The one and the many
Ernest E. (1995) The one and the many. In: Steffe L. & Gale J. (eds.) Constructivism in Education. Lawrence Erlbaum, New York: 459–486. Available at http://cepa.info/4160
Table of Contents
The one: Constructivism as a unity
Three research paradigms
The many: The diversity of constructivism
Epistemologies or research paradigms?
Seven educational paradigms and their metaphors
Comments on the Spivey chapter
Comments on the Saxe chapter
Wood, Cobb, and Yackel
The one again: A synthesis of the concerns of constructivism
Individual or social construction?
Is constructivism one or many schools of thought? This is not as trivial a question as it sounds. If one, then perhaps almost everybody belongs, which does not tell us much. It leads to such outcomes as neobehaviorists claiming that we are all constructivists now. Neobehaviorists should not be dismissed too lightly because the models of cognition in the work of Ausubel, Gagne, and others are subtle and complex. Ausubel (1968) said long ago that “the most important single factor influencing learning is what the learner already knows. Ascertain this, and teach him accordingly” (p. i). Coupled with his work on meaningful versus discovery learning, this seems to place him close to some weak form of constructivism. But Lewin (chap. 23) has demonstrated that a lack of clarity in representing constructivism may allow its appropriation by the most authoritarian of pedagogies. One of the central tasks of this chapter is to clarify what is and what is not constructivism.
In contrast, if there are many types of constructivism, as I suggest elsewhere (Ernest, 1991b), there is the risk of wasting time by worrying over the minutiae of differences. But accepting that the differences between various constructivist positions are significant, I am left on the horns of a dilemma. At one end, (almost) all is constructivism, especially if, as Rubin (chap. 19) says, constructivist epistemology is the only paradigm that survives. At the other extreme, there are almost as many varieties of constructivism as there are researchers.
Suggested by this dilemma, I use the pre-Socratics’ contrast of the one and the many as my organizing theme. It fits well with the aim of this chapter: to offer both an analysis and synthesis of half of the preceding contributions. Synthesis constructs unity (the one), whereas analysis turns out to suggest diversity (the many). Starting with “the one,” I consider what different positions in constructivism have in common. Then I consider “the many” – analyzing some salient differences between different constructivist positions (and a few others). Finally, I offer a synthesis, “the one” again, identifying what seem to be deeply shared themes, problems and points of growth for constructivism, and alternative epistemologies in education.
As a preliminary, there is a term that can benefit from clarification, namely epistemology. This has two main contexts of use: psychological and philosophical. In psychology, epistemology concerns theories of knowledge growth and development, the structures of knowledge constructed by individuals, and theories and the general conditions of learning. Less common, but equally important, is a concern with the nature, structure, and development of knowledge in the large (which I shall call conventional knowledge), and its relationship with the individual knower and that knower’s learning.
It is worth pointing out that persons’ beliefs in psychology are regarded as stronger than their knowledge, in view of the personal investment in them. After all, one might know about three mutually exclusive educational research paradigms, but only believe in one of them. Overall, in psychology, epistemology is about the genesis and nature of knowledge, including learning. In the main, this concerns what Popper (1959) called the context of discovery.
In philosophy, epistemology is a synonym for the theory of knowledge, which is understood to concern the logical categories of knowledge and its justificational basis. There is concern for both the warranting of the subjective knowledge of a single knower and that of conventional knowledge. The traditional emphasis is with knowing whether declarative sentences are true or false, which has led foundationist programs to try to establish bodies of objective knowledge as absolutely true. But there is also an ancient sceptical tradition, currently enjoying a revival, that regards this goal as mistaken and unattainable (Lakatos, 1962; Rorty, 1979).
In philosophy, knowledge is distinguished from beliefs, which are unwarranted knowledge claims. Unlike in psychology, belief is weaker than knowledge.[Note 1] Overall, the emphasis in philosophical treatments of epistemology is on the logical classification and justification of knowledge, or what Popper (1959) called the context of justification.
Some constructivist researchers, such as Noddings (1990), described radical constructivism as postepistemological. This points out that radical constructivism rejects foundationism, and is a successor to traditional epistemology paralleling the successor movements postmodernism and poststructuralism. However, even in traditional terms, antifoundational epistemology is possible, and the warranting of knowledge can be addressed without presupposing absolute foundations (Rorty, 1979; for mathematics: Ernest, 1991a, in press; Kitcher, 1984). Thus, it is not necessary to abandon the term epistemology to be antifoundationist.
The one: Constructivism as a unity
What binds together the various forms of constructivism is what Spivey (chap. 16) terms the metaphor of carpentry or architecture (or construction work). This is about the building up of structures from preexisting pieces, possibly specially shaped for the task. The metaphor describes understanding as the building of mental structures, and the term restructuring, often used as a synonym for accommodation or conceptual change, contains this metaphor. What the metaphor of construction does not mean in constructivism is that understanding is built up from received pieces of knowledge. The process is recursive (Kieren Pirie, 1991), and so the “building blocks” of understanding are the product of previous acts of construction. Thus, the distinction between the structure and content of understanding can only be relative in constructivism. Previously built structures become the content in subsequent constructions.
The metaphor of construction is not unique to constructivism in learning theory discussed here. There is also constructivism in sociology (Berger & Luckmann, 1966), educational evaluation (Lincoln & Guba, 1985), the philosophy of mathematics (e.g., Brouwer’s intuitionism), and Soviet art (Tatlin, Pevsner, & Cabo), all of which are based on the metaphor.
The metaphor of construction is contained in the first principle of constructivism as expressed by von Glasersfeld (1989a): “knowledge is not passively received but actively built up by the cognizing subject” (p. 182). von Glasersfeld terms this the principle of trivial constructivism, constituting those positions based on this principle alone. In this section, I simply refer to constructivism, signifying the shared and undifferentiated constructivist perspective of learning that arises from following the previous principle. (It can also be derived by uniting the various different constructivist positions and washing out their differences.)
Simple minded as this basic form of constructivism might seem to be, it represents a significant step forward from naive empiricism or classical behaviorism. It recognizes that knowing is active, that it is individual and personal, and that it is based on previously constructed knowledge. Just getting student teachers to realize this, by reflecting on “child methods” in mathematics or alternative conceptions in science, represents a significant step forward from the naive transmission view of teaching and passive – receptive view of learning many arrive with. However, a passive-receptive view of learning is not dead among professionals or administrators in education. Many government-driven curriculum reforms, in Britain at least, assume that the central powers can simply transmit their plans and structures to teachers who will passively absorb and then implement them in “delivering the curriculum.” Such conceptions and strategies are deeply embedded in the public consciousness, although it may be no accident that they also serve authoritarian powers (Ernest, 1991a).
The previous remark shows that even theories of learning cannot be regarded as politically neutral. Politics arises in a number of chapters, and, as von Glasersfeld (chap. 20) says, to teach “to imbue the young with the accepted knowledge and beliefs of the community is a political endeavor.” Likewise, by acknowledging the power differentials between writers and responders in some situations, Spivey is talking micropolitics. Just as we accept that epistemology is inescapable in education, so, too, is politics.[Note 2]
Politics is about ideology and conflict. We are in an era in which politicians and political parties greatly impugn and disparage their opponents. There is a new barbarism afoot in politics. In their open competition for power, almost no holds are barred. In my view, this brings the whole democratic process into disrepute. Certainly ethical values like modesty, decency, honesty, and mutual respect have gone by the board. Surely we in education and the public service, if anyone, should strive to maintain these values. What has this to do with constructivism and alternative epistemologies in education? It has to do with giving respect to those positions with which we disagree.[Note 3] For example, even if we think that trivial constructivism misses out on the crucial epistemological issues raised by radical constructivism, it deserves credit for its strengths. Of course this does not mean that we should withhold any rational criticism of that position: To engage critically with a position is to recognize its significance, and thus pay it respect.
Three research paradigms
Educational researchers such as Gage (1989) and Schubert (1986) have identified three research paradigms: (a) the scientific or neopositivistic paradigm, with its emphasis on quantitative methods; (b) the interpretative or constructivist paradigm, with its emphasis on qualitative methods (sometimes called “Alternative Paradigms Research,” presumably an umbrella under which the orientation of this collection falls); and (c) the critical paradigm, based on critical theory, and concerned to promote social justice and social change. This last paradigm is due to Habermas and others, and was popularized by Carr and Kemmis (1986). Of course many researchers are eclectic and combine elements of more than one of these. For example, much middle-of-the-road research at the annual Psychology of Mathematics Education conferences draws on elements of the first two paradigms, perhaps without being overly clear about epistemological underpinnings.
Now the question is, do we identify ourselves with the second paradigm and reject the other two paradigms as mistaken? After all, they both assume naive realist ontologies and epistemologies.[Note 4] Do we acknowledge that there are multiple and equally legitimate perspectives, even if they are not all equally successful according to our criteria?[Note 5] Do we engage in paradigm wars, or do we opt for co-existence (Gage, 1989), acknowledging that each of the paradigms is valuable and deserving of respect, even if we disagree strongly with some of the underlying assumptions? My view is that we should acknowledge the significance and legitimacy of all three paradigms, while criticizing those assumptions and consequences of the neopositivist paradigm that we find deficient.
A danger for constructivism, although by no means realized in the chapters I am commenting on, is an overly child-centered, romantic progressivism. Constructivism, conceived in a loose and emotive way, can become associated with a sentimental view of the child (Walkerdine, 1984). Lewin (chap. 23) describes the romantic myth early on in his chapter, and the problem to which it leads (viz., the overshielding of the child from social influence). Wood, Cobb, and Yackel (chap. 22) are also on their guard against the romanticism that can accompany progressive teaching, which sanctions anything the child does as expressions of his or her individual creativity. Discovery learning, beginning in the 1960s, was often bound up with a romanticism that, in the end, was not wholly productive for learners. We must guard against constructivism becoming identified with this position. Several of the chapters, focus on the need to let learners construct their own meanings, yet for the teacher and peers to interact with learners to negotiate a passage toward socially accepted knowledge. They propose that the teacher should steer children away from nonproductive solutions, but not steer children toward productive solutions. The latter presupposes that the teacher is in possession of “the truth,” rather than someone aware of the conventional nature of knowledge. Elsewhere I have analyzed the ideology of romanticism in progressive education in greater depth, indicating that it often rests on an absolutist epistemology. In such circumstances, there is a perception that there are “right answers” toward which to steer children (Ernest, 1991a). The epistemological contrast between this absolutist position and the fallibilism of radical constructivism immediately reveals an area where different versions of constructivism make a significant difference in practice.
The many: The diversity of constructivism
The way that I have used the portmanteau title of constructivism applies to much modern research on learning. However, it thereby includes a whole range of different research paradigms and epistemologies, as the last section began to show. For example, many well-established researchers pursuing generously funded research projects based on abso‑lutist epistemologies and neopositivist methodologies now term themselves constructivists. I want to draw out the differences between variant constructivist positions here. In doing this, a number of questions seem worth considering.
Epistemologies or research paradigms?
An immediate terminological issue concerns whether the different constructivist positions should be termed epistemologies or research paradigms. Although the conference on which this book is based was called “Alternative Epistemologies in Education,” there is a parallel movement in education called “Alternative Paradigms Research.” This regards each research paradigm as providing an overarching theoretical world view, including an epistemology. Although the difference is largely terminological, I prefer this description, calling an overall theoretical research perspective a research paradigm that includes an epistemology.
What then characterizes a research paradigm? A number of scholars have identified a research paradigm as composed of an ontology, an epistemology, and a methodology (Sparkes, 1991). In view of the educational concerns of this book, I want to add to this a pedagogy. Thus, in the analysis that follows, I distinguish the following components of an educational paradigm:[Note 6]
Ontology: a theory of existence concerning the status of the world and what populates it.Epistemology: (a) a theory of the nature, genesis, and warranting of subjective knowledge, including a theory of individual learning; (b) a theory of the nature, genesis, and warranting of knowledge (understood as conventional or shared human knowledge), as well as a theory of truth.?Methodology: a theory of which methods and techniques are appropriate and valid to use to generate and justify knowledge, given the epistemology. Pedagogy: a theory of teaching – the means to facilitate learning according to the epistemology.
The last component is the most problematic part of the paradigm because teaching fundamentally concerns technique for facilitating learning in the service of a given set of values.[Note 8] Thus, pedagogy is inescapably, if tacitly, value laden (echoing von Glasersfeld’s comment about the political nature of education). There is also the possibility of overlap between methodology and pedagogy, especially in the case of the constructivist teaching experiment (see, e.g., Steffe & Wiegel, 1992). Engaging a child in a task to elicit evidence of his or her mathematical understanding is equally involving the child in a learning activity. However, despite this instance of overlap, there is logical distinction between pedagogy and methodology that is worth maintaining, especially as in some paradigms, such as traditional empiricism, the epistemology rules out any possibility of intersection.
Beyond these explicitly stated dimensions of educational paradigms, can there be other aspects of difference? In addition there could be hidden metaphors, images and values belonging to the paradigm. Shotter (chap. 3) elicits and contrasts the underlying metaphors of von Glasersfeld’s radical constructivism and Gergen’s social constructionism. Likewise, Wood et al. contrast the “two incompatible images of what it is to be a human being” that underpin information-processing theory, on the one hand, and radical or social constructivism, on the other hand. Similarly, I focus on two issues that I consider to be central in distinguishing educational paradigms (viz., the underlying metaphors for the mind and the world; Ernest, 1992). Because these probably underpin the explicit theories of the mind and the epistemology of these research paradigms, they are of central importance and worthy of attention.
Seven educational paradigms and their metaphors
Seven different paradigms are considered, and all but the first is related in some way to the positions indicated in the chapters discussed here. They are: traditional empiricism, information-processing theory, trivial constructivism, sociocultural cognition, radical constructivism, social constructivism, and social constructionism. It must be admitted at the outset that none of the chapters on which I am commenting specifies with any completeness all four dimensions of the underlying educational paradigm, nor do they detail the assumed metaphors of mind and world. Instead, the chapters apparently aim to: (a) briefly indicate some aspects of the educational paradigm, typically the constructivist learning theory part of the epistemology; (b) illustrate with a few carefully chosen cases the findings of empirical research and accounts of teaching experiments, and to relate them to the underlying epistemology or paradigm; and (c) abstract from the former, or simply outline inferred practical pedagogical outcomes and principles.
Although this is not a wholly accurate account, it makes the point that the specification of these paradigms and their metaphors depends on an act of construction on my part. Relating the positions adopted in this volume to such a classificatory framework requires me to construct the unsaid in a text (Spivey, chap. 16). Indeed, to give a comprehensive account of one educational paradigm would require the space of several monographs, rather than the single chapter that the authors have available. In some cases, my proposed classificatory scheme turns out to be a procrustean bed onto which the positions do not fit neatly. Clearly any attributions in the following text must be offered tentatively. Nevertheless, the range of concepts that I have elucidated provides a useful framework for analysis, even if wanting in several respects.
Traditional empiricism represents the historical backdrop against which the other paradigms have developed. The empiricist metaphor of mind is that of an empty bucket, a blank page, a tabula rasa waiting to be filled with sense impressions or the results of reasoning. It may also be that of a mirror, passively reflecting reality. The associated metaphor of the world is that of absolute Newtonian physical space, in which permanent material objects are positioned and move. Rorty (1979) interrelated these two images with his metaphor of the mind as a “mirror of nature.” The mind as mirror reflects the disposition of objects in the material world. By introducing the relationship between mind and world, Rorty necessitates a consideration of knowledge and truth. By mirroring nature (sometimes with flaws, imperfections, and distortions), the mind is appreciating a true (or false) picture of the world. Thus, there is an associated theory of truth and knowledge: the correspondence or “picture,” theory of truth, and knowledge as a true representation or description of the world, or of necessary connections of ideas, as in the case of mathematics.
Modern science has repudiated Newtonian absolute space in favor of Einsteinian space, but the former inhabited by material objects lingers on as a tacit metaphor of the world in this perspective. Thus, the ontology of naive realism of science persists: The world of things we experience are out there, part of an ultimate reality. The epistemology is objectivist: True knowledge of the state of affairs in the world is possible, and certainty in mathematical knowledge it is also attained. The associated theory of learning takes knowledge to be received ready made by the learner.
Empiricism is not represented by any of the chapters because it is a neopositivistic research paradigm. In its simplest form, it assumes a naive transmission view of teaching as its pedagogy, and a passive–receptive view of learning. This view of learning ascribes student errors and misconceptions as due to faulty memorization or recall, or to carelessness in application. Such a view is deeply embedded in the public consciousness, and unarticulated it is the “default” paradigm that many student teachers start with, as I suggested earlier.
Information processing might be regarded as one of the simpler forms of constructivism. It appears to accept von Glasersfeld’s first principle, which is common to all constructivist positions, but rejects the second principle, with its far-reaching epistemological consequences. However, I argue that it falls short of being even a form of trivial constructivism.
The information-processing paradigm is a broad church that includes the psychology of Ausubel (1968), Anderson (1984), repair theory, and many of the positions adopted by researchers in cognitive science and related psychologies. It is largely based on the metaphor, and sometimes the conscious model, of the mind as computer. This actively processes information and data, calling up various routines and procedures, organizing memorization and retrieval of data. It can even be heuristically programmed (i.e., modifying its outputs as it learns from experience). The computer metaphor is fruitful because it has led to important analyses of human problem solving (Newell & Simon, 1972) and “buggy” error analysis (Brown & Burton, 1978), with important outcomes for the psychology of mathematics education.
The most common associated metaphor of the world is that of Newtonian absolute space populated by material objects. In general, information-processing theory represents a shift from the traditional empiricist metaphor of mind as passive to a complex mechanical (or rather electronic) metaphor of mind as computer, but there is usually no shift in the underlying metaphor of the world.
In summary, the ontology is the naive realism of science: The world of things we experience are out there. They are part of an ultimate reality. The epistemology is objectivist: True knowledge of the state of affairs in the world may be possible, as is certainty in mathematical knowledge. However, where this position differs from empiricism is in its theory of learning. It represents a significant step forward from empiricism and classical behaviorism because it recognizes that knowing involves active mental processing, it is individual, and it is based on previously acquired knowledge. Thus, learning is not just a passive absorption of information. Rather, it is more interactive, involving the selection, processing, and assimilation of information according to the state of mind of the learner.
An important outcome of this perspective in terms of learning theory (and pedagogy) is that it accounts for student “error patterns” in mathematics (Ashlock, 1982), and similarly for misconceptions in science.
Does information-processing theory constitute a form of constructivism? It is evidently close to one, on the grounds that its account of learning seems to satisfy von Glasersfeld’s first principle – that knowledge is not passively received, but built up actively. However, as indicated earlier, this principle means that the construction of knowledge is recursive, and it builds on previously constructed knowledge instead of passively received information or knowledge. But the metaphor of mind as computer means that, at a basic level, incoming information or knowledge must simply be received by the cognizing subject, in preconstituted form, and that any complex response or elaboration that may follow its reception builds from this. This is not a recursive process of construction, but assumes instead the existence of objective, preexisting knowledge at a basic level. As one of its exponents said, alongside cognitive science, information-processing theory is “the study of how humans process information, and includes the acquisition, storage and retrieval of knowledge” (Mayer, 1982b, p. 3). The rhetoric of this quotation reveals the underlying presupposition that some knowledge learned by humans is information that is admitted from the outside, not constructed from within. Thus, in the final analysis, information-processing theory contradicts von Glasersfeld’s first principle, and hence is not a form of constructivism.
Accepting that information-processing theory is not a form of constructivism, it is clear that a weak form of constructivism can be developed from it simply by fully accepting von Glasersfeld’s first principle. This is termed trivial constructivism, and combines the principle that all individual human knowledge is constructed by each individual with the other assumptions of the information-processing paradigm described previously. Thus, the underlying metaphors of mind and world are almost the same. However, a difference is that the mind is an ideal “soft” computer, namely, the brain. Thus, the data it processes are self-con‑structed all the way down to the basic level of electrochemical nerve impulses.
Now the question arises, to what extent is this position reflected in chapters in this book? It is most evident in the chapter by Spiro et al. (chap. 6), which discusses advanced knowledge acquisition in complex and ill-structured domains. Constructivism is defined thus: “(a) understandings are constructed by using prior knowledge to go beyond the information given; and (b) the prior knowledge that is brought to bear is constructed, rather than received intact from memory, on a case-by-case basis.” The emphasis here is clearly on the first of von Glasersfeld’s epistemological principles of constructivism, accepted fully. In addition, the use of computer-based hypertext instructional systems and simulations suggests the computer metaphor of mind of this position. The Spiro et al. chapter is consistent with this position, although the fit is only partial. The use of computers as an instructional medium is quite independent of a computer metaphor of mind, and should not be used as evidence of it. A striking counterexample is given by Papert (1980), who is widely regarded as a constructivist.
There is an inherent weakness – an instability even – built into trivial constructivism because it is difficult for the dual aspects of its epistemology to co-exist. On the one hand, all individual knowledge is constructed. On the other hand, there is a realm of objective knowledge, which would include the truths of mathematics and facts about the world. But how can such knowledge be known by any individual if his or her knowledge is a personal construction? It must be that an individual can construct truths to be able to know such knowledge. Thus, the individual constructs truths about the world and mathematics. But in this case, an individual’s constructions are correct representations of external states of affairs, via sense organs. This means that knowledge is constructed to match the world, or the eternal verities of mathematics, and not as a recursive construction based on previous constructions satisfying inner constraints. In short, constructed truths can only be known as such by means of information from the world. Thus, there is at the very least an antinomy – if not a direct contradiction – at the heart of weak constructivism.
Trivial constructivism deals with this by being a local, as opposed to a global, paradigm. That is, it accepts traditional epistemology concerning knowledge, and only tries to account for the knowledge representations of individuals.
Comments on the Spivey chapter
Of the chapters that I am commenting on, Spivey shares some features with trivial constructivism, although I believe she regards herself as a radical constnictivist. A significant quote is that, “for now. a repre‑ sentation of meaning is conceptualized by theorists as consisting mainly of interconnected concepts or semantic propositions … although eventually it may be characterized instead in more physiological terms, such as the firing of neurons.” The unproblematic assumption that subjective meaning can be equated with an objectivist linguistic representation, or with a neurophysiological description, sits best with the scientific realism of trivial constructivism, and is inconsistent with a fallibilist epistemology. However, on the basis of slender indications like this, it is not possible to categorize the Spivey chapter as either trivial or radical constructivist.
More importantly, perhaps, there is an elaborate and valuable theory of learning applied to the production and reading of text, independently or in cooperation with others. There is no indication of a methodology, but there is the basis of a pedagogy that combines reading and writing in a potentially important way. Spivey’s chapter focuses on local (as opposed to global) theoretical issues of the processes of writing and reading, and represents a valuable contribution to a constructivist theory of this area.
The sociocultural cognition paradigm may be regarded as based on the metaphor of the mind as game player and strategist. Like any metaphor, this is only a simple, underpinning image, and none of the connotations of triviality sometimes associated with games is intended. This image extends the mind as a computer metaphor because of the rational rules, scripts, and procedures that can describe the game playing and strategies. But it transcends that metaphor because it also admits social interaction and its sociocultural contexts as essential. Game playing involves deliberate, player-initiated moves and co-player responses. As well as building on rules and strategies (and modifying them), game playing is essentially goal orientated, so this perspective also admits the importance of goals. The sociocultural cognition metaphor of the world remains that of absolute Newtonian space, but it is a version of Newtonian space including human society.
The pedagogy of this position could hinge on the central notion of apprenticeship, or “legitimate peripheral participation” (Lave & Wenger, 1991). This involves admitting novices into social practices at the periphery, and then letting them take up full player roles as they develop mastery. This is typical of informal learning and craft apprenticeship. But the pedagogy can also involve specially designed social activities, which allow novices to gain mastery cumulatively over a range of simplified and specially designed domains of knowledge and activity. This is more relevant to formal education systems.
Comments on the Saxe chapter
There is a loose parallel between this paradigm and the position described in Saxe (chap. 15). Although the ontology and epistemology are not specified, symbolic cultural artifacts and knowledge of them is a dimension of the epistemology, which goes beyond the positions considered earlier. Learning has an overt cultural dimension, and may even be situation-specific. The inclusion of learner goals also constitutes an important innovation. Thus, the assumed model of the learner seems to transcend that of information-processing theory because of this recognition of the learner’s existent and emergent goals, and the learner’s conscious use of symbolic representations back and forth across the boundary of personal-public domains. The pedagogy is eclectic, building on all of these features, but with an emphasis on social aspects of the teaching- learning situation.
In commenting on other chapters, Becker and Varelas (chap. 24) emphasize the Vygotskian perspective, which combines a top-down flow of knowledge (cultural knowledge, via the teacher) with one that is bottom-up (personally constructed knowledge brought by the learner). This polarity is present, at least in large part, in Saxe’s chapter. Both the emergent goals and the semiotic artifacts are central to Vygotsky and to the subsequent development of activity theory.
Saxe also exemplifies his pedagogy in the form of an instructional game that stresses both emergent goals and semiotic artifacts, and employs translational processes between different modes of representation. Although explicitly utilizing the underlying metaphor of mind of sociocultural cognition in his pedagogy, all this demonstrates is the consistency of Saxe’s stated position with this paradigm. More revealing is the importance he attributes to social context and the socially situated nature of much of knowledge and learning. There is the important recognition in his chapter that learners inhabit multiple social practices, which are sources of varying knowledge. This has major implications for the education of all, but especially disadvantaged, children. The conclusion drawn is to develop a culturally embedded mathematics curriculum. This has the advantage of drawing on children’s existing cultural knowledge, although it will not be uniform, as Saxe recognizes.
The use of learner’s cultural knowledge raises the problem of transfer of mathematical skills. Although transfer is noted in the pedagogical example, it is from one mode of representation to another, not from one social practice to another. In fact, there may be underrecognition of the larger social context of the pedagogical game employed – notably the contract between teacher/researcher and children. This includes the children’s need to satisfy the more nowerful adults – to operate within assumed social parameters of time, place, and behavior limitations. This constitutes a single discursive practice, namely that of school, with its unique language, knowledge, and power relationships. Thus, opening the door to the social context of learning raises important new psychological considerations, but it also dramatically increases the complexity of the phenomena to be described.
Overall, Saxe’s chapter offers what may be an important beginning in bridging anthropology and constructivist epistemology, and linking them to pedagogy. Whether it exemplifies sociocultural cognition or radical constructivism is not clear from the necessarily abridged account in this book. There are no pointers as to the assumed epistemological status of knowledge, absolutist or otherwise. Certainly elsewhere Saxe (1991) affirmed his adherence to constructivism, but he left unanswered the question of which variety of constructivism is meant.
Although it originates with Piaget and is anticipated by Vico, in its modern form radical constructivism has been most fully worked out in epistemological terms by von Glasersfeld, in a series of publications over the past 15 years. In methodological terms, the leading figure in the area of mathematics education has perhaps been Steffe.
Definitionally, radical constructivism is based on both the first and second of von Glasersfeld’s principles. The second profoundly affects the world metaphor, as well as that of the mind: “The function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality” (von Glasersfeld, 1989a, p. 182). Consequently, “From an explorer who is condemned to seek ‘structural properties’ of an inaccessible reality, the experiencing organism now turns into a builder of cognitive structures intended to solve such problems as the organism perceives or conceives” (von Glasersfeld, 1983, p. 50).
Although it is controversial, and the caveat is needed that there are a number of different forms of radical constructivism (Ernest, 1991b), my claim is that the underlying metaphor for the mind or cognizing subject is that of an organism undergoing evolution, patterned after Darwin’s theory, with its central concept of the “survival of the fit (or fitter).”[Note 9] This is indicated in Piaget’s notion of adaptation to the environment, and his explicit discussion of cognitive evolution (Piaget, 1972).[Note 10]
According to the evolutionary metaphor, the cognizing subject is a creature with sensory inputs, furnishing data that are interpreted (or rather constructed) through the lenses of its cognitive structures; it is composed of a collection of those structures, all the while being adapted, and has a means of acting on the outside world. The cognizing subject generates cognitive schemas to guide actions and represent its experiences. These are tested according to how well they “fit” the world of its experience. Those schemas that fit are tentatively adopted and retained as guides to action. Cognition depends on an underlying feedback loop.
Thus, on the one hand, there is an analogy between the evolution and survival of the fitter of the schemas in the mind of the cognizing subject and the whole of biological evolution of species. Schemas evolve and, through adaptation, come to better fit the subject’s experienced world. They also split and branch out, and perhaps some lines become extinct. On the other hand, the organism as a whole is adapting to the world of its experiences, largely through the adaptation of its schemas.
It is difficult to isolate the underlying metaphor of the world of this position because it is implicated in that of the mind. It is experienceable, but not knowable, in any ultimate sense, just as the Kantian world of phenomena floats above the unattainable substratum of the “noumena.” It is like the environment or world surrounding an animal: It is real and resists and constrains the animal, but is not known by the animal (including humans) over and above the ways that the animal’s schemas fit or fail to fit the world. No match between these schemas and the world is possible, nor could it be verified if it did exist.
Overall, radical constructivism is neutral in its ontology, making no presuppositions about the existence of the world behind the subjective realm of experience. The epistemology is wholeheartedly fallibilist, sceptical, and anti-objectivist. The fact that there is no ultimate, true knowledge possible about the state of affairs in the world, or about such realms as mathematics, follows from the second principle, which is one of epistemological relativity. As its name implies, the theory of learning is radically constructivist all knowledge being constructed by the individual on the basis of his or her cognitive processes in dialogue with his or her experiential world. The pedagogy is multifaceted, but at its heart lies sensitivity to individual construction.
Radical constructivism is a rich theory that is giving rise to a whole body of fruitful and innovative research, as this volume indicates. Indeed, its importance to mathematics and science education can hardly be overestimated. Indeed most, if not all, of the chapters I am commenting on adopt a radical constructivist position, or some development or variant of it. Radical constructivism represents the state of the art in epistemological theories for mathematics and science education. Uniquely, it represents an educational paradigm that has been fully developed: The ontological, epistemological (in both senses), methodological, and pedagogical dimensions have all been extensively treated in the recent literature. Such enthusiasm does not mean that radical constructivism should not also be subject to critical scrutiny; only through the dialectic of criticism and response can it grow to greater richness and strength.
One central criticism that might require a clearer exposition of the relevant aspects of the position, or some revision, is as follows. The account of the cognizing subject emphasizes his or her individuality, separateness, and primarily cognitive representations of his or her experiences. His or her representations of the world and other human beings are personal and idiosyncratic. Indeed, the construal of other persons is driven by whatever representations best fit the cognizing subject’s needs and purposes. None of this is refutable. But such a view makes it hard to establish a social basis for interpersonal communication, for shared feelings and concerns, let alone for shared values. By being based on the underlying evolutionary metaphor of the mind, there is a danger that interpersonal relationships are seen as nothing but competitive – a version of the “law of the jungle.” After all, this is but another way of phrasing “the survival of the fit.” Yet society and its functions, in particular education, depend on articulated and shared sets of concerns and values – values that are most evidently subscribed to by radical constructivists. Thus, the paradigm needs to accommodate these issues by balancing knowing with feeling, and acknowledging that all humans start as part of another being, not separate.
However, von Glasersfeld has shown in his treatments of aspects of radical constructivism that it is possible to elaborate the position extensively to answer, if not rebut, much of this criticism. Each individual’s knowledge of other persons and, hence mediately, the realm of the social can be consistently construed as constructions of the individual knower. Such an epistemology is self-consistent, and does not fall prey to facile critiques. Similarly, being ontologically neutral, radical constructivism is not solipsistic, as some critics have claimed.
As I stated previously, most if not all of the chapters that 1 am commenting on are considered by the authors to be located in the radical constructivist paradigm. Several are concerned with accommodating the social dimension, which has been underdeveloped in past accounts of the position. Thus, at this point I discuss the chapters of von Glasersfeld; Duit; Wood, Cobb, and Yackel; and Driver.
Thus far, I have delayed mentioning what is perhaps the greatest central strength of radical constructivism: its detailed account of the process of individual construction of knowledge. Ultimately, a psychological theory must stand or fall by its ability to account for psychological phenomena. In this respect, radical constructivism is peerless in being able to describe the psychological processes by which individuals form concepts, elaborate those concepts into conceptual structures, and restructure them into more powerful and encompassing schemas by means including what Piaget (1972) termed reflective abstraction. Piaget’s own work opened up this area of investigation, and it has continued in the work of Steffe, von Glasersfeld, and colleagues (see Steffe et al., 1983).
Von Glasersfeld (chap. 20) focuses on the elaborate processes of reflective abstraction, whereby the cognitive subject uses his or her cognitive processes and powers to reconstruct his or her conceptual structure. He describes “bootstrap” operations that allow the construction of concepts or operations of ever higher levels of abstraction, thus addressing one of the perennial challenges for psychology. A persuasive account of concept formation is presented, involving the development of a “fit” between the cognizer’s schemas and the world of experience via trial and error, or conjecture and test, in an iterated, dialectical cycle.
Von Glasersfeld argues that much use of language, even the recognition of pluralities, essentially depends on operations of the mind. The careful analysis offered adds to the accounts available, and is quite convincing, but convincing that language is embedded in a complex conceptual structure (including mental operations), which to me was not in dispute. Later, in a discussion of consciousness, he refers to thought being structured by language. These remarks suggest an area of needed growth and development for radical constructivism: to account more fully for the mind-forming role of language (and the concomitantly key role of the social in shaping individuality). How can such things as the inherited Language Acquisition Device, and the face-reading/person-recognizing and social cue-reading capacities of babies, which modern research suggests precedes experience and predisposes to certain types of specifically human learning, be accommodated?
One criticism of von Glasersfeld’s account of radical constructivism should be rebutted here. Lewin (chap. 23) argues that radical constructivism cannot account for an agreed body of mathematical knowledge. He argues persuasively that the “happy agreement” over what constitutes mathematical knowledge is due to a social process implicated in the epistemic process. However, elsewhere von Glasersfeld has conceded this, and has argued that conventional knowledge of mathematics emerges from interpersonal “fit.” In this book, he argues that the basic rules and conventions of mathematics are what confer happy agreement on the overall body of mathematical knowledge, just as they do in chess. Thus, this criticism is to be rejected.[Note 11]
Duit (chap. 14) makes a parallel point in his discussion of epistemology: “scientific conceptions are also human constructions (i.e., inventions of mankind and not pieces of objective truth).” He distinguishes trivial and radical varieties of constructivism on the grounds that the former has an accumulation or assimilation view of the growth of science, whereas the later requires conceptual change, including revolutions, cognitive conflict, and accommodation on both the micro- and macroscale. Explicit parallels with the philosophy of science are drawn, such as dependence on metaphor, the variability of conceptions or beliefs, and fallibilist epistemology.
Duit consistently carries his radical constructivist epistemology into the realm of methodology by problematizing research knowledge and acknowledging that researcher conceptions or construals of learner conceptions requires a hermeneutic cycle. He does not make the mistake of epistemologically nonradical varieties of constructivism in privileging the researchers’ knowing.
Although largely identified with this position, Duit is also critically aware of areas of weakness in radical constructivism. For example, there is a tendency toward isolated individualism, which needs to be balanced by elements of social constructivism. Thus, radical constructivism under- stresses the “being developed” pole of knowledge acquisition (i.e., the role of social guidance), and overfocusses on the “self-development” pole.
Duit suggests a number of aspects of pedagogy that follow from his radical constructivist position. He points out the need to recognize not only student cognitions, but also other conceptions and beliefs: their metaknowledge or personal philosophy of science, their view of learning, and their attitudes. He stresses a range of strategies and emphases, including taking the metacognitive/personal philosophy dimension seriously not only for learners, but in teacher education as well.
Driver (chap. 21) offers an account of the learning and teaching of science that stresses not only the constructed basis of personal knowledge, but the fallibilist epistemological consequences of radical constructivism, which she claims are not yet taken seriously enough. One of her innovations is to offer three concentric perspectives of constructivism. First, and innermost, is the traditional personal construction perspective, describing an individual knower’s sense making and construction. Second, surrounding this core, is the interpersonal construction of knowledge through social interaction, both child-child and adult-child. Third, surrounding the others, is the construction of science as public knowledge, the larger sociocultural context in which individual (and interpersonal) acts of construction take place. The contrast between these different perspectives is one of the central issues of importance for the future development of constructivism.
In exploring the pedagogical implications of constructivism, Driver brings up a number of further vital issues. These include constructivist views of scientific knowledge (in the large) as constructed, and constructivist views of the learner, the curriculum, and teaching, which parallel the proposals of Duit and others. Driver suggests that her perspectives of constructivism respectively impact on:
Raising awareness of unique personal conceptions in teaching and curriculum design;The effect of discussion, with students providing each other with scaffolding for the developing of their thinking and the opportunity to reorganize their own ideas through talk and listening; andThe leading of children from their own meanings to socially agreed definitions, concepts, and theories.
Wood, Cobb, and Yackel
Wood, Cobb, and Yackel (chap. 22) again identify themselves as radical constructivists, although much of their concern is to elaborate the underdeveloped social dimension of this position. Thus, the symbolic interactionist perspective is drawn on, and it is argued that neither the social nor the cognitive domain should be given overall priority. Wood et al. acknowledge that students learn from their own sensory experiences of objects, from their own conceptual operations and actions, but also from reflective abstractions on their interpretations of others’ activities and social interactions.
The epistemological position adopted is explicitly nonabsolutist and nonobjectivist. “Objective mathematical knowledge” is taken-as-shared knowledge (i.e., socially constructed and socially agreed knowledge). This epistemological position is also consistently applied to the researchers’ own knowledge-producing activity.
Wood et al. provide a detailed pedagogy by means of episodes and reflections from the Purdue project. They stress that the teacher initiates students into an interpretative stance, and that the latter’s mathematical experiences are socially situated. They regard the use of manipulatives and representations as valuable, provided what they represent is problematized; they argue that perceptions of such externals alone cannot reveal what the teacher regards as their mathematical content.[Note 12] The social dimension further enters in because real negotiation, communication, and interaction (whether overt or covert) are essential to teaching-learning in context.
They point to the existence of folk mathematics, and beyond the stereotyped associations it sometimes has, and argue that ontologenetically and phylogentically informal maths are sources of pure, academic maths. Thus, activities that are personally meaningful to learners at various conceptual levels, and that tap their informal or folk knowledge – not formal mathematics – should provide the pedagogical starting point. However, they do not address the problem that cognitions of street versus school mathematics, say, are socially situated and part of the cognitive apparatus of different social practices (raising the problems of transfer that Saxe indicates).
Their classroom cameos provide an implicit teaching approach that involves creating situations children find problematic, stimulating them to resolve contradictions, getting past obstacles, and so on. The collaboration they require of learners goes beyond resolving cognitive conflicts to also achieving consensus by developing taken-as-shared meanings, as well as autonomy. Wood et al. suggest that their pedagogy depends on changes in the beliefs and conceptualization of the teacher’s role to: (a) facilitator of learning; (b) creator of conditions for conflict resolution, meaning negotiation, and mutual perspective taking; and (c) asker of questions that prompt children to move toward socially accepted meanings.
Overall, it can be seen that, apart from von Glasersfeld’s contribution to the core, cognitive dimension of radical constructivism, the other chapters commented on in this section elaborate the social aspect of the position. This is also the concern of the next paradigm.
Social constructivism regards individual subjects and the realm of the social as interconnected. Human subjects are formed through their interactions with each other, as well as by their individual processes. Thus, there is no underlying metaphor for the wholly isolated individual mind. Instead, the underlying metaphor is that of persons in conversation – persons in meaningful linguistic and extralinguistic interaction and dialogue (Ernest, in press; Harré, 1989). Mind is seen as part of a broader context: the “social construction of meaning.” Likewise, the social constructivist model of the world is that of a socially constructed world that creates (and is constrained by) the shared experience of the underlying physical reality (Ernest, 1990). The humanly constructed reality is all the time being modified and interacting to fit ontological reality, although it can never give a true picture of it.
Adopting persons in conversation as the underlying metaphor of social constructivism gives pride of place to human beings and their language in its account of knowing. Following the seminal work of Vygotsky, symbolic interactionism and activity theory language is regarded as the shaper of, as well as the summative product of, individual minds. Increasing attention is being given to the impact of language in much psychological research in the psychology of mathematics education, such as the cognitive role of linguistic features such as metonymy and metaphor. It is increasingly recognized that much instruction and learning takes place directly through the medium of language. Even manipulative or enactive learning, emphasized by Piaget (1972) and Bruner (1963), takes place in a social context of meaning and is mediated by language and the associated socially negotiated understandings.
In summary, the social constructivist research paradigm adopts a sophisticated realist ontology (i.e., there is a world out there supporting the appearances we have shared access to, but we have no certain knowledge of it). It is based on a fallibilist epistemology that regards conventional knowledge as that which is lived and socially accepted. The associated learning theory is constructivist, with an emphasis on the essential and constitutive nature of language and social interaction. The methodology is eclectic, but recognizes that all knowing is problematic and there is no privileged vantage point. Likewise, the pedagogy is eclectic, aware of the interactive and inseparable effects of the micro and macro social contexts, and the internal construction of self, beliefs, and cognitions.
Although none of the chapters in this volume fits into this paradigm, with the possible exception of Becker and Varelas (chap. 24), it is clear from the previous discussion that a number of radical constructivists are incorporating elements of social constructivism in their positions. In particular, Driver, Wood et al., and Confrey all offer contributions that suggest that the distinction between these two paradigms is becoming less clear cut as the social dimension is more fully accommodated by radical constructivism.
Although my account of this last paradigm is brief, and perhaps incoherent, it is included to round out the range of paradigms sketched because it is represented in this volume in Gergen’s (chap. 2) and Shotter’s (chap. 3) chapters. Social constructionism resembles social constructivism, but prioritizes the social above the individual. Mind is regarded as the introjected social dimension. To put it another way, evidence of the mental is to be found in social performance and public display.
The metaphor of mind is that of dialogue or drama, with individuals represented as actors with parts to play in the drama. This metaphor reflects the importance ascribed to texts in this perspective. The world metaphor is that of social reality. Thus, the ontology is that of a universe of persons residing in the world. In other respects, the paradigm resembles that of radical and social constructivism, including a fallibilist epistemology, except that pedagogically it is less developed because its practical import has been worked out for psychological therapy, rather than for education.
The metaphors for the mind and the world of the different paradigms are summarized in Table 26.1. One of the major distinctions lies in the underlying metaphor for the world (and the concomitant epistemology). The first four paradigms are based on the metaphor of Newtonian absolute space for the world. With it comes an absolutist epistemology and a neopositivist paradigm of research. This locates the “problematic” of epistemology exclusively in the immediate object of inquiry (i.e., in the mind of the learner). Thus, these research paradigms do not require any reflexivity or doubts about the researcher’s constitutive role in knowledge and meaning making. This is a stance that is increasingly questionable in the social sciences.
In contrast, the last three paradigms do not regard the world as something that can be known with any certainty. They problematize the whole relationship between the knower and the known, and accept that no certain knowledge is attainable by humans. This humility with regard to epistemology, knowledge, and the results of the methods employed in research process resonates with much of current thinking in philosophy, the humanities, and social sciences. However, it does mean that neither research paradigm nor methodology can be employed mechanically in the quest for knowledge, but that every such approach is fraught with epistemological difficulties and stands in need of justification.
ParadigmMetaphor for the MindMetaphor/Model of the WorldTraditional EmpiricismBlank page, mirrorNewtonian absolute spaceInformation-Processing TheoryMachine, computerNewtonian absolute spaceTrivial Constructivism“Soft” computerNewtonian absolute spaceSociocultural CognitionGame player and actorSociety in Newtonian spaceRadical ConstructivismEvolving, adapting organismSubject’s experiential worldSocial ConstructivismPersons in conversationSocially constructed worldSocial ConstructionismDialogue, actors in a dramaSocial reality
Table 26.1: Contrasting metaphors for the mind and world of five research paradigms in education
Another important feature that emerges from Table 26.1 is that of the varying complexity of the underlying metaphors of mind. Some of the earlier metaphors (following the order presented) offer an inanimate model of mind. Simplifying assumptions are essential in science, but so, too, is the recognition of complexity. If the metaphor of mind adopted accords the cognizing subject something less than human status, there is a risk of neglecting the richness of human thought, feeling, values, reflection, planning, purposes, and goals.
Revealing the underlying metaphors, as I have attempted to do, may help clarify and make public the tacit assumptions of research paradigms. Philosophers like Pepper (1948) and MacCormac (1990) proposed that underpinning any epistemology are basic metaphors taken from commonsense understanding. According to this view, each research paradigm comes with a stock of assumed images, metaphors, and unspoken conventions and assumptions that remain tacit and implicit, but that permeate and underpin the shared understandings of a research community. But,
Bachelard regards the common-sense mind’s reliance on images as a breeding ground for epistemological obstacles. However, epistemological obstacles may also arise from successful scientific work that has outlived its value…. The views and attitudes that constitute epistemological obstacles are often not explicitly formulated by those they constrain but rather operate at the level of implicit assumptions or cognitive or perceptual habits. (Gutting, 1990, p. 135)
Some of the metaphors of mind and world are indeed leftovers from successful scientific theories of the past. By uncovering them, the outcome may be to clear the path to a better evaluation of those theories and to buttress the more powerful and practical forms of constructivism.
Finally, it is clear from the previous discussion that one of the crucial differences between paradigms underlying epistemology is whether it is absolutist or fallibilist. Table 26.2 shows the paradigms classified according to whether the epistemology is fallibilist or absolutist. In addition, another important aspect is shown – the human focus – be it individual alone, social alone, or a combination of individual with social. To ensure that no category is left empty, traditional sociology has been added, which has a social focus but an absolutist view of knowledge, at least in science and mathematics. One problem with the classification in Table 26.2 is that radical constructivism is a developing position, with some versions close to social constructivism, and hence not focused on the individual alone.
Underlying EpistemologyAbsolutistFallibilistIndividual Focus AloneInformation-processing theoryTrivial constructivismRadical constructivismSocial and Individual FocusSociocultural cognitionSocial constructivismSocial Focus AloneTraditional sociologySocial constructionism
Table 26.2: Research paradigms classified according to the individual-social and absolutist-fallibilist distinctions.
The one again: A synthesis of the concerns of constructivism
Individual or social construction?
Perhaps the overriding concern of the discussed chapters is the individual- social dichotomy. Rubin (chap. 19) contrasts the mental construction position, which locates the construction of social reality within an individual’s mental construing processes, and the (socially) constructed convention position, which locates meaning in community consensus born of ongoing dialogue. He goes on to stress the necessity for convergence between mental representation and social convention.
Piagetian constructivism seems to emphasize internal cognitive processes at the expense of social interaction in the learner’s construction of knowledge. However, the majority of authors and commentators stress the need for constructivism to accommodate the complementarity between individual construction and social interaction. von Glasersfeld has already shown that mathematical knowledge is taken as shared through agreed rules and conventions, explicitly opening the door to the influence of social interaction. But further elaboration of radical construc‑tivism is needed to recognize the fundamental implication of the social in the construction of the individual.
In my comparison of the different paradigms and positions adopted in the chapters, one contrast stands out. Although I have distinguished seven different paradigms, virtually all of the positions adopted are variants of radical constructivism. This weakens the force of my classificatory scheme somewhat because the metaphors I have attributed to this position, especially that of mind, are not in evidence in most of the chapters discussed. The varieties of radical constructivism, and the number of attempts to incorporate the social dimension shown in the chapters discussed, suggest that it is a progressive research program (Lakatos, 1970), evolving to better describe the broad range of phenomena involved. Thus, radical constructivism is adapting to accommodate the criticism it has met, especially concerning its possible neglect of the sociocultural dimensions of learning and knowledge. This development removes much of the sting of the criticism. It also removes much of the difference between that position and social constructivism. Therefore, perhaps a new unity is forming around radical constructivism. Needless to say, such a move leaves intact the crucial epistemological difference between radical constructivism (and social constructivism and constructionism) and absolutist positions such as trivial constructivism.
Ultimately, the import of an educational paradigm concerns its implications for practice, notably in pedagogy. However, in my view, there is little in any pedagogy that is either wholly necessitated or ruled out by the other elements of an educational paradigm (Goldin, 1990). This is because pedagogy is predicated on a set of values reflected in the following questions. What are the aims of education? What selection from the stock of cultural knowledge is valuable to teach? What forms of human organization and interaction fit with the values? What view of the child or person, with what rights and powers, is associated with the values? A pedagogy is merely a theory of techniques for achieving the ends of communicating or offering the selected knowledge or experiences to learners in a way consistent with these values. The other elements of a paradigm are consistent with a wide range of pedagogical approaches, therefore the pedagogy of the paradigm is likely to be eclectic.[Note 13]
Many of the paradigms other than traditional empiricism suggest the following pedagogical implications. The need and value for:
sensitivity toward and attentiveness to the learner’s previous constructions;diagnostic teaching attempting to remedy learner errors and misconceptions; perturbation and cognitive conflict techniques as part of this;attention to metacognition and strategic self-regulation by learners;the use of multiple representations of mathematical concepts;awareness of the importance of goals for the learner, and the dichotomy between learner and teacher goals; andawareness of the importance of social contexts, such the difference between folk or street mathematics and school mathematics (and an attempt to exploit the former for the latter).
Beyond these pedagogical emphases, a number of stronger implications follow from radical (and social) constructivism:
Knowledge as a whole is problematized, not just the learner’s subjective knowledge, including mathematical knowledge and logic.Methodological approaches are required to be much more circumspect and reflexive because there is no “royal road” to truth or near truth.The focus of concern is not just the learner’s cognitions, but the learner’s cognitions, beliefs, and conceptions of knowledge.The focus of concern with the teacher and in teacher education is not just with the teacher’s knowledge of subject matter and diagnostic skills, but with the teacher’s beliefs, conceptions, and personal theories about subject matter, teaching, and learning.Although we can tentatively come to know the knowledge of others by interpreting their language and actions through our own conceptual constructs, the others have realities that are independent of ours. Indeed, it is the realities of others along with our own realities that we strive to understand, but we can never take any of these realities as fixed.’’An awareness of the social construction of knowledge suggests a pedagogical emphasis on discussion, collaboration, negotiation, and shared meanings more, as evidenced in Wood et al. (chap. 23).
A central issue is that of reflexivity. Some of the chapters herein have caused me to reflect on myself as learner and researcher. Auerswald (chap. 25) offers a parody of four styles of critique that led me to be self-aware of myself as writer, thinker, and researcher – raising the question of what preconceptions, presuppositions, and genre assumptions I bring to bear in this task of critique, or in writing in general. Likewise, in her complex and persuasive analysis of writing, reading, and co-writing, Spivey again confronts me with my here-and-now activity of writing and reading. Raising levels of self-awareness like this is surely a vital activity from a constructivist perspective.
In addition to personal self-awareness, there is the need for reflexivity in epistemology. Fallibilistic epistemology requires humility in knowledge claims at every level, including both educational research and mathematics. This must be recognized in the methodology employed, with its limited aspirations for lasting knowledge.
Reflexivity also arises concerning the respect offered to different research paradigms. If we want learners to share and value each others’ perspectives, it is important that we are consistent and offer each other the same respect (unlike politicians).
Finally, it is worth remarking that the form and organization of this volume, like the conference from which it originates, are consistent with its theme and contents. It is collaborative, dialogical, and is building on previous understandings. Such reflexivity in research is refreshing, but all too rare.
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Belief is stronger than knowledge in psychology in terms of subjective, personal commitment. Knowledge is stronger than beliefs in philosophy in terms of justificactory warrant. Thus, two different relations of “stronger than” are involved. In fact, metaphysical beliefs or presuppositions in philosophy play a similar role to psychological beliefs. Indeed, Polanyi (1958) proposed a philosophical theory of personal knowledge that is precisely concerned to add commitment to subjective knowledge.
Politics, or power, is a theme that enters into most of the commentators’ chapters: Rubin on sexual harassment; Auerswald’s utopian social vision; Becker and Varelas on the issue of power and epistemologies, the negative side of taken-as-shared knowledge; Lewin on the individual as constructed in Foucauldian sites of power/knowledge in discursive practices. Thus, one way or another, our attempts to treat constructivism in education seem to open the door to sociopolitical issues in a way that cognitive psychology never did in the past.
It also has to do with being consistent with the values we preach for the classroom: cooperation, listening, mutual respect, and so on.
In fact, the third paradigm assumes a sophisticated realist ontology and epistemology.
But then again, if we adopt a consistent stance, it follows on the metalevel also that there are multiple and equally legitimate sets of criteria, and that ours are not privileged above any others.
It might be argued that pedagogy has no place in a research paradigm, on the grounds that it is concerned with the researchers’ knowledge, not with facilitating learning. This can be answered on epistemological grounds. But, more significantly, a central concern of constructivism in mathematics and science education is to improve practice, which is reflected in the chapters of this volume. Hence, it is important to include the component of pedagogy in an educational paradigm.
The difference between subjective knowledge and conventional knowledge resides in the type of warrant. The former is warranted by an individual’s experience, and the latter satisfies socially shared criteria.
Methodology also concerns a theory of technique for creating and validating knowledge according to the epistemology.
Two points: (a) As Rotman (1977) pointed out, Piaget draws on both Lamarckian and Darwinian ideas of evolution in his metaphor of mind because cognitive characteristics can be acquired. (b) I avoid the “survival of the fittest” because it suggests that survivors have some intrinsic supremacy, and smacks of teleology. The “fit” or “fitter” are merely those organisms that happen to survive a series of circumstances of which others fail to do so. (I am grateful to von Glasersfeld for pointing out the importance of this distinction to me.)
Piaget’s constructivism is also based on a hydrodynamic metaphor, with the cognizing subject seeking equilibration, like the adjustment of water levels toward equilibrium.
In my view, this represents a significant advance on Piaget’s position, who seems to believe that there is a unique set of mathematical knowledge structures that emerge from the mind, resulting in this “happy agreement.”
This represents another widely overlooked weakness of discovery learning, resulting from its underlying absolutist epistemology.
It may be that, to accommodate the values underpinning a pedagogy, the four-component model of an educational paradigm should be expanded to include a fifth component – a set of values.
I am grateful to Steffe for suggesting this consequence.
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