Introduction THE CONTRADICTIONS IN THE CONSTRUCTIVIST DISCOURSE

Stuart Rowlands

Centre for Teaching Mathematics, University of Plymouth.

Robert Carson

Educational Foundations, Montana State University.

Introduction

Constructivism is the most influential ‘theory’ of learning in education. Inmathematics education in particular, many and if not most educators would claim tobe constructivist in one form or another. The word ‘theory’, however, is inappropriate,for constructivism is not a structured framework of ideas or concepts that candetermine an approach to mathematics education (a notable exception is the workdone by David Tall, 1991, and colleagues on the learning of advanced mathematicalconcepts within a Piagetian framework). The point is that constructivism is more abelief system than it is a theory. It is a ‘world-view’ (weltanschuung) and a BroadChurch (Matthews, 2000) that encompasses ethics, politics, ethnography andcurriculum design and development.

The difficulty in writing about constructivism is that it is very hard to define or ‘pin-down’. Despite the (literally) thousands of scholarly articles on mathematics andscience education informed by a ‘constructivist perspective’, there are no text-bookson constructivism as such – no A-B-C of its fundamental principles – just thousandsof articles with hundreds of ‘buzz-words’ (e.g. attentional frame by von Glasersfeld,1995; label by Mason, 1999) that are rarely referenced or duplicated. Of course, thereare a few exceptions (e.g. David Wheeler’s mathematization in Jaworski, 1994) buteven some of these have been interpreted differently by different people (perhaps thiswould be a confirmation for the radical constructivist!).

Constructivism’s popularity seems largely due to the consensus that the learner is nota passive recipient of knowledge but that knowledge is ‘constructed’ by the learner insome way. Von Glasersfeld (1995) calls this ‘trivial constructivism’ and many articlesin mathematics and science education claim to be ‘constructivist’ because of the view

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that learning is an active and not a passive affair. Indeed, many of the few critics ofconstructivism (e.g. Irzig, 2000) have argued that you don’t have to be a constructivistto reject the passive recipient (or empty vessel) metaphor. Constructivism as apsychology – as a field of study that views learning as a construction process - may beuncontroversial. The controversy begins when constructivism confuses what it is thatis being constructed (mental representation) with knowledge itself. One of the moststriking features of constructivism is its denial that knowledge is anything over andabove either mental representation or consensus.

If an A-B-C textbook on constructivism were possible, then one would expectknowledge defined in terms of mental representation under ‘Radical constructivism’and knowledge defined in terms of consensus under ‘Social constructivism.Unfortunately the situation is not so simple, which may account for why such an A-B-C textbook has not been forthcoming. As will be demonstrated below, in theirconsideration as to what knowledge is, many radical constructivists switch from‘mental representation’ to ‘consensual domain’ and back again without realising thatthere are contradictions in doing so. It seems as if these contradictions are overlookedbecause of an incessant drive to deny the status of knowledge – to deny thatknowledge is anything over and above what is subjective (mental representation) orintersubjective (consensual domain). Even social constructivists are unsure as to whatconstitutes the ‘social’ and how it relates to the individual. As Lerman (1994, 1999),has indicated, social constructivism in the main is merely radical constructivism’sinterpretation of the social – the ‘consensual domain’ added as an appendage to theway the individual interprets the world – and Lerman (1994) points out that if thesubject is seen as the autonomous meaning-maker then any reference to the ‘social’has no ‘bite’. We suggest that if knowledge is seen in terms of something over andabove mental representation and consensus then the contradiction between theindividual meaning-maker and the ‘social’ might be resolved.

To understand how the existing ideas of the student can accommodate new ideastaught by the teacher would involve examining in some way the subjective ideas ofthe student. Similarly, to understand how a group of students arrived collectively at asolution to a problem would involve what may be described as an intersubjectiveconsideration. In both cases, however, consideration must be given to the content of

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what is being learnt or solved, such as what concepts are involved and their relationto other concepts in the body of knowledge, or what inferences need to be made, etc.That is, if we wish the students to make the mathematics their own (without referenceto content would be like Woolgar and Latour observing the behaviour of twoscientists in the laboratory without any reference to the content of the science thatmust have some influence on their behaviour. See Phillips, 1998, and Irzig, 2000, onthis). There are three approaches to examining a discipline such as mathematics: asubjective, intersubjective and objective approach. The subjective approach mayexamine the ideas of a mathematician, the intersubjective approach may examine theresponse of the mathematician’s peers to his or her ideas, but if both approaches are tomake sense and not collapse into contradiction and misapprehension, then both haveto refer to the mathematical concepts involved.

Education is to do with knowledge and any consideration as to what knowledge is hasfar reaching pedagogical implications. For example, if knowledge is nothing morethan what is constructed by the individual, then the learner is never wrong - whateverhas been constructed has made sense and whatever makes sense is knowledge! If truthis whatever the learner considers to be the case, then there is no body of knowledge,no ‘subject-matter’ that can be taught as such.

There is no world ‘out there’ we can know; none we can talk about.

Traditionally, epistemology – the ‘theory of knowledge’ that goes back to the AncientGreeks – was a subject exclusive to philosophy and attempted to answer the questionas to what is knowledge of the world and how is this knowledge possible. Althoughepistemology is still a philosophical subject, it has become a topic central to educationresearch. If teaching and learning has to do with knowledge, then epistemology has tobe involved. What is knowledge and how it is possible has implications for teaching,learning and the content of what is being taught/learnt. Perhaps the most dominant (orat least the most referenced) epistemology in education is the one expressed by vonGlasersfeld:

1a. Knowledge is not passively received either through the senses or by way ofcommunication;

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1b. Knowledge is actively built up by the cognising subject.

2a. The function of cognition is adaptive, in the biological sense of the term, tendingtowards fit or viability;

2b Cognition serves the subject’s organization of the experiential world, not thediscovery of an objective ontological reality. (von Glasersfeld, 1995, p.51)

Of the first, von Glasersfeld (1998) states that this is ‘trivial constructivism’ andbecause it doesn’t challenge ‘traditional epistemology’ he added the qualifier radical.Jaworski states that this is an expression of the Ausubelian form ‘that the learner’snew understandings are dependent on prior knowledge and experiences’, (Jaworski,1994, p.16). In one sense, this is quite obvious: you cannot expect a class or a pupil tounderstand a topic in mathematics without having the necessary mathematicalbackground. Unfortunately, just how new understandings are dependent on priorknowledge has not been made at all clear in the literature. ‘Prior knowledge andexperiences’ seems to be recall of previously learnt mathematics in order to solve acurrent problem but with no explanation as to how the prior knowledge was learnt inthe first place. All we seem to get is basic recall. For example, After needlesslyspending days on an activity involving measurement:

I held up the ruler that Joan had been holding a few days before and asked ifanyone knew what it was or how to use it. Mark said it was called a ruler andthat his dad used one to measure things (Hendry, 1996, p.12).

Praise be for Mark and his dad, for without them the target-concept may never havebeen reached and the class may never have guessed what was in Hendry’s head!Although understanding may be regarded as an active process, knowledge as beingactively built by the learner is controversial (although some constructivists, e.g.Staver, 1998, and some of its critics, e.g. Irzig, 2000, say that it isn’t controversial).This will be discussed below in the context of whether knowledge can becommunicated. What is agreed generally amongst constructivists and non-constructivists alike is that the second principle is controversial. A comprehensivephilosophical analysis of the second principle can be found in Nola (1998),summarised by Irzig (2000) and a deconstruction of von Glasersfeld’s radicalconstructivism can be found in Suchting (1998). The point we would like to stress

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here is that the second principle seems to give license to replacing the ‘top-down’curriculum (for example, a mathematics syllabus that is taught in school) with a child-centred ‘bottom-up’ pedagogy in which mathematics becomes, with all its entirety,that which is constructed by the child. Any reference to mathematics as an objectivebody of knowledge independent of consensus is to be condemned as ‘absolutist’ (seeRowlands et al, 2001).

In the defence of constructivism in science education, Staver elaborates the fourprinciples of von Glasersfeld. The first three, Staver states, are non-controversial:Knowledge is actively built up from within by a thinking person: knowledge is notpassively received through the senses or by any form of communication. Second,von Glasersfeld described the importance of social interaction in the construction ofknowledge. Social interactions between and among learners are central to thebuilding of knowledge by individuals. Third, the character of cognition isfunctional and adaptive. Cognition and the knowledge it produces are a higher formof adaptation in the biological context, in which the functional concepts of fit andviability - two concepts which we know well and embrace in evolutionary theory -also characterise knowledge. Fourth, von Glasersfeld described what the purpose ofcognition is, and what it is not. Cognition’s purpose is to serve the individual’sorganisation of his or her experiential world; cognition’s purpose is not thediscovery of an objective ontological reality (Staver, 1998, p.503).

If knowledge is not received by any form of communication (first principle) then howis social interaction central to the building of knowledge (second principle)? Ifknowledge is a product of cognition (third principle) then how can mathematicssimply be a product of the student’s cognition if mathematical knowledge has takencenturies to develop? As of the fourth principle: in what possible sense can we saythat cognition’s purpose is the discovery of an objective ontological reality? Whatdoes ‘discovering’ an ontological reality mean, and would we recognise anontological reality once we have discovered it? The proposition ‘There is an externalworld that exists prior to consciousness’ separates the realist from the idealist and hasbeen the subject of much heated debate in the history of philosophy (see Suchting1986). This is also the central issue in constructivism and one that is central toStaver’s article:

The continuing debate about constructivism as an epistemology and, morefundamentally, about truth as correspondence versus coherence provides anopportunity to examine the relative importance of the two issues that Osbornementioned. Scientists value parsimony and building on prior work. Which do

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scientists value more? As an epistemology, constructivism should be preferred ongrounds of parsimony because it contains one less assumption - the realistpresupposition; it does not assume a priori the existence of an external world whichis separate from human perception (p.514).

Do scientists value parsimony? The Copernican system is often cited by many as anexemplar of ‘simplicity’ because it contains fewer assumptions than the Ptolemaicone (the latter contains assumptions such as epicycles in order to sustain it). Actually,for centuries, this was not the case and was never the reason why the latter replacedthe former! There were more difficulties besetting the Copernican system then thereever were besetting the Ptolemaic one (initially, for example, the former still had toemploy epicycles in order to give correct predictions) and these troubles werepersevered with for centuries until the system was perfected (Chalmers 1982). Soshould we accept constructivism on the grounds of ‘parsimony’ - that it does notcontain the assumption of an external world prior to consciousness? Despite any loveof parsimony that the scientist might have, the practice of science presupposes theexistence of the external world that is prior to consciousness simply because science isan attempt to say something about the world and is not an attempt to give coherenceto our experiences (Chalmers 1982; Matthews 1994, 1998b; Nola 1998; Thomas1994). In fact, the concepts and theoretical objects (see Suchting 1986) of science arecontrary to making sense of experience. For example (and one that is inspired bySuchting 1986), a ball that is thrown upward is on the one hand ‘going up’(unproblematic as regards making sense of experience) but is also in free-fall (‘free-fall’ is a theoretical object contrary to making sense of experience). On the one handwe have student ‘preconceptions’, ‘misconceptions’, ‘alternative frameworks’ or‘intuitive ideas’ of force and motion (making sense of experience) as reported inhundreds of research papers (see Rowlands et al. 1999) and in contradiction we havethe well-defined meaning of force in Newtonian mechanics (which models thephysical world. See Hestenes 1987, 1992; see also Wittgenstein’s Tractatus,propositions 6.341-6.361).

The removal of the assumption that the external world exists does not makeconstructivism simpler, in fact it does quite the reverse: constructivism will continueto be debated with the likelihood of more and more outrageous statements beingadded to what presently exists.

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Von Glasersfeld’s (1995), and Staver’s (1998), attack on realism is in essence anattack on the correspondence theory of truth. The correspondence theory goes back toAristotle and defines truth in terms of the adequacy of thought to reality. According tothe correspondence theory, knowledge of the world is structured according to the stateof affairs that it depicts. For example, if I were to say ‘the cat is on the mat’ then I amnot just expressing a thought but saying that things are as I say they are - and theproposition is true iff the cat is actually on the mat! The correspondence theoryimplies a belief in the external world. For example, according to Searle (1995) theproposition ‘there is no money in my pocket’ only makes the sense it does because ofthe belief in the existence of money. To deny the existence of the external world, or todeny any possibility of knowing the external world, would be to deny thecorrespondence theory as plausible. However, to imply the converse would be tocommit an error: the denial of the correspondence theory is not necessarily a denial ofthe existence of the external world and, which is more to the point, nor is itnecessarily a denial that knowledge of the external world is possible. Von Glasersfeldand Staver do not deny the possibility that the external world exists, but they deny thatknowledge of this world (‘ontic’ or ‘ontological’ reality) is possible. Von Glasersfeldand Staver’s fundamental error is the belief that a refutation of the correspondencetheory is also, or must entail, a refutation of realism. Following Nola, Irzig (2000)makes the point that a ‘minimal realist’ could believe that there exists a worldindependent of consciousness without necessarily holding onto the correspondencetheory of truth.

The correspondence theory may be tenable with respect to everyday propositions andeveryday states of affairs such as cats on mats, but the correspondence theory as atheory of the way science explains the physical world is fraught with difficulty (For acritique of the correspondence theory see Chalmers 1982, and Suchting 1986. Adefence of the theory can be found in Searle 1995). For example, what state of affairsdoes the proposition ‘an object in the absence of force moves in uniform motion’depict? No one on Earth has ever seen an object in the absence of force. Theproposition ‘the hydrogen atom has one electron’ is embedded in the theoreticalmodel of the hydrogen atom rather than describing an observed state of affairs (ofcourse, an experiment may be seen as a creation of a state of affairs to show that

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nature can be described by these means). Implicit in von Glasersfeld’s main argumentis that it is because the correspondence theory is untenable then the external world(‘ontological reality’) is unknowable (hence legitimising the claim that knowledge isof experience and not of the world). If the correspondence theory is untenable when itcomes to science, then some account must be given as to what the theoretical objectsof science actually refer to. For von Glasersfeld (1995), however, they don’t refer toanything but are merely ‘fictions’ that can ‘explain anything you want to explain’ –but then how would von Glasersfeld reconcile Chalmers’s (1982) point that such‘fictions’ can be seen colliding with smoke particles in the phenomenon of Brownianmotion, or that Kekule’s molecular rings can be seen almost directly under an electronmicroscope or that Hertz was able to express Maxwell’s equations ‘in a visible and

Chalmers (1982) explains, acceleration is on the one hand an

abstract mathematical concept and yet it is what objects do! If I were to drop a penwhat would it do? For one thing it will not go up, for another it will acceleratedownwards. The pen obeys the laws of nature and cannot be explained in terms ofmaking sense of experience (unless perhaps you are Aristotle, but then Armstrongwalking on the moon was calculated using Newtonian physics, not Aristotle’s). Whatlaws of nature? The laws of nature as expoused in many physics textbooks. If thephysical world was other than it is, then the concepts of science would be different towhat they are. So-much-so that if an alternative culture were to come up with aninverse-cubed law for gravitation, then it can be shown that nature is such that it willnot obey this law (unlike the inverse-square law)!

Von Glasersfeld’s two principles are much quoted in the mathematics educationliterature and, it seems, with the downplay of mathematics as a discipline very muchin mind. For example, with reference to the second principle, Jaworski (1994) arguesthat if there are any ‘absolutes’ regarding triangles, then developing experience andknowledge tells us nothing about what they are - cognition has to do with viabilityand ‘not the discovery of an objective ontological reality’. However, this is quitesimply wrong! My experience of the proof that the area of all flat triangles is half baselength times height tells be something about any possible flat triangle which is‘absolute’ (fixed and unchanging). In fact, Jaworski is committing a fundamentalcontradiction: any intervention by the teacher (for example, a teacher constraining astudent’s thinking that there are two possible answers to the area of a triangle by

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posing a contradiction [see ch.7]) would be from the teacher knowing that the studentis wrong and what it is for the student to be right (that any flat triangle has one andonly one value for the area). According to Jaworski, if we each know ‘only what wehave individually constructed’ (p.17), then we can never know of anything outsidewhat we have individually constructed, which includes other people (what they sayand do being a question of how we perceive it). This is solipsism, the theory thateither the self is all that exists or it is only the self that can be known, becausereference cannot be made to anything outside of personal experience (many radicalconstructivists, e.g. Steffe, 1999, deny solipsism by stating that they do not deny theexistence of the external world. Von Glasersfeld, 1995, argues that solipsism is atheory of being whereas radical constructivism is a theory of knowing. However, ifknowledge is defined solely in terms of making sense of experience, then the externalworld might as well be denied). But Jaworski wants her cake and eat it becauseexperience ‘includes the interactions with others’ (p.17). On the one hand, ‘others’ aremerely part of my experience yet because of my interaction (with parts of myexperience, which can be labelled ‘others’) we no longer have solipsism butapparently a world containing other people. Jaworski leaps from the world of theindividual to the social world without realising the implications of such a leap – it ismerely taken for granted that such a leap is unproblematic. For example, ‘If thesecond principle implies there is no world outside the mind of the knower, it could, asLerman (19a) points out, imply that ‘‘we are certainly all doomed to solipsism’’.Lerman reassures us that this is actually not the case since the second principlerecognises experience, which includes the interactions with others in the world aroundus’ (Jaworski, 1994, p.17). As you will see below, Lerman has changed his mindsince 19 and no longer ‘reassures us’ with respect to radical constructivism.In the introduction to the influential book ‘Radical Constructivism in MathematicsEducation’, von Glasersfeld states what may be regarded as fundamental to the radicalconstructivist perspective:

Language frequently creates the illusion that ideas, concepts, and even wholechunks of knowledge are transported from a speaker to a listener. This illusion isextraordinarily powerful because it springs from the belief that the meaning ofwords and phrases is fixed somewhere outside the users of the language. Perhapsthe best way to dismantle the illusion is to remember or reconstruct how one cameto form the meanings of words and phrases when one was acquiring language in the

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first place. Clearly it could only be done by associating bits of language one heardwith chunks of one’s own experience – and no one’s experience is ever exactly thesame as another person’s. Thus, whatever another says or writes, you cannot but putyour own subjective meanings into the words and phrases you hear. Given that welive in a community of other language users, our subjective meanings tend, ofcourse, to become intersubjective, because we learn to modify and adapt them sothat they fit the situations in which we interact with others. In this way we manageto achieve a great deal of compatibility. But to prove compatible, individualmeanings do not have to be identical. Indeed, throughout our lives we now and thendiscover that the meaning we have associated with a certain word is not quitecompatible with the use others make of that word. This may serve to remind us –especially when we act as teachers – that new concepts and new knowledge cannotsimply be passed to another person by talk, because each must abstract meanings,concepts, and knowledge from his or her own experience. (von Glasersfeld, 1991,p.xiv)

Many pertinent points and criticisms could be made about this passage. For example,the appeals to our memory of personal experience (introspection, something that vonGlasersfeld appeals to quite a lot. For example, ‘A dictionary will in many casesresolve the problem – and, in doing so, confirm the illusion that meanings are, afterall, fixed entities that do not depend on individual usage. But a moment’s thought onhow anyone acquires the meaning of a word would reveal that it is an illusion’, vonGlasersfeld, 1996, p.6. An exercise would be to count the number of introspectiveappeals in von Glasersfeld’s, 1995, book - the best way to ‘dismantle the illusion’ thatwe can communicate being just one of them). This appeal to personal experienceassumes that personal experience is the only legitimate basis upon which knowledgeclaims can be made, which it is if knowledge is defined as making sense ofexperience. What is striking about this passage is that it contains an irony and aglaring contradiction.

The irony is that the meanings of words and phrases within mathematics educationand indeed other disciplines are fixed (many ‘schooled’ or ‘academic’ concepts arewell-defined, Howard, 1987), but the so-called ‘illusion’ that we can communicatedoes not spring from the belief that meanings are fixed outside the user – it springsfrom the belief that we share a common meaning – and a common meaning is onlypossible because of the existence of the external world. For example (and one takenfrom Searle, 1995), you can understand what I mean if I say ‘there is no money in mypocket’ because of our shared belief in the existence of money.

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Meanings within science and mathematics are fixed (with respect to some domain).However, what I mean by ‘force’, for example, may be different to what you mean by‘force’ and both may be different to what is meant by ‘force’ within the Newtoniansystem – and any difference can only be ascertained by comparison with how force isdefined and understood within the Newtonian system (an excellent example of thedifference between how force is conceptualised by text-book authors and how force isdefined within the system can be found in Warren, 1979. A rare examination of thelogical structure of Newtonian mechanics together with student intuitive ideas can befound in Hestenes, 1992). The essential point here is that the passage above contains aglaring contradiction, a kind of switch from one statement straight into it’s opposite.We have, on the one hand, the association of language with our own subjectiveexperience – and no one’s experience, we are told, is ever quite the same as anyoneelse’s - yet on the other hand we live in a community in which our subjectivemeanings tend to become intersubjective. But then, how does von Glasersfeld knowthis? The contradiction is stark: we have private experience and subjective meaning asif you are the only entity in existence (or rather, you might as well be, since all wehave here is private experience and no external world to speak of) and in one fellswoop we have on the other hand the existence of a social world (presumably one thatcan be said to exist independently of private experience) in which we haveintersubjective meaning. How can radical constructivism refer to the intersubjectivewhen it reduces everything to the subjective? However, if everything is reduced tohow you perceive it and as an addition we have the social domain (theintersubjective), then you might as well include the physical as an addition as well.However, it is the physical world – the world that lies outside of the individual’ senseperception – that von Glasersfeld claims we can have no knowledge of!

If knowledge, language, concept and meaning are all reduced to private experience,then the intersubjective becomes problematic. Yet many constructivists will beginwith private experience and glibly bring in the social aspect and not realise that theyhave undermined their reduction to private experience. All this simply because theauthority of truth has to be personal or social. An exemplar is Confrey (1991): ‘Inrejecting the idea of Platonist truths whose existence is independent of humanity, theconstructivist relies on explanation based in the interplay between social negotiationof meanings and individual creativity and genius’ (p.114) and is quite happy to speak

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of meanings as lying within individual experience and, at the same time, createdwithin a ‘culture of mathematicians’ or even society as a whole. Cobb et al (1991)even go so far as to state that they find it impossible to give an adequate explanationof how students cognitise independently of social norms that are negotiated andconstructed: ‘The norms (and consensually constructed mathematical knowledge)constrain the activity that creates the norms. Conversely, individuals’ activity createsthe norms that constrain that activity.......We thus acknowledge that social context isan integral aspect of an individual’s cognitions without reifying mathematics as aready-made body of cultural knowledge that is somehow internalised from without byindividuals.’ (p. 163). Despite any sense that can be made of individual activity‘restrained’ by the ‘norms’ it creates, if the social context is an integral aspect of anindividual’s cognition, then why can’t the social context also include mathematics asa ‘ready made’ body of cultural knowledge?

A consensual domain presupposes a domain outside private experience and once bothare evoked then it must be stipulated how one relates to or ‘impinges’ on the other.The contradiction is that if knowledge is reduced to private experience (mentalrepresentation) then you cannot refer to the consensual domain as if it were outsidewhat Lerman (1994) refers to as the ‘autonomous meaning-maker’. If everythingwithin the consensual domain is interpreted subjectively, then the notion ofintersubjectivity becomes redundant.

In many publications, von Glasersfeld emphasises the following: ‘The meanings ofwords – and this also applies to every sign and every symbol – must be constructed byeach user of the language individually, and this construction is based solely on thesubjective experience of the particular person. Hence it stands to reason that theinterpretation of a word or a text will always remain an essentially subjectiveoperation’ (von Glasersfeld, 1998, p.26/27). From this particular quote vonGlasersfeld then stresses that ‘shared’ meanings are only an ‘impression’. However, ifthe meanings of words are our own subjective construction, then how iscommunication possible? No problem it seems for the radical constructivist! Forexample, Jaworski (1994) argues that viability and fit (as opposed to describing orreferring to an ‘ontological reality’) applies not only to the construction of meaningbut also to shared meaning. But if meaning is a private construction, then is shared

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meaning possible? She states ‘A match in meaning between teacher and student cannever be known, even if it were achieved. It is quite surprising that meanings areshared at all, yet in many cases people do appear to understand each other’ (Jaworski,1994, p.23). If meanings are reduced to personal constructions then it is not onlysurprising that meanings are shared, it would be impossible to ascertain whether wedo in fact share the same meanings. If meanings are reduced to personal constructsthen a shared meaning implies an infinite regress within each personal construct. Forexample:

‘How could I ever know that what I understand by what you say is the same as whatFor the radical constructivist:

‘I do not actually know if my interpretation of what you say is the same as what youmean; however, I can always assume that my interpretation is the same as what youmean because languages are learnt through similar experiences. I can always changemy assumption if there is a perturbation between what I assume you are saying andwhat I previously understood as being said. In other words, understanding what isbeing said is dependent upon the way the individual constructs a meaning of what isbeing said, and if what is being said is unfamiliar then an idiosyncratic interpretationis possible’.

Now, this is fine if this is restricted to whether an individual understands anotherindividual, but a ‘shared meaning’ also entails the other person assuming his or herinterpretation of what I say is the same as what I mean, and here is the regress: if weare not to talk in ‘parallel’ then, in turn, I must assume that your interpretation of whatI say is the same as what I mean and you must assume that my interpretation of whatyou say is the same as what you mean and so on ad infinitum.

Can you imagine what this would be like if a shared meaning were possible in aconversation between more than two people! The only way out of the regress is toassume that any conversation shares a common content. For example, and oneadapted from Irzig (2000): I might believe that today is Sunday and you might denythat today is Sunday. We have two different mental representations but they share thesame content expressed by the proposition ‘Today is Sunday’ (which is true or falsedepending on what day it is). For Irzig, mental representation is private, subjectiveand has to do with the individual’s intentional state, whereas content is public and

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intersubjective. He makes the point that we might disagree on the content of aparticular sentence, but it does not follow that ‘my meaning’ is any different to yours.With respect to intentional states, Searle (1995) argues that, in many cases, we do nothave a collection of ‘I’ intentional states but a ‘we’ intentional state that is atomistic –it cannot be broken down into a number of ‘I’ components.

In the context of shared meanings, von Glasersfeld states ‘The conceptual structuresthat constitute meanings or knowledge are not entities that could be used alternativelyby different individuals. They are constructs that each user has to build up for him- orherself. And because they are individual constructs, one can never say whether or nottwo people have produced the same construct. At best one may observe that in a givennumber of situations their constructs seem to function in the same way, that is, theyseem compatible’ (von Glasersfeld, 1996, p.5, emphasis added). In other words, itmay just so happen that we have formed compatible conceptual structures, althoughthere is no way of knowing this apart from perturbations. This is scepticism and doesnot offer in a way to see how shared meaning is possible!

Jaworski (1999) adds the ‘social dimension’ to radical constructivism by stating the‘third principle’ which ‘derives from the sociology of knowledge, and acknowledgesthat reality is constructed intersubjectivity, that it is socially negotiated betweensignificant others who are able to share meanings and social perspectives of acommon lifeworld’ (p.156). Adding a ‘social dimension’ is qualitatively different toasserting that a shared meaning is possible. Here ‘reality’ is something that is sociallynegotiated, not privately constructed. Yet Jaworski states that the third principle is notan ‘extra’ but rather a qualification of the second! The third, however, is not aqualification but a contradiction! To state this is to have apparently missed out on anongoing debate between Leslie Steffe, who argued that radical constructivism iscompatible with the socioculturism of Vygotsky, and the socioculturist Steve Lermanwho argued that the social dimension cannot be reduced to the autonomous maker ofmeaning. Steffe is a radical constructivist who views learning as the capability of theindividual to change his or her conceptual structures in response to perturbation, yethe also emphasises ‘social interaction as a primary means of engendering learning’(Steffe and Tzur, 1994, p.8). With reference to Steffe’s attempt to reconcile theindividual with the social, Lerman states:

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As long as there is a separation between the subject and the world, including otherpeople, one has to go all the way with solipsism, or give it up. When the source ofknowledge and of meaning is the individual, social interactions are on the sameplane as physical interactions; they are filtered, or refracted, through theperceptions of the receiver. In some sense the possibility of cognitive conflict is lesslikely through social interactions than physical ones. I can imagine that I couldchallenge my belief that I can walk through a wall and I would probably receive arather strong perturbation. When In attempt to challenge my belief thatconstructivism is misguided by discussing ideas with radical constructivists, noperturbation is set up; I interpret their disagreements as their not listening to me...The ‘problem’ of the social is no problem at all if one accepts that socialinteractions are indeed on the same plane as physical interactions and both areseparate from the autonomous meaning-maker’. (Lerman, 1994, p.45).

Radical constructivism is rather like Kant’s ‘transcendental enterprise’ – a heuristicthat explained concept formation in terms of how the mind structures raw senseperception. Kant’s heuristic is a form of idealism because (and this is very, verycrudely put) any objectivity (e.g. ‘this stone is heavy’) is conferred by the mind (‘thisstone can be predicated ‘‘heavy’’’) onto the ‘manifold’ of the senses (‘wow! Thisstone as I perceive it seems heavy to me’). Von Glasersfeld (1998) refers to‘ontological reality’ as analogous to Kant’s thing-in-itself which lies beyondperception. Although neither of them deny the existence of the external world (andKant at least tries to prove the existence of the external world), the external worlddoes not feature in their explanation as to how knowledge is possible. Fine as aheuristic, but it would be a contradiction to then refer to the social world – a worldthat is external to what is given in sense perception. You have to go all the way withsolipsism, or give it up!

‘Contrary to what is maintained in one current interpretation [no reference given], anoption of this kind does not hail the return of the ‘‘sovereign subject’’ in all its might,be this in the form of idealism or solipsism. Were constructivism to advocate thisposition, it would scarcely merit a moment’s thought

Larochelle and Bednarz, 1998,

p.5, emphasis added). We seem to live in an age of ‘flat refusal’ – to deny what youwill at all costs despite what has been said and the compelling arguments that happento be inconvenient. ‘Of course, the role of prior knowledge in learning is only half thepicture. If I believed that there were no mechanism by which the external environmentcould impose itself in some way upon a person’s conceptions, I certainly would not bewriting this page, nor would I be concerned with any other social, collective activity. I

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would be condemned, or delivered, depending on one’s point of view, to my own

Konold, 1991, p. 140). One can believe that there is a

mechanism by which ‘ontological reality’ can impose itself, but von Glasersfeld’spoint is that we can never know (or refer) to what that mechanism is. He does notdeny an ontological reality, he denies we can ever have knowledge of it – so we areleft with our own internally isolated world (I say ‘we’, of course, according to thisdictum, you may not actually exist as far as I know).

And of course, any sceptic who doubts the existence of the external world can beinvited to drive into a brick wall at a speed proportional to his or her disbelief(Chalmers, 1982)! Lerman’s main argument is that constructivists do not recognisethat meanings are carried in practices and that cognition is ‘situated’ – a socioculturalview that does not acknowledge the separation between the subject and the world.1However, according to Steffe (1999) in a reply to Lerman, social interaction entersinto radical constructivism ‘at its very core’ (p.2) and is not incorporated as anafterthought. But this is exactly how the social is incorporated! 2 If everything isreduced to subjective judgement and interpretation, then that reduction must includesocial interaction because, for the radical constructivist, nothing is over and abovehow the individual has made sense of experience – including experience of socialinteraction. Steffe (1999) criticises Lerman (1994) for the claim that the subjectivityof radical constructivism makes it a solipsistic position. For Steffe (1999), radicalconstructivism is not solipsism because (a) perturbations make it possible for theindividual to distinguish between what is ‘in here’ from what is ‘out there’ (p.4) –without perturbations, everything would be the knowing ‘I’ and (b), it recognises ‘areality other than one’s own’ with its feet ‘firmly on the ground’ (p.7). But ifknowledge is making sense of experience, as the radical constructivist claims, thenhow is it possible to distinguish between making sense of experience and the worldthat is independent of experience? If perturbations have made it possible for Steffe todistinguish between ‘in here’ from ‘out there’ then perhaps he ought to tell us what is‘out there’. Remember that for von Glasersfeld, no distinction can be made – for if itcould, then we have a ‘discovery of an ontological reality’ which he denies. However,just when you think that Steffe is adopting realism in the defence of constructivism

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(‘Because the other subject is not unlike ourselves, we are obliged to attribute a realityto that other that is distinct from our own reality’ p.4/5), he uses relativism (theintersubjective as the most reliable level of ‘experiential reality’) to distinguishbetween knowledge (‘that we want to trust as though it were objective’ p.5 - i.e. let uspretend knowledge is objective) and illusion. 3 So Steffe wants us to pretend thatknowledge is objective, to regard it as subjective and to use the intersubjective todispel illusion – not a simple (coherent) theory at all!‘Post-epistemology’

According to Jaworski (1994), constructivism is ‘post-epistemological’ because itsays nothing about the ‘status’ of knowledge: it says nothing as to whether a statementis true or false. Consequently, a teacher cannot ‘challenge’ a student’s ‘mis-conception’, for there is only the student’s conception.

Now, in case that you are thinking that this may be a clever teaching strategy thatenables the facilitation of the student’s awareness of what is true or false, Jaworski ismaking a claim about the status of knowledge itself. Her claim, and one that is arguedby many prominent constructivists such as von Glasersfeld, is that knowledge itselfcannot be predicated true or false because it is context specific. This is tantamount tosaying that knowledge isn’t possible because it doesn’t actually exist. For example,‘the way we segment the flow of our experience, and the way we relate the pieces wehave isolated, is and necessarily remains an essentially subjective matter. Hence,when we intend to stimulate and enhance a student’s learning, we cannot afford toforget that knowledge does not exist outside a person’s mind’ (von Glasersfeld, 1996,p.5, emphasis added). Of course, knowledge does imply a knower, but vonGlasersfeld is claiming much more than this! According to Matthews (1998a), ‘thesephilosophical aspects of constructivism are frequently taken for granted, or assertedwithout argument or awareness of the tradition or depth of debate that has occurredaround them. For instance two leading constructivists have recently written that ‘‘theauthority for truth lies within each of us’’. This claim, which goes back at least toProtagoras in the 4th Century BC, if true, is truly breath taking in its cultural andepistemological ramifications. But the claim is made without any argument, or any

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consideration of its obvious flaws’ (p.x). Brown (1999), for example, makes a similarclaim

Mason follows Gattegno in seeing truth gravitating around personal awareness, thatis, truth is located in the mind of the individual. (p.14),

but does not locate this claim in the history of philosophy. ‘The history of philosophyis a footnote to Plato’ (A. N. Whitehead) and the issues that surround relativism, thatcentral ingredient of constructivism, have been debated since Plato and Protagoras.Brown (1999), for example, shifts the emphasis from the traditional view ofmathematical meaning as independent of individual human performance to thepersonal awareness of mathematics, but without any reference to any tradition in thephilosophy of mathematics. In so doing, logical necessity (that certain conclusionsfollow from the premises) is swept aside and student error becomes a question oflanguage and activity (for a critique of relativism and logical necessity see Rowlandset al, 2001). It therefore comes as no surprise that, for Brown (1994), mathematics is astyle of activity rather than something to be learnt.

‘Does the external world exist and if so, how is knowledge of this world possible?’ isa fundamental question of philosophy in which the various philosophical schools overthe centuries tried to resolve (Suchting 1986). Constructivism, on the other hand, is afairly recent and influential view of learning that asserts answers to this fundamentalquestion but with little reference made to the history of research that went into thisquestion. Of course, a constructivist could always claim that constructivism is atheory of learning and not a theory of knowing: that it is a psychological theory abouthow beliefs are developed rather than what makes beliefs true, that it makes noontological claim concerning the external world and that it is ‘post-epistemological’.But any learning-theory must have a core epistemology (something that Piaget wellrecognised) and this core outward must necessarily involve philosophicalconsiderations (Matthews 1998b). Thomas (1994) is a radical constructivist who has‘no desire to attack radical constructivism’ (p.33) yet recognises ‘the denial of thepossibility of knowledge of the world’ (p.33) as a ‘radical’ deficiency:

With a properly circumscribed claim, we can claim to know the world. What else?If we had only our own observations to go on, then we might reasonably beconcerned whether our knowings were indeed of the world. But since our scientific

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knowledge is concordant with that of others within common understandings of theworld, there is no practical room for doubt that it is the common world that weunderstand and have knowledge of. It is all very well for von Glasersfeld to be‘post-epistemological’, but to insist that it is not the world that we know is morelike the post-modernist cutting off the branch on which one sits. Amusing ascreative writing, but not to be taken seriously or taught to children. The verydiscussion of constructivism relies on common understanding; it is important forconsistency and for teaching that it does not lapse into a self-contradictoryabsurdity comparable to that of a proselytizing solipsism (Thomas, 1994; p.33).Of course, if von Glasersfeld was sincere in his post-epistemological view then hewould not be writing articles or trying to argue any position at all. Radicalconstructivism is a belief system, the paradox is that a true belief in radicalconstructivism would compel its believers to fall mute. While its advocates are legionand vociferous within the world of education academe, they would wish those at the‘chalk-face’ to fall mute. Consider the following: A boy is shown a 45, 45, 90 triangleand a 30, 60, 90 triangle and he responds that one is not a triangle. Now, if we were tosay that the boy is wrong, then Jaworski would say that ‘this is to make judgementsabout truth without taking into account the circumstances which the statements (thatthis is not a triangle) fits. Consider, for example, a geometrical object with anglesadding up to more than 180. We might be tempted to say that this could not be atriangle, meaning a plain triangle. However, a triangle on the surface of a spherecould fit the criterion. The context in which the statement is made is crucial to thevalidity of the statement, and it is very difficult to say therefore when any statement istrue without knowing the context’ (p.20). Unfortunately, Jaworski does not give thecontext why the boy thought one of the triangles isn’t a triangle, but we can safelyassume that the boy wasn’t thinking of non-Euclidean triangles. The boy’smisconception (with respect to all Euclidean triangles, or if that sounded too Platonic,then with respect to any possible triangle can be constructed within the Euclideanplane) has nothing to do with the context of triangles on surfaces on spheres, despiteJaworski’s insistence that we have to know the context. One the one hand, Jaworski‘demands’ that the context be known, yet there is no hint of specificity in thisexample. The boy is wrong, and constructivists would disempower the teacher toexplain why.

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Impossibility of transmitting ‘knowledge’

We may think of many examples that would render the idea that ‘new’ knowledge isconstructed from previous knowledge acceptable. Solving quadratic equations wouldmake little sense without a knowledge of linear equations and teachers may oftenrefer to previously ‘learnt’ material as a cue to a new concept, but constructivists haveconsistently failed to make explicit the process of utilising familiar concepts in theconstruction of new concepts With the exception of the work done by Tall (1991) andcolleagues, ‘utilising familiar concepts’ has been the teacher asking the class or achild to recall some fact so as to solve a problem (e.g. Hendry, 1996). To ask hownew knowledge relates to previous knowledge is not only a cognitive question butalso a question of specificity regarding the domain of knowledge – how the conceptswithin that domain relate to each other (e.g. see Hestenes, 1992, on mechanicseducation). In other words, to ask how a newly acquired concept relates to an alreadyexisting cognitive structure (or ‘schema’) is a psychological question, but anyspecificity regarding the learning of an academic or ‘schooled’ concept transforms thepsychological question into an epistemological one. A consideration of the question‘how is knowledge possible?’ can enable us to link the learner with a discipline, a linkbetween the psychology of learning with the way the discipline relates to the world.Although some constructivists have explored this link, the majority, it seems, woulddeny such a link as (shock! horror!) ‘Platonist’!

According to Jaworski (1994), constructivism challenges the transport metaphor thatunderlies much of educational discussion. The idea that knowledge cannot betransmitted can be a very useful heuristic. If knowledge is seen as a constructionprocess then there is a possibility of approaching a solution to the learning paradox:how can anyone get to know that x is the case if the learning process requires arecognition that x is the case. The problem is that constructivism has becomes radical– the construction process is no longer a heuristic but an insistence.

Schifter’s (1996) introduction to the book ‘What’s Happening in Math Class?Envisioning New Practices Through Teacher Narratives’, contrasts the practice ofmathematics teaching of twenty years ago with the consensus for a ‘new practice ofmathematics instruction’ based on ‘changing social needs and two decades of research

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in cognitive psychology’ (p.1). Implicit throughout is the assumption that nothing canbe transmitted or explained by the teacher. This has been made explicit in the book bySimon (1996), who states that, according to the constructivism of von Glasersfeld,‘teachers (or students) cannot give their understandings to another person. This ideacan be validated easily: Interview students who have all listened to the same lectureand you will hear quite discrepant ideas of what was discussed’ (p.39, emphasisgiven). The absurdity of this quote is twofold. Firstly, if it is all a question ofinterpretation, then how can anyone distinguish between ideas are discrepant withthose that are not? Secondly, if it is all a question of interpretation, then which idea ofthe author has been ‘validated easily’?

The view that communication is inherently impossible is popular with constructivists,but it is patent nonsense. The majority of people in an audience may not interpret themeaning as intended by the speaker, but that is not to say that understanding what thespeaker is trying to say is impossible. Consider the kind of communication that takesplace when a large team of engineers at Boeing designs the 747 and the variousmanufacturing parts are subcontracted out. Working off the engineer’s plans the partsare produced and later assembled. The thing flies. If there are problems, then theycan be traced to particular mistakes and misunderstandings and this would involveverbal communication. To say that communication is impossible, or necessarilyarbitrary may say more about the speaker than the referent in that particularconversation.

Constructivism seems to be the anxiety not to transmit anything. Yet concepts have tobe defined, explained, elaborated etc. if there is to be a ‘cognitive response’ to what istaught. We would argue that the fundamental question for mathematicseducationalists has to be ‘what is the best way to enable a class to understand – tomake their own – a framework of concepts that constitutes the subject-matter?’ By‘understand’ we mean in the manner of Skemp’s (1971) relational understanding ofmathematics as a body of knowledge.

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REFERENCES

Brown, T.: 1994, ‘Towards a Hermeneutical Understanding of Mathematics andMathematics Learning’, in (P. Ernest, ed.) Constructing Mathematical Knowledge:Epistemology and Mathematical Education, Falmer, London.

Brown, T.: 1999, ‘Mathematics, Language and Derrida’ in L. Brown (ed.) MakingMeaning in Mathematics, Annual Proceedings for the British Society for Researchinto Learning Mathematics, QED & BSRLM.

Chalmers A.: 1982, What is This Thing called Science? O.U. Press, Milton Keynes.Cobb, P., Wood, T., Yackel, E.: 1991, ‘A Constructivist Approach to Second GradeMathematics’, in (E. Von Glasersfeld, ed.) Radical Constructivism in MathematicsEducation, Kluwer, Dordrecht.

Confrey, J.: 1991, ‘Learning to Listen: A Student’s Understanding of Powers of Ten’,in (E. Von Glasersfeld, ed.) Radical Constructivism in Mathematics Education,Kluwer, Dordrecht.

Hendry, A. M.: 1996, ‘Math in the Social Studies Curriculum’, in (D. Schifter, ed.)What’s Happening in Math Class: Envisioning new Practices Through TeacherNarratives, vol. 1, Teachers College Press, NY.

Hestenes, D.: 1987, New Foundations of Classical Mechanics, Reidel, Boston.Hestenes, D.: 1992, ‘Modeling Games in the Newtonian World’, Am. J. Phys., 60(8),p.732-748. August.

Howard, R. W.: 1987, Concepts and Schemata, Cassell Educational, London.Irzig, G.: 2000, ‘Back to Basics: A Philosophical Critique of Constructivism’, Science& Education, 9(6), 621-639.

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Jaworski, B.: 1994, Investigating Mathematics Teaching: A Constructivist Enquiry,Falmer, London.

Jaworski, B.: 1999, ‘Tensions in Teachers’ Conceptualizations of Mathematics andof Teaching’, in (L. Burton, ed.) Learning Mathematics: From Hierarchies toNetworks, Falmer, London.

Konold, C.: 1991, ‘Understanding Students’ Beliefs About Probability’, in (E. VonGlasersfeld, ed.) Radical Constructivism in Mathematics Education, Kluwer,Dordrecht.

Larochelle, M and Bednarz, N.: 1998, ‘Constructivism and Education: BeyondEpistemological Correctness’, in (M. Larochelle, N. Bednarz and J. Garrison, eds.)Constructivism and Education, Cambridge University Press, Cambridge.

Lerman, S.: 1994, ‘Articulating Theories of Mathematics Learning’, in (P. Ernest, ed.)Constructing Mathematical Knowledge: Epistemology and Mathematical Education’,Falmer, London.

Lerman, S.: 1999, ‘A Response to Steffe’s Reply to Lerman on Intersubjectivity: ACase of Interpretations of ‘Social’, Chreods, 13, February. The ManchesterMetropolitan University (http://s13a.math.aca.mmu.ac.uk).

Mason, J.: 1999, ‘The Role of Labels in Promoting Learning from Experience AmongTeachers and Students’, in (L. Burton, ed.) Learning Mathematics: From Hierarchiesto Networks, Falmer, London.

Matthews, M., R.: 1994, Science Teaching: The Role of History and Philosophy ofScience. Routledge, London

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Matthews, M. R.: 1998b, ‘Introductory Comments on Philosophy and Constructivismin Science Education’ in (M. R. Matthews ed.) Constructivism in Science Education.Kluwer, Dordrecht.

Matthews, M. R.: 2000, ‘Editorial’, Science & Education, 9(6), 491-505.

Nola, R.: 1998, ‘Constructivism in Science and in Science Education’ in (M. R.Matthews ed.) Constructivism in Science Education. Kluwer, Dordrecht.

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Searle, J., R.: 1995, The Construction of Social Reality. Penguin, London.Shifter, D.: 1996, ‘Introduction: Constructing Meaning for the Rhetoric ofMathematics Education Reform’ in (D. Schifter, ed.) What’s Happening in MathClass: Envisioning new Practices Through Teacher Narratives, vol. 1, TeachersCollege Press, NY.

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p.501-520.

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Von Glasersfeld, E.: 1996, ‘Introduction: Aspects of Constructivism’, in (C. T.Fosnot, ed.) Constructivism: Theory, Perspectives, and Practice, Teachers CollegePress, New York.

Von Glasersfeld, E.: 1998, ‘Why Constructivism must be Radical’, in (M. Larochelle,N. Bednarz and J. Garrison, eds.) Constructivism and Education, CambridgeUniversity Press, Cambridge.

Warren, J.: 1979, Understanding Force, John Murray, London.Notes

1

Lerman does not agree with Suchting’s (1992, see 1999) ‘irritating and pedantic

attack’ (Lerman, 1994, p.47) that radical constructivism is inconsistent andincoherent. On the contrary, says Lerman, ‘it is a strong and consistent position’ albeita limited one. Yet he states that radical constructivism is incoherent when it attemptsto incorporate a social view of knowledge. How can constructivism be ‘strong andconsistent’ if it cannot reconcile the social? We urge a reading of

2

Steffe (1999) justifies his claim by saying that cognitive processes are, at the same

time, a result of autoregulation and interaction with the environment. He states thatradical constructivism is compatible with the Vygotskian approach that individuallearning is dependent on social interaction yet radical constructivism is not identicalto Vygotskianism because the former also includes interaction with the physicalenvironment. However, quite apart from avoiding the issue as to how the socialimpinges on the individual, Steffe makes reference to the physical environment –something that von Glasersfeld denies that we can ever have knowledge of!

3

The ‘social’ in Steffe’s (1999) radical constructivism is treated in terms of relativism(a lá consensus) and in terms of a ‘second order model’ of an observer constructing anobserved subject’s knowledge. For Steffe, knowledge is still making sense of

experience, but in a second order model it is making sense of someone making senseof their experience (rather in the manner of a teacher making sense of a pupil’s

reasoning), and since this is done through a process of social interaction then secondorder models constitute intersubjective knowledge and are ‘social models’. Lerman’s(1999) reply is that an individual consciousness is not an observers’ formulation andthat Steffe offers a very weak sense of intersubjectivity – a ‘between people’ ratherthan how meaning is carried in practices.

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