CEPA eprint 3655

Social constructivism and the psychology of mathematics education

Ernest P. (1994) Social constructivism and the psychology of mathematics education. In: Ernest E. (ed.) Constructing mathematical knowledge: Epistemology and mathematics education. Falmer Press, London: 68–79. Available at http://cepa.info/3655
Table of Contents
Introduction: The Problem
Background Traditions
Social Constructivism with a Piagetian Theory of Mind
Social Constructivism with a Vygotskian Theory of Mind
Introduction: The Problem
It might be said that the central problem for the psychology of mathematics education is to provide a theory of learning mathematics. That is, to give a theoretical account of learning which facilitates interventions in the processes of its teaching and learning. Piaget’s Stage Theory, for example, inspired a substantial body of research on hierarchical theories of conceptual development in the learning of mathematics in the 1970s and 1980s (e.g., in the study of Hart and colleagues 1981). Piaget’s constructivism also led to the currently fashionable radical constructivist theory of learning mathematics, (von Glasersfeld 1991; Davis et al. 1990). Part of the growth in popularity of radical constructivism is due to its success in accounting for the idiosyncratic construction of meaning by individuals, 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; those of assimilation and accommodation. Radical constructivism also has the appeal of rejecting absolutism in epistemology, something often associated with behaviourist and cognitivist theories of learning (Ernest 1991a, 1991b).
It is widely recognized that a variety of different forms of constructivism exist, both radical and otherwise (Ernest 1991b). However it is the radical version which most strongly prioritizes 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 criticized 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 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. Possibly as a consequence of this feature, 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 characterized as social constructivist. Thus there is a lack of consensus about what is meant by the term, and what its underpinning theoretical bases and assumptions are. The aim of this chapter is to clear up some of this confusion by clarifying the origins and nature of social constructivism, and indicating some of the major differences underlying the use of the term. In particular, I shall argue that a major division exists between two types of social constructivism according to whether Piagetian or Vygotskian theories of mind and learning are adopted as underlying assumptions. However to clarify what social constructivism means it is necessary to go back further to its disciplinary origins.
Background Traditions
Although there are few explicit references to social construction in the work of symbolic interactionists and ethno-methodologists such as Mead, Blumer, Wright Mills, Goffman and Garfinkel, their work is centrally concerned with the social construction of persons and with interpersonal relationships. They emphasize conversation and the types of interpersonal negotiation that underpin everyday roles and interactions, such as those of the teacher in the classroom. Mead (1934) in fact 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 book elaborated the theory that our knowledge and perceptions of reality are socially constructed, and that we are socialized in our upbringing to share aspects of that received view. They describe the socialization 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, Fleck and more recently Knorr-Cetina, Latour, Restivo, and others working in the strong programme in the sociology of knowledge (Bloor 1976). This tradition drew upon the work of Durkheim, Mannheim, Marx, and others, and its primary object is to account for the social construction of scientific knowledge, including mathematics (Restivo 1988). Thus the emphasis is on the social institutions and processes that underpin the construction of scientific and mathematical knowledge, and in particular, those that underpin the warranting of knowledge. Recently, there has been further work in this tradition (e.g., by Restivo and Collins) in developing a social theory of mind. This draws on the work of Mead and Vygotsky.
Not long after the development of these sociological traditions, in the 1970s social constructionism became a recognized movement in social psychology through the work of Harré, Gergen, Shotter, Coulter, Secord, 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 starting point shared by the social constructionists, but elaborated by different researchers in different ways, is that of Vygotskian theory of mind. This is based on the notion that thought in its higher manifestations is internalized speech or conversation. In consequence, one of the special features of social constructionism in social psychology is the explicit use of conversation as a central metaphor for mind, as well as for interpersonal interaction (e.g., Shotter 1993).
Within the field of 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 Soviet Activity Theorists (Vygotsky, Luria, Leont’ev, Gal’perin, Davydov), and the dialogue theorists, or socio-linguisticians, for want of a better term, including Volosinov, Bakhtin, Lotman, Wertsch, as well as socio-cultural theorists such as Lave, Wenger, Rogoff, Cole and Saxe.
The term ‘social constructivism’ does not seem to appear in philosophy until the late 1980s, when the growing interdisciplinarity of sociological and social psychological studies, and their terminology, spilled over into philosophy. A social constructivist tradition in spirit, if not in name, can be identified in philosophy, with its basis in the late work of Wittgenstein (1953). However, some scholars, such as Shotter, trace this tradition or at least its anticipations, back to Vico. There are strands in various branches of modern philosophy which might be termed social constructivist. Ordinary language and speech act philosophy, following on from Wittgenstein and Ryle, including the work of Austin, Geach, Grice, Searle and others, makes up one such strand. 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 its social structure and contexts. Although not known as social constructivist, this strand traces the historical social construction of these traditions and ideas. In social epistemology there is the work of Toulmin, Fuller and others, exploring how scientific knowledge is socially constructed and warranted. Finally, in the philosophy of mathematics there is a tradition including Wittgenstein, Lakatos, Bloor, Davis, Hersh, and Kitcher concerned with the social construction of mathematical knowledge. Ernest (1991a, in press) surveys this tradition, and represents one of the few, perhaps the only philosophical approach to mathematics which explicitly adopts the title of ‘social constructivism.’
In the early 1970s the ‘social construction (of the 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.
It appears, however, that the term ‘social constructivism’ first appeared in mathematics education from two sources. The first is the social constructivist sociology of mathematics of Restivo, which is explicitly related 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 all have called themselves social constructivist. Unfortunately 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. Instead he focused more on the social and cultural contexts of the teaching and learning of mathematics. Social constructivism became a more widely recognized position following Ernest (1990, 1991a, 1991b). However, a number of other authors have used and continue to use the word in different ways, such as Bauersfeld (1992) and Bartolini-Bussi (1991). Beyond mere terminology, 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. For example, the approach termed ‘socio-constructivism’ used by Yackel et al., (in press) which is discussed below, can be regarded as social constructivist.
Thus it can be said that social constructivism originates in sociology and philosophy, with additional influences from symbolic interactionism and Soviet psychology. It subsequently influenced modern developments in social psychology and educational studies, before filtering through to mathematics education. Because of these diverse routes of its entry, probably combined with its assimilation into a number of varying paradigms and perspectives 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, and that the individual constructs (or appropriates) his or her meanings in response to his or her experiences in social contexts. This description is vague enough to accommodate a range of positions from a slightly socialized version of radical constructivism, through socio-cultural and sociological perspectives, perhaps 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 characterized 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. Does language express thought, as Piagetians might view it, or does it form thought, as Vygotsky claims?
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 mathematics education, the second question is more immediately 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 and cognitively based theories (e.g., Piagetian or radical constructivist theories), on the one hand, and socially based theories (e.g., Vygotskian theories of learning mathematics), on the other.
Although this is a significant distinction, an important feature shared by radical constructivism and the varieties of social constructivism discussed here is a commitment to a fallibilist view of knowledge in general, and to mathematical knowledge in particular. This is discussed elsewhere (Ernest 1991a, 1991b, in press; Confrey 1991).
Social Constructivism with a Piagetian Theory of Mind
A number of authors has 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 but to add on social aspects of classroom interaction to it. That is, to prioritize the individual aspects of knowledge construction, but to acknowledge the important if secondary place of social interaction. This is apparently the strategy of a number of developments in radical constructivism which seem to fall under this category in all but name (e.g., Richards 1991). Likewise Confrey (1991) espouses a radical constructivist position which incorporates both social interaction and socially constructed knowledge. Researchers whose positions might be located here commonly prefer to describe their perspectives as constructivist as opposed to social constructivist.
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 interpersonal, and concerns social interaction and negotiation between persons. This second framework is extended far enough by some theorists to account for cultural items, such as mathematical knowledge. A number of researchers has 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.’ Earlier Bauersfeld described this theory as that of the ‘triadic nature of human knowledge’ ‘the subjective structures of knowledge, therefore, are subjective constructions functioning as viable models, which have been formed through adaptations to the resistance of “the world” and through negotiations in social interactions.’ (Grouws et al. 1988: 39) 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 symbolic interactionism, but he retains a cognitive (radical constructivist) theory of mind complementing his interactionist theory of interpersonal relations.
Another group of researchers is Cobb, Wood and Yackel of the Purdue Project, who describe their position as constructivist, but also emphasize the social negotiation of classroom norms. They have on occasion used the term ‘socio-constructivist’ for their position (Yackel et al., in press.) Cobb (1989) has explicitly written of the adoption of multiple theoretical perspectives and of their complementarity. These researchers explicitly draw upon the idea of the complementarity of cognitive aspects and acculturation. ‘[W]hen we talk of students’ constructive activities we are emphasising the cognitive aspect of mathematical learning. It then becomes apparent that we need to complement the discussion by noting that learning is also a process of acculturation’ (Cobb et al. 1992: 28, my emphasis). Thus it seems appropriate to identify them as complementarist.
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 illustrates [omitted here]). 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, ibid.: 86–7)
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, some researchers, such as Lerman (1994), argue 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.
In my view, the complementarist forms of social constructivism described above leave intact some of the difficulties associated with radical constructivism. There are first of all many of the problems associated with the assumption of an isolated cognizing subject (Ernest 1991b). Radical constructivism might 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 cognizing 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) proposes that some of the difficulties associated with radical constructivism might be overcome by replacing its Piagetian theory of mind and conceptual development by a Vygotskian theory of mind and language. The outcome might best be seen as a form of social constructivism. More recently, in explicitly taking leave of radical constructivism, Lerman (1994) extends his critique. He argues that any form of social constructivism which retains a radical constructivist account of individual learning of mathematics must fail 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, argues for the value of the co-existence of a Piagetian form of social constructivism, and for the necessity of multiple perspectives. Thus, on the basis of divided opinion and opposing arguments it cannot be legitimately claimed that the Piagetian forms of social constructivism has been refuted or shown to be unviable.
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 consctructionist 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 his colleagues at Nottingham 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 emphasizes mind, interaction, conversation, activity and social context as forming an interrelated whole. Going beyond Weinberg and Gavelek’s brief but suggestive sketch, Bartolini-Bussi 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 the 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 internal processes) in social contexts. These contexts are shared forms-of-life and located in them, shared language-games (Wittgenstein 1953). For this version of social constructivism there is no underlying metaphor for the wholly isolated individual mind. It draws 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, internalized 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 or meanings (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 of these are accommodated in one cyclic pattern of appropriation, transformation, publication, conventionalization. This provides descriptions of both the development of personal knowledge of mathematics in the context of mathematics education (paralleling Berger and Luckmann’s socialization cycle), and describes the formative relation between personal and ‘objective’ mathematical knowledge in the context of academic research mathematics (Ernest 1991a, 1993b, 1994, in press). Such a theory has the potential to overcome the problems of complementarity discussed above.
It is important to distinguish Vygotskian from radical constructivist varieties of social constructivism, for progress to be made in theoretical aspects of the psychology of mathematics education. The adoption of a Vygotskian version is not, however, a panacea. Piagetian and post-Piagetian work on the cognitive aspects of the psychology of mathematics education remains at a more advanced stage and with a more complete theorization, 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 semiotic transformation skills in working with symbolic 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; andthe import of the overall social context of the mathematics classroom as a complex, organized form of life including (a) persons, relationships and roles, (b) material resources, (c) the discourse of school mathematics, including both its content and its modes of communication (Ernest 1993a, in press).
Let me end on a cautionary note. Having delineated a fully social form of social constructivism based on a Vygotskian theory of mind, it is tempting to lump many social and cultural theories together, under this one umbrella. However, significant differences remain between such theories as the sociological theory of mind of Restivo and Collins, Soviet Activity Theory, socio-cultural forms of cognition (Lave, Lave and Wenger), social constructionism in social psychology (Harré, Gergen, Shotter), and Post-structuralist theories of mind (Henriques et al. 1984; Walkerdine 1988). These theories may have more in common with the Vygotskian form of social constructivism than with Piagetian forms. But to lump these diverse perspectives together would lead to far more confusion and inconsistency than that which I have been trying to dispel in this chapter with my analysis of social constructivism.
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