MUSINGS
ON
MATHEMATICAL
CREATIVITY
PETER LILJEDAHL, BHARATH SRIRAMAN
Thisconversationbegan when I introducedthe two authors
in a hotel lobby in Cap Roig (Spain) after the close of
CERME4[I]. The conversation continued by e-mail well
after the evening had ended, (ed.)
Bharath:I think that the notion of mathematicalcreativity among mathematiciansis distinctly different from the
notion of creativityfound in the mathematicseducationliterature.What do you think? Do we even have an agreed
upon definition of what mathematicalcreativity is? Is this
importantat all?
Peter:I too wish to discuss the definingof creativity.This
certainly is important. The writings in my dissertation
expressa slightly differentparsingof creativitythanthe one
you propose (i.e., among mathematicians, psychologists,
teachers,mathematicseducators).
Bharath: I would like to give some background to the
AHA! momentthat you talked about (Liljedahl,2004). The
literatureseems to attributethis constructto Wallas (1926)
andhis famousbook Theart of thought.Some authorserroneously attributeit to Hadamardand Poincaré due to the
popularizationof theirwritings.
However,this constructwas developedwithinGestaltpsychology in Germanyin the very earlypartof the 20th century
Koehler,andKoffka.Some historianspush it
by Wertheimer,
backto the late 19thcenturyto the writingsof Mach (a physicist turnedphilosopherinterestedin the physiologyof sensory
perception). There is evidence of written communication
betweenPoincaré,Hadamardandthe Gestaltists,whichcould
lead one to inferthatHadamardwas influencedby the developments and the terminology within Gestalt psychology.
Hencethe use of the 3- or 4-stage model fromthen on.
I really liked your critiqueof the reductionistattemptof
viewing creativity as a confluence of person, process and
product,as well as your critiqueof the misconceptionsthat
occurwhen using creativeor creativity.Shouldwe focus our
conversationon the definition of creativity by considering
each aspect (person, process and product) separately and
thenanalyzewhetherthis is consistent(or inconsistent)with
variousconfluencemodels thatcombinethe threealong with
societal variables proposedby researchersin psychology?
Our attempt could perhaps be to construct a definition of
creativityfor the particulardomainof mathematicsand then
seeing whethergeneraldefinitionshold up?
Incidentally,Ervynck(1991) describedmathematicalcreativity in terms of three stages. The first stage (Stage 0) is
referredto as thepreliminarytechnicalstage, whichconsistsof
some kindof technicalor practicalapplicationof mathematicalrules and procedures,withoutthe user having
any awarenessof the theoreticalfoundation,(p. 42)
The second stage (Stage 1) is that of algorithmic activity,
which consists primarilyof performingmathematicaltech-
niques, such as explicitly applying an algorithmrepeatedly.
The thirdstage (Stage 2) is referredto as creative (concep-
tual, constructive)activity;truemathematicalcreativity
occurs and consists of non-algorithmicdecision making:
The decisions thathave to be takenmay be of a widely
divergentnatureand always involve a choice, (p. 43)
Although Ervynck (1991) tries to describe the process by
which a mathematicianarrivesat the questions throughhis
characterizationsof Stage 0 and Stage 1, his descriptionof
mathematicalcreativity is very similar to that of Poincaré
and Hadamard. In particular, his use of the term "nonalgorithmic decision making" seems to be analogous to
Poincaré's use of the "choice"metaphor.
Peter:As I turnmy mindback to ourconversationthus far
many ideas come to the fore. To begin with, you correctly
identify my reductionistattemptat defining creativity as a
critique.I find that many definitions (and authorsof definitions) rely too much on rhetoric.The discoursesthatemerge
are like a sieve - they are easily graspedbut they don't hold
much water.Attemptingto fit actualinstancesof (perceived)
creativityinto these discourses,the definitionsare often too
confining (even rigid). One of the findings that emerged
from my own work with mathematicians on their AHA!
experiences (and I defined the AHA! experiencein the four
stage Gestaltist way of initiation-incubation-illuminationverification as discussed in Wallas, 1926) was that the
experiencewas self defining.Thatis, I did not have to define
the experience for the mathematiciansin orderfor them to
know that they had one. As it turnedout, this was also true
for undergraduatemathematicsstudents,as well as pre-service elementaryschool teachers.
I wonderif we cannotsay the samethingfor creativity- cresuch a positiondoes not
ativityis self-defining.Unfortunately,
advanceouremerging(or at least attemptsat emerging)definition,but it does allow us to includeall thingsthatarecreative
andexcludeall thingsthatarenot.Oneof the downfallsof such
a relianceon setf-definitionis the curseof discourse(oras I call
it, discurse).How much of mathematicians'understandingof
is a productof the discourseon
creativity/invention/discovery
theseideaswithintheirdomain?Forexample,you cite a mathematician (in Sriraman,2004) as stating that "Opportunity
knocksbut you have to be able to answerthe door"(p. 32). I
had a mathematicianstate"Chancewill favourthose who are
Aren'tbothof thesemathematicians
parrotingLouis
prepared."
Pasteur'sphrase"Chancefavours the preparedmind"?For
anotherexampleconsiderthe followingtwo statements:
PerhapsI could best describe my experience of doing
mathematicsin terms of enteringa darkmansion. One
goes into the first room, and it's dark,completely dark.
One stumbles aroundbumping into the furniture,and
For the Learningof Mathematics26, 1 (March,2006)
FLM PublishingAssociation, Edmonton,Alberta,Canada
17
gradually,you learn where each piece of furnitureis,
and finally, after six months or so, you find the light
switch.Youturnit on, andsuddenly,it's all illuminated.
(Wiles, 1997)
It is like going into an unfamiliar hotel room late at
night without knowing even where to switch on the
light. You stumble aroundin the darkroom, perceive
confused black masses, feel one or the other piece of
furnitureas you are gropingfor the switch. Then, having found it, you turn on the light and everything
becomes clear. (Polya, 1965, p. 54)
My final point is that whatever definition we arrive at it
needs to be relativistic.As Hadamardpoints out,
Between the work of a studentwho tries to solve a difficult problem in geometry or algebra and a work of
inventionthere is only a differenceof degree, (p. 104)
If we use the wordcreationinsteadof inventionin this quotation, the implicationsbecome interesting.Creationimplies
creativity.Does this meanthatinventionpresupposescreativity?I thinkso. Butdoes it also implythatcreativitypresupposes
invention?I don'tthinkso, andthis leadsto my stanceon relativisticas opposedto absolutistviews of creativity.
Bharath:Yourthoughtson having a relativisticdefinition
as opposed to an absolutist definition are interesting and I
will commenton this shortly.
First,I will commentfurtheron the Gestaltistsandthe fact
thatredundanciesaboundin extantdescriptionsof creativity.
I was recently re-reading Schoenfeld (2002) and, interestingly enough, he points out that insight and structurewere
central concerns of the Gestaltists. This article also recalls
the famous story of Poincaré taking a day trip after having
struggled on a problem for a while and his experience of
stumblingupon the solutionjust as he boardsthe bus. However, Schoenfeld's commentsafterthis anecdoteare:
Poincaré's story is typical, both in substanceand methods. Withregardto substancethe outline of the story is
the basic tale of Gestalt discovery: One works as hard
as possible on a problem,lets it incubatein the subconscious, has an insight,and verifies it. Similarstories are
told concerning the chemist Kekulé's dreaming of a
snake biting its tail, and realizingthatbenzene must be
ring-like in structure,and of Archimedes (in the bath)
solving the problemof how to determinewhetherKing
Heron'scrownis puregold, withoutdamagingthe crown
itself. Withregardto method,whatPoincaréofferedis a
retrospectivereport.(Schoenfeld,2002, p. 438)
In otherwords, the researchcarriedout by the Gestaltists
was by and large retrospective and relied on mathematicians and scientists reportingon theirthinkingafterthe fact.
Numerouscritiquesare availableon the unreliabilityof this
researchmethod.However,the featuresof creativethinking
as proposed by the Gestaltists were meaning, insight and
structure. Any definition should include these elements
(whetherthey are 'measurable'or how they are identifiedis
a completely differentmatter).
If we move away from the domain of mathematicsto the
general literature on creativity, numerous definitions are
18
found. Craft (2003) used the term "life wide creativity"to
describe the numerouscontexts of day-to-daylife in which
the phenomenonof creativity manifests. Otherresearchers
have described creativity as a natural"survival"or "adaptive" response of humansin an ever-changingenvironment
(Gruber,1989; Ripple, 1989).
Craft (2003) points out that it is essential we distinguish
"everydaycreativity"such as improvisingon a recipe from
"extraordinary
creativity",which causes paradigmshifts in a
of
specific body knowledge. It is generally accepted that
works of "extraordinarycreativity"can be judged only by
experts within a specific domain of knowledge (Csikszentmihalyi, 1988, 2000; Craft, 2003). For instance Andrew
Wiles' proof of Fermât'sLast Theoremcould only be judged
by a handfulof mathematicianswithin a very specific subdomainof numbertheory.
Throughoutschool levels or even the beginning undergraduate level, I normally do not expect works of
extraordinarycreativity. However, I do think it is feasible
for studentsto offer new insights/solutionsto a mathematics
problem.Getting back to definitions, in psychology the literaturedefinescreativityas the abilityto produceunexpected
original work, which is useful and adaptive(Sternbergand
Lubart,2000). Otherdefinitionsusually impose the requirement of novelty, innovationor unusualnessof a responseto
a given problem(Torrance,1974).
Numerousconfluencetheoriesof creativitydefinecreativity
as a convergenceof knowledge,ability,thinkingstyle,motivational and environmentalvariables (Sternbergand Lubart,
1996,2000), an evolutionof domainspecificideasresultingin
a creativeoutcome(GruberandWallace,2000). Mostrecently,
PluckerandBeghetto(2004) offeredan empiricaldefinitionof
creativitybasedon a surveyandsynthesisof numerousempirical studiesin the field. Theydefinedcreativityas
the interplaybetween ability and process by which an
individual or group produces an outcome or product
that is both novel and useful as defined within some
social context, (p. 156)
Coulda synthesisof these numerousdefinitionsof creativity
lead to a working definition of mathematicalcreativity at
both the professional and school levels? Wouldone definition work for both levels? At the professional level
mathematicalcreativitycan be defined as:
1. the ability to produce original work that significantlyextendsthe body of knowledge(whichcould
also includesignificantsynthesesandextensionsof
known ideas)
2. opens up avenues of new questionsfor othermathematicians.[2]
To illustrate the first point, Wedderburn'stheorem that a
finite division ring is a field is one instanceof a unification
of apparentlyrandomfragmentsbecause the proof involves
algebra, complex analysis and number theory. Hewitt's
(1948) paper on rings of continuousfunctions led to unexplored possibilities and questions in the fields of analysis
and topology that sustained other mathematicians for
decades. This would be an example of the latter.
Most descriptionsof creativityalso includethe elementof
chance. Shouldthis be includedin the definition?Although
Poincaréattributedhis particularbreakthroughin Fuchsian
functions to chance, he did acknowledge that there was a
considerableamountof previousconscious effort, followed
by a periodof unconscious work. Hadamard(1945) argued
thateven if Poincaré's breakthroughwas a resultof chance
alone, chance alone was insufficient to explain the considerablebody of creativework creditedto Poincaréin almost
every areaof mathematics.
One question then is how does (psychological) chance
work? I conjecture that the mind throws out fragments
(ideas), which are products of past experience. Some of
these randomfragmentscan be juxtaposedand combinedin
a meaningfulway. For example, if one reads a complicated
proof consisting of a thousand steps, a thousand random
fragments may not be enough to construct a meaningful
proof. However,the mind chooses relevantfragmentsfrom
these random fragments and links them into something
meaningful(such as Wedderburn'sproof).
I agree with your relativistic stance on any definition of
creativity.It would be difficult(butnot impossible)for a student(atthe school levels) to meet the criteriain the proposed
definition.Instead,could we define mathematicalcreativity
at the school levels as:
1. the process that results in unusual (novel) and/or
insightful solution(s) to a given problemor analogous problems,and/or
2. the formulationof new questions and/orpossibilities that allow an old problemto be regardedfrom
a new angle (this bears resemblance to Kuhn's
ideas). (Sriraman,[2])
Peter This final definition that you offer is more palatable
than others I have seen; it has the relativistic quality that I
have been seeking. It also has an absolutist overtone with
regardsto "solution(s)".Perhapsthis is necessary. I don't
know,but I would like to explore it for a moment.
Almost all of the definitions make some reference to a
product,a solution, or an outcome. What is it that a product/solution/outcomegives us that makes it so important?I
suggest thatthe answerto this questionlies in Schoenfeld's
observation/criticism
presentedabove. Withouta productall
is
a
remains
that
story;a story withoutan ending.I thinkthat
the centralityof producthas to do with having a tangibleand
objective artifact by which to judge the art by which to
the
process. But, crejudge the process by which to study
about
is
it
about
ative process isn't
process. Our
product,
of
most
this
access
to
private processes does not
inability
withoutcreatingsomecreative
we
be
Can
fact.
this
change
thing? I don't think we can, but this does not make them
one and the same. The creative process is both inseparable
and unrecognizable within the creative product. So, the
questionremains- what is mathematicalcreativity?
Bharath:Perhapsthis very enterpriseof trying to define
this constructfalls withinyourdescriptionof a sieve - we are
engagedin the discourse,using wordsthatalludeto it, but are
failing to define it adequately.Given the reductionistpitfalls
of our enterprisehere, I have to resort to the words of the
practicalyet mysticalChineseTaoist,ChuangTzu (1968):
Heavenand earthwere bornat the same time I was, and
the ten thousand things are one with me. We have
alreadybecome one, so how can I say anything?But I
have just said thatwe are one, so how can I not be saying something?The one and what I said about it make
two, andtwo and the originalone makes three.If we go
on this way, then even the cleverest mathematician
can't tell where we'll end, much less an ordinaryman
. . . Betternot to move, but to let things be! (p. 43)
Notes
[1] Fourth Congress of the EuropeanSocietyfor Researchin Mathematics
Education,Sant Feliu de Guixols, Spain, 17-21 February,2005.
[2] An article entitled 'Are mathematicalgiftedness and mathematicalcreativitysynonyms?A theoreticalanalysisof constructs',by B. Sriramanis in
press with TheJournal of SecondaryGiftedEducation.
References
ChuangTzu (1968, Watson,B., translator)The complete works of Chuang
Tzu, New York,NY, ColumbiaUniversity.
Craft,A. (2003) 'The limits to creativity in education: dilemmas for the
educator',BritishJournal of EducationalStudies51(2), 113-127.
Csikszentmihalyi,M. (1988) 'Society, culture,and person:a systems view
of creativity', in Sternberg,R. (éd.), The nature of creativity: contemporary psychological perspectives, Cambridge, UK, Cambridge
UniversityPress, pp. 325-339.
Csikszentmihalyi, M. (2000) 'Implications of a systems perspective tor
the study of creativity', in Sternberg,R. (éd.), Handbookof creativity,
Cambridge,UK, CambridgeUniversityPress, pp. 313-338.
Ervynck,G. (1991) 'Mathematicalcreativity', in laii, v. tea.;, Aavancea
Mathematical Thinking,Dordrecht,The Netherlands, Kluwer Academic Publishers,pp. 42-53.
Gruber,H. (1989) 'The evolving systems approachto creative work , in
Wallace, D. and Gruber,H. (eds), Creativepeople at work: twelve cognitive case studies, Oxford,UK, OxfordUniversityPress.
Gruber,H. and Wallace, D. (2000) 'The case study method and evolving
in
systems approachfor understandinguniquecreativepeople at work',
Sternberg,R. (éd.), Handbookof creativity,Cambridge,UK, Cambridge
UniversityPress, pp. 93-115.
Hadamard,J. (1945) Essay on the psychology of invention in the mathematicalfield, Princeton,NJ, PrincetonUniversityPress.
Hewitt,E. (1948) 'Rings of real-valuedcontinuousfunctions', Transactions
of the AmericanMathematicalSociety 64, 45-99.
Liljedahl, P. (2004) TheAHA! experience: mathematicalcontexts, pedaFraser
gogical implications, Unpublisheddoctoral dissertation,Simon
Canada.
British
Columbia,
Burnaby,
University,
ana
Polya, G. (1965) Mathematicaldiscovery: on understanding,learning
teachingproblemsolving (vol. 2), New York,NY, Wiley.
Plucker, J. and Beghetto, R. (2004) 'Why creativity is domain gênerai,
why it looks domain specific, and why the distinctiondoes not matter',
in Sternberg,R., Grigorenko, E. and Singer, J. (eds), Creativity:from
Assopotential to realization,Washington,DC, AmericanPsychological
ciation, pp. 153-168.
Ripple, R. (1989) 'Ordinarycreativity', Contemporarytducatwnai Psychology 14, 189-202.
Schoenfeld, A. (2002) 'Researchmethods m (mathematics)education , in
English, L. (éd.), Handbookof international research in mathematics
education,Mahwah,NJ, LawrenceErlbaumAssociates, pp. 435-487.
Sriraman,B. (2004) 'The characteristicsof mathematicalcreativity , me
MathematicsEducator14(1), 19-34.
Sternberg.R. and Lubart,T. (2000) 'The concept ot creativity: prospects
and paradigms', in Sternberg,R. (éd.), Handbook of creativity, Cambridge, UK, CambridgeUniversityPress, pp. 93-115.
Sternberg, R. and Lubart,T. (1996) 'Investing in creativity , American
Psychologist 51, 677-688.
Torrance,E. (1974) Torrancetests of creative thinking: norms-technical
manual,Lexington,MA, Ginn.
Wallas,G. (1926) Theart of thought,New York,NY, HarcourtBrace.
Wiles, A. (1997, NOVA, 1997, updated2000) Theproof, Aired on PBS on
October 28, 1997, http://www.pbs. org/wgbh/nova/proof/wiles.html,
accessed January5th, 2003.
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