceived outcomes which lead to the restructuring of knowledge, to improve its fit with experience. The shaping effect of experience, to use Quine's metaphor, must not be underestimated. For this is where the full impact of human culture occurs, and where the rules and conventions of language use are constructed by individuals, with the extensive functional outcomes manifested around us in human society. (Ernest 1991a: 72)

However this conjunction [of social and radical constructivist theories] raises the question as to their mutual consistency. In answer it can be said that they treat different domains, and both involve social negotiation at their boundaries (as Figure 4.1 [in original text] illustrates). Thus inconsistency seems unlikely, for it could only come about from their straying over the interface of social interaction, into each other's domains. ...there are unifying concepts (or metaphors) which unite the private and social realms, namely construction and negotiation. (Ernest 1991a: 86-87)

In commenting on work that combines a (radical) constructivist perspective with an analysis of classroom interaction and the wider social context, Bartolini-Bussi (1991: 3) remarks that "Coordination between different theoretical frameworks might be considered as a form of complementarity as described in Steiner's proposal for TME: the principle of complementarity requires simultaneous use of descriptive models that are theoretically incompatible." However, Lerman (1994) argues that there is an inconsistency between the subsumed social theories of knowledge and interaction, and radical constructivism, in this (or any) complementarist version of social constructivism.

I, too, am now inclined to think that there are severe difficulties associated with the form of social constructivism which builds on radical constructivism. There are first of all many of the problems associated with the assumption of an isolated cognizing subject (Ernest, 1991b). Radical constructivism can be described as being based on the metaphor of an evolving and adapting, but isolated organism – a cognitive alien in hostile environment. Its world-model is that of the cognising subject's private domain of experience (Ernest 1993c, 1993d). Any form of social constructivism that retains radical constructivism at its core retains these metaphors, at least in some part. Given the separation of the social and individual domain that a complementarist approach assumes, there are also the linked problems of language, semiotic mediation, and the relationship between private and public knowledge. If these are ontologically disparate realms, how can transfer from one to the other take place?

Lerman (1992) proposed to rescue (as he saw it) radical constructivism by replacing its Piagetian theory of mind and conceptual development with a Vygotskian theory of mind and language, in what might be seen as a form of social constructivism. However, in taking leave of radical constructivism Lerman (1994) has recently extended his critique, and now argues that any form of social constructivism which retains a radical constructivism account of individual learning of mathematics inevitably fails to account adequately for language and the social dimension. Bartolini-Bussi (1994), however, remains committed to a complementarist approach, and although espousing a Vygotskian position herself, argues for the value of the co-existence of a Piagetian form of social constructivism, and the necessity for multiple perspectives.

Social Constructivism with a Vygotskian Theory of Mind

In a survey of social constructivist research in the psychology of mathematics education Bartolini-Bussi (1991) distinguishes complementarist work combining constructivist with social perspectives from what she terms social constructionist work based on a fully integrated social perspective. Some of her attributions of individual projects to these approaches might be questioned. For example, I would locate the diagnostic teaching approach of Alan Bell and colleagues in a cognitively-based post-Piagetian framework, not one of social constructivism. Nevertheless, the distinction made is important. It supports the definition of a second group of social constructivist perspectives based on a Vygotskian or social theory of mind, as opposed to the constructivist and complementarist approaches described in the previous section.

Weinberg and Gavelek's (1987) proposal falls within this category, since it is a social constructivist theory of learning mathematics explicitly based on Vygotsky's theory of mind. A more fully developed form of social constructivism based on Vygotsky and Activity Theory is that of Bartolini-Bussi (1991, 1994), who emphasises mind, interaction, conversation, activity and social context as forming an interrelated whole, and indicates a broad range of classroom and research implications and applications.

In Ernest (1993a, 1993c, 1993d, 1994, In-press) I have been developing a form of social constructivism differing from my earlier version (Ernest 1990, 1991a) because it similarly draws on Vygotskian roots instead of Piagetian constructivism in accounting for the learning of mathematics. This approach views individual subjects and the realm of the social as indissolubly interconnected, with human subjects formed through their interactions with each other (as well as by their individual processes) in social contexts. These contexts are shared forms-of-life and located in them, shared language-games (Wittgenstein). This version of social constructivism has no underlying metaphor for the wholly isolated individual mind, drawing instead upon the metaphor of conversation, comprising persons in meaningful linguistic and extra-linguistic interaction. (This metaphor for mind is widespread among 'dialogists' e.g. Bakhtin, Wertsch and social constructionists e.g. Harré; Gergen, Shotter.)

Mind is viewed as social and conversational because of the following assumptions. First of all, individual thinking of any complexity originates with and is formed by internalised conversation; second, all subsequent individual thinking is structured and natured by this origin; and third, some mental functioning is collective (e.g. group problem solving). Adopting a Vygotskian perspective means that language and semiotic mediation are accommodated. Through play the basic semiotic fraction of signifier/signified begins to become a powerful factor in the social (and hence personal) construction of meaning (Vygotsky, 1978).

Conversation also offers a powerful way of accounting for both mind and mathematics. Harré (1979) has elaborated a cyclic Vygotskian theory of the development of mind, personal identity, language acquisition, and the creation and testing of public knowledge, all in one cyclic pattern of appropriation, transformation, publication, conventionalisation. This provides descriptions of both the development of personal knowledge of mathematics in the context of mathematics education (paralleling Berger and Luckmann's socialisation cycle), and describes the formative relation between personal and 'objective' mathematical knowledge in the context of academic research mathematics (Ernest 1991a, 1993b, 1994, 1998). Such a theory has the potential to overcome the problems of complementarity discussed above.

Conclusion

Given the current growth of interest social constructivism, it is important to distinguish Vygotskian from radical constructivist varieties, for progress to be made in theoretical aspects of the psychology of learning mathematics. However, although it is increasingly favoured, the adoption of the Vygotskian version is not a panacea. Piagetian and post-Piagetian work on the cognitive aspects of the psychology of learning mathematics education remains at a more advanced stage and with a more complete theorisation, research methodology and set of practical applications. Nevertheless, Vygotskian versions of social constructivism suggest the importance of a number of fruitful avenues of research, including the following:

· the acquisition of transformation skills in working with semiotic representations in school mathematics;

· the learning of the accepted rhetorical forms of school mathematical language, both spoken and written;

· the crucial role of the teacher in correcting learner knowledge productions and warranting learner knowledge; and

· the import of the overall social context of the mathematics classroom as a complex, organised form of life including (a) persons, relationships and roles, (b) material resources, (c) the discourse of school mathematics, both content and modes of communication (Ernest 1993a, In press).

References

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Bauersfeld, H. (1994) Theoretical Perspectives on Interaction in the Mathematics Classroom, in Biehler et al. (1994), 133-146.

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Bartolini-Bussi, M. G. (1991) Social Interaction and Mathematical Knowledge, in Proceedings of PME-15 Assisi, Italy, 1, 1-16.

Bartolini-Bussi, M. G. (1994) Theoretical and Empirical Approaches to Classroom Interaction, in Biehler et al. (1994), 121-132.

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Driver (In press) Constructivist Approaches to Science Teaching, in L. P. Steffe, Ed., Alternative Epistemologies in Education, London: Erlbaum.

Ernest, P. (1990) Social Constructivism as a Philosophy of Mathematics: Radical Constructivism Rehabilitated?, Poster paper, PME 14 Conference, Mexico, July 1990.

Ernest, P. (1991a) The Philosophy of Mathematics Education, London: Falmer.

Ernest, P. (1991b) Constructivism, The Psychology of Learning, and the Nature of Mathematics: Some Critical Issues, in Proceedings of PME-15 (Italy), 2, 25-32. (Reprinted in Science and Education, 2 (2), 1993, 87-93.)

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Ernest, P. (1993c) Metaphors for Mind and World, Chreods 6, 3-10.

Ernest, P. (1993d) Constructivism and the Problem of the Social, in C. Julie, D. Angelis, Z. Davis, Eds, Political Dimensions of Mathematics Education: Curriculum Reconstruction for Society in Transition, Johannesburg, South Africa: Maskew, Miller and Longman, 1993, 121-130.

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Paul Ernest

University of Exeter

PErnest@ex.ac.uk

NOTE

‘What is social constructivism in the psychology of mathematics education?’ was presented at PME-18, and appeared in Ponte, J. P. da, and Matos, J. F. Eds Proceedings of the 18th Annual Conference of the International Group for the Psychology of Mathematics Education, Lisbon, Portugal: University of Lisbon, 1994, Vol. 2, 304-311. A revised version was published as ‘Social constructivism and the psychology of mathematics education’, in Ernest, P. Ed. Constructing Mathematical Knowledge: Epistemology and Mathematics Education, London, Falmer Press, 1994, 62-72.