Constructivism, Mathematics and Mathematics Education

Author(s): Stephen Lerman

Source: Educational Studies in Mathematics, Vol. 20, No. 2 (May, 1989), pp. 211-223
Published by: Springer
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 STEPHEN LERMAN

CONSTRUCTIVISM, MATHEMATICS AND

MATHEMATICS EDUCATION

ABSTRACT. Learning theories such as behaviourism, Piagetian theories and cognitive

psychology, have been dominant influencesin education this century. This article discusses
and supports the recent claim that Constructivismis an alternativeparadigm, that has rich
and significant consequences for mathematics education. In the United States there is a
growing body of published research that claims to demonstrate the distinct nature of the
implications of this view. There are, however, many critics who maintain that this is not the
case, and that the researchis within the currentparadigmof cognitive psychology. The nature
and tone of the dispute certainlyat times appearsto describea paradigmshift in the Kuhnian
model. In an attempt to analyse the meaning of Constructivismas a learningtheory, and its
implicationsfor mathematicseducation, the use of the term by the intuitionistphilosophersof
mathematicsis comparedand contrasted. In particular,it is proposed that Constructivismin
learning theory does not bring with it the same ontological commitmentas the Intuitionists'
use of the term, and that it is in fact a relativistthesis. Some of the potential consequencesfor
the teachingof mathematicsof a relativistview of mathematicalknowledgeare discussedhere.

Constructivism has been described (e.g. Kilpatrick, 1987) as consisting of

two hypotheses:

(I) Knowledge is actively constructed by the cognizing subject, not passively received from
the environment.

(2) Coming to know is an adaptiveprocessthat organizesone's experientialworld;it does not

discover an independent,pre-existingworld outside the mind of the knower.

The first of these is becoming generally accepted, certainly by mathematics

educators, and is seen to be a useful and productive hypothesis when
thinking about listening to children and their mathematical learning. The
second is more controversial and perhaps worrying, since it appears to lead
us immediately into problems on two levels: firstly, whether it is ever
possible to understand what anyone else is saying or meaning, that is,
problems of private languages, and secondly, what kind of meaning can
thus be given to what we all accept as known, that is, the nature of
knowledge in general and of mathematical knowledge in particular. One
might suggest that we content ourselves with hypothesis (1), call ourselves
'weak' constructivists, (or the more pejorative term 'trivial'), and leave
debate of hypothesis (2) to philosophers, and conferences, with the implica-
tion that it is not really relevant to the business at hand, the teaching of
mathematics. (Those accepting both hypothesis have been called 'radical
EducationalStudies in Mathematics20: 211-223, 1989.
? 1989 KluwerAcademicPublishers.Printedin the Netherlands.
212 STEPHEN LERMAN

constructivists').This is ratherunsatisfactory,though, since the connections

between hypothesis (1) and (2) seem to be quite strong. After all, in
mathematics, and in philosophy, we are accustomed to pursuing the
consequencesof an hypothesis, despite their sometimes disturbingnature.
In fact it is crucialthat the second hypothesisis consideredhere, precisely
because of the significanceof the nature of mathematical knowledge for
epistemology and philosophy in general. From Plato, through Descartes,
Leibnitz, and Kant to modem philosophers,mathematicalknowledge has
served two essential functions at least: first, the ultimate test of the
adequacy of the philosophicalideas proposed is whether they can include
and explain mathematicaltruths;second, the apparentlytimeless,certain,a
priori, tautologicalnature of mathematicalpropositionsform the paradigm
of knowledge. If one can establish the validity of propositions about
'justice','good', or 'freedom'with the kind of certainty that mathematical
propositionsappear to exhibit, the classical problemsof philosophy can be
solved. Thus it is often the case that philosophersbegin with mathematical
knowledge, and constantly refer to mathematicalconcepts, at the heart of
their ideas. Plato increasinglyused mathematicalforms to characterisehis
theory, perhaps because there seem to be so many differentforms of, for
instance 'table'; Leibnitz'ssuccess with his notation for the calculus led to
the proposal that such a notation should be developed for all reason, and
Kant characterisedSpace and Time as transcendentalcategories.Develop-
ments in science in the last three centuries have reinforced the role of
mathematicsas the last bastion of certainty.If, however, one can presenta
case for the fallibility and relativity of mathematical knowledge, and of
such concepts as 'proof' and 'truth', this has fundamentalimplicationsfor
philosophy. Bloor (1976) was so successfulin proposing his relativistthesis
which focussed on mathematicsand logic, that debate about his ideas have
drawn in many of the major British philosophers today (e.g. Hollis and
Lukes, 1982).
One way out, discussed and criticised in depth by Stove (1982), is to
continue as before, but to surroundwords such as proof, truth, etc. with
inverted commas, as 'proof' and 'truth'. In that way, we can slightly
weaken our claims in mathematics,but continue to use familiar words. It
is as if we are saying "This is true, within the confines of present notions
of truth".Or it can be taken to mean that the term is completelydevalued.
Stove demonstrates how Popper, Lakatos, Kuhn and Feyerabend use
invertedcommas, and puts forwardthe thesis that all four philosophersare
in fact irrationalists,as seen by their use of this punctuation. I will return
to this issue below, but a comment of Bloor's (1982) is relevant here,
 CONSTRUCTIVISMAND MATHEMATICSEDUCATION 213

although he gives it in a differentcontext:

Are believers in a flat earth the only ones amongst us with the right to operate with the
distinction between 'up' and 'down'? (p. 321)

Both 'weak' and 'radical' constructivism are concerned with learning

theory, and find their modern roots in Piaget's genetic epistemology.
However Piaget was attemptingto resolve the ancient problemsof how one
comes to know anything, and the relationshipbetween the individual and
an objective world, and was proposing an alternative to empiricism or
platonism.Thus the questions that were amongst those asked and discussed
at the Eleventh InternationalConferenceon the Psychology of Mathemat-
ics Education,what does constructivismimply for mathematics,and does it
have any implicationsfor mathematicseducation, arise directly.
Much concern and disquiet has been expressedin recent years with the
rigidity, appropriatenessand applicabilityof the Piagetian stages of devel-
opment. I suggest that part of Piaget's motive in constructinghis elaborate
theories of mental schema was his desire to establish objectivity, since his
epistemologicalalternativeto empiricismand platonismplaces the roots of
knowledge in the individual, and thus borders on private thoughts and
language. I will attempt to show that radical constructivismreturns to
geneticepistemology,but takes the full consequencesof Piaget'sphilosophy
without feeling the need to establish objectivity in this sense.
In this article I will examine what radical constructivismmight mean in
mathematics,and propose some implicationsfor mathematicseducation. In
the former discussion I will suggest that the present use of the word
'constructivism'has some similaritieswith its historical use in connection
with Intuitionism, but is fundamentally different. I will then go on to
propose that radicalconstructivismis a relativistepistemology,which does
not leave us unable to say anything to each other, but actually endows
philosophicaldiscussionswith content and controversy.In the latter discus-
sion, I will not avoid the issue of the connections between theory and
practice, in the sense of the question "How can different epistemologies
lead to distinct and testable hypotheses for the teaching of mathematics?"

CONSTRUCTIVISM IN MATHEMATICS

i. Constructivismin Intuitionism
In the philosophyof mathematics,constructivismis usually used in connec-
tion with Intuitionism. Indeed Kline (1980, p. 241) uses the terms inter-
changeably. I will call this constructivism Cl. Whilst the origins of
214 STEPHEN LERMAN

intuitionist ideas in mathematicscan be traced back at least to Kronecker

(1823-1891) and perhaps to Descartes, one looks to Brouwer, whose
doctoral dissertation 'On the Foundations of Mathematics'was written in
1907, for the essence of this epistemologicalposition. A clear descriptionof
the use of this notion of Ci is given by Weyl (1963):
An existentialstatement,such as 'thereexists an even number',is not considereda proposition
in the proper sense that asserts a fact. An 'infinitelogical summation'such as is called for by
a statement of this kind (1 is even or 2 is even or 3 is even or... ad infinitum) is evidently
incapable of execution. '2 is an even number',this is a real proposition (provided 'even' has
been defined recursively...); 'there exists an even number' is nothing but a propositional
abstract derived from that proposition. If I consider an insight a valuable treasure,then the
propositional abstract is merely a document indicating the presence of a treasure without
disclosing its location. Its only value may lie in the fact that it causes me to look for the
treasure.It is a worthlesspiece of paperas long as it is not endorsedby a real propositionsuch
as '2 is an even number'. Whenever nothing but the possibility of a construction is being
asserted, we have no meaningfulproposition;only by virtue of an effective construction, an
executed proof, does an existential statement acquire meaning. In any of the numerous
existentialtheorems in mathematics,what is valuable in each case is not the theorem as such
but the construction carried out in its proof; without it the theorem is an empty shadow.
(p. 51)

Clearly,there are alreadydifferencesbetweenthis notion of constructivism,

C,, and the one which is discussedtoday in mathematicseducation, which
I shall call C2. The epistemology of the Intuitionists focusses on what
mathematics ought to be, which methods, statements, proofs etc. are
acceptable, in the construction of mathematics, in order to achieve the
certainty that was sought by these philosophers of mathematics. Genetic
epistemology, the origins of the present notion C2, focuses on the activity
of constructionas the process by which the individuallearns, and by which
knowledgeis created. Mathematicsas such, is taken for granted by Piaget,
his concernwas not with valid or invalidmathematicalstatements,but with
how the individualgains that knowledge. There are some similaritiesalso.
Heyting (1956, in Benacerrafand Putnam, 1964) writes, in a manner that
sounds much like the speech of constructiviststoday:
... a mathematicaltheorem expressesa purely empiricalfact, namely the success of a certain
construction. "2 + 2 = 3 + 1" must be read as an abbreviation for the statement: "I have
effected the mental constructionsindicated by "2 + 2" and by "3 + 1" and I have found that
they lead to the same result". (p. 61)

The role of language is one of the distinguishingfeatures between C, and

C2. For the Intuitionists,languageis secondaryto thought, and only serves
to communicate that thought to others. Thought, which is right intuition,
is the essence, and language is an imperfect device for communication. I
will discuss more fully below, what essential function language might serve
in an elaborationof C2,but brieflya concept is identifiedby its use, it gains
 CONSTRUCTIVISM AND MATHEMATICS EDUCATION 215

its meaningfrom the sharedsocial interpretationwhich is its use, and hence

language, which itself is socially negotiated, and finds its meaning only in
its use, is integrallyconnected with the notion of a concept.
Other major differencesare revealed in an examination of the ontology
of the Intuitionists.As with C2, the Intuitionistsdo not rely on a notion of
transcendentalexistenceof mathematicalentities. They maintain that there
may be such objects, that exist independentof acts of human thought, but
their existence "is guaranteed only insofar as they can be determinedby
thought.... Faith in transcendentalexistence, unsupportedby concepts,
must be rejectedas a means of mathematicalproof" (Heyting in Benacerraf
and Putnam 1964, p. 42). Similarly,faith in the existenceof a real objective
world without establishing some means of connecting with it, must be
rejectedas providingus with certainty.However, it is just this certaintythat
distinguishes C2 from Cl. The programme of the Intuitionists was the
Euclideanone of resolvingthe problemsin the foundationsof mathematics,
and establishingthe certaintyof mathematicalknowledge.They maintained
that one must rejectclaims to any transcendentalexistence,but they needed
some certainfoundationsfrom which to build the structureof mathematics
with constructive proofs. Their method, in fact, was to weaken but not
reject Kantian intuitionism, which relied on transcendentalcategories of
time and space, and which had been fundamentally threatened by the
inventionof non-Euclideangeometries,by "abandoningKant's apriorityof
space, but adheringthe more resolutely to the apriority of time" (Bouwer
1912 in Benacerrafand Putnam 1964, p. 69). They then appeal to intuition
of the integersas the 'basal intuition of mathematics'.Kroneckeris said to
have declared, in an after-dinnerspeech, "God made the integers;all the
rest is the work of man." It is interestingto note that Brouwer,Heyting and
their supporterswere known as radical Intuitionists, whereas others were
prepared to accept, for instance, the real numbers rather than just the
integers,as God-given and the rest as human construction.Whetherthese
latter were known as 'weak' intuitionists, or 'trivial'intuitionists,I do not
know.
To summarise,C, has similaritiesto C2in that both rejecttranscendental
knowledgeof the real world as providingproof, and are content to leave its
existence an open question. Both assert the centrality of the notion of
construction in concept-formation, and indicate the little value of pure
existence theorems,whether in mathematicalstatements, or concepts. The
fundamentaldifferenceis that Intuitionism is an epistemology of mathe-
matical knowledge,concernedwith a programmeto establish the certainty
of mathematics, based on the apriority of time and subsequentlyof the
216 STEPHEN LERMAN

integers, to be grasped by intuition. C2 is, in my interpretation,a more

complete and consistent view of coming to know, of knowledge and of
mathematics.

ii. ConstructivismC2 and Mathematics

For the Intuitionists, as for the Logicists and the Formalists at the time,
certainty was the aim. They were trying to achieve a revised Euclidean
programmefor mathematics,which of course had implicationsfor knowl-
edge in general, since mathematics was seen as the last bastion of the
absolutists. C2 has no such teleology, it is not a philosophy built on trying
to achieve certain goals. It suggests that we examine the consequences,
honestly, of hypothesis (2) above, that coming to know does not discover
an independent,pre-existingworld outside the mind of the knower.
What, then, does 'coming to know' discover?What is meant by 'coming
to know'? If the second constructivisthypothesis implies that there is no
world outside the mind of the knower, an implication that many seem to
assume, then we are certainlyall doomed to solepsism. However, I suggest
that this is not the case. The second hypothesis recognises experiences,it
does not cast doubt on the idea that we all interactin some way with people
and the world around us. When we ascribemeaningand significance,when
we interpretand attempt to explain, when we propose theories based on
those experiences,we are organizingour experientialworld. The hypothesis
implies that we are not 'discovering'the way the world works, in the sense
that America was there and inhabitedand then 'discovered'by Europeans,
or whoever. In fact we cannot talk about the way the world actually,
certainly and in a timeless way, works, simply because we cannot have
knowledge that what we are describing is just what is. There is no
Archimedianposition from which to view our concepts and theories. But
this has always been the case, and it has not halted people from developing
theories, discussing explanatory power, or comparing evidence. Far from
making one powerless,I suggest that researchfrom a radicalconstructivist
position, is empowering. If there are no grounds for the claim that a
particulartheory is ultimatelythe right and true one, then one is constantly
engaged in comparing criteria of progress, truth, refutabilityetc., whilst
comparing theories and evidence. This enriches the process of research.
If it is accepted that the knowledgewe have of the way the world works
is not forced upon us by that world (empiricism), nor do we have this
knowledge innately (platonism), then what we know becomes conjecture,
theory and hypothesis.It may be well-establishedconjecture,it may even be
 CONSTRUCTIVISM AND MATHEMATICS EDUCATION 217

just about inconceivablethat things could be otherwise, but this still does
not provide certainty, and all our deductions and reasoning must adjust
accordingly. Further, the theories, concepts and constructs are culturally
and temporally relative. One need only think of the Greek notion of the
centre of the earth being the centre of the universe, or modem creationist
interpretationsof the developmentof life, to illustrate this.
Thus, I suggest that what has to be abandonedwith the rejectionof the
belief that we are discoveringan independentpre-existingworld outside the
mind of the knower, is only that the knowledgewe have can in any way be
described as certain and ultimately true. What we lose is certainty and
absoluteness,we do not lose the whole purpose in searching.There is still
the possibility of makingjudgements,of using terms such as true and false,
up and down, better or worse, more or less fruitful etc., and I will clarify
in what sense one can use these terms below. Certaintyis perhaps only a
psychologicalnecessity,or an emotional necessity, not a logical one in any
sense. Loss of certainty means that different theories and conjecturesare
comparable, examinable, and equally valid, until one establishes some
acceptablecriteriaof 'better'. Certaintyhas a tendency to lead one to say
"That's it, no more discussion, we have the answer". Fallibilism, a view
which acceptsthe potential refutationof all theories, and counter-examples
to all concepts, allows one to ask how does one know that this answer is
better than that one, what might constitute a notion of 'better',might they
not both be possible, as with Euclideanand non-Euclideangeometries,or
arithmeticswith or without the Continuum Hypothesis.
I have attemptedto demonstratethe radicalconstructivism,as I interpret
it, is a relativist epistemology. I have suggested that it does not leave us
unable to use the terms'truth'and 'falsity', for instance, or make the whole
process of investigation worthless. It is perhaps an interesting empirical
question whether scientists, or in our case mathematicians,find relativism
an empoweringphilosophy within which to work, or whether they find a
form of platonism more fruitful, (this idea was first proposed by Reuben
Hersh). One of the ways in which we may justify preferringone theory or
explanationover another, is the comparativefruitfulnessof those theories.
Of course these notions are themselves problematic and have different
interpretationsin differentparadigms,but this is not the place to discuss
these issues. I have engaged in that debate elsewhere (Lerman, 1986). In
suggestingthat this question,whethermathematiciansfind one theory more
empowering than another, is an interesting one for research, I wish to
indicate that the relativistthesis, to be consistent, must be reflexive,that is,
to adopt a critical and fallibilist position on relativism itself. One of the
218 STEPHEN LERMAN

criticisms, misguided I believe, that Kilpatrick made of the radical con-

structivists (Kilpatrick, 1987) was that they were not reflexive in their
advocacy of what I have described as relativism.
The relativistposition poses difficultquestions, particularlyin relation to
mathematics,amongst which are the following:
(a) How can one account for the apparent successes of mathematics?
After all, buildings generally stay up, satellites reach their destinations,
arithmetic,even with impredicativedefinitions, seems to work.
(b) Feyerabendclaims that 'anythinggoes' (e.g. 1978), includingastrol-
ogy, creationism,etc. Is there a way of preferring?
(c) Can there be any communication,or are all languagesand concepts
private?
First, as a general comment, it is my view that one consequence of
abandoning certainty and accepting fallibilism is that questions such as
these are generated, and this is in itself a support for the relativist
programme.
One can suggest that (a) gives clear evidenceof the mathematicaltheories
being true, in the sense of correspondencewith the real world. It is certainly
one of the wonders of mathematics,and one of the best teachers of these
phenomena in recent years has been Professor Morris Kline. Thus it is
instructive to read the final chapter in his Mathematics- The Loss of
Certainty (Kline, 1980), where he surveys the comments of the great
mathematiciansand scientists on this enterprise.He, and they, confirm the
thesis of his book, that one can only express wonderment,and a sense of
the power of this human invention, which many do in almost mystical and
religious terms, but this still does not make it possible, or indeed necessary
to claim any absolute truth for the theories. After all there are revolutions,
even in mathematicalthought. On the other hand, Feyerabend'sclaim that
'anything goes' results from his, in my view justified, rejection of any
absolute criteria for preferringone theory from another. We can give up
absolute criteria, however, and still claim the mathematicsprogresses, or
that one theory is to be preferredto another, even if only in hindsight, by
accepting the relativism of such criteria, in relation to the scientific and
cultural community in which they arise. To return to Stove's critique of
Popper, Kuhn, Feyerabend and Lakatos as irrationalists,one can accept
his arguments,that they are inconsistent in their theories, and that in the
end they are led to irrationalism,in the sense that for instancePopper holds
on to a progressiontowards truth, and Lakatos to the rational reconstruc-
tion of scientifichistory, despite their rejectionof certainty. Stove goes on
to reject all their ideas, because they are irrationalists,and to develop an
 CONSTRUCTIVISM AND MATHEMATICS EDUCATION 219

alternative,empiricistepistemology, that provides the certainty that Stove

requires.However,pointing out their inherentirrationalismcan also lead to
a strengthening of the relativist thesis, that forms such a large part of
Popper's and Lakatos's work, and yet is opposed by their total pro-
grammes. C2, as a relativist thesis, does not imply that we cannot use
cognitive terminology,merelythat we have to referterms such as truth, and
proof, and better, to a particularphilosophical frame of reference.
Finally, on the question of ontology, or 'what is', again certain knowl-
edge of an objectively-existingreal world cannot be achieved.This is not to
deny that the real world exists, only that even if it does, we can have no
way of having "knowledge"of it, if we demand that this knowledgehas to
be certain and absolute. As discussed above, neither the empiricistanswer
nor the platonist one are adequate. One can pursue this ultimately to
Descartes' "Cogito, ergo sum", but only if one is searching for absolute
knowledge. Objectivityrests in the public nature of language, of concepts,
of theories and hence of knowledge. These can change, as they are social
constructions, publicly negotiated concepts, but relative to a particular
culture, in a time and a place, they function as objective knowledge,
without ascribingto them a transcendentalexistence. Bloor's highly illumi-
nating discussion of objectivitytakes the issue to the heart of mathematics,
what we mean by the existence of mathematical objects (Bloor, 1977).
Indeed he takes the views of Frege, one of the major figuresin presenting
the neo-platonist image of mathematics,to illustrate the social nature of
mathematicalknowledge. Frege uses the equator, the axis of the earth and
the centre of mass of the solar system as examples of objective but
non-physical entities. Bloor points out how these are just social construc-
tions, invented by people to function as structuringand orderingconcepts.
In the Ptolemaicsystem, the centre of the earth served the same role, as the
centre of the universe which consisted of concentric circles around the
circular earth. Bloor comments that Frege would be as horrified by
'sociologism' as he was by 'psychologism',as he termed Mill's empiricist
philosophy, the latter being the idea that concepts gain meaning in the
individualmind, and the formerin the social mind, as it were. Nevertheless
Frege's descriptionof mathematicalobjectivityappearsto fit the character-
isation of mathematicsas a social construction,changeableand negotiated.
As Wittgenstein(1967) puts it:
If humans were not in generalagreed about the colour of things, if undeterminedcases were
not exceptional, then our concepts of colour could not exist.' No: - our concept would not
exist. (para 351)

This is not the place to pursuethese argumentsfurther,around (a) and (b),

220 STEPHEN LERMAN

but I have given some indicationof the directionsuch discussionscan take,

and have given a more complete account of these issues elsewhere(1986).

On 'PrivateLanguages'
In this present discussion, however, it is most important to try to answer
(c), since without a means of communication,teachingis certainlya wasted
and futile effort, and this highlights some of the worries people express
when consideringC2. I will attempt to do this with referenceto the concept
'understanding',as this is a major concern for mathematicseducators,and
hence will lead to the final section on the implications for mathematics
education.
Discussions about children's'understanding',how we examine and iden-
tify 'understanding',and ultimately what 'understanding'means, form a
central core of researchin mathematicseducation. We talk about whether
a student doing such and such would demonstrate understandingof the
mathematicstaught, as if there is some inner phenomenon, called 'under-
standing', which may be a 'correctunderstanding'or an 'incorrectunder-
standing', and particular behaviour on the student's part would identify
which one for us. Here is the essence of the difficulty,since if all under-
standings are private and individual constructions, no student behaviour
will allow me to do anything other than make my own privateconstruction
about what the student 'understands'of my 'understanding'of the concept
or idea in question. The difficulty,however, may exist only in an absolutist
epistemology.If 'addition'has a transcendentalexistenceas a concept, then
the student either has that concept, or not, there are no partial stages.
The job of the teacher is then to discover whether the student has the
'correct understanding'in its totality, otherwise it would not constitute
'understanding'.
If one abandons the absolutist epistemology, the discussion changes.
Considerthe familiarphilosophicalexample, what is meant by saying that
a child has learnedand understandsthe concept 'hat'. This comes about by
pointing out instances of 'hat', objects that have the use implied by that
term. When the child points to a hat and says "Hat", we confirmthat this
is correct. When the child points to a tea-cosy and says "Hat" we have to
explain that this object has a differentuse, and is not a hat. It is in the use,
according to the public, objective notion of 'hat', that we can apply the
word 'understand',and it has no application without this public connec-
tion. (Of course someone, initially called eccentric, and later perhaps a
person who sets a new trend, may put the tea-cosy on their head, and call
 CONSTRUCTIVISM AND MATHEMATICS EDUCATION 221

it a hat, and our concept will have to undergo a public change.) Wittgen-
stein says the following about understanding(Wittgenstein, 1974):
Do I understandthe word 'perhaps'?- And how do I judge whetherI do? Well, somethinglike
this: I know how it's used, I can explain its use to somebody, say by describingit in made-up
cases. I can describe the occasions of its use, its position in sentences,the intonation it has in
speech.- Of course this only means that 'I understandthe word "perhaps"'comes to the same
as: 'I know how it is used etc.'; not that I try to call to mind its entire applicationin order to
answer the question whether I understandthe word. (p. 64)

To summarise, the shift from behaviourismto cognitive psychology fo-

cussed attention on teachingfor understanding,but the problemsof how to
carry this out, and how to identify that 'it' had happened, remained as
ongoing and major ones for mathematicseducation. It is suggested here
that central to the difficultyis our notion of 'understanding',tied as it is to
the idea of certain and absolute concepts. According to this view, the
process of coming to understanda concept is one that takes place in the
mind of an individual,and the final step of achieving that full understand-
ing of a timeless, universalnotion is a very private, almost mystical one. It
is certainly beyond the power of any outsider, such as a teacher, to know
that the process has taken place in full.
The difficultyof privatelanguagesthus arises with absolutistepistemolo-
gies, and not with relativism and the present use of constructivism C2,
contrary to the usual discussions. Accepting hypothesis (1) alone, "that
knowledge is actively constructed by the cognizing subject, not passively
receivedfrom the environment",does not ease any of our difficulties,and
may in fact aggravatethem. On its own, this hypothesis rejectscoming to
know throughempiricalmeans, but leaves us with a view of knowledgethat
cannot be actively constructedby the individual,since knowledge is objec-
tive in an absolutist sense. We need to be able to continue with our belief
that if we create the right environment,in the classroom, in our teaching,
learning and understanding will take place. Accepting hypothesis (2),
however, forces us to re-examinewhat is meant by knowledge, and locates
objectivity in the social domain, not the transcendental.Concepts are
public, as their meaning is their use, and so too is understanding.
In the first section, the differencesbetween the Intuitionists use of the
term constructivismand the presentuse were compared,in order to clarify
some of the issues in this paradigmshift in mathematicseducation, if that
is what is taking place. In this section, I have attempted to show that
epistemology, mathematicalknowledge, and learning theories are interde-
pendent areas of study. Piaget's genetic epistemology was a philosophical
theory, resulting from his rejection of platonism and empiricism, and it
placed the question of the nature of knowledge in the study of its
222 STEPHEN LERMAN

acquisition. Radical constructivismC2 is a re-examinationof these ideas,

and proposes that the programmehas distinct consequencesfor mathemat-
ics education. I have attemptedto support this view, and illustrateit by the
discussion of the notion 'understanding'.In the final section, the issue of
consequencesof a theory for practice is discussed.

CONSTRUCTIVISM C2 IN MATHEMATICS EDUCATION

In discussing the nature of scientific revolutions, Popper proposed that

there are 'crucial experiments'that finally reveal which of two competing
theories is the correct one. Both Lakatos and Kuhn pointed out that this
was a naive idea, and that, for instance, had the Michelson-Morley
experimentnot supported Einsteinianmechanics,it would only have been
declaredthat the equipmentwas unsatisfactory,or the experimenthad been
carried out incorrectly.Kuhn's analysis of scientificrevolutions examined
the issue of paradigms, the hard-core of theories, and the nature of the
conflicts between scientific communitiesat the stage or paradigm shifts. I
suggest that it would thereforebe naive of us, in mathematicseducation, to
expect to pick on a 'crucialexperiment'to establish or refute C2.
In order to maintain, however, that C2 is an alternativeand competing
paradigm,it must at least be shown that there are potentially rich theories
and ideas that are distinct to this view. In this article I have attempted to
show, by drawing on the second hypothesis, one can identify a notion of
understandingthat is distinct from that of cognitive psychology, and is
particularlyaccessiblefrom the point of view of mathematicseducation, by
its very nature.
ElsewhereI have discussed other possible influences of alternativeper-
spectivesof the natureof mathematicalknowledgeon aspects of mathemat-
ics education. I have suggested that a relativist view of mathematical
knowledge, and I have attempted to show that C2 is a relativistview, has
implications for teaching styles (Lerman, 1983, 1986), for the way the
curriculumis developing in relation to problem-solvingand investigations
(Lerman, 1987), and also draws issues of social values and politics into the
mathematicsclassroom (Lerman 1988). These are theoreticaldevelopments
of the consequencesof C2, and theory formation is an integral part of the
developmentof researchprogrammes.Indeed without such theory forma-
tion, proposing consequences of hypothesis (2) in particular,radical con-
structivismis perhaps vulnerableto the strong criticismsof e.g. Kilpatrick
(1987), and to attemptsto subsumethe innovative researchin the construc-
tivist paradigmunder that of earlier ideas.
Finally, there remains the question of how theory and practice relate.
 CONSTRUCTIVISM AND MATHEMATICS EDUCATION 223

This is an ancient question, but again one that, I suggest, is a consequence

of epistemologies with an absolutist teleology. The ideas developed here,
taking inspiration from Wittgenstein, propose that theories and concepts
are rooted in practice, and obtain their meaning from use. They gain their
objectivity in their public nature, in that theories written down become
public property, subject to dispute, negotiation and adaptation. Their
objectivitydoes not lie in their being the ultimate truths. Thus, there is in
general a problem of the relationshipbetween theory and practice,but not
for ConstructivismC2.

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Departmentof Computingand Mathematics,

South Bank Polytechnic,
BoroughRd.,
LondonSEI OAA,
England