Constructivism, the psychology of learning, and the nature of mathematics: Some critical issues
Ernest P. (1993) Constructivism, the psychology of learning, and the nature of mathematics: Some critical issues. Science & Education 2: 87–93. Available at http://cepa.info/2948
Table of Contents
The nature of mathematics
The primacy of the individual over the social
Constructivism is one of the central philosophies of research in the psychology of mathematics education. However, there is a danger in the ambiguous and at times uncritical references to it. This paper critically reviews the constructivism of Piaget and Glasersfeld, and attempts to distinguish some of the psychological, educational and epistemological consequences of their theories, including their implications for the philosophy of mathematics. Finally, the notion of ’cognizing subject’ and its relation to the social context is examined critically.
Constructivism has become one of the main philosophies of mathematics education research, as well as in science education and cognitive psychology. I use the term ’philosophy’ deliberately, for I do not believe that constructivism is well enough defined to be termed a theory. Neither its key terms, nor the relationships between them are sufficiently well or uniformly defined for the term ’theory’ to be strictly applicable. The multiple ways in which ’constructivism’ is understood are shown by Malone and Taylor (1989) whose brief survey distinguishes between ’personal’, ’radical’, ’socio’, ’pragmatic’ and ’C1 and C2’ varieties of constructivism.
The growing, importance of this philosophy means that a critical discussion of its theoretical basis and practical implications is necessary. A further reason is the risk of teachers and researchers both dividing into camps of non-believers and believers, and the latter identifying constructivism with the child-centred educational ideology of ’progressivism’. My aim is to ask a number of questions from a critical but non-partisan standpoint. What is constructivism? What are its implications for the psychology of mathematics education, and for epistemology and the philosophy of mathematics? To avoid ambiguity I shall first consider Piaget’s constructivism, and then Glasersfeld’s radical constructivism.
It is primarily the influence of Jean Piaget which has established constructivism as a central philosophy in the psychology of mathematics education. His constructivism has a number of components including an epistemology, a structuralist view, and a research methodology, although I do not claim that these are either exhaustive or independent.
Piaget’s epistemology has its roots in a biological metaphor, according to which the evolving organism must adapt to its environment in order to survive. Likewise, the developing human intelligence also undergoes a process of adaptation in order to fit with its circumstances and remain viable. Personal theories are constructed as constellations of concepts, and are adapted by the twin processes of assimilation and accommodation in order to fit with the human organism’s world of experience. Indeed Piaget claims that the human intelligence is ordering the very world it experiences in organizing its own cognitive structures. “L’intelligence organize le monde en s’organisant elle-meme” (Piaget, 1937, cited in Glasersfeld, 1989a: 162).
Piaget’s structuralism involves a belief that in organizing itself, the human intelligence necessarily constructs a characteristic set of logicomathematical structures. Ultimately, these are the three mathematical mother structures distinguished by Bourbaki. Piaget posits an invariant sequence of stages through which an individual’s cognition develops, in constructing these structures (Piagetian Stage Theory).
Piaget’s methodology centres on the use of the clinical interview. In this procedure an individual subject is required to perform certain carefully designed tasks in front of, and with prompting and probing from an interviewer. A series of sessions are likely to be needed for the researcher to develop and test her/his model of the subject’s understanding concerning even the narrowest of mathematical topics.
Each of these three components of Piaget’s constructivism has strengths and weaknesses and important educational implications (although I shall leave epistemology until the next section). Piaget’s structuralism is the weakest of these three areas, and can be regarded as inessential to constructivism (but not to Piaget) so that some recent accounts of constructivism discuss only the epistemological and methodological aspects (Noddings 1990). The two main features are (1) the assumption that the normal development of the human intelligence necessitates the construction of the three logico-mathematical Bourbakian mother structures, and (2) Piagetian Stage Theory, which I shall not discuss here. The Bourbakian account of mathematics in terms of three mother structures (algebraic, ordering and topological structures) represented an advance in the formulation of modern structural mathematics mid-20th century. But it is neither complete nor timeless. It represents mathematical knowledge as a single, fixed hierarchical structure founded in set theory and logic and developing level by level, not to greater abstraction but to greater complexity. This structure is neither necessary nor sufficient to represent the bulk of modern mathematics. The account leaves out combinatorial mathematics which has grown dramatically in import with the development of the electronic computer, including, for example Chaology. In short, the Bourbakian approach, for all its undoubted strengths, is only one way of conceptualizing mathematics, and does not have the essential and foundational character it was taken to have by Piaget. Thus this structuralist feature of Piaget’s theory can be rejected. Elsewhere, I provide a more complete critique of Piaget’s structuralism (Ernest, 1991).
Piaget’s clinical interview method is undoubtedly an important contribution to research methodology in the psychology of mathematics education. With its accompanying methodological assumptions it is among the most widely used approaches of today. Quite rightly so, when in-depth information about an individual’s thinking and cognitive processing is required. However, as even its title suggests, the clinical interview is not the invention of Piaget. Depth psychology, sociological and anthropological interview approaches all offer parallels from outside Piagetian psychology. Indeed the very title ’ethnomethodological approaches’, widely used in educational research, explicitly refers to another disciplinary tradition. Piaget deserves credit for introducing the clinical interview methodology to the psychology of mathematics education. But it cannot be claimed to result uniquely from his constructivism, given that broadly similar approaches also result from widely different theoretical bases.
Ernst von Glasersfeld has extended the foundational work of Piaget significantly, developing a well founded and elaborated constructivist epistemology. He bases this on the following two principles.
Principle A: The ’Trivial’ Constructivism Principle: “knowledge is not passively received but actively built up by the cognizing subject”. Glasersfeld (1989b, page 182)
Principle B: The ’Radical’ Constructivism Principle: “the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality”. op. cit.
Principle A has important psychological and educational implications. It means that knowledge is not transferred directly from the environment or other persons into the mind of the learner. Instead, any new knowledge has to be actively constructed from pre-existing mental objects within the mind of the learner, possibly in response to stimuli or triggers in the experiential world, to satisfy the needs and wants of the learner her/him- self. An immediate consequence is that the transmission model of learning, as assumed in the crudest forms of the ’lecture method’, is seen to be a grossly inadequate model. Although lectures may succeed in communicating to learners, the underlying mechanism is far more complex than that implied by the ’transmission model’. One major difference is that individual learners construct unique and idiosyncratic personal knowledge, even when exposed to identical stimuli. This model of learning has had a profound impact on research on the psychology of mathematics education in the past decade, and also underpins many recent developments in teaching.
Despite this great influence on the twin practices of research and teaching, two important caveats concerning Principle A should be stated. First, although it may help to reconceptualize the teaching of mathematics, the Principle does not strictly imply or disqualify any teaching approach. Rote learning, drill and practice, and passive listening to lectures can, as they always have, give rise to learning. The activity that is necessitated by the Principle takes place cognitively, and so visible inactivity on the part of the learner is irrelevant. Some teaching techniques may be more or less efficient than others (although I suspect that this is only the case given also certain learner and contextual factors). However, this is not the issue here. The ’trivial’ constructivist view of learning does not rule out any teaching techniques in principle. Nor does it equate to the ’discovery method’ or problem-solving teaching approaches, as Goldin (1989) also argues, although it can be used to support them.
Secondly, the model of learning and the concomittent psychological and educational implications of Principle A are not the unique consequences of it. The unique and idiosyncratic nature of learners’ mental constructions also follow, to a very large extent, from other psychological approaches such as those of Ausubel, Kelly, and others, as well as from Repair Theory, information processing and other cognitive science approaches; from sociological theories such as those of Mead, Schutz, Berger and Luckmann; and from the hermeneutic and interpretivist research paradigms. Once again, although constructivism (and in particular, Principle A) may have been a great stimulus in mathematics education research on idiosyncratic learner meaning-construction, it is only one of many theories in the social sciences giving rise to comparable theoretical insights.
Principle B adds a further important epistemological dimension to constructivism (transforming it into ’radical constructivism’). It claims that all knowledge is constructed, and that none of it tells us anything certain about the world, nor presumably, any other domain. This is not entailed by Principle A, for it is consistent to assume that objective truth exists, but that the cognizing subject constructs idiosyncratic personal representations of it, following Principle A. This is trivial constructivism’s claim, which accepts A but rejects B. Principle B has been criticized for leading to the denial of the existence of the physical world (Goldin, 1989; Kilpatrick, 1987). However, this is an incorrect conclusion (see Ernest, 1990). Radical constructivism is consistent with the existence of the world. All that it denies is the possibility of any certain knowledge about it. Glasersfeld (1989b) has explicitly made the point that radical constructivism is ontologically neutral.
The implication of Principle B that there is no certain knowledge is very important, both philosophically and educationally. Applied to mathematics, important consequences can be shown to follow, provided that additional premises are assumed, notably a set of values (Ernest, 1990, 1991). However, the claim I wish to make is that Principle B, whether alone or in combination with Principle A, does not in my view lead to any new practical implications for education and psychology, beyond those of trivial constructivisrn. The one possible exception to this is the adoption of a sceptical epistemology, although what this adoption signifies in practi‑cal terms is not clear. Any good modern scientific practice must be ready to admit that its theories are refutable, following Popper, whether in the domain of subjective or objective knowledge. Hence even trivial constructivists should accept the refutability of scientific knowledge.
Even if it is allowed that there are stronger implications for practice than I have indicated, it can be said that such implications are by no means unique to Principle B, or indeed to any formulation of radical constructivism. There is a long sceptical tradition in philosophy which denies the existence of certain knowledge, which in the modern era finds expression in the work of Dewey and the pragmatists; Wittgenstein, Putnam, Rorty; and which constitutes a central tradition in the philosophy of science, in the work of Kuhn, Feyerabend, Toulmin and Lakatos. In the sociology of science and knowledge the dominant paradigm takes all knowledge to be a fallible social construction. Likewise in post-structuralist and post-modernist thought, such as that of Derrida, Foucault and Lyotard, no true knowledge of the world is assumed. Thus the denial of certain knowledge is far from unique to radical constructivism.
The nature of mathematics
Brouwer’s Intuitionism is a view of the nature of mathematics sometimes associated with radical constructivism. Steffe (1988) argues that these views are consistent, and implies that intuitionism is the appropriate philosophy of mathematics to consider. There are indeed parallels, for Brouwer argues for the subjective construction of mathematics, and maintains that mental constructions have philosophical priority over linguistic forms. However the weakness of Brouwer’s position is the assumption that all the many subjective constructions of mathematics by different persons lead to the same body of knowledge. This is an essentialist view which assumes that a given structure of knowledge, notably mathematics, must emerge from any individual’s consciousness. It also assumes that there is a unique body of mathematical knowledge, and so that any individual’s construction, if correct, must produce a part of it. These conclusions contradict both principles. However, my intention is not to demonstrate that radical constructivism is inconsistent. As Lerman (1989) shows, radical constructivism and intuitionism are distinct and independent.
Glasersfeld does not commit this error, for his treatment is closer to the ’full-blooded’ conventionalism of Wittgenstein (1978). Glasersfeld (1989c) draws the analogy between mathematics and chess, arguing that the seeming necessity of mathematical theorems follows from the social acceptance of its rules and definitions. However, the radical constructivist account of other persons is that they are hypothetical entities constructed by the cognizing subject to account for certain apparent regularities in its experiential world (Glasersfeld, 1989b). Since the warrant for accepting a mathematical theorem is therefore that it conforms to socially accepted rules of justification, like those of chess, its certainty can be no stronger than any link in the overall chain of hypotheses and reasoning. The existence of other persons is part of the constellation of assumptions. Therefore mathematical knowledge rests on foundations at most as firm as the existence of certain hypothetical entities, notably persons. Given Principle B, this is a less than certain basis for mathematical knowledge.
I see this outcome as a strength of radical constructivism. A growing tradition in the philosophy of mathematics (in the works of Bloor, Davis, Hersh, Kitcher, Lakatos, Putnam, Tymoczko and Wittgenstein) regards mathematical knowledge as uncertain and fallible, contrary to traditional absolutism. Radical constructivism is consistent with this new tradition (see Ernest, 1990, 1991). But although these parallels offer support, they also mean that this conclusion does not follow uniquely from Principle B.
The primacy of the individual over the social
Following the rational tradition in philosophy since Descartes the radical constructivist account begins with the assumption of a cognizing subject. The genesis of all knowledge is described in a narrative based on a epistemological/biological metaphor. This posits an homunculus, a miniature being at the core of the cognizing subject, which is necessitated by both principles. This subject is the active creator of knowledge, and the adaptive organiser of the experiential world. However, a weakness is that radical constructivism assumes the subject to be unproblematic. There is no question as to its origin or constitution: is it unitary and indivisible?
Post-structuralist psychology, founded on the work of Foucault, Lacan and others, elaborated in Henriques et al. (1984), and applied to mathematics by Valerie Walkerdine, questions the unity and unproblematic nature of the cognizing subject. It regards the individual-social dualism that it necessitates as a false dichotomy, viewing the individual as constituted by social and discursive practices, including positioning in power relations. Other psychological traditions also deny the inviolable primacy of the cognizing subject, including the symbolic interactionism of Mead, the Soviet psychology of Vygotsky, and the Activity Theory of Leont’ev and Davydov. Thus a problem for radical constructivism is to account for the cognizing subject itself, which cannot be taken as unproblematic. A range of theoretical perspectives suggests that the subject is at least partly constituted by the social context, and that language plays a more central part in the formation of mind than radical constructivism allows.
The secondary role accorded to the realm of the social is also a problem. Other persons are regarded is subjective constructs, and so the social domain, the source of parental and human love, of language, culture and mathematics, has no intrinsic primacy. To regard the social as secondary to the pre-constituted cognizing subject is again problematic, and this difficulty is not adequately resolved by radical constructivism.
Radical constructivism leads to significant consequences, but in each case these also follow from other theories. So what is unique about it? Although a number of theories lead to similar consequences in methodology, learning theory, sceptical philosophy and fallible philosophy of mathematics, none offers an account so inclusive and wide ranging. Hence a unique strength of radical constructivism is its breadth. This is important, for a single broad theory is more powerful than a set of narrow ones. Perhaps for this reason, despite the problems raised here and elsewhere, constructivism remains one of the most fruitful philosophies of mathematics education research today.
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