Contributions from cognitive research to mathemati

ContributionsfromCognitiveResearchtoMathematicsandScienceEducation*aninvitedtalkgivenattheWorkshoponResearchinScienceandMathematicsEducationJanuary20–24,1992Winterton,SouthAfricabyWilliamJGeraceDepartmentofPhysicsUniversityofMassachusettsAmherst,MA01003[InD.Grayson(Ed.),ProceedingsoftheWorkshoponResearchinScienceandMathematicsEducation(pp.25-44).Pietermaritzburg,SouthAfrica:TeeanemPrinters(Pty)Ltd.(1992).]*WorksupportedinpartbyNSFGrant#MDR–9050213ContributionsfromCognitiveResearchtoMathematicsandScienceEducation1[Note:Duringthetalkonwhichthispaperisbased,thereweremanyquestions,comments,andobservationsbytheWorkshopparticipants.Thesehavebeenparaphrased,andappearinitalicstodistinguishthemfromthespeaker’sreplies.]Ithasbeenrequestedthatwemakethesepresentationsasinteractiveaspossible.Thatisfinebymeandismuchmoremystyle,sopleasefeelfreetointerruptmewithanyquestionsyoumayhave.Bywayofbeinginteractive,Iwouldliketostartwithaquiz:KKKKNNNNNNOOOOOO“Whatdoyouseeinthispicture?”Whatdoesthisfiguremean?Justthis:fromfarawayyoucanmakeouttheimportantwords:wisdominthefirstinstance,andknowledgeinthesecond.Ifyouaretoocloseyoucan’tperceivethebigpicturebutratherfocusonthesmall,individuallettersusedtoformthebigmessage.Thesmalllettersareusedtospellknowledgeinthefirstcase,andinformationinthesecond.Thisisparalleltotheforest-and-treesanalogy.MyapologiestoDouglasHofstadter,whoistheauthorofabookentitledGödel,Escher,Bach[1]whereIgottheideaforthisfigure.Perhapssomeofyouhavereadit.Ibelievethatitisavaluablebooktoreadandrecommendthatyoutryit.2GeraceWhatIhavetriedtodointhisfigureispresentarepresentationofwhatIconsidertobethefundamentalproblematthemomentwithmathematicsandscienceeducation.ItisperhapsbestexplainedbyaquotefromT.S.Eliot’sChorusesfromtheRock:Whereisthewisdomwehavelostinknowledge?Whereistheknowledgewehavelostininformation?Thepointhereisthatweinscienceandmathseducationhaveallowedourselvestobecomeinformationdispensers.Weshouldresistthattrendandrestorethegoalofimpartingwisdom.Thistalkisdividedintofourbroadareas.Theseare:(1)Generalremarksaboutcognitivescience(2)Hardevidencefromcognitiveresearch(3)Somedescriptivecognitivemodels(4)Contributionsfromcognitivesciencetomathsandscienceeducation.GeneralremarksaboutcognitivescienceIshould,perhaps,pointoutattheoutsetthattheviewsexpressedinthistalkaremyown;Imakenoclaimthattheyareauthoritativeorevenrepresenttheconsensusviewofcognitiveresearchers.Iwillbeginmygeneralremarksbyaddressingthemotivationfordoingcognitivescience.AsfarasIamconcerned,amajormotivationiswhatIalludedtointheintroductoryremarks.Thismightbedefinedasinformationpollution.Asinotherendeavors,itistimeforusinscienceandmathseducationtotakeamore“ecological”perspective.Wearebeinginundatedwithalotoffacts,andtheprevailingviewisthatstudentsoughttoknowthesefacts.Ithinkthatisthewrongwaytogoaboutit.Whatyouwanttodoistreatyourstudentsasthinkers.Teachthemtoteachthemselves.Makethemlearners,andthenyoudon’thavetoworryaboutteachingthemallthefacts;they’lldothatthemselves.Whatistheaimofcognitivescience?Itistoachieveanunderstandingoftheprocessesunderlyingknowledgeacquisitionandutilization.Wedevisemodelstohelpexplaintheprocessesoflearningandproblemsolving.Forusinthefieldofpedagogy,researchonContributionsfromCognitiveResearchtoMathematicsandScienceEducation3cognitiveprocessesoffersthepotentialfordevelopingeducationalpracticesthataremoreefficientand/orapplicabletoawideraudience.Whoarethepractitionersorcontributorstocognitiveresearch?Thisisveryimportantbecause,inmyopinion,you,aseducators,shouldbeplayingasignificantrole.Thereisagroupofpeoplewhocallthemselvescognitivescientists,butthosepeoplearegenerallyinpsychologydepartmentsandtendtobeconcernedaboutthingslikeeyemovementsandneuronsinthebrain.Cognitivescienceisaverybroad-basedspectrum,rangingallthewayfromthatkindofpsychologyrightonuptopedagogy.You(aseducators)cannotaffordtowaitforsomespecialistorsomeexperttocomealongandtellyouhowyoushouldteachorwhatyoushouldteach.Youshouldbeengagedindiscoveringthoseanswersyourselves.Youareascompetentasanybodytomakethosediscoveries,andpeoplearemakingthemallthetime.Onethingthatiscleartomeisthatcognitivescienceisnotgoingtobearmuchfruituntilcontentpeopleandeducatorsgetverystronglyinvolvedintheresearchprocess.Whatisthenatureofcognitivescience?Atpresent,itisveryqualitative.Allwehavearegeneral,qualitativedescriptionsofcertainkindsofprocesses.Preciseorquantitativemodelsarenotyetpossible.ArtificialIntelligence(AI)wouldliketohavesuchquantitativeandproceduralmodelsoflearningbecausethesemodelscouldthenbeusedtomakeacompu