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Creativity: The Essence of Mathematics

Eric L. Mann Purdue University

for the gifted mathematics student, early mastery of concepts and skills in the math-ematics curriculum usually results in getting more of the same work and/or mov-ing through the curriculum at a faster pace. testing, grades, and pacing overshadow the essential role of creativity involved in doing mathematics. talent development requires creative applications in the exploration of mathematics problems. traditional teaching methods involving demonstration and practice using closed problems with predetermined answers insufficiently prepare students in mathematics. Students leave school with adequate computational skills but lack the ability to apply these skills in meaningful ways. teaching mathematics without providing for creativity denies all students, especially gifted and talented students, the opportunity to appreciate the beauty of mathematics and fails to provide the gifted student an opportunity to fully develop his or her talents. in this article, a review of literature defines mathematical creativity, develops an understanding of the creative student of mathematics, and dis-cusses the issues and implications for the teaching of mathematics.

“Themovingpowerofmathematicalinventionisnotreasoningbutimagination.”—AugustusdeMorgan(1866,p.132)

Background

In1980,theNationalCouncilofTeachersofMathematics(NCTM)identifiedgiftedstudentsofmathematicsasthemostneglectedseg-mentofstudentschallengedtoreachtheirfullpotential.In1995,theNCTMtaskforceonthemathematicallypromisingfoundlittlehadchangedinthesubsequent15years(Sheffield,Bennett,Beriozábal,DeArmond,&Wertheimer,1999).Thedefinitionofmathematicalgiftednessvariesdependingontheidentificationtoolsusedandtheprogramoffered.Regardlessofthedefinitionused,findingstudents

EricMannisVisitingAssistantProfessorintheDepartmentofCurriculumandInstructionatPurdueUniversity.

Journal for the Education of the Gifted.Vol.30,No.2,2006,pp.236–260.Copyright©2006PrufrockPressInc.,http://www.prufrock.com

Essence of Mathematics 237

withmathematicalgiftednessisachallengeforbotheducatorsandsociety.Oftengiftednessinmathematicsisidentifiedthroughclass-roomperformance,testscores,andrecommendations.Yet,researchsuggeststhatahighlevelofachievementinschoolmathematicsisnotanecessary ingredient forhigh levelsofaccomplishment inmath-ematics(Hong&Aqui,2004;Mayer&Hegarty,1996;Pehkonen,1997;Sternberg,1996).Thisapparentdetachmentbetweenschoolmathematicsandmathematicalaccomplishmentsindicatesthatsometalentedstudentsareoverlookedbycurrentpracticesinschool.

Polya (1962) defined mathematical knowledge as informationand know-how. Of the two, he regarded know-how as the moreimportant, defining it as the ability to solve problems requiringindependence,judgment,originality,andcreativity.Agiftedstudentof mathematics possesses all of these characteristics and needs theopportunitytousethemwhensolvingchallengingproblems.Callsfor renewed emphasis on mathematics education accompany eachnew round of published test results, but often such efforts stressremediationoradditionalpracticeratherthanthedevelopmentofa mathematical frame of mind. Frequently, all students, includingthosewhometorexceededthetestgoals,simplyreceivemoreofthesamemethodsof instructionthatyieldedtheresultsunderexami-nation. If mathematical talent is to be discovered and developed,changesinclassroompracticesandcurricularmaterialsarenecessary.These changes will only be effective if creativity in mathematics isallowedtobepartoftheeducationalexperience.

The visionary classrooms described by leaders in the NCTMenablestudentsto

confidentlyengageincomplexmathematicaltasks...drawonknowledgefromawidevarietyofmathematicaltopics,sometimesapproachingthesameproblemfromdifferentmathematicalperspectivesorrepresentingthemathematicsindifferentwaysuntiltheyfindmethodsthatenablethemtomakeprogress.(NCTM,2000,p.3)

For many adults, this vision is unlike the math classrooms theyrememberfromtheiryouth.Timewasspentlearning from the mas-ter where the teacher demonstrated a method with examples, and

Journal for the Education of the Gifted238

thestudentspracticedwithsimilarproblems(Pehkonen,1997).Forthese adults, the concept of mathematics is of “a digestive processrather than a creative one” (Dreyfus & Eisenberg, 1996, p. 258).However,mathematicsisnotafixedbodyofknowledgetobemas-teredbutratherafluiddomain,theessenceofwhichisthecreativeapplicationsofmathematicalknowledgeinthesolvingofproblems(Poincaré,1913;Whitcombe,1988).

Defining and Measuring Mathematical Creativity

Anexaminationoftheresearchthathasattemptedtodefinemath-ematical creativity found that the lack of an accepted definitionfor mathematical creativity has hindered research efforts (Ford &Harris 1992; Treffinger, Renzulli, & Feldhusen, 1971). Treffinger,Young,Shelby,andShepardson(2002)acknowledgedthattherearenumerousways toexpresscreativityand identifiedmore than100contemporary definitions. Runco (1993) described creativity as amultifacetedconstruct involving“divergentandconvergentthink-ing, problem finding and problem solving, self-expression, intrin-sicmotivation,aquestioningattitude,andself-confidence”(p. ix).Haylock(1987)summarizedmanyoftheattemptstodefinecreativ-ity.Oneview“includestheabilitytoseenewrelationshipsbetweentechniquesandareasofapplicationandtomakeassociationsbetweenpossiblyunrelatedideas”(Tammadge,ascitedinHaylock,1987,p.60). The Russian psychologist Krutetskii characterized creativityinthecontextofproblemformation(problemfinding),invention,independence, and originality (Haylock, 1987; Krutetskii, 1976).Others have applied the concepts of fluency, flexibility, and origi-nality to mathematics (Haylock, 1997; Jensen, 1973; Kim, Cho,&Ahn,2003;Tuli,1980).Inadditiontotheseconcepts,Holland(ascitedinImai,2000)addedelaboration(extendingorimprovingmethods)andsensitivity(constructivecriticismofstandardmeth-ods).Torrance(1966)offeredthefollowingdefinitionofcreativity:

Creativityisaprocessofbecomingsensitivetoproblems,deficiencies,gapsinknowledge,missingelements,dishar-monies,andsoon;identifyingthedifficult;searchingforsolutions,makingguessesorformulatinghypothesesabout


thedeficiencies;testingandre-testingthesehypothesesandpossiblymodifyingandre-testingthem;andfinallycommu-nicatingtheresults.(p.8)

Indevelopinganoperationaldefinitionofmathematicalcreativity,Singh(1988)appliedTorrance’sdefinitionofcreativitytothefor-mulationofcauseandeffecthypothesesinmathematicalsituations.

Balka (1974) introduced criteria for measuring mathematicalcreativeability.Headdressedbothconvergentthinking,character-izedbydeterminingpatternsandbreakingfromestablishedmind-sets, and divergent thinking, defined as formulating mathematicalhypotheses,evaluatingunusualmathematicalideas,sensingwhatismissingfromaproblem,andsplittinggeneralproblemsintospecificsubproblems.InreviewingBalka’scriteria,breakingfromestablishedmindsetswasadefiningfeatureintheeffortsofotherstounderstandthecreativemathematician.Haylock(1997)andKrutetskii(1976)believed that overcoming fixations was necessary for creativity toemerge.Both,likeBalka,focusedonthebreakingofamentalsetthatplaceslimitsontheproblemsolver’screativity.Limitsarealsoestab-lishedwhencreativityandsystematicapplicationsareconfused.Inan earlier work, Haylock (1985) discussed the difference betweencreativity and being systematic in mathematical problem solving.By applying learned strategies, a student can systematically applymultiplemethodstosolveaproblembutneverdiverge intoacre-ativestrategy,neverexploringareasoutsidetheindividual’sknowncontent-universe. To encourage the development of mathematicalcreativity,educatorsneedtoenablecreativeexplorationandrewardstudentswhoseektoexpandtheircontent-universe.

The Essence of Mathematics

Theessenceofmathematicsisthinkingcreatively,notsimplyarrivingattherightanswer(Dreyfus&Eisenberg,1966;Ginsburg,1996).Inseekingtofacilitatethedevelopmentoftalentedyoungmathemati-cians,neglectingtorecognizecreativitymaydrivethecreativelytal-entedundergroundor,worseyet, cause themtogiveup the studyofmathematicsaltogether.HongandAqui(2004)studiedthedif-ferences between academically gifted students who achieved high


grades in school math and the creatively talented in mathematics,students with a high interest, who were active and accomplishedinmathbutnotnecessarilyhighachievinginschoolmath.Astheywereexaminingdifferences,theirstudydidnotincludestudentswithstrengthsinbothareas.HongandAquifoundsignificantdifferencesin cognitive strategies used by the two groups with the creativelytalentedbeingmorecognitively resourceful.Neithergroupof stu-dentsshouldbeneglected,yetChing(1997),asupervisorofteach-ingpractices,foundhiddentalenttoberarelyidentifiedbytypicalclassroompractices.Traditionalteststoidentifythemathematicallygifted,suchasthecommerciallyavailableachievementtestsorstateassessments,donotidentifyormeasurecreativity(Kimetal.,2003)butoftenrewardaccuracyandspeed.Thesetests identifystudentswhodowellinschoolmathematicsandarecomputationallyfluent(HongandAqui’sacademicallytalented),butneglectthecreativelytalentedinmathematics.BrodyandMills(2005)reportthatthetal-entsearchmodelsofidentificationhavebeenproventobevalidasapredictorofacademicachievement,aswellasachievementlaterinlifeforthosestudentswhoqualify.Servicesandopportunitiespro-videdtothesestudentsmatchtheidentificationsystemusedandareappropriatefortheirneeds.Thesestudentstypicallyhavesupportivefamiliesandadvantagedhomes(Brody&Mills)andareinvitedtoparticipateinthesearchbasedontheiracademicperformance.HongandAqui’sresearchsupportstheneedtolookdeeperformathemati-caltalent.Encouragingmathematicalcreativityinadditiontocom-putationalfluencyisessentialforchildrentohaveaproductiveandenjoyablejourneywhiledevelopingadeepconceptualunderstand-ingofmathematics.Forthedevelopmentofthemathematicaltalent,creativityisessential.

Mathematical creativity is difficult to develop if one is lim-ited to rule-based applications without recognizing the essence ofthe problem to be solved. Köhler (1997) discussed an experimentbyHollenstein inwhichonegroupofchildrenworkedonamathexercisepresentedinthetraditionalmethod.Theseproblemswerecompleteorclosedinthattheywereconstructedsothatasinglecor-rectanswerexisted(Shimada,1997).Asecondgroupwasgiventheconditions on which the first group’s exercise were based and was


asked to develop and answer problems that could be solved usingcalculations. The open-ended nature of the task given to the sec-ond group did not limit them to a set number of problems. Thisgroupcreatedandansweredmorequestionsthanwereposedtothefirstgroup,calculatedmoreaccurately,andarrivedatmorecorrectresults. Researchers at Japan’s National Institute for EducationalResearch conducted a 6-year research study that evaluated higherordermathematicalthinkingusingopen-endedproblems(problemswithmultiplecorrectanswers).Inaround-tablereviewofthestudy,Sugiyama,fromTokyoGakugeiUniversity,affirmedthisapproachasameanstoallowstudentstoexperiencethefirststagesofmath-ematicalcreativity(Becker&Shimada,1997).

Doing What Mathematicians Do

Doing what mathematicians do as a means of developing math-ematicaltalent(asopposedtoreplicationandpractice)isconsistentwiththeworkatTheNationalResearchCenterfortheGiftedandTalented(Reis,Gentry,&Maxfield,1998;Renzulli,1997;Renzulli,Gentry,&Reis,2003,2004).Emphasisisplacedoncreatingauthen-tic learning situations where students are thinking, feeling, anddoingwhatpracticingprofessionalsdo(Renzulli,Leppien&Hays,2000).Thefundamentalnatureofsuchauthentichigh-endlearningcreatesanenvironmentinwhichstudentsapplyrelevantknowledgeandskillstothesolvingofrealproblems(Renzullietal.,2004).

Solutionstorealproblemsalsoentailsproblemfinding,aswellasproblemsolving.Kilpatrick(1987)describedproblemformula-tionasaneglectedbutessentialmeansofmathematicalinstruction.Real-worldproblemsarenotpresentedinatextbookorbyateacher.Theyareill-formedandrequireonetoemployavarietyofmethodsandskills.Inadditiontoequationstosolveandproblemsdesignedtoconvergeononerightanswer,studentsneedtheopportunitytodesignandanswertheirownproblems.InhisCreativeMathematicalAbility Test, Balka (1974) provided the participants with math-ematical situations from which they were to develop problems.Mathematicalcreativitywasmeasuredbytheflexibility,fluency,andoriginalityoftheproblemstheparticipantsconstructed.Byworking


withthesetypesofmathematicalsituations,studentscanbeencour-aged to use their knowledge flexibly in new applications. Flexibleapplicationsofknowledgerequiremorethanknowledgeofacquain-tance,orasimpleknowledge of,whichistheentrylevelofknowledgeidentifiedbyWilliamJames(ascitedinRenzullietal.,2000;Taylor&Wozniak,1996). Jamesconsideredconceptualknowledgetobeahigher levelofknowledge.Hereferredtothis levelasknowledge about,awayofknowingthatisbasedonacontinuityofexperiences(Taylor&Wozniak).Conceptualknowledge lieswithinauthenticmathematical experiences provided to students rather than simplereplicationofdemonstratedmethods.

Factualandproceduralknowledgeisnecessarytodevelopprofi-ciencyinmathematics,butresearchsuggeststhatconceptualunder-standingisequallyasimportant(Bransford,Brown,&CockingascitedinNCTM,2000,p.21;Schoenfeld,1988).Davis,Maher,andNoddings(1990)listedfourtypesofmathematicalexperiencesthatchildrenneedintheclassroom.Thefirstexperienceischaracterizedby teacher demonstration, followed by student drill and practice.Thistypeofexperience isthemostprevalent intheclassroombuthaslimitedusefulnessindevelopingdeepmathematicalunderstand-ing.Inthereport,all Students reaching the top,theNationalStudyGroupfortheAffirmativeDevelopmentofAcademicAbilityfoundthatdrillandpracticemayactuallybeworkingagainstthetransferof learning to applications not replicated by the drills (LearningPointAssociates,2004).Schoenfeldreportedsimilarfindingsinhisresearchinwhichhefoundthatstudentsfailtoconnectdrill-and-practicelearningwithreal-worldproblems.

To allow for a transfer of learning, students need more thandrillandpractice; theyneedtounderstandthemathematicalcon-ceptsbeyondthepracticeexercises(Davisetal.,1990).BassokandHolyoak (1989) conducted an experiment with 12 high-abilityninth-gradestudentsinanacceleratedscientificprogramtoinvesti-gatethetransferoflearningbetweenalgebraandphysicsproblemswiththesameunderlyingstructure.Theirresultsshowedthat90%ofthestudentswhoweretrainedtosolvethephysicsproblemswereunabletomakethetransferoflearningtosolvesimilarbutunfamil-iaralgebraproblems.However,72%ofthealgebrastudentssuccess-


fullytransferredtheirunderstandingofthemathematicalconceptstosolvethephysicsproblems.Thedifferenceintransferabilitymayhavebeencompromisedbythecontextoftheproblem-solvingsitua-tionsinwhichthestudentsweretrained(LearningPointAssociates,2004),withthealgebrastudentsbeingabletorecognizesimilaritiesinproblemstructuremorereadily.Davisetal.(1990)alsodescribedgeneralreadiness-buildingexperiencesdesignedtopreparestudentstorecognizemathematicalsituations.Open-endedexperiencesalsoprovide students opportunities to demonstrate their conceptualunderstanding.Bothof thesetypesofexperiences lendthemselvesto the transfer of learning from classroom to real-word situations.Learningmathematics,therefore,involvesmuchmorethanmemo-rizingarithmeticalfactsandmasteringcomputationalalgorithms;itentails incorporatingexperiencesandconceptualunderstandingtosolvingauthenticmathematicalproblems

The Role of Risk Taking in Mathematics

The new open-ended assessments used by many state Departmentof Education officials often place little value on creative solutions.Problems with test scoring in Connecticut’s 2003–2004 masterytestsillustratesomeoftheissueswherestrictguidelinesfocusingonaccuracyarethenorm.“Thereisanarttoscoring...thereissubjec-tivity...ourworkistoremoveasmuchofthatvariableaspossible”according to Hall, CTB/McGraw-Hill’s director of hand-scoring(Frahm, 2004, p. A1). While accuracy is important, strict empha-sisonaccuracywhenassessingachild’sconceptualunderstandingofmathematicsdiscouragestherisktakerwhoappliesherorhisknowl-edgeandcreativitytodeveloporiginalapplicationsinsolvingaprob-lem(Haylock,1985).Suchanindividualwouldbeinthecompanyof Poincaré, Hadamard, and Einstein, all eminent scientists andmathematicianswhoconfessedtohavingproblemswithcalculations(Hadamard,1945).

Mayer and Hegarty (1996) reported converging evidence thatstudents leave high school with adequate skills to accurately carryout arithmetic and algebraic procedures but inadequate problem-


solvingskillstounderstandthemeaningofwordproblems.Agoodmathematicalmindiscapableofflexiblethoughtandcanmanipu-lateandinvestigateaproblemfrommanydifferentaspects(Dreyfus&Eisenberg,1996).Proceduralskillswithoutthenecessaryhigherordermathematicalthinkingskills,however,areoflimiteduseinoursociety.Thereis littleuseforindividualstrainedtosolveproblemsmechanically as technology is rapidly replacing tedious computa-tional tasks (Köhler,1997;Sternberg,1996).Oftenthedifferencebetweentheerrorsmadebyeminentmathematiciansandstudentsofmathematicsisafunctionoftheirinsightintoandappreciationofmathematics,nottheircomputationalskills(Hadamard,1945).

With the increased emphasis on accountability from the NoChild Left Behind Act of 2001, teachers are under even morepressure to teach to the test rather than to work toward develop-ing in their students a conceptual understanding of mathematics.Encouragingstudentstotakerisksandlookforcreativeapplicationsreintroducesvariability inscoringthatassessmentteamsarework-ing to eliminate. Discouraging risk taking limits student exposuretogenuinemathematicalactivityanddampensthedevelopmentofmathematicalcreativity(Silver,1997).Forsubstantialandpermanentprogressinachild’sunderstandingofmathematics,anappreciationof “the difficult-beautiful-rewarding-creative view of mathematics”(Whitcombe,1988,p.14)mustbedeveloped.However,ratherthandevelopinganappreciationformathematicsbyfocusingonqualitiesofmathematicalgiftedness,teacherswhoonlyemphasizealgorithms,speed,andaccuracyprovidethecreativestudentnegativereinforce-ment.Thus,manytalentedstudentsdonotenvisionthemselvesasfuturemathematiciansorinotherprofessionsthatrequireastrongfoundation in mathematics (Usiskin, 1999). Failing to encour-agecreativityinthemathematicsclassroomdeniesallchildrentheopportunitytofullydeveloptheirmathematicalunderstanding.Forthemathematically talented, lackofcreativitymaydelay,orworseyet,preventtherealizationoftheirpotentialtocontributetonewunderstandingsoftheworldaroundusthroughtheadvancementofmathematicaltheory.


The Enjoyment Factor

In a recent undergraduate course on the teaching of mathemat-ics, futureelementary school teacherswereasked todescribe theirmostmemorablechildhoodexperienceinschoolmathematics.Theoverwhelmingmajoritydescribedanunpleasantexperience(Mann,2003).“Wehaveknownforsomeyearsnow...thatmostchildren’smathematicaljourneysareinvainbecausetheyneverarriveanywhere,andwhatisperhapsworseisthattheydonotevenenjoythejourney”(Whitcombe,1988,p.14).Itisdifficulttodevelopanunderstand-ingofmathematicsiftheefforttoinspirethisknowledgeisuninter-esting.Confirmingthis,Csikszentmihalyi,Rathunde,andWhalen(1993)foundthatenjoymentiscentraltocapturingachild’sinterestanddevelopinghisorhertalent.UsingAmabile’s(1989)ingredientsofcreativity,Starko(2001)alsodiscussedtheroleofinterestinintrin-sicmotivationforthedevelopmentofcreativity.Thegreaterachild’sintrinsicmotivation, thegreater the likelihoodofcreativeapplica-tionsanddiscoveries.Yet,intrinsicmotivationishighlydependentonsocialenvironment(Amabile,1989)andthesocialenvironmentofaclassroomisdependentontheteacher.Whenteachersdonotlookbeyondthewronganswer,theyconveythebeliefthatmathisdividedintorightandwronganswers(Balka,1974;Ginsburg,1996)andmayrejectcreativeapplications,fosteringaclassroomenviron-mentthatdiscouragesdevelopingcreativity.

Time and Experience

Creativityneedstimetodevelopandthrivesonexperience.Drawingfromcontemporaryresearch,Silver(1997)suggested,“creativityisclosely related to deep, flexible knowledge in content domains; isoftenassociatedwithlongperiodsofworkandreflectionratherthanrapid, exceptional insight; and is susceptible to instructional andexperiential influences” (p. 75). Poincaré’s (1913) essay on mathe-maticalcreationalsodiscussedtheneedforreflection.Hedescribedhisdiscoveryofthesolutiontoaproblemonwhichhehadworkedforaconsiderableamountoftimearrivingasasuddenillumination


ashesteppedontoabusonageologicexcursion.Thisilluminationwas“amanifest signof long,unconsciouspriorwork . . .which isonlyfruitful,ifitisontheonehandprecededandontheotherhandfollowed by a period of conscious work” (p. 389). This period ofincubationappearstobeanessentialaspectofcreativityrequiringinquiry-oriented, creativity-enriched mathematics curriculum andinstruction(Silver,1997).Whitcombe(1988)describedanimpov-erished mathematics experience as one in which instruction onlyfocusesonutilitarianaspectsofmathematicsandiswithoutappro-priateinterest-stimulatingmaterialandtimetoreflectneededbythestudent.Suchexperiencesdenycreativitythetimeandopportunitiesneededtodevelop.

HongandAqui’s(2004)divisionofmathematicaltalentintotheacademicallygiftedandcreativelytalentediscritical intheconsid-erationoftalentdevelopment.Theacademicallygiftedstudentmayexcelintheclassroombydemonstratinghighachievement,or“school-housegiftedness,”thatisvaluedintraditionaleducationalsettings.Thesestudents’abilitiesremainrelativelystableovertime(Renzulli,1998).Thoseacademicallygiftedinmathematicsareabletoacquiretheskillsandmethodologiestaughtoftenatamuchmorerapidpacethan for less able students and perform well on standardized test-ing.Theacademicallygiftedusuallydemonstratetheirmasteryoftheutilitarian aspects of mathematics, but neither speed nor accuracyin computation or the analytical ability to apply known strategiestoidentifiedproblemsaremeasuresofcreativemathematicaltalent.Hadamard(1945)describedindividualshelabeled“numericalcal-culators” as “prodigious calculators—frequently quite uneducatedmen—whocanveryrapidlymakeverycomplicatednumericalcalcu-lations...suchtalentis,inreality,distinctfrommathematicalabil-ity”(p.58).Thus,itispossibletobeconsideredacademicallygiftedinmathematicsbutlackcreativemathematicaltalent.

While speedof informationprocessing is important in testingsituationsinwhichstudents’mathematicalthinkingisassessedusingstandardizedtests,itislessimportantwhenamathematicianspendsmonthsorevenyearstryingtoworkoutaproof(Sternberg,1996).Although current tests of number or numerical facility emphasizespeedwithstress imposedbyseveretimelimitsandaccountability


ontheaccuracyofthesolutions(Carroll,1996),thenextgenerationofmathematiciansmustbeshownthe“wellspringsofmathematics;creativity,imagination,andanappreciationofthebeautyofthesub-ject”(Whitcombe,1988,p.14).Inananalysisofcognitiveabilitytheory and the supporting psychological tests and factor analysis,Carrollnotedthatdespitesixtosevendecadesofwork,therelation-shipsbetweenthediscreteabilitiesmeasuredbypsychometrictestsand performance in mathematics remains unclear. Restricting thesearchformathematicaltalenttotheacademicallygiftedwhoper-formwellontimedstandards-basedassessmentsdeniesopportuni-tiestothecreativelytalentedthatgoundiscoveredbecauseoflowerlevelsofclassroomachievementorlimitededucationalexperiences.

Understanding the Creative Student of Mathematics

NCTM’staskforceonthemathematicallypromising(Sheffieldetal.,1999)characterizedourpromisingyoungmathematicsstudentsinlightoftheirability,motivation,belief(self-efficacy),andopportu-nity/experience,allconsideredvariablesthatmustbemaximizedinordertofullydevelopastudent’smathematicaltalent.Davis(1969)considereddevelopingcreativityinstudentsofmathematicsintermsofthreemajorparameters:attitudes,abilities,andtechniques(meth-ods of preparing and manipulating information). While 26 yearsseparate these efforts, they offer similar recommendations. Skillswiththetechniquesofthedisciplinedeveloponlythroughoppor-tunityandbelief inone’sability.Renzulli’s(1978,1998)modelofgiftednessdefinesthreelearnerattributes(above-averageability,taskcommitment,andcreativity)inthreeoverlappingringssignifyinganinterdependenceof thesequalities toproducegiftedness.Of thesethreequalities,therearetwo(taskcommitmentandabove-averageability) thatmirrorDavis’parameters.Overlaying the twomodelsyields a conceptual framework in which mathematical creativitycanbeconsidered.Thechildpossesseshisorherinnateabilitythatremainsdormantifnotdevelopedwiththeappropriatechallengesandexperience.Teachersandparentsmusthelpdeveloptaskcom-mitmentbycreatingopportunities forpurposefulandmeaningful


experiences, by fostering an understanding of techniques throughinstruction and modeling, and by establishing a creative environ-mentthatencouragesrisktakingandcuriosity.

If any of these elements are missing, creativity in mathemat-ics may not develop. Classrooms in which teachers do not acceptalternativeviewsandinwhichtheroteapplicationofskillsisvaluedwillprovidetheworldwithstudentsthatonlyhavethecapabilitytoapplytechniques inknownsituations.Thesestudentswill strugglewhentheyencounterunknownsituationsinwhichoriginality,cre-ativity,andproblemsolvingarenecessary.

Underdeveloped Talent

Onemaywonderhowmanypotentialcreativemathematiciansarelost when weak analytical skills prevented them from progressingthrough the levels of mathematics offered in our educational sys-tem.Limitinguseofcreativityintheclassroomreducesmathematicstoasetofskillstomasterandrulestomemorize.Doingsocausesmany children’s natural curiosity and enthusiasm for mathematicsto disappear as they get older, creating a tremendous problem forthemathematicseducatorswhoaretryingtoinstilltheseveryquali-ties(Meissner,2000).Sternberg(1996)referredtocommentsfrommathematicianswhosuggestthat:

...performanceinmathematicscourses,uptothecollegeandevenearlygraduatelevelsdoesnoteffectivelypredictwhowillsucceedasamathematician.Thepredictionfailureisduetothefactthatinmath,asinmostotherfields,onecangetawaywithgoodanalyticbutweakcreativethinkinguntilonereachesthehighestlevelsofeducation.....However,itiscreativemathematicalthinkingthatisthemostimportant...(p.313)

Itisimportantthatteachersworktodevelopmathematicalcreativityasthechildbeginshisorhereducationaljourney.Haylock(1997)suggests that the pupil’s mathematical experience and techniqueslimittheircreativedevelopment.Yet,Hashimoto(1997)foundthat,


ingeneral,mostclassroomteachersthinkthereisonlyonecorrectanswerandonlyonecorrectmethodtosolveamathematicsprob-lem.Iftaughtthatthereisonlyonerightanswer,onlyonecorrectmethod, a student’s concept of mathematics as an application ofmathematictechniquesisreinforced.Köhler(1997)illustratedthispoint in a discussion with an elementary classroom teacher abouta studentwhohadarrivedat thecorrectanswer inanunexpectedway:

Whilegoingthroughtheclassroom,thatpupilaskedme[theteacher]whetherornothissolutionwascorrect.i was forced[italicsadded]toadmitthatitwas.Thatiswhatyougetwhen you don’t tell the pupils exactly what to do[italicsadded]....Theteachernowreproacheshimselffornot hav-ing prevented this solution[italicsadded].Heisobviouslyinfluencedbyaninsufficientunderstandingofwhatismath-ematics,bytheimageofschoolasaninstitutionforstuffingofbrains...(p.88)

Teachersoftenencouragestudentstoexplore,question,interpret,andemploycreativityintheirstudiesofotherdisciplinessuchaslan-guagearts,science,orthesocialsciences;yet,thisexampleillustratesmanyteachersfocusontheuseofrules-basedinstructionformath-ematics.Iftheinstructionfocusesonrotememorizationratherthanmeaning,thenthestudentwillcorrectlylearnhowtofollowaproce-dure,andwillviewtheprocedureasasymbol-pushingoperationthatobeysarbitraryconstraints.Withoutaconceptualunderstandingoftheunderlyingconceptsandprinciplesnecessaryforcreativeappli-cations,studentsmayovergeneralizefrom“bitsandpieces”ofpriorknowledgeandapplyprocedurescorrectlyininappropriateproblemsituationsarrivingatcomputationallycorrect“wrong”answers(Ben-Zeev,1996).Studentsmaygaincomputationalskills,yethavelittleorincompleteunderstandingsoftheapplicationsforwhichtheseareappropriate.Pehkonen(1997)suggestedthattheconstantemphasisonsequentialrulesandalgorithmsmaypreventthedevelopmentofcreativity, problem-solving skills, and spatial ability. The develop-mentofmathematicalcreativitydeservesthesameemphasisofferedtothecreativedevelopmentinotherdisciplines.


Incorporating Creativity in the Teaching of Mathematics

Teachers must be prepared to appreciate the beauty and creativityofmathematics.Theymustexploretheworldofmathematicsbeforetheycanhelptheirstudentsdiscoverit.Itiseasyforteacherstoforgetthevalueofthestruggletheymayhaveencounteredastheylearnedmathematicsaschildrenandfallintoateachingpracticethatinvolvesdemonstrationbyteacherandreplicationbythestudent(Pehkonen,1997).Yet, forstudentstoexperiencethetrueworkofmathemati-cians,thestruggleisnecessaryastheydiscoverandapplymathematicaltheorytosolveproblems.Poincaré(1913)believedthattruemathe-maticianshadanintuitivesensethatguidedthemincreativeapplica-tionsofmathematics.Jensen(1973)referencedBrunerindescribingacreativeactasonethatproducessurpriseonlyrecognizablebythoseprepared to see it and explained “creativity demands readiness andunderstandingoftheproblembothbytheproducer[thestudent]andbythosewhowoulduseandappreciatethecreativeact[theteacher]”(p.22).ItisthereforenecessaryforteachersofmathematicstohavePoincaré’sappreciationformathematicsandapreparedmathemati-calmindtohelpstudentsdevelopanunderstandingofthebeautyofmathematics.Itisalsonecessaryforteacherstoencouragethedevel-opmentofmathematicalcreativity.

Thereissignificantpowerinlearningconceptually.Thispowercomesfromtheabilitytorecombineandrelateconceptsinavarietyof settings, as opposed to factual learning, which has applicationswithinthecircumstancesastheyexist(Skemp,1987).Mathematicsisapowerful tool thatcanbeusedatvarying levelsofcomplexityinalmosteveryoccupation.Yet,manystudentsleaveschooldislik-ingmathematicsandwiththebeliefthattheyjustcannotdomath.Whitcombe’s(1988)modelofthemathematicalmindisbasedonhisbeliefthatmathematicalmindsfunctionefficientlywhenthreeaspectsofmathematicsareinvolved:algorithms(logical),creativity(intuitive),andbeauty(speculative).Intuitionandspeculationfunc-tionataconceptuallevel,whilealgorithmsarearule-basedapplica-tionofmathematics.Yet,“algorithmsconstitutethemajorityofthemathematicaldietofmanyofourchildren, . . .[theyare]theleastimportantasmachinescandoitbetterandfaster,...[they]arethe


least importantbecausetheyareboring,”(Whitcombe,p.15)andtheyaretheleastimportantastheyofferthestudentnosenseofthestructureofmathematics.Accuracyisimportant,butaccuracywith-outunderstandingisofminimaluse.Therightanswertothewrongproblemisaspotentiallyharmfulonaconstructionsiteorinahos-pitalwardasthewronganswertotherightproblem.

Crosswhite(1987)definedtheprocessof“bottomlineproblemsolving”and“bottomlineteaching”astheunwrittengamestudentsandteachersplay.Thestudentspatientlywaitthroughtheteacher’slessonpresentationknowingthat,intheend,thepreferredmethodofsolvingtheproblemwillbepresented.Thestudentthenbecomesaccountableonlyforareplicationoftheprocesswithsimilarprob-lems.Creativityplaysnoroleinthiskindofteachingandlearning.Tofullyimplementthechangesneededininstruction,teacheredu-cationprogramsmustchange.Thisisalongprocess,astheteachersofteachersmustalsomaketheshift.Forcurrentteachers,anunder-standingoftheroleofcreativityinmathematicsisanimportantfirststep,butcurricularmaterials,classroomandadministrativesupport,andtrainingareallneededforprogresstocontinue.

Thedevelopmentofmathematicalcommunicationsskillsisnec-essaryforcreativitytoberecognized,appreciated,andshared.Theimpactonacreativestudentwhothinkssymbolicallybutisconsis-tentlyaskedtoexplaininwrittenororallanguagemaybesignificantbecausethoughtsandunderstandingmaybelostinthetranslation.Whendescribingacharacteristicofmentalactivityassociatedwithcreativemathematicalthought,SirFrancisGaltonsaid

Itisaseriousdrawbacktomeinwriting,andstillmoreinexplainingmyself,thatIdonotsoeasilythinkinwordsasotherwise.Itoftenhappensthatafterbeinghardatwork,andhavingarrivedatresultsthatareperfectlyclearandsat-isfactorytomyself,whenItrytoexpresstheminlanguageIfeelthatImustbeginbyputtingmyselfuponquiteanotherintellectualplane.Ihavetotranslatemythoughtsintoalan-guagethatdoesnotrunveryevenlywiththem.Ithereforewasteavastdealoftimeinseekingforappropriatewordsandphrases,andamconscious,whenrequiredonasudden,ofbeingoftenveryobscurethroughmereverbalmaladroitness,


andnotthroughwantofclearnessofperception.Thatisoneofthesmallannoyancesofmylife.(ascitedinHadamard,1945,p.69)

Teachers who require each step of the problem to be written out,eachanswerjustifiedbothinspokenorwrittenlanguage,aswellasinmathematicalsymbols,maynotunderstandtheprocessesusedbycreative,intuitivestudents.However,therearemanystoriesofmath-ematical discoveries lost or delayed because the mathematician’swork was not effectively communicated. After Riemann’s death in1866,abriefnotefoundinhispapersdealingwiththedistributionofprimenumbershasbecomethefocusofthecareersofmanymath-ematicians. Riemann’s note simply stated “These properties of ζ(s)arededucedfromanexpressionofitwhich,however,Ididnotsuc-ceedinsimplifyingenoughtopublishit”(ascitedinHadamard,p.118).Thelackofanyotherreferencestohisfindingsunderscorestheneedforeffectivecommunicationskills.Whiletheabilitytoexplain,justify,anddefendone’sworkisimportant,formostitisalearnedskill. In languagearts, childrenare taughthowtowritepersuasiveessays or creative short stories and how to evaluate the writing ofothers; a similar investment todevelopmathematical communica-tionskillsisnecessaryfortheexpressionofmathematics.

Teaching practices need to shift to a more balanced applica-tionofWhitcombe’s(1988)modelofthemathematicalmindthatrecognizescreativityandthebeautyofmathematics,aswellastherule-basedalgorithmsthatdominatemostmathematicsclassrooms.Ratherthanoveremphasizerules,algorithms,andconvergentthink-ing to produce a single right answer, instruction should center onmathematical thought. Dreyfus and Eisenberg (1996) found thatmathematicalthoughtismorethanabsorbingsomepieceofmath-ematicsorsolvingsomemathematicalproblem:“itiscloselyassoci-atedwithanassessmentofelegance”(p.255).Termssuchasbeautyand elegance are as difficult to define as creativity, yet each createsavisionofmathematicsasextendingbeyondalgorithms.“Inmath-ematics,factsarelessimportantthaninotherdomains;ontheotherhand, relationships between facts, relationships between relation-shipsandthusstructure,aremoreimportantthaninotherdomains”(Dreyfus&Eisenberg,p.265).Dienes(2004)comparedtheworkof


amathematicianwiththatofanartist.Bothlabortoconstructinaneffort tocommunicate theirunderstandingwithothers.Bothplaywithideas,combiningtheminvariouswaysandasdifferentstruc-tures until something emerges that the individual finds satisfying.Withouttheopportunityforcreativeplaywithaproblem,students’problem-solvingskillsarelimitedtotherecallofmethodscreatedbyothers(Bronsan&Fitzsimmons,2001).

Whenstudentsbegintoexplorethestructureofmathematics,theybegintoexplorethebeautyofthedomainanddevelopasenseofmathematics.Poincaré(1913)describedthismathematicalsenseasmathematicalintuitionortheabilitytoseethewholeandtofindharmony and relationships gained through study and experience.These experiences can lead to students’ mathematical growth. Thepupil’sinsightcanonlybefacilitatedbyachallengingproblemthatissufficientlydemanding,aswellassufficientlyaccessible.Theempha-sisinteachingmathematicsshiftsfromreplicationofdemonstratedmethods to allowing the student the right to make mistakes andexplorealternativeroutes,therebyopeningnewperspectives(Köhler,1997).Ratherthanclosedproblemswithasinglesolution,studentsshouldbeprovidedopen-endedproblemswitha rangeofalterna-tive-solutionmethods(Fouche,1993).Somemethodstosolvetheproblemsmaybetoosimple,somemaybeoutofreachofthechild,whilestillothersarewithinthechild’sgrasp.Encouragingachildtoreachbeyondthe familiarandprobedeeper intotherelationshipsandstructuresofaproblemistheessenceofteachingmathematicscreatively.

Summary

Achild’sgrowthinmathematicsinvolvesmorethanjustmasteringcomputational skills. Identification of mathematical talent usingonly speed and accuracy of computation would qualify hand-heldcalculatorstobecalledtalentedmathematicians.Mathematicaltal-entrequirescreativeapplicationsofmathematicsintheexplorationofproblems,notreplicationoftheworkofothers.Problemsolvingistheheartofgenuinemathematicalactivity,yetthesupplyofcur-


ricular materials designed to support a problem-solving approachtomathematicalinstructionissmallincomparisontothematerialsalignedwithaprocedural,mechanicalpointofview(Silver,1997).

IntheUnitedStates,thecombinedeffortsoftheNCTMandthe National Science Foundation have begun to yield appropriatecurricularmaterialstodevelopmorecreativetalentsinmathematics.Yet,thejourneyisonlybeginning.Methodsofassessingmathemati-cal creativity, teacher accountability, and greater emphasis withinteacher education programs on teaching for conceptual under-standingandpedagogicallyrelevantcontentknowledgeareneeded.Without such a shift, the education community perpetuates theaccepted practices of the childhood classroom experiences for ournextgenerationofteachers.Schoolboardmembersandadministra-torsneedtoencourageandsupportinnovativemethodsofteachingmathematics.Thiseffortneedstobemorethanapolicystatement;funding for materials and training, promoting mathematics as acreative endeavor within the community, providing specialists inmathematicsandgiftededucation,andsupportandencouragementfor classroom teachers are all needed. Classroom teachers shouldexaminetheirteachingpracticesandseekoutappropriatecurricularmaterialstodevelopmathematicalcreativity.Thechallengeistopro-videanenvironmentofpracticeandproblemsolvingthatstimulatescreativity, while avoiding the imposition of problem-solving heu-risticstrategies(Pehkonen,1997)thatwillenablethedevelopmentof mathematically talented students who can think creatively andintrospectively(Ginsburg,1996).

References

Amabile,T.M.(1989).Growing up creative: nurturing a lifetime of creativity.Buffalo,NY:C.E.F.Press.

Balka,D.S.(1974).Creativeability inmathematics.arithmetic teacher, 21,633–636.

Bassok,M.,&Holyoak,K.J.(1989).Interdomaintransferbetweenisomorphictopicsinalgebraandphysics.Journal of Experiment Psychology: learning, Memory, and cognition, 15(1),153–166.


Becker,J.P.,&Shimada,S.(Eds.).(1997).the open-ended approach: a new proposal for teaching mathematics.Reston,VA:NationalCouncilofTeachersofMathematics.

Ben-Zeev,T.(1996).Whenerroneousmathematicalthinkingisjustas“correct”:Theoxymoronofrationalerrors.InR.J.Sternberg&T.Ben-Zeev(Eds.),the nature of mathematical thinking(pp.55–80).Mahwah,NJ:LawrenceErlbaum.

Brody,L.E.,&Mills,C.J.(2005).Talentsearchresearch:Whathavewelearned?High ability Studies, 16,97–111.

Bronsan,P.A.,&Fitzsimmons,J.A.(2001).Creativity.InL.S.Grinstein&S.I.Lipsey(Eds.),Encyclopedia of mathematics edu-cation(pp.164–166).NewYork:RoutledgeFalmer.

Carroll,J.B.(1996).Mathematicalabilities:Someresultsfromfac-toranalysis.InR.J.Sternberg&T.Ben-Zeev(Eds.),the nature of mathematical thinking(pp.3–25).Mahwah,NJ:LawrenceErlbaum.

Ching,T.P.(1997).Anexperimenttodiscovermathematicaltal-entinaprimaryschoolinKampongAir.international reviews on Mathematical Education, 29(3),94–96.RetrievedMarch10,2003,fromhttp://www.fizkarlsruhe.de/fix/publications/zdm/adm97

Crosswhite,F.J.(1987).Cognitivescienceandmathematicseduca-tion:Amathematician’sperspective.InA.H.Schoenfeld(Ed.),cognitive science and mathematics education (pp.253–277).Mahwah,NJ:LawrenceErlbaum.

Csikszentmihalyi,M.,Rathunde,K.,&Whalen,S.(1993).talented teenagers: the roots of success and failure.NewYork:CambridgeUniversityPress.

Davis,G.A.(1969).Thinkingcreativelyinadolescence:Adiscus-sionofstrategy.InR.E.Grinder(Ed.),Studies in adolescence (pp.538–545).NewYork:Macmillan.

Davis,R.B.,Maher,C.A.,&Noddings,N.(1990).Suggestionsfortheimprovementofmathematicseducation.InR.B.Davis,C.A.Maher,&N.Noddings(Eds.),constructivist views on the teaching of mathematics. Journal for research in Mathematics Education Monograph no. 4 (pp. 187–191). Reston, VA:NationalCouncilofTeachersofMathematics.


deMorgan,A.(1866).SirW.R.Hamilton.Gentleman’s Magazine and Historical review, 1,128–134

Dienes,Z.P.(2004).Mathematics as an art form. an essay about the stages of mathematics learning in an artistic evaluation of mathe-matical activity.RetrievedSeptember1,2004,fromhttp://www.zoltandienes.com/Mathematics_as_an_art_form.pdf

Dreyfus,T.,&Eisenberg,T.(1996).Ondifferentfacetsofmath-ematicalthinking.InR.J.Sternberg&T.Ben-Zeev(Eds.),the nature of mathematical thinking(pp.253–284).Mahwah,NJ:LawrenceErlbaum.

Ford,D.Y.,&Harris,J.J.,III.(1992).Theelusivedefinitionofcre-ativity.Journal of creative Behavior, 26,186–198.

Fouche,K.K.(1993).Problem solving and creativity: Multiple solu-tion methods in a cross-cultural study in middle level mathematic.Unpublished doctoral dissertation, University of Florida,Gainesville.

Frahm,R.A.(2004,March21).Feverishlyfixinganerror:CompanyracestocorrectinaccuratescoringofConnecticut’smasterytest.the Hartford courant,p.A1.

Ginsburg,H.P.(1996).Toby’smath.InR.J.Sternberg&T.Ben-Zeev(Eds.), the nature of mathematical thinking(pp.175–282).Mahwah,NJ:LawrenceErlbaum.

Hadamard,J.(1945).the psychology of invention in the mathemati-cal field.Mineola,NY:Dover.

Hashimoto,Y.(1997).Themethodsoffosteringcreativitythroughmathematical problem solving. international reviews on Mathematical Education, 29(3),86–87.RetrievedMarch10,2003,fromhttp://www.fiz-karlsruhe.de/fix/publications/zdm/adm97

Haylock,D.W.(1985).Conflictsintheassessmentandencourage-mentofmathematicalcreativityinschoolchildren.International Journal of Mathematical Education in Science and technology, 16,547–553.

Haylock,D.W.(1987).Aframeworkforassessingmathematicalcreativityinschoolchildren.Education Studies in Mathematics, 18(1),59–74.

Haylock,D.(1997).Recognizingmathematicalcreativityinschool-children.international reviews on Mathematical Education,


29(3),68–74.RetrievedMarch10,2003,fromhttp://www.fiz-karlsruhe.de/fix/publications/zdm/adm97

Hong,E.,&Aqui,Y.(2004).Cognitiveandmotivationalcharac-teristicsofadolescentsgifted inmathematics:Comparisonsamongstudentswithdifferenttypesofgiftedness.Gifted child Quarterly, 48,191–201.

Imai,T.(2000).Theinfluenceofovercomingfixationinmathemat-ics towards divergent thinking in open-ended mathematicsproblemsonJapanesejuniorhighschoolstudents. international Journal of Mathematical Education in Science and technology, 31,187–193.

Jensen,L.R.(1973).the relationships among mathematical creativity, numerical aptitude and mathematical achievement.Unpublisheddoctoraldissertation,UniversityofTexasatAustin.

Kilpatrick,J.(1987).Problemformulating:Wheredogoodprob-lemscomefrom?InA.H.Schoenfeld(Ed.),cognitive science and mathematics education(pp.123–148).Mahwah,NJ:LawrenceErlbaum.

Kim,H.,Cho,S.,&Ahn,D.(2003).Developmentofmathematicalcreativeproblemsolvingabilitytestforidentificationofgiftedinmath.Gifted Education international, 18,184–174.

Köhler,H.(1997).Actingartist-likeintheclassroom.international reviews on Mathematical Education, 29(3),88–93.RetrievedMarch10,2003,fromhttp://www.fizkarlsruhe.de/fix/publica-tions/zdm/adm97

Krutetskii,V.A.(1976).the psychology of mathematical abilities in school children.Chicago:UniversityofChicagoPress.

LearningPointAssociates.(2004).all students reaching the top: Strategies for closing academic achievement gaps. a report of the national study group for the affirmative development of academic ability.RetrievedOctober18,2004,fromhttp://www.ncrel.org/gap/studies/allstudents.pdf

Mann,E.L.(2003).the impact of mathematical experiences as a child on the teaching practices of elementary school teachers.Unpublishedmanuscript,UniversityofConnecticut,Storrs.

Mayer,R.E.,&Hegarty,M.(1996).Theprocessofunderstandingmathematicalproblems.InR.J.Sternberg&T.Ben-Zeev(Eds.),


the nature of mathematical thinking(pp.29–54).Mahwah,NJ:LawrenceErlbaum.

Meissner,H.(2000,July–August).creativity in mathematics edu-cation. Paper presented at the meeting of the InternationalCongressonMathematicalEducation,Tokyo,Japan.

NationalCouncilofTeachersofMathematics.(1980).an agenda for action.Reston,VA:Author.

NationalCouncilofTeachersofMathematics.(2000).Principles and standards for school mathematics.Reston,VA:Author.

NoChildLeftBehindAct,20U.S.C.§6301(2001).Pehkonen,E.(1997).Thestate-of-artinmathematicalcreativity.

international reviews on Mathematical Education, 29,63–66.RetrievedMarch10,2003,fromhttp://www.fiz-karlsruhe.de/fix/publications/zdm/adm97

Poincaré,H.(1913).the foundations of science.NewYork:TheSciencePress.

Polya,G.(1962).Mathematical discover: on understanding, learning and teaching problem solving (Vol.1).Hoboken,NJ:JohnWiley&Sons.

Reis,S.M.,Gentry,M.,&Maxfield,L.R.(1998).Theapplicationofenrichmentclusterstoteachers’classroompractices.Journal for Education of the Gifted, 21,310–324.

Renzulli,J.S.(1978).Whatmakesgiftedness?Reexaminingadefi-nition.Phi delta Kappan, 60,180–184,261.

Renzulli,J.S.(1997).How to develop an authentic enrichment cluster.Storrs:UniversityofConnecticut,NationalResearchCenterontheGiftedandTalented.RetrievedNovember12,2002,fromhttp://www.gifted.uconn.edu/sem/semart01.html

Renzulli,J.S.(1998).Thethree-ringconceptionofgiftedness.InS.M.Baum,S.M.Reis,&L.R.Maxfield(Eds.),nurturing the gifts and talents of primary grade students (pp.1–27).MansfieldCenter,CT:CreativeLearningPress.

Renzulli,J.S.,Gentry,M.,&Reis,S.M.(2003).Enrichment clusters: a practical plan for real-world, student-driven learning.MansfieldCenter,CT:CreativeLearningPress.

Renzulli,J.S.,Gentry,M.,&Reis,S.M.(2004).Atimeandplaceforauthenticlearning.Educational leadership, 62(1),73–77.


Renzulli,J.S.,Leppien,J.H.,&Hays,T.S.(2000).the Multiple Menu Model: a practical guide for developing differentiated cur-riculum.MansfieldCenter,CT:CreativeLearningPress.

Runco,M.A.(1993).creativity as an educational objective for disadvan-taged students (RBDM9306).Storrs:UniversityofConnecticut,TheNationalResearchCenterontheGiftedandTalented.

Schoenfeld,A.H.(1988).Whengoodteachingleadstobadresults:Thedisastersof“well-taught”mathematicscourses.Educational Psychologist, 23,145–166.

Sheffield, L. J., Bennett, J., Beriozábal, M., DeArmond, M., &Wertheimer,R.(1999).Reportofthetaskforceonthemath-ematicallypromising.InL.J.Sheffield(Ed.),developing math-ematically promising students (pp. 309 – 316). Reston, VA:NationalCouncilofTeachersofMathematics.

Shimada,S.(1997).Thesignificanceofanopen-endedapproach.InJ.P.Becker&S.Shimada(Eds.),the open-ended approach: a new proposal for teaching mathematics(pp.1–9).Reston,VA:NationalCouncilofTeachersofMathematics.

Silver,E.A.(1997).Fosteringcreativitythoughinstructionrichmathematicalproblemsolvingandproblemposing.international reviews on Mathematical Education, 29, 75–80. RetrievedMarch10,2003,fromhttp://www.fiz-karlsruhe.de/fix/publica-tions/zdm/adm97

Singh,B.(1988).teaching-learning strategies and mathematical cre-ativity.Delhi,India:MittalPublications.

Skemp, R. R. (1987). the psychology of learning mathematics.Mahwah,NJ:LawrenceErlbaum.

Starko,A.J.(2001).creativity in the classroom: Schools of curious delight.Mahwah,NJ:LawrenceErlbaum.

Sternberg,R.J.(1996).Whatismathematicalthinking?InR.J.Sternberg&T.Ben-Zeev(Eds.),the nature of mathematical thinking(pp.303–318).Mahwah,NJ:LawrenceErlbaum.

Taylor,E.I.,&Wozniak,R.H.(1996).Pure experience, the response to William James: an introduction.RetrievedSeptember3,2004,fromhttp://psychclassics.yorku.ca/James/TaylorWoz.htm

Torrance,E.P.(1966).torrance test on creative thinking: norms-tech-nical manual (Researched.).Lexington,MA:PersonalPress.


Treffinger,D.J.,Renzulli,J.S.,&Feldhusen,J.F.(1971).Problemsintheassessmentofcreativethinking.the Journal of creative Behavior, 5,104–112.

Treffinger,D.J.,Young,G.C.,Selby,E.C.,&Shepardson,C.(2002).assessing creativity: a guide for educators (RM02170).Storrs:UniversityofConnecticut,TheNationalResearchCenterontheGiftedandTalented.

Tuli,M.R.(1980).Mathematical creativity as related to aptitude for achievement in and attitude towards mathematics(Doctoraldissertation,PanjabUniversity,1980).DissertationAbstractsInternational,42(01),122.

Usiskin,Z.(1999).Themathematicallypromisingandthemath-ematicallygifted.InL.J.Sheffield(Ed.),developing mathemati-cally promising students (pp. 57–69). Reston, VA: NationalCouncilofTeachersofMathematics.

Whitcombe, A. (1988). Mathematics: Creativity, imagination,beauty.Mathematics in School, 17,13–15.

Author Note

Correspondence concerning the article should be addressed toEric Mann, Purdue University, Department of Curriculum andInstruction,100N.UniversityStreet,BeeringHallRm3122,WestLafayette, IN 47907. E-mail correspondence should be sent [email protected],Dr.LindaSheffield,Dr.M.KatherineGavin,Dr.TerryWood,andDr.MarciaGentryfortheirreviews,comments,andsuggestionsonthisarticle.

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Media Themes and ThreadsIampleasedtoannouncethereturnof“MediaThemesandThreads”to the Journal for the Education of the Gifted. My name is KristieSpeirsNeumeister,andIwillbetheeditorforthissection.Ireceivedmydoctorateineducationalpsychology,withanemphasisongiftedand creative education, from the University of Georgia in 2002. IamcurrentlyanassistantprofessoratBallStateUniversity,whereIteachundergraduatecoursesineducationalpsychologyandgradu-atecoursesingiftededucation.Iamexcitedabouttheopportunitytoeditthissectionofthejournal.

Veteran readers of the journal will be familiar with “MediaThemesandThreads,”asitappearedinpastvolumesofJEG;how-ever,fornewerreaders,Iwantedtoprovideadescriptionofthesec-tion.Thepurposeofthesectionistofeaturereviewsofrecentmediaproductspertainingtogiftededucation,includingcurrentpublica-tions of books, videos, and instructional DVDs and CD-ROMs.Threereviewsarepublishedinthiscurrentissue.

RebeccaNordinhaswrittenathoroughreviewofGaryDavis’snewbook,Gifted children and Gifted Education: a Handbook for teachers and Parents. This book provides a foundational overviewofthemostsalientissuesingiftededucationandwouldbeagreatresourcesforparentsandteachersofgiftedchildrenalike.

Middleschoolhasbeenreferredtoasthe“blackhole”ofgiftededucationduetoa lackofprogrammingandoptionsforstudents.SusanRakow’sbook,Educating Gifted Students in Middle School: a Practical Guide, reviewedbyJamieMacDougall,expertlyrespondstothisassertionbyprovidingadescriptionofeffectiveprogrammingoptionsforgiftedmiddleschoolstudentsandalsoemphasizingtheneedforstrongguidanceandcounselingcomponentsatthislevel.

Finally,FeliciaDixonreviewedLaurenceColeman’srecentaddi-tionto thefield,nurturing talent in High School: life in the fast lane,inwhichheaccountstheexperiencesofhighschoolstudentsattendingaresidentialacademyforgiftedlearners.Colemanoffersauniquecontributiontothefieldwiththisinsider’sperspectiveonlivedexperiencesofgiftedadolescents.

Ihopeyouwillfindthesereviewsinformativeasyoucontinueto search for new information on parenting and educating gifted


students.Shouldyoubeinterestedinreviewinganewmediapub-licationforJEGorifyouwouldliketohaveaspecificcurrentpub-lication reviewed, please send your inquiries directly to me at theaddressbelow.

Dr.KristieSpeirsNeumeisterDepartmentofEducationalPsychology

TeachersCollegeBallStateUniversity

Muncie,IN47306Phone:765-285-8518

Fax:765-285-3653E-mail:[email protected]

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