neuroscience and early childhood math education: a...
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NeuroscienceandEarlyChildhoodMathEducation:
ABlueprintforBetterBridgesAssessingtheefficacyofanewresearchdesignforestablishingconnectionsbetweencognitiveneurosciencefindingsandpreschoolmathlearning
JuliaL.Niego
MastersofScienceThesis
NeuroscienceandEducationProgram
BiobehavioralSciencesDepartment
May2011
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“Themoretheschemataaredifferentiated,thesmallerthegapbetweenthenewandthe
familiarbecomes,sothatnovelty,insteadofconstitutinganannoyanceavoidedbythesubject,
becomesaproblemandinvitessearching.”
JeanPiaget
ProjectDescription
Alargebodyofliteraturedescribestheriftbetweenneuroscienceandeducation,twotheoreticallyrelatedfieldsthathavestruggledtoestablishcommonground.Researchershavepointedtoadisconnectbetweenthepromiseofneurosciencedataandactualapplicationtotheclassroom.Thisprojectconsidersthewaysthatcognitiveneuroscienceandeducationcanforgecommongroundtoamelioratetheseproblems.Buildingontheclaimthataconnectionbetweendataandtheclassroomwillnecessitatebi‐directionalandcollaborativecommunicationbetweenscientistsandeducators,aclinicalinterviewmethodhasbeenchosenasthetoolforuncoveringteachers’experience‐basedmentalmodelsoflearningprocessesintheirstudents’brains.Drawingontheneedforcommonmodelsandlanguage,theseinterviewsarebasedaroundvideosegmentsofchildrenlearningmathintheearlychildhoodclassroom.Theoverarchinggoaloftheprojectistoengendermeaningfuldiscussionbetweeneducationalneuroscienceresearchersandteachersinordertoestablishimportantconnectionsbetweendataonthebrainandlearningasittrulytakesplaceinthepre‐schoolenvironment.Interviewswerecarefullyrecordedandanalyzedinaninterpretivefashiontoidentifyimportantvariablesandobservationcategoriesforfutureresearch.
BackgroundandTheoreticalApproach
NeuroscienceandEducation
Atfirstglance,thefieldsofeducationandneurosciencemightseem
intrinsicallylinked.Afterall,learningtakesplaceinthebrain.Education,perhaps
nowmorethanever,recognizestheneedforaclearerunderstandingofhow
studentslearn.Overthepastdecades,neuroscienceresearchhasmadesignificant
advances,uncoveringnewandexcitingdataonthelearningbrain.
Yet,theliteraturetellsadifferentstory.Neuroscienceandeducationare
theoreticallyrelatedfieldsthathavestruggledtoforgecommonground.Sincethe
90’s,declaredbytheUSCongresstobethe“DecadeoftheBrain,”muchoftheinitial
zealfordirectapplicationofneurosciencedatahasfaded.Earlyattemptstouse
neurosciencetheoriesintheclassroomresultedinmisunderstandingsand
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questionableeducationalpoliciesbasedonhypedideassuchassynaptogenesis,
criticalperiodsandenrichment(Bruer,1997;Mayer,1998).
Thishasledtheresearchcommunitytoreactwithappropriatecautionand
skepticismaboutfurtherattemptstomergeneuroscienceandeducation.Bruerhas
beenattheforefrontofthispositionsincehis1997paper“Neuroscienceand
Education:ABridgeTooFar”(Bruer,1997;Bruer,2006).Hemakestheclaimthat
thetheoreticaldistancebetweenneuroscienceandeducationmaysimplybetoo
great.Thecoreconceptsusedbyneuroscientistsaredifferentfromthoseusedby
psychologistsandclassroomteachers,leadingtoinevitablemisunderstandings.
Brueralsoquestionshowasuperficialfascinationwithsynapsesandbrainimages
causesteacherstooverlookasubstantialbodyoflargelyuntappedcognitive
psychologyresearch(Bruer,2006).Heposesthatresearchersshouldinstead
devoteeffortstoshorterbridgesbetweeneducationandcognitivepsychology,
cognitivepsychologyandneuroscience(Bruer,1997).
Mayer(1998)pointsoutthatthereisaninherentprobleminthegoalof
tailoringneurosciencefindingsforeducatorsinordertobaseeducationalpractices
onneuroscience.Instead,heposesthatthebridgebetweencognitiveneuroscience
andeducationalpsychologyneedstobea“two‐waystreet,”whereeducational
theoryguidesandinformsneuroscienceresearch.
Yet,neurosciencehascontinuedtoadvance,andeverydaynewfindings
abouthowstudents’brainslearntoread,write,domath–andhandlethestressof
toomuchhomework–areshowinguponbestsellertablesatthebookstore,covers
ofpopularmagazinesandtelevisionscreens.Thelargerfieldofneurosciencehas
branchedandmergedwithotherdisciplines,leadingtosubfieldsofcognitive,social
anddevelopmentalneuroscience.In1999NRCdeclared“Neurosciencehas
advancedtothepointwhereitistimetothinkcriticallyabouttheforminwhich
researchinformationismadeavailabletoeducatorssothatisitinterpreted
appropriatelyforpractice.”
Overthelastdecadethe“neuroscienceandeducationargument”(Bruer,
1997)hascontinuedtoreverberateamongresearchersineducation,psychology
andneuroscienceastheyregardthepromiseofnewdataonthebrainforadvancing
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thescienceofeducationandlearning.Researchershaveposednewsolutionsfor
buildingstrongerbridgesthatcan“spanthechasm”betweenneuroscienceand
education(AnsariandCoch,2006;Goswami,2006,Bruer,2006).
Asoberconsensusremainsinthecurrentliteraturethatdescribinghowdata
onstructuralandfunctionalchangesinthebraincanactuallybeusedtoinform
educationalpolicyandpracticeisadifficulttask(AnsariandCoch,2006;Goswami,
2006,Bruer,2006).Forone,researchersidentifyalackofabilityonthepartof
teacherstoacquireandrecognizerelevantdata,leadingtoacceptanceof“neuro‐
myths”and“Brain‐Based”learningpackagesthatarenotsupportedbyscience.
AnsariandCoch(2006)refertoquestionablemediareportsandoversimplified
claimsaboutleft/rightbrainlearners,“exercising”thewholebrain,and“brain‐
buttons”thatinundatedschoolsoverthelastdecade.
Anotherproblemisteachers’lackofknowledgeandskillsnecessaryto
interpretneurosciencedata.Goswami(2006)notesthattheprogressin
neurosciencelabsislargelytheoretical,andwithoutafirmunderstandingof
hypothesis,theoryandestablishedmodel,teachersarenotabletoestablishhowdata
fitsintothe“bigpicture.”Furthermore,Goswamistatesthatteachersoftenview
scientificresearchastooconcernedwithestablishingrigorinpreciseexperimental
manipulationsandcomplainthatscienceresearcherssimplyprovidetoomuchdata.
EducationalNeuroscience
Takentogethertheseobstaclesstackuptoadifficultproject,yethintsof
promiseareemerginginrecentliterature.Researchershavebeguntocollaborate
acrossthedisciplinesofneuroscienceandeducationandanewinterdisciplinary
fieldhasemerged.Thisisevidencedbythegrowthofnewresearchcentersand
educationalprogramssuchastheCentreofEducationalNeuroscienceatCambridge,
theMBESociety(http://www.imbes.org)andgraduateprograminMind,Brainand
EducationatHarvardUniversity,theundergraduateprograminNeuroscienceand
EducationwithintheEducationDepartmentatDartmouthCollege,andthegraduate
programinNeuroscienceandEducationwithintheBiobehavioralScience
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DepartmentatTeacher’sCollege,ColumbiaUniversitywherethepresentproject
wasdeveloped.
Thefieldofeducationalneurosciencehasbecomeanexcitingnewareaof
educationalresearch.Acentralpositionwithinthisfieldistheacknowledgment
thatattemptsatdirectapplicationofneurosciencedatatotheclassroomisindeeda
“bridgetoofar”(Bruer,1997;Goswami,2006).Insteadtheaimofnewinvestigation
isfindingpossiblewaysto“buildstrongbridges”(DeSmedt,Ansari,Grabner,
Hannula,ScheiderandVerschaffel,2010)throughmultiplebi‐directionaland
reciprocalinteractionsbetweenresearchersineducationandcognitive
neuroscience.
ThismodelcanbeseenasincorporatingBruer’s(1997,2006)emphasison
theimportanceofcognitivepsychologytobotheducationandneuroscience,aswell
asMayer’s“two‐waystreet”(Mayer,1997).Theemergenceofcognitive
neuroscienceasasubfieldofneurosciencewasdrivenbythelinkingof
connectionistmodelsofthoughtandbehaviordevelopedbycognitivepsychologists
(seeAnderson,2005forafullreview)withnewneurosciencetechniquesfor
creatingimagesoftheactivebrain.Throughvarioustechniquesinvolving
electromagnetic(MEG,EEG,ERP)andhemodynamic(PET,MRI,fMRI)
measurement,dataiscollectedandusedtobuildmodelsorpicturesofbrain
structures,connectionsandpathways(Campbell,2006).
Educationalneuroscienceresearchersholdthatcognitivepsychology
informedby,andinforming,cognitiveneuroscienceshouldmakeupthecoreof
educationalneuroscience(Berninger&Corina,1998;Bruer,1997;Campbell,2006;
Geake&Cooper,2003).Cognitiveneuroscienceresearchfocusesontheneural
mechanismsunderlyinghumanbehaviorandcognition,whichmatchescloselywith
educationalresearchonlearningandinstruction.Bruer(1997)pointedout,asone
noteofoptimismin“ABridgetooFar,”cognitivemodelscanbeseenasalocusof
similaritybetweenresearchinneuroscienceandeducation.Afterall,cognitive
neuroscienceisbuiltonmodelsofbrainprocessesprovidedbycognitive
psychology,andthesemodelsareasfundamentaltocognitiveneuroscienceasthey
aretotheappliedscienceoflearning.
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Furthermore,inidentifyingsolutionsforbuildingbridgesbetween
neuroscienceandeducation,bi‐directionalcollaborationisavitalelement.Ansari
andCoch(2006)describeaneedfor“practicalmechanisms”tosupportandfoster
meaningfulintegrationbetweenthebrain‐labandtheclassroomsothatteachers
willbemoreinformedconsumers.Thismustbeaccomplishedthroughawareness
andunderstandingofsimilaritiesanddifferencesbetweenresearchinneuroscience
andeducation,andbyestablishingpointsofcontactbetweenscientistsand
educatorsincludingbidirectionaldialogueandcollaborativeexperimentdesign.
Goswami(2006)urgesthatanewgenerationofmulti‐disciplinary
researchersmustdedicatetheircombinedexpertiseinbothneuroscienceand
educationtopresentinghighqualityknowledgeonthebrainindigestibleformand
interpretingneurosciencefromtheperspectiveofandinthelanguageofeducators.
Thisrequiresthatsuchresearchersspendtimewitheducatorsinordertoestablish
waysthatmeaningfulcommunicationcanbeestablished.Commonmodelsof
students’thinkingneedtobeestablishedbetweenresearchersandteachersinorder
tosupportandsustaintrueandequalcollaboration,andtofostersuccessful
exchangeofrelevantdatabetweendisciplines.Anactiveprojectmustalsobe
underwaytodevelopacommonlanguagebetweenneuroscientistsandeducatorsso
thateachunderstandsoneanotherclearly,andsothatneithersideisputoffby
misinterpretedtheoriesorjargon.AsMayer(1998)recognized,drawingon
teachers’experienceandintuitionsaboutstudents’learningpresentsnewhopefor
collaborativedesignofcognitiveneuroscienceresearchthatwillbetrulymeaningful
toeducators.
CognitiveNeuroscienceandMathematics
Amidthegrowingfieldofcognitiveneuroscience,mathematicslearninghas
presentedanareaofparticularpromise.InProceedingsforthe30thConferenceofthe
InternationalGroupforthePsychologyofMathematicsEducationheldinPraguein
2006,Campbelldescribesabuildingfrustrationonthepartofmatheducatorswith
purelytheoreticalmodelsofstudent’learning.Hediscusseshisowndesireasa
matheducatortoseeinsidehisstudent’sbraininordertounderstandhowitis
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changinginresponsetoinstruction,andwhatishappeningwhenbreakdownsin
understandinginevitablyoccur.Campbelloutlineshowadvancesincognitive
neuroscienceholdnewpossibilitiesforfillinginthesegapsineducators’
understandingofstudents’learning.
InarecentpositionpaperbasedonissuespresentedattheEARLIAdvanced
StudyColloquiumonCognitiveNeuroscienceandMathEducation,heldinBelgium
in2009,DeSmedtetal(2010)statethatbringingcognitiveneurosciencetobearon
mathematicslearninghasledtoveryproductiveresearchinrecentyears(Ansari,
2009;Dehaene,2009;Lipton&Spelke,2005;Lemer,Dehaene,Spelke,&Cohen,
2003).Theauthorspointouthowever,thatneuroimagingstudiesaimedat
describingrelationshipsbetweenbrainactivityandinstructionalpracticearestill
scarce.
Inordertoexploresuchconnections,neurosciencemustbeguidedby
educationalandpsychologicaltheories.ReferencingresearchbydeJongetal
(2009),DeSmedtetaloutlinespecificquestionsprominentineducationalresearch
thatcoulddrivefuturecognitiveneuroscienceresearch,suchaslearningfrom
multiplerepresentations,cognitiveloadandtheroleofaffectiveprocessesin
learning.Theauthorsconcludethateducationalresearcherscanplayapivotalrole
byidentifyingimportantvariableswhichneedtobeincorporatedintoeducational
neuroscienceinvestigations.
EducationalNeuroscienceandEarlyChildhoodMath
Math education in the preschool years has been a source of question and
controversyfordecades.In2002,theNationalCouncilofTeachersofMathematics
and theNational Association for the Education of Young Children collaborated to
produceajointpositionstatementrecognizingthefailureofUSstudentstoachieve
mathematicalprowessconsistentwithpeers inothercountriesandadvocatingan
increasedattentiontoearlychildhoodmatheducation(NationalAssociationforthe
EducationofYoungChildren&NationalCouncilofTeachersofMathematics,2002).
Whilemanyeducatorsnowagreethatmatheducationshouldbeginearly,thereisa
lackofunderstandingofhowchildren’smathematical thinkingdevelops, andhow
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besttosupportitintheclassroom(Ginsburg,Jang,Preston,VanEsselstyn,&Appel,
2004).
Agrowingbodyofresearchineducationanddevelopmentalpsychologyhas
investigated the development of mathematical thinking and best practice for
teaching math to young children (Piaget, 1970; Clements, 1999; Ginsburg, Jang,
Preston, VanEsselstyn & Appel, 2004; Seo & Ginsburg, 2004). Ginsburg (1988,
1989,1997,1998,2007)hascontributedmuchtothisburgeoningfieldbybuilding
on Piaget’s method of cognitive clinical interviews as a research tool to assess
children’s mathematical knowledge, problem‐solving strategies and overall
understanding of math concepts at various stages of development. This has
provided researchers and educators with new ways to “enter the child’s mind
(Ginsburg, 1997)” and observe mathematical thinking in order to improve math
teachingandassessment.
Inrecentyearstherehasbeenacallforpreschoolmathcurriculathatare
researchbased(Clements,2007;Clements&Samara,2007).Questionshavearisen
regardingtheappropriatenessofteachingmathematicstoveryyoungchildren,the
bestmodeofinstruction,andtheroleofconcretemanipulativesinmathlearning
(Ginsburg,BoydandSunLee,2008).Finally,despitemisinterpretationofearly
discoveriesaboutneuronsandsynapses,cognitiveneuroscienceconfirmsthatthe
earlyyearsoflifeareveryimportantinachild’smathematicaldevelopment,serving
asafoundationonwhichshewillbuildsubsequentknowledge(Baroody,2007,
Spelke&Dehaene,1999,Xu,Spelke,&Goddard,2005).Clearly,matheducatorsand
educationalneuroscienceresearcherswanttoachieveafullerandmoreaccurate
understandingofhowtheyoungbraindevelopsmathematicallyinordertoknow
howtobestsupportbothtypicalandatypicalmathlearning.
Evenso,thereisalackofcognitiveneuroscienceresearchleadingto
insightfulexplanatorymodelsofchildren’slearningprocessesthathaveproven
relevantforapplicationintheearlychildhoodmathclassroom.Partofthereason
forthisisthatthesemodelsdonotresultfromthecarefullycontrolledexperiments
favoredbycognitivepsychologyandneuroscience.Asanypreschoolmathteacher
willtellyou,3and4‐yearoldsarespontaneous,unpredictable,andficklelearners.
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Higher‐levelcognitiveprocesses,evenintheadultbrain,arebynaturenon‐linear,
fulloffeedbackloopsandprocessesthatcompeteandcoordinate.Modelsofthese
processesthatcanbeunderstoodandusedbyteachersmustbetrueexplanatory
modelsthatareiconicandanaloginnaturebeingbuiltupfrom“moreprimitiveand
familiarnotions(Clement,2000).“
Inconclusion,whileeducationalneuroscienceresearchdoesnotasyet
provideafullorapplicablemodelofchildren’smathlearning,thefieldismuch
closertothispossibilitythanever.Withinthecurrentliteratureisablueprintfor
newresearchmethodsthatcanbeginconstructingthismodel.Itisclearthatsuch
researchmustinvolve(1)bi‐directionalcollaborationbetweenearlychildhood
mathteachersandeducationalneuroscienceresearchers(2)involveopportunities
foropendiscussion,reflectionandexchangeofideasand(3)createcommonground
formeaningfulcommunicationaboutresearchquestions,includingashared
languageandtheoreticalframework.
ProjectAim
Theaimofthisprojectwastodevelopandassesstheinitialefficacyofa
researchmodelfordetermininghowandinwhatwayscognitiveneuroscience
researchcanbemademeaningfultoearlychildhoodmatheducators.Thespecific
goalsoftheprojectwere:
(1) Todesignaresearchmodelthatwouldincorporaterecommendationsinthe
currentliteratureforbidirectionalcollaboration,commonmodelsand
commonlanguageinordertofostermeaningfulcommunicationbetween
educationalneuroscienceresearchersandteachers
(2) Togatherpilotdataontheefficacyoftheresearchmodelforengagingor
changingteacher’scurrentmentalmodelsofstudent’slearning
(3) Toidentifyimportantvariablesandobservationcategoriesforuseofthe
researchmodelinfutureinvestigation
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ResearchModelDesign
Theprimarygoalofthisprojectwasthecarefulconstructionofaresearch
designthatcouldexplorehowcognitiveneuroscienceonchildren’smathlearning
couldeffectivelyengageteachers.
BidirectionalCollaboration
Thefirsttaskwastochoosearesearchtechniquethatwouldallowcollection
ofdataonteachers’authenticthoughtsandideasthroughbi‐directional
communicationbetweenresearcherandteacher.
Acognitiveclinicalinterviewmethodwaschosenastheresearchtechniquethat
wouldbestaccomplishthisgoal.SinceitwasdevelopedbyPiaget(1955,1975)todescribe
internalthoughtstructuresinchildren,theclinicalinterviewmethodhasbranchedintoa
varietyoftechniquesforresearchinpsychologyandeducation(Strauss,1993),including
matheducation(Clement,2000;Ginsburg,1989,1997;Ginsburg&Opper,1988;Ginsburg,
Jacobs&Lobez,1998;Ginsburg,Jang,Preston,VanEsselstyn&Appel,2004;Ginsburg,
Boyd,&SunLee,2008).Thestrengthoftheclinicalinterviewmethod,asopposedtonon‐
clinicaldatagatheringtechniques,isthatitcanbeusedtocollectandanalyzedataon
mentalprocessesatthelevelofasubject’sauthenticideasandmeanings,andtoexpose
hiddenstructuresandprocessesinthesubject’sthinkingthatcouldnotbedetectedbyless
open‐endedtechniques(Clement,2000).
Astheclinicalinterviewmethodisbynatureanopen‐endedtechnique
severalissuesregardingreliabilityofdatawillbeaddressed.Clement(2000)
outlinesimportantconsiderationsforresearchersusingthetechniqueinorderto
fosterreliability.First,hesuggestspreparingsetprotocolsforinterviewsthatcan
beusedwithrepeatedsubjects.Second,videotapinginterviewscreatesarich
recordofverbalandbehavioraldatathatcanbereexaminedandshared.Finally,a
combinationofgenerativeandconvergentinterviewmethodsallowsinterviewsto
befirstanalyzedinanexploratoryfashionsothatobservationcategoriescanbe
developedforlatercodingofdata.
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Inhisownresearchconstructingmodelsofproblemsolvinginalgebra
students,Clementurgestheneedtobeginwithgenerative,interview‐basedcase
studies.Theseareanalyzedandtheoreticalhypothesestaketheformofmodelsthat
aregroundedinnaturalisticobservationofbehaviors.Asdataiscollectedand
analyzed,theseinitialmodelsarerevisedinabi‐directionalresearchtechnique
incorporatingbothbottomup(inductive)andtop‐down(deductive)methods.
Throughthisprocess,investigatorsbecomesensitizedregardingwhattolookfor,
andobservationcategoriescanbedevelopedforfuturedata,providingthe
foundationforthedesignofconvergentstudies.
Basedontheserecommendations,thefollowingelementswereincorporated
intotheresearchdesign:
Threesetprotocolsweredevelopedforeachclinicalinterview.Thechoiceof
questionsfortheprotocolsdrewontheseminalworkofPiaget(1975),aswell
adaptationsofthesemethodsdevelopedbyClement(2000),Strauss,(1993)and
Ginsburg,(1989,1997).Thefirstprotocol,(Interview1)wasdesignedtocollect
teachers’uniquepathtoteaching,educationalbackground,andpersonalphilosophy
orapproach.Thesecondprotocol(Interview2)includessixquestionsthatrelateto
discussionofaspecificteachingmoment.Thefinalprotocol(Interview3)asksa
seriesoffourquestionsthatinviteteacherstorelatefindingsofcognitive
neurosciencetotheteachingmoment,theirownclassroomsandpersonal
philosophies,aswellastodescribewaysthattheneuroscienceinformation
impactedorchangedtheiropinions/beliefsaboutearlymathlearning.Itis
importanttonotethat,whilequestionsfortheseprotocolswerepre‐set,the
interviewerwasabletoreactresponsivelytoanswersastheywerecollectedby
askingfollow‐upquestionstoclarifyandextendtheinvestigation(Clement,2000).
TheseprotocolsandquestionswillbediscussedinfurtherdetailintheMaterials
andMethodssectionofhispaper.Ascriptforeachprotocolisalsoincludedinthe
Appendix.
Eachclinicalinterviewwasvideorecorded.Repeatedreviewofrecorded
interviewswouldallowdatatobeinitiallyanalyzedinanopen‐endedexploratory
fashioninordertodevelopobservationcategoriesforfutureanalysisandcoding.
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CommonModels
Thesecondtaskinthegoalofassemblingtheresearchdesignwastoidentify
commonmodelsofstudentlearninginordertoallowmeaningfuldiscussion
betweenresearcherandteachers,andguidetheincorporationofneurosciencedata
intothisdiscussion.
Itisnoteworthythatevenwithallthenewdatacomingfromresearch,there
isamarkedlackofinsightfulexplanatorymodelsofstudents’learningprocesses.
Partofthereasonforthisisthatchildren’scognitiveprocessesarecomplex,highly
influencedbyenvironmentandexperience,andfullofinteractionsbetween
development,instructionandsocialandemotionalfactors.Furthermore,teachers’
experienceswiththeseprocessesarenotderivedfromstudyofdevelopmentand
cognition,butontheirtimespentintheclassroom,teaching,observingand
assessingtheirstudents.
Clementassertsthatsubjects,inthiscaseteachers,dohavepre‐existing
knowledgestructuresandreasoningprocessesthatguidetheirthoughtsandbeliefs
aboutstudents’learning.Thesementalmodelshavestrongeffectsonteachers’
ideasandapproaches.Strauss(1993)claimsthatwithoutdeterminingthenatureof
teachers’mentalmodelsaboutchildren’smindsandlearning,newinformationwill
notbeabletoengageorchangethesemodels.Usingasemi‐structuredclinical
interviewtechniquehecollectedandanalyzedteachers’answerstoquestionsabout
whattheywoulddoinspecificteachingsituations,uncoveringmentalmodelsthat
hecouldthenconnecttoinformationprocessingmodelspresentedbycognitive
psychology(i.e.workingmemory,elaborativeprocessing).
Inasimilarway,thisprojectaimedtouseclinicalinterviewtofirstaccess
thesepre‐existingmodelssothattheycouldbeutilizedindiscussionofstudent
learning,andlaterconnectedtomodelspresentedbycognitiveneuroscience.In
ordertoguidemeaningfuldiscussionofstudents’learning,aproblemspaceneeded
tobeestablishedwithinwhichbothresearcherandeducatorcouldobserve,reflect
onandanalyzestudentinstruction,behavior,strategy,andperformance.By
incorporatingtheelementsthatteachersnaturallychosetodescribelearning
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processes,newinformationcouldbeintroducedonthelevelofateachers’unique
model.
Acase‐basedresearchapproachwaschosenasthetechniquethatwouldbest
accomplishthisgoal.Widelyusedinteachereducationandeducationalresearch,
case‐basedinstructionisapractice‐anchoredlearningpedagogythatcanbeusedto
provideteachercandidatesandresearcherswithmultipleopportunitiestoconsider
complexitiesofteachingandlearning(Andrews,2002;Kilbane,2008;Rosen,2008;
Steinkuehleretal,2002).Kilbane(2008)emphasizesthatcasesillustratelifeinthe
classroomasitactuallyis,notasitshouldbe.Recentresearchonthecase‐based
methodhasrevealedthatcomputercaseswhichoffervideovignettesoflearning
momentscombinedwithinteractivediscussionquestionshavethegreatestimpact
onthequalityofteacher’sreflectivenarratives(Rosen,2008).Teachersareasked
towatchandreflectonvideo‐recordedteachingmomentsastheyhappeninreal
time,thusallowingthemtodescribelearningprocessesontheleveloftheirown
personalmodelsoftheseprocesses.
Videovignettesofearlychildhoodmathteachingmomentswerechosenfrom
theVITAL(videointeractionsforteachingandlearning)archivesofTeacher’s
College,ColumbiaUniversity.Lee,GinsburgandPreston(2009),intheir
descriptionoftheproject,statethatimprovingearlychildhoodteacherpreparation
isthemostpressingneedforearlychildhoodmathematicseducationintheUnited
States.VITALprovidesprospectiveteacherswith“engagingandintellectually
stimulatinghands‐onandminds‐onlearningexperiencesthatsupplementthe
traditionaltextbookandreadings."ThepresentdesignextendstheuseofVITALas
atoolforcollectingdataonteachers’knowledge.
Thequestionsforthesecondprotocol(Interview2)werebasedonthese
learningvignettes,andimbeddedatpointsthroughoutthevideosothatteachers
wereabletodrawonspecificteachingmomentstheyhadjustwatchedwhen
reflectingonanswerstothequestions.Finally,theneuroscience‐basedmodelsof
learningpresentedtoteacherswereconstructedbasedonthesevideocasesin
ordertorelatetothespecificelementsandvariablescentraltoeach.Thesevideo
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vignettesandthecorrespondingneurosciencesupplementswillbediscussedin
furtherdetailintheMaterialsandMethodssectionofthispaper.
CommonLanguage
Thefinaltaskinassemblingtheresearchdesignwastoestablishacommon
languagetoallowformeaningfuldiscussionbetweenresearcherandteachers,and
guidetheincorporationofneurosciencedataintothisdiscussion.
Abasicfamiliaritywithbehavioralandeducationaltermswasassumedfor
subjectsduringconstructionofinterviewprotocolsandpresentationof
neuroscienceinformation.Carewastakentostripawayanypsychologicaljargon,
mathematicalterminology,andneurosciencelanguagethatmaynotbealreadypart
ofsubjects’vocabulary.Throughoutinterviews,closeattentionwaspaidtothe
languageandtermsusedbyanindividualsubject.Anyfollow‐upquestionsor
discussionremainedconsistentwiththisvocabularyandterminology,anddidnot
introducenewlanguage.
Itwasestablishedthatneuroscienceinformationwouldbepresentedto
teachersintwoways.First,aone‐pagesummaryofrecentcognitiveneuroscience
findingsrelevanttothespecificteachingmomenttheywatchedwasgivento
teacherstoread.Thiswrittensummarywasaccompaniedbyasimpleiconic
drawingofthebrain,withkeyareasdiscussedinthewrittensummaryclearly
labeled.Additionalpicturesandsymbolicelementsservedtoillustratethe
neuroscienceinformationinawayconsistentwiththefiguresanddiagramsone
mightfindinpre‐serviceteachereducationmaterials.Inbothofthese
supplementarymaterials,vocabularydidnotexceedthatwhichateachermight
comeintocontactwithincourseworkmaterials.Forexamplebrainareaswere
discussedasbeinginthe“front,”“back”or“side”ofthebrainratherthan“frontal”
“pre‐frontal”“occipital”or“temporal.”Neurosciencesupplementswillbediscussed
infurtherdetailinthematerialsandmethodssectionofthispaper.
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CollectionofPilotData
Thesecondgoalofthisprojectwastotesttheefficacyoftheresearchmodel
bycollectingpilotdata.Interviewswereplannedoverthecourseofthe2010–
2011academicyearwithearlychildhoodmatheducatorsfromvariouseducational
andphilosophicalbackgroundsthathadbeenteachingbetweentwoandfiveyears
inavarietyofpre‐schoolprograms.Itwasestablishedthateachinterviewwouldbe
videorecordedusingaMacintoshMacBookPro.Recordingsofinterviewswouldbe
editedandorganizedusingtheiMovieprogram.Finally,videoswouldbeexported
toQuicktimeandstoredonanexternalharddriveforlaterreviewandanalysis.
MaterialsandMethods
ProjectDesign
Subjectswererandomlyassignedtooneofthreeresearchvignettes:Number
Sense,Arithmetic,orPattern.Eachvignettewascomposedofthreeclinical
interviewsincluding(1)ashort,structuredinterviewaskingteacherstoprovide
informationontheirbackground,pathtoteaching,andphilosophicalapproach(2)a
semi‐structuredinterviewbuiltaroundvideocasesofearlymathteachingmoments
(3)amoreopen‐endedreflectiveinterviewfollowingpresentationofneuroscience
supplements.ResearchvignetteswerepresentedasaPowerPointPresentationona
MacintoshMacBookcomputer.Allclinicalinterviewswerevideorecordedusingthe
iMovieprogramonthesamecomputer.
ResearchVignettes
Thethreemathtopicschosenforthevignettesshareasolidfoundationof
relevantliteraturefromthefieldsofeducation,developmentalpsychology,cognitive
psychologyandmostrecently,cognitiveneuroscience.Thesetopicsarealsocentral
toquestionsguidingcurrentinvestigationintothedevelopmentofmathematical
thinkinginthebrain.
Eachvignetteincludeduniquevideocasesandneurosciencesupplements.
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VideoCases
AllvideoswerechosenfromtheVITAL(VideoInteractionsforTeachingand
Learning)vaultatTeacher’sCollege,ColumbiaUniversity.Eachislessthanthree
minutesinlength,andwasdividedintothreesegmentsshorterthanoneminute.
Eachvideowaschosentopresentateachingmomentthataimedatbuilding
orassessingachild’s/children’sunderstandingofeithernumbersense,arithmetic
orpattern.Belowarebriefdescriptionsofeachvideocase:
NumberSense
• CaseshowedteacherataprivateManhattanpreschoolinherownclassroomwithagroupofpreschoolstudents
• TeacheragreedtobevideorecordedwhileteachinglessonsadaptedfromtheBigMathForLittleKidscurriculum,whichisbasedoncurrentresearchabouthownumbersenseisbuilt
• Activityinvolvedaskingagroupofchildrentofind“fours”ofamongrepresentationsofdifferentnumbersandquantitiesonmulti‐coloredcardsspreadonthefloor:numeralswithandwithoutnumberwords,lines,dots,andpicturesofobjects.
o CLIP1:teacherintroducestasktochildreno CLIP2:veryyoungchildiscalleduptofindafour,usesatrial‐
and‐errorapproachuntilguidedtopickupcorrectcardbycolor
o CLIP3:olderchildcomesforwardandquicklyfindsacorrectcard
Arithmetic
• CaseshowedgraduatestudentatTeachersCollegeconductingateachinginterviewwithapreschoolstudentinaManhattanpreschoolclassroom
• Graduatestudentrecordedinterviewastraininginusingtheclinicalinterviewmethodtoassessandsupportchildren’slearningofarithmetic
• Activityinvolvedaskingchildtocountusingsmallgreenblocks.Usingagameof“addingapples,”childwaspresentedwithtwoadditionproblems
o CLIP1:Childcountstheblocksshehas(six)arrangedintworowsofthree
o CLIP2:Interviewgiveschildsevenmoreblocks,sheincorporatesthemintorowsandcountsthirteencorrectly
16
o CLIP3:Interviewergiveschildonemoreblock,sheincorporatesit,andincorrectlycountsthirteenagain
Pattern
• CaseshowedgraduatestudentatTeachersCollegeandteacher(PrincipalInvestigator)conductingateachinginterviewwithapreschoolstudentinteacher’sownclassroominMontessori‐basedprivateprogressivepreschool
• Graduatestudentrecordedinterviewastraininginusingtheclinicalinterviewmethodtoassessandsupportchildren’slearningofpattern/earlyalgebra
• Activityinvolved(1)askingchildtowatchteacherconstructanABABpatterninvolvingalternatinggroupsofobjectswhichshefirstdescribeswithlanguageand(2)askingchildtobuildasimilarpatternbyherselfwhichtheteacherdescribeswithlanguage
o CLIP1:Childwatchesteacherasshe“thinksofpatterninhermind,”describespatternverbally(“Threecameos,twobluestones…”)andthenconstructsit,placingcameoshorizontallyandstonesvertically
o CLIP2:Teacherdescribesnewpatternforchildto“holdinhermind:”(“Twocameos,threebluestones…)
o CLIP3:Childassemblespattern,preservingcorrectnumberingroupsandABABstructure,butbeginningwithbluestonesandplacingallgroupshorizontally
ClinicalInterviews
ProtocolsforInterviews1and3wereidenticalforallresearchvignettes.
(SeeAppendixAandC).
Thesecondclinicalinterviewprotocol(Interview2)soughttoprobe
teachers’in‐the‐momentreactionstoearlymathteachingmoments.Foreachofthe
threevignettes,numbersense,arithmeticandpattern,videoclipswereeditedinto
threesegmentswhichwereembeddedintoPowerPointslides.Segmentswere
basedonnaturaldivisionsintheteachingmoment,i.e.task‐setup,taskpresentation
bytheteacher,andtaskperformancebythestudent.
Aftereachclip,teacherswereaskedtwoquestions,foratotalofsix(See
AppendixB).Thoughthereisslightvariationintheorderingofthequestionsbased
ondifferencesineachvideo,thequestionsfollowabasiccontinuum:Thefirstsetof
17
questionsinvitesreflectiononstudentbehavior,suchasengagementandinteraction
withmaterials.Thesecondsetaskssubjectstoevaluatetheconceptualteaching
method,i.e.theconcepttheteacherwasattemptingtoteach,andtheapproachused
toteachit.Thelastsetofquestionsaimedatteachers’intuitionsregarding
cognition,includingthechild’sdevelopingmathematicalabilitiesandstrategiesin
solvingthetasks.
Inthisway,Interview2wasconsistentwiththecognitiveclinicalinterview
methodofprobingknowledgestructureinacareful,step‐upfashion.Questions
werescaffoldedtodescribeteachers’relativeplaceonacontinuumofknowledge
spanningbehavioral,educational,psychological,developmental,andcognitivelevels
ofanalysisregardingteachingandlearning.
Foreachvignette,Interview2includedadifferentconsiderationnotedafter
thelastsetofquestionsonthestudent’scognitiveprocess:
Fornumbersense,theinterviewerremainedparticularlyopentoexploration
ofwhythesecondchildwasabletoperformthetaskoffindingfoursomuchmore
quicklyandeasilythanthefirstchild.
Forarithmetic,theinterviewerremainedparticularlyopentoexplorationof
theuseofblocksinsolvingtheproblem,includingwhythechildarrangedtheblocks
thewayshedid,andhowthisaffectedherperformance/success.
Forpattern,theinterviewerremainedparticularlyopentoexplorationof
whythechildpreservedtheoverallstructureofthepattern,butdidnotpreserve
theorderdescribedwithlanguageandneglectedspatialdetails.
NeuroscienceSupplements
Theinformationpresentedintheneurosciencesupplementsprovided
teacherswithasimplifiedsummaryofcurrentandsoundcognitiveneuroscience
findingsrelatedtothetopicsofnumbersense,arithmeticandpattern.Ratherthan
givingteachersacompletemodeloraprescriptionforteaching,thesesupplements
insteadaimedtoestablishaproblemspacewithinwhichbothteacherand
researchercouldmeettoaskandanswerquestionsaboutthespecificteaching
momenttheywatched.
18
Neuroscienceinformationwaspresentedintwoways:(1)asaone‐page
printedsummaryofresearchfindingsand(2)asaniconicdiagramofthebrainwith
relevantlabels,picturesandsymbols.
Theprintedsupplementsfirstsummarizewhathascometobegenerally
agreeduponwithinthefieldofimagingresearchregardingthebrainareasactivein
thetypeofmathtaskpresentedinthevideocase.Nextthesupplementsbriefly
outlinehowbrainpathwaysandconnectionsmightsupportperformanceofthese
mathematicaltasksduringthepreschoolyears.Finally,thesupplementspresent
teacherswithasuggestionorthrustoftheresearchtoguidespeculationandopen
thewayforfurtherquestionsanddebate.Thesedescriptionsaresimpleand
strippedofanyneuroscientificlanguagethatmaybeunfamiliartoteachers.For
example,“takingpicturesofadultsandchildren’sbrains”isusedinsteadof“PET”or
“fMRI.”Additionally,byusinglanguagesuchas“researcherswanttoknow,”
“researchersposethat,”“somestudiessuggest,”and“children’sbrains…seemto
have,”thesupplementsaimedatcreatingspaceforthecollaborativeconstruction,
throughactivebi‐directionaldialogue,ofmodelsincorporatingcognitive
neurosciencefindingsintoateacher’sownexperience‐basedintuitivemodelof
mathlearning.
Thepicturesupplementssupportedprintedinformationinavisualformat.
Thesediagramsweresimilartofiguresexplaininglearningprocessesthatteachers
mightencounterinapre‐serviceorgraduateteachereducationtextbook.The
picturesupplementsshowedasimplepicture/picturesofthebrainthatwere
biologicallyaccuratewithwell‐definedgyributwithoutcoloredregionsor
superimposedoutlines.Brainareasdescribedintheprintedsupplementwere
labeledclearlywithwords,i.e.vision,generalreasoning,language,workingmemory.
Simpleandcolorfulimagesweregatheredonlineandlaidoutingeneralproximity
tocorrespondingbrainregions.Forinstance,acardwiththeword“Five”was
placedbesidelabeledlanguageareasinthelefthemisphere,whilealineofrubber
duckswasplacedbesidevisuo‐spatialareasintherighthemisphere.
Itisimportanttonotethattheseneurosciencesupplementswerecarefully
constructedbytheprincipalinvestigator,basedonnearlytwoyearsofstudying
19
peer‐reviewedliteratureandgraduatecourseworkincognitivepsychology,
developmentalpsychology,cognitiveneuroscienceandeducation.Informationwas
drawnfromnumerous,wide‐spreadandqualifiedsources,andpresentedinaway
thatthePrincipalInvestigatorfeltwouldbemeaningfulandeasilyunderstoodby
teachers,basedonfiveyearsofexperienceteachinganddevelopingmathcurricula
inanearlychildhoodclassroom.
Carewasputintopresenting“highqualityknowledgeonthebrainin
digestibleformandinterpretingneurosciencefromtheperspectiveofandinthe
languageofeducators(Goswami,2006).”Thus,theinformationintheneuroscience
supplementsdoesnotinvolvedirectconnectionsbetweenpiecemealfindingsand
recommendationsforteaching.Rather,thestatementsintheneuroscience
supplementsrepresentthecurrentthrustoftheliteratureonmathlearningandthe
brain,builtinturnonafoundationofeducational,developmentalandcognitive
psychology.
Furthermore,themodelsarepresentedintheircurrent‐andincomplete‐
theoreticalstate,withcorrespondingquestions,ambiguitiesandgapsinknowledge.
Thiswasdoneinhopesthatcurrentmodelscouldbeanalyzedandextended
collaborativelybybothteacherandresearcher.Acompletetheoreticalbackground
ofthedevelopingtheoriesonmathlearningpresentedintheneuroscience
supplementsisbeyondthescopeofthispaper.Evenso,afullreferencesection,
includingmanysolidreviews,hasbeenprovidedafterneurosciencesupplementsin
AppendixD.
PilotDataAcquisition
Subjects
SixteacherswererecruitedfromarangeofNewYorkCitypreschools,
includingpublicPre‐K,private,andprogressiveprograms.Allteacherswerefemale,
andhadbeenteachinginearlychildhoodbetweentwoandsevenyears.Subjects
20
wereeithercurrentlyenrolledinorhadcompletedanEarlyChildhoodEducation
degreeprogram.Noteachershadcompletedcourseworkinneuroscience.
Subjectswereassignedrandomlytoonofthethreeresearchvignettes,so
thattwointerviewswerecollectedforNumberSense,twoforArithmetic,andtwo
forPattern.Partlyduetothedesiretokeepinterviewsessionsnaturalandnot
overly“scientific,”verbalconsentwasobtainedpriortointerview.Teacherswere
informedthatpilotdatawasbeinggathered,andthatwrittenconsentwouldbe
obtainedbeforeanyrecordedmaterialwasshownpubliclyoronline,andbeforeany
personalquoteswerepublished.
Methods
Interviewswereconductedatquiet,familiar,yetneutrallocations,including
emptyschoolclassrooms,orroomsinresidencesotherthanthesubjects’home.
Teacherswerecomfortablyseatedwithinterviewerbeforethecomputerandbriefly
toldthattheywouldbewatchingsomevideosofpreschoolchildrenlearningmath,
andreflectingonwhattheysaw.Noise‐blockingheadphoneswereprovidedwhile
watchingvideoclips.
Interview1wasconducted.SubjectsthenwereshownthePowerPoint
Presentation.Slide1informedthesubjectthattheywouldbewatchingapre‐school
agedchild/pre‐schoolchildreninaManhattanpreschool.Slides2–4showedvideo
clips1–3.Betweeneachclip,subjectswererecordedansweringtwoquestionsto
completeInterview2.PowerpointpresentationsaswellastheiMovieprogram
werekeptinseparatewindowsonthedesktoptoallowforeasytransitionbetween
presentationandrecording.
Subjectswerepresentedwithneurosciencesupplementsuponcompletionof
Interview2.Printedmaterialwasinpaperform,andthepicturewasshownasthe
lastPowerPointslide.Subjectsweregivenasmuchtimeasneededtogetageneral
senseofthematerial,andwereassuredthattheydidnotneedtomemorizeor
repeatinformation.Oncesubjectsconfirmedthattheyfeltsufficientlyfamiliarized
withtheneurosciencesupplements,Interview3wasconducted.
21
Aspreviouslymentioned,Interview3wasamoreopen‐endedandreflective
clinicalinterviewthatattemptedtofindmatchesbetweenteachers’intuitionsand
neuroscienceinformation,evidenceofmisconceptionsaboutneuroscience,andany
teacherperspectivesorknowledgeregardingquestionsorholesincurrenttheories
ofmathlearningintheyoungbrain.
Inordertoexploretheseconnections,follow‐upquestionswereaskedto
furtherexposeteachers’thoughtsandbeliefs.Specificconsiderationsforeach
researchvignetteareoutlinedbelow.
NumberSense
• Howdoteacher’sintuitionsabouttheimportanceofbuildingastrongnumbersenseinpreschoolchildrenmatcheducationalresearch?
• Whatintuitionsdoesteacherhaveregardingdifferentkindsofmathematicalinformationrepresenteddifferentwaysinthebrain?
• Whatevidencedoesteacher’sreflectionrevealofbrainmythsaboutdifferentkindsoflearners(i.e.visualorverbal)or“exercisingthewholebrain?”
• Particularopennesswasgiventoanyintuitionsabouthowtobestsupportthebuildingofnumbersenseasaconcept
• Considerationwasgiventothefactthatteachershadwatchedimplementationofaresearch‐basedcurriculum:BigMathForLittleKids(Greenes,Ginsburg&Balfanz,2004).
Arithmetic
• Howdoteacher’sintuitionsabouthowandwhentosupportarithmeticskillsmatcheducationalresearch?
• Whatintuitionsdoesteacherhaveregardingthedevelopmentofarithmetic,andhowthisinteractswiththeneedforconcreteobjectsandtheuseofsymbols?
• Whatevidencedoesteacher’sreflectionrevealofbrainmythsaboutpurelyconstructivistenrichmentmodelsorabuiltin“mathcenter”inthebrain?
• Particularopennesswasgiventoanyintuitionsabouthowandwhentheshifttolanguage‐basedmathfactshappensorshouldhappentobestsupportmathematicaldevelopment.
• Considerationwasgiventothefactthatteachershadwatchedateachingmomentbuiltontheclinicalinterviewmethod,whichallowedfreeuseofconcretematerialsforproblemsolving,butwasalsoteacherdirected.
22
Pattern
• Howdoteacher’sintuitionsabouttheteachingpatternsupportcurrentfindingineducationalresearchaboutchildren’suseofvisuo‐spatialskillsinmathtasks?
• Whatintuitionsdoesteacherhaveregardinghowsuchtasksdrawonlanguage,mathfactsandvisuo‐spatialreasoning?
• Whatevidencedoesteacher’sreflectionrevealofbrainmythsaboutright/leftbrainlearners?
• Particularopennesswasgiventoanyintuitionsabouttheroleofvisuo‐spatialabilitiesinbuildingthedeepmathematicalunderstandingneededforhigher‐levelmathskillssuchasalgebraicthinking.
• Considerationwasgiventothefactthatteachershadwatchedateachingmomentthatwasbasedonaprogressivevisuo‐spatialmethodofmathinstruction(theMontessorimethod).
AftercompletionofInterview3,teacherswerethankedfortheirparticipationanddismissed.
PilotDataAnalysis
Theresearchdesignyieldedasmallandcohesivesetofpilotdata,whilestill
allowingforatleastonedegreeofcomparisonbothacrossandwithinvignettes.The
sixinterviewsubjectswerecurrentlyteachinginavarietyofpre‐schoolclassrooms,
includingpublicPre‐K,privateandprogressiveprograms.Theyhadvaried
backgroundexperiences,rangingfromdaycarecenterstoprivateandlabpreschools.
Theyalsohadvariededucation,spanningpartiallycompletedcourseworkforaBAin
EarlyChildhoodwithpendingcertification,toanMAinEarlyChildhoodwith
certification.Subjectswererandomlyassignedtooneofthreeresearchvignettes.
Thus,sixinterviewswereanalyzed,twoforeachresearchvignette:NumberSense,
ArithmeticandPattern.
Thegoalofpreliminarydataanalysiswasto(1)determinetheefficacyofthe
researchdesigninengaging/changingteacher’smodelsofstudents’thinking(2)
identifyimportantvariablesandobservationcategoriesforfurtherresearch.
Interviewswereconsideredasindividualcases.Recordedinterviewswerewatched
repeatedlyinordertoorganizeanswerstointerviewquestions.
23
Inordertoaddresstheoverallefficacyoftheresearchdesigninengaging
teacher’smentalmodels,answerstoInterview2weretranscribedandcompared
acrossallsubjects.AspreviouslydiscussedinMaterialsandMethods,Interview2
questionsproceededonacontinuum,callingonteachers’independentabilitiesto
utilizebehavioralobservation,educationalandmathematicslanguage,
developmentalpsychology,andfinallycognitivepsychologyintheiranswers.Three
observationcategoriesweresetup,basedontheoverarchinglevelofanalysisthat
questionsinvitedteacherstouseintheirreflectiononthevideocasesegments.
(1) Areteachersabletocomfortablyanswerquestionsbasedonobservationsof
children’sbehaviorinagiventeachingmoment?
(2) Areteachersabletoeasilyidentifytheconceptbeingtaughtanddiscusswhy
theywoulduseasimilar/differentapproach?
(3) Willteacher’sdiscussionofstudents’developingmathabilitiesandstrategies
revealanexplanatorymodelofchildren’scognitiveprocessesduringthe
teachingmoment?
Forthefinalquestion,answerswerereviewedtodetermineifateacher’s
intuitivemodelwasabletoprovideanadequateexplanationregardingspecific
considerationsforeachmathtopic.ForNumberSense,whywasthesecondchild
wasabletoperformthetasksomuchmorequicklyandeasily?ForArithmetic,what
wasthesignificanceofthechild’suseofblocks?ForPattern,whydidthechild
preservethestructureofthepattern,butnottheorderprescribedinlanguage,or
thespatialorientationofthematerials?
Inordertoassesshowtheneuroscienceinformationpresentedinthe
supplementsengagedteachers’currentmodels,answers1‐3ofInterview3were
transcribedandcomparedacrossallsubjects,includinganswerstofollow‐up
questions.Answerswereorganizedaccordingtothelevelatwhichthedeveloping
modelsofmathematicalthinkinginpreschoolstudents,presentedineducational
neuroscience,matchedteachers’ownintuitivemodelsasdescribedinInterview2.
Threeobservationcategoriesweresetup,basedonthespecificconsiderations
24
guidingtheinterview,whichareoutlinedinMaterialsandMethods.These
considerationsaimedtoidentify:
(1) Howteachers’intuitivemodelsmatchthefoundationaldevelopmentalor
educationalpsychologytheoriesonthemathtopic.
(2) Howteachers’intuitivemodelsmatchthemodelsofmathematicalthinking
presentedbycognitivepsychology,onwhichcognitiveneurosciencemodels
arebased.
(3) Evidenceofbrainmythsspecificallyrelevanttothemathtopic.
Inordertoidentifyhowneuroscienceinformationmightimpactorchangea
teacher’scurrentmodelofmathematicalthinking,answerstoInterview3,including
question4,werereviewedinordertogathercasedescriptivesonhowthe
neuroscienceinformation(1)resolvedquestionsorambiguitiesregardinga
student’sthoughtprocess,(2)changedateachers’opinionsorbeliefsabout
children’smathematicalthinking,or(3)gaveateachernewinsightintostudents’
thinking.
Toassesstheabilityoftheresearchdesigninestablishingacommon
languagebetweenresearchersandteachers,interviewswerethoroughlycombed
for(1)termsthatwerenoteasilyunderstoodbyteachers,(2)instanceswhere
teacherswereabletosubstitutetheirowndefinitionsofforgottenorunfarmiliar
terms,(3)phrasesthatexemplifymeaningfulcommunication,includingteachers’
descriptionsofexperience‐basedintuitionsandactivesense‐makingof
neuroscienceinformation.
Finally,allinterviewsweresearchedforanyinstanceswhereteacher’s
intuitionscouldbeseenasguidingcurrentquestionsineducationalneuroscience
research.
25
PreliminaryResultsandDiscussion
Forthepurposeofdiscussion,subjectsarelabeledTeacher1–Teacher6.
Teachers1and2participatedintheNumberSensevignette.Teachers3and4
participatedintheArithmeticvignette.Teachers5and6participatedinthePattern
Vignette.
Interview2:TeacherMentalModels
Behavior
Basedonthesixinterviewsconductedforthisproject,mostteacherswere
comfortablereflectingonstudent’sbehavior,andallbutonewereabletoassess
studentengagementandthesignificanceofspecificbehaviorstothemathtask
beingtaught.Teacher1explainedthatshedidnotknowifstudentswatchingthe
lessonwereengagedfromtheirbehavior.Allotherteachersdescribedstudents’
being“quiet,”or“fixated,”“orienting,”“listening,”andansweringquestions“right
away.”Allteacherswereabletoexplainwhyastudentpickedupa“red”/”orange”
card,identifytheskillastudentwasdemonstratingbycountingblocks,andrelate
thesignificanceinachild’sassemblyofapattern.
ConceptualTeachingApproach
Allteacherswereabletodiscussthemathconceptbeingtaught,aswellas
theiralignmentwiththeteachingmethod.Differencesdidemergebetween
vignettesonteachers’abilitytonamethemathconceptanduseeducationalor
mathematicallanguage.Forinstance,bothteachersstruggledtonametheconcept
ofNumberSense.Teacher1said,“Whatfourlookslike…aconceptualunderstanding
thatnumbersarerepresentedindifferentways.”Sheadmittedthattheirwasaterm
forthisconcept,andwasembarrassedthatshecouldnotrememberit.Teacher2
answered,“Differentnumbers,differentwaystheyarerepresented...waysyoucan
showfour.”TeachershadnoproblemintheArithmeticvignette,bothreplying
“addition,”rightaway.TherewassomeuncertaintyinthePatternvignette,duein
bothcasestothedifferentelementsincorporatedinthepatterntask(counting,
ordering,spatiallayout),thoughbothteachersidentifiedthatthetaskinvolved
26
“patterns.”Teacher5described“unitsofnumber,patterns,grouping,visual
patterning,”whileTeacher6describeda“numericalrepeatingpattern–groupsof
numbers.”
BothsubjectsintheNumberSensevignetteclaimedthattheywouldteach
thisconceptdifferently.Teacher1statedthatthelessoncouldbetterbetaughtin
smallergroups,withalongerandmoreflexibletimecourseforeachstudentto
answerquestions.Thiswouldallowtheteachertobetterassessstudent
understanding.Shealsoclaimedthatmoresensory/tactilesupportwasneeded.
Teacher2claimedthatratherthanleadingchildrentotheanswerwithclues(the
teacherguidedthechildtopickupanorangecard),theteachershouldhavelether
pickupawrongcard,andthendiscusswhyitdidnotshowfour.Bothteachersin
theArithmeticvignetteclaimedthattheywouldteachthisconceptinasimilarway,
usingmanipulatives,thoughTeacher2statedshewouldstartwithasmaller
numberofblocksfirsttoseewherecountingabilitiesbrokedown.Bothteachersin
thePatternvignetteclaimedtheywouldusethesameteachingmethodofmodeling
thepatternwithlanguagebeforeaskingthechildtolayitout.Teacher5addedthat
shewouldneedtofirstassesswhatindividualpatternabilitiesthestudenthad
beforepresentingalltheelementstogetherinonetask.
Cognition
TeachersanswerstothefinalquestionsinInterview2revealedmixed
abilitiesindiscussingstudents’cognitiveprocesses,includingthemathabilitiesand
mentalstrategiesusedtosolvemathtasks.ForNumberSense,thequestions
probedteachers’intuitionsaboutwhythesecondchildwasabletosolvethetask
morequicklyandeasilythanthefirst.Teacher1claimedthatthetaskdrewon“rote
counting”and“onetoonecorrespondence,”andposedthatthesecondchildhadhad
previousexposuretosymbolsandmoreexposuretosimilaractivities,andthuswas
abletopickupthecardwithlessscaffolding.Teacher2claimedthatthechild
neededtoknow“whatthenumberfouris,”havea”literacyofhowtoshowfour,”and
neededtobeabletocounttofour.Sheposedthatthesecondchildwaspaying
closerattentiontotheactivityandhadalreadyscannedthecard.Sheoutlinedhis
27
strategyaslooking,counting,matchingdotstonumbers....”seeingfour,countingfour
andrecognizingthesymbol.”
ForArithmetic,thequestionsprobedteachers’intuitionsregardingthe
child’suseofblocksinsolvingthetask.Teacher3claimedthattheadditiontask
requiredthechildtousecountingskills,and“remember”theamount13,thoughshe
admittedshedidnotknowthecorrectterminologyforthisskill.Teacher3claimed
thatthechild’sstrategyinvolvedaddingtheblocksintoherowngroupto“make
themhers”andthencounting.Sheclaimedthatthechildsimplymessedupinher
counting,butthatherstrategywassound.Teacher4claimedthatthechilddrewon
onetoonecorrespondence,aswellasvisuo‐spatialskillsinarrangingtheblocksin
rows.Sheconcludedthatthechild’sstrategyledhertosolvethetaskincorrectly,
guessingthatperhapsshe“gotstuck”onthenumber13.
ForPattern,thequestionsprobedteachers’intuitionsregardingwhythe
childpreservedtheoverallstructureofthepattern,butdidnotfollowtheorderas
prescribedintheteacher’slanguage,northespatialorientationofthematerials.
Teacher5claimedthatthechildneededtobeabletocounttothree,understand
whattwoandthreemean,andhavesomefamiliaritywithpattern.Sheposedthat
thechildlistenedandrememberedthepatterninherheadandthenstartedfrom
thebeginninginassemblingthepattern.Teacher5admittedthatshedidnotknow
whythechilddidnotreproducetheexactorderandspatiallayout,guessingthatshe
“focusedmore”onthepattern.Teacher6alsoclaimedthatthechildneededto
understandnumbers,haveaconceptofpattern,andbeabletocount.Sheguessed
thatthechildsimply“associatedtwowithcameosandthreewithstones,”and“let
thatbewhatwasfloatinginhermind…Shewasn’tthinkingaboutorder.”
Interview3:ImpactofNeuroscienceInformation
AnswerstoInterview3revealeddifferencesbetweensubjectsregardinghow
individuals’intuitionsmatchededucational,developmentalandcognitive
psychologytheoriesregardingspecificmathconcepts.Whileexaminationofeach
interviewisbeyondthescopeofthisdiscussion,theinterviewsconductedinthe
NumberSensevignetteserveasanexampleofthesedifferences.
28
Teacher1feltthattheneuroscienceinformationpresentedwasaboutinitial
developmentofmathconcepts,anddemonstratedtheimportanceofmaking
connectionsbetweendifferentkindsofmathinformation.Shewentontodescribe
howdifferentpartsofthebrainare“active,”andmathinformationneedstobe
“dispersed”and“connected”throughoutthebrain.Teacher1statedthattheidea
that“mathconceptsarestoredindifferentareasofthebrainwouldsuggesttomethat
youwanttoexposeorconnectthosemathconceptstoasmanyareasasyoucan.”
Ultimately,shefeltthatthissupportedherintuitionthatmoresensoryandtactile
informationshouldbeincorporatedintotheteachingofthisconcept.Inlightofthis,
shefeltthatthelessonshehadreliedmostlyonlanguage,butlackedinboth
adequatetime,andsensoryinteractionwithconcretematerials.
Incontrast,Teacher2feltthattheneuroscienceinformationsupportedher
intuitionthatthroughoutmathematicaldevelopment,“allchildrenaredevelopingin
brainfunctionsdifferently.”Thereforesomeareasarestrongerthanothers.Teacher
2feltthissupportedherintuitionthatchildrenlearnindifferentwaysand“ifone
wayisn’tworking”ateachercan“gowiththeother.”Ratherthanseeingan
importanceinteachersguidingstudentstobuildconnections,shefeltthatmath
learningshouldbelessteacher‐directed.Aslongasinformationisprovidedasboth
“numbersandtallies”thechildrenwill“figureouttheconnectionfromthere.”
Researchonthetopicofnumbersenseconfirmsthattheconceptunderlies
latermathematicalabilities(Baroody,2010;Dehaene,1997;Dehaene,Piazza,Pinel
&Cohen,2003).Cognitivepsychologyandneuroscienceliteratureshowsthat
mathematicalinformationisrepresentedindifferentareasofthebrain(Delazer&
Benke,1997;Eger,Sterzer,Russ,Giraud,&Kleinschmidt,2003;Lemer,Dehaene,
Spelke&Cohen,2003)andthatsymbolicinformationismappedontomoreabstract
numericalknowledgethroughexperiences,instructionandpractice(Gilmore,
Indefrey,Steinmetz&Kleinshmidt,2001;Lipton&Spelke,2005;Piazza,Pinel,
LeBihan&Dehaene,2007;Strauss,2003).
StudiessuchasthatofBaroody,EilandandThompson(2009)have
investigatedthesuccessofvariousmethodsofinstruction(structuredorsemi‐
structureddiscovery,directinstructionandunstructuredpractice)forfostering
29
preschooler’snumbersense.InlinewiththeintuitionsofTeacher1,recent
researchisshowingthatthebrainlearnscomplexconceptsbestwhentheyare
taughtandexperiencedthroughvarioussensorystimuli(Rains,Kelly,&Durham,
2008;Tokuhama‐Espinosa,2008b;TokuhamaEspinosa,2010).
TheintuitionsofTeacher2showedsomepossibleinterferencefrom
misconceptionsaboutthesignificanceofmathematicalinformationbeing
representedindifferentareasofthebrain.Heranswersseemtoshowsome
evidenceofbrainmythsregardingvisualversusverballearners,andisolatedparts
ofthebrainbeing“stronger”thanotherslikemuscles.Workingfromherown
model,whichlikelywasimpactedbyherbackgroundinSpecialEducation,she
interpretsthefindingstomeanthat“stronger”partsofthebraincanbeusedif
othersaren’t“working.”
Educationalresearchdoeslendsomesupporttotheideathatdifferentiated
instruction(allowingstudentstolearnattheirownlevelandpace)makessense
givenstudents’differentintelligencesandcognitivepreferences(Tomlinson,1999,
Tomlinson&McTighe,2006).However,theeducationalneuroscienceliterature
cautionsthatdespiteitssuccessinclassrooms,moreresearchneedstobe
conductedinordertoestablishwhydifferentiatedinstructionappearstowork
basedonbrainresearch.Differentiationbasedondifferentintelligencesand
cognitivepreferencesremainsintelligentspeculation(Tokuhama‐Espinosa,2010).
Furthermore,thinkingaboutthewaysdifferentchildrenmighttakeinand
learnnewinformationisaseparateconsiderationfromhowconceptual
understandingisfosteredandachieved.Inotherwords,thoughchildrenmight
showstrengthsorweaknessesforvariouskindsofmathematicalinformation,this
doesnotmeanthatbuildingstrongknowledgeofandconnectionsbetweenallof
thesetypesofinformationisnotstillvitalforconceptualdevelopment.
ReviewinganswersforInterview3yieldedsubstantialevidenceof
neuroscienceinformationhavingimpactedorchangedteachers’currentmodels.
ThereflectionsofTeacher6provideanexampleofhowtheneuroscience
informationresolvedambiguitiesinherentinthevideocase.Atfirst,thesubject
couldnotfigureouthowthevideodemonstratedthestudentfavoringorrelyingon
30
visuo‐spatialabilitiesovermathfactsandlanguageinsolvingthepatterntask;
ratherthetwoseemedtobeworkingintandem.
Throughconsiderationofwhatshehasseeninherownclassroom,Teacher6
wasabletoconstructanewmodelofhowvisuo‐spatialabilitiesinteractwithmath
factsandlanguagetoyieldadeeperunderstandingofnumericalpatterns.She
describeshowachangeoccurredinherstudent’sunderstandingofonehundredas
theycountedthedaysofschoolandhungnumbersinrowsoftenonthewall.As
soonastheywereableto“see”theremainingnumberstoonehundredontheir
hands,theirguessesabouthowmanydayswereleftbecamemuchmoreaccurate.It
wasasifthephysicalandvisualconnectiontotheideahelpedtheminternalizeit.
SheconcludedthataboutninetypercentofthemathworkinherMontessori‐based
classroomisindeedbuiltontheideathatmathematicalunderstandingmustinvolve
visualrepresentationsofmathematicalpatterns,suchasatriangleofcoloredbarsof
beadsthatrepresentnumbersonethroughten.Asthesebeadsaremanipulated,
arranged,andphysicallymatchedtonumberedtiles,solidmathematical
understandingisbuilt.
ThereflectionsofTeacher3provideanexampleofhowtheneuroscience
informationchangedteachers’opinionsorbeliefsregardingthemathtopic.She
relatedhow,inherownclassroom,shehadalwaysusedbothsymbolsandconcrete
objectsforchildrenofanyagewhenteachingbeginningaddition,aslongasthey
knownumbersandamounts.Afterconsideringthedevelopmentalshiftthatoccurs
fromreasoningandworkingmemorytosymbolsandmathematicallanguage,the
subjectmused,“Nowthatwe’retalkingaboutit,Ineverthoughtofthereasonofwhy
wewereusingthem…that’sjustwhatyoudowhenyou’reteaching.Ididn’tthink
aboutthewaychildren’smindsworktothatextent…Wow,childrenmaybethreeand
underareworkingonjustwhattheycanholdintheirminds.”
ThereflectionsofTeacher2provideanexampleofhowtheneuroscience
informationgaveteachersnewinsightintostudent’sthinking.Atfirst,sheclaimed
thatnumbersenseasaconceptwasnotsomethingthatwasexplicitlytaughtinher
emergent,child‐centeredclassroom.Afterconsideringhowmathematical
informationisrepresentedinthebrainandconnectedintotheconceptofnumber
31
sense,sherealizedthatherstudents’discoveriesduringblockplaydemonstrated
how“differentareasprocessdifferentrepresentationsofnumbers.”Theyseeand
handlequantities,uselanguagetocount,comparemoreorless,andusemathto
“figurethingsout.”Inthiswaythereis“numbersenseallovertheplace.”Teacher2
saidtheneuroscienceinformationremindedherthat“sometimesyouarenot
understandingwhatkidsareunderstanding…,”that“[neuroscience]isanotherwayof
observingthechild.”
CommonLanguage
Inthesixinterviewsconductedtherewerefewinstanceswhereteachers
werenotabletounderstandterms.Theflexibilityoftheclinicalinterviewallowed
theresearchertorepeatandre‐wordphrasesslightlysothatteacherswereableto
succeedinunderstandingandansweringallquestions.Onecuriousfindingwasthat
allsixteacherspausedforsomelengthwhileansweringquestion5ofInterview2,
whichaskedthemtolistthemathabilitiesthatthechildneededtohavetosolvethe
task.Atsomepointduringtheiranswer,allsixteachersalsorepeatedtheterm
“mathabilities”tothemselves,whichaidedtheminthinkingofafewmore.
Therewereseveralinstancesofteachersnotbeingabletorememberor
accessthecorrecteducationalormathematicaltermforanabilityorconcept,but
thisdidnotdetertheminsubstitutingadequatedefinitionsintheirownwords.As
examples,Teacher2used“whatnumberfouris…aliteracyofhowtoshowfour,”in
theplaceofnumbersense.Teacher1used“whatfourlookslike,”whenshecannot
recalltheproperterm,likelyinplaceofthetermsubitize.Teacher3admittedshe
wasnotsureof“terminology”formathabilitiesinvolvedinaddition,butwasableto
identify“countingskills”and“rememberingtheamountof13,”whichservesasa
fairdefinitionforcardinality.
Mostimportantly,reviewoftheinterviewsyieldedmanyphrasesthat
exemplifymeaningfulcommunicationbetweeneducatorandresearcherregarding
students’mathematicalthinkingasitrelatestocognitiveneurosciencefindings.As
Teacher1stated,knowingthatmathconceptscanbestoredindifferentareasofthe
brain“[suggests]tomethatyouwanttoexpose–orconnect–thosemathconceptsto
32
asmanyareasasyoucan.”Asteacher3articulated,theneuroscienceperspectiveis
“veryexciting,eventhoughit’sscientific…teachingissuchanintuitive
science…youngerchildrenrespondmoretoobjectsbecausetheyusetheirworking
memory,reasoning,whattheyknow…ItjustconfirmsthattheywaythatIapproach
teachingmathisawaythatworksforchildrenofthisage.”AsTeacher5stated,“I’m
interestedtoknowmore…it’snotenoughtohelpmebeabetterteacher.Someofitis
whatIalreadyknow…so,youknowthesethingsareindifferentpartsofthebrain,
and…soIshoulddowhatbetter?”
Foreachofthethreemathtopicsexploredintheseinterviews,important
questionsemergedduringteacherinterviewsthatcanbeseenasparticularly
relevantinthattheytouchuponareasattheforefrontofthebodyofresearchon
whichtheneurosciencesupplementsdrew(FullReferencesinAppendixD).For
NumberSense,thesequestionsconcerntheteacher’sroleinbuildingstrongnumber
senseinstudents.Whatistheimportanceofadequatetimeforstudentstoexplore
materialsandsolvetasks?Howimportantissensoryortactileinformationin
buildingthisconcept?Whataspectsofnumbersensedochildrenbuildontheir
own,andwhichrequiredirectinstructionforconnectionstobemadebetween
differentkindsofmathematicalinformationasitisstoredinthebrain?
ForArithmetic,thesequestionsconcernwhenandhowtheshiftfrom
relianceonconcreteobjectstovisualandlanguageabilitiesoccursorshouldoccur.
Atwhatageindevelopment(somewherebetweenthreeandfiveaccordingto
teachersinterviewed)dochildrennaturallymakethisshift?Howdoeslanguage
learninginteractwiththisshift,ie(asTeacher4observedinherownmulti‐lingual
classroom),whydoEnglishlanguagelearnersseemtohaveanadvancedabilityto
solveadditionproblemsusinglanguage?
ForPattern,thesequestionsconcerntheroleofvisuo‐spatialabilitiesin
understandingofpattern.Howshouldteachingofpatternincorporateandbuild
differentelementsofpattern(language,counting,numbersense,sensory,visuo‐
spatial)intoinstruction?Whatevidenceistherethatstudentsneedthevisuo‐
spatialelementinordertorecognizeandunderstandmathematicalpatterns,a
33
valuablecomponentofalgebraicandabstractmathematicalthinking,ina
meaningfulway?
Discussion
Accordingtothepreliminarydata,thisresearchdesignachievedtheaimof
establishingcommonmodelsandlanguageinordertoengendermeaningful
discussionbetweenresearcherandteachers.Videocasesservedasafunctional
problemspacewithinwhichinterviewerandintervieweecouldmeettoexplore
varioustopicsinearlychildhoodmathematics.Theclinicalinterviewtechnique
servedit’spurposeingatheringteacher’sintuitivementalmodelsatthelevelof
theirownthoughtsandbeliefs,andprovidedarichsourceofdescriptivedataon
howneuroscienceinformationengaged/impactedteachers’currentmodels.Finally,
analysisofthisdatarevealedobservationcategoriesthatcanbeusedforuseofthe
researchmodelinfurtherstudy.Interestinginteractionsbetweenteachers’
intuitionsandgapsinneuroscientificknowledgepointthewaybacktothebrainlab.
Whilethisstudywasdesignedasagenerativemethod,theresultsopenthe
wayforapplicationofmoreconvergentmethods.Theusetheobservation
categoriesandvariablesdiscussedabovecancontinuetorefineongoinggenerative
research,viabothinductiveanddeductivemethods(Clement,2000).
Infurtherstudies,answerstoInterview2maybeassignedvaluesaccording
toTeacherEfficacyandTeacherBeliefscodingschemessimilartothoseemployed
incurrenteducationalresearchusingvideocasestoassessteacherknowledge(see
Rosenfeld,2010forareview).
Arubriccouldalsobedesignedtoratetheimpactofneuroscience
informationonteacher’scurrentbeliefs.Commonelements,elementsthatappear
inmorethanfiftypercentofinterviews,couldbecollectedforteachers’modelsand
language.Inthisway,alargersetofdatacouldbeanalyzedwithincreased
reliability.
Ultimately,thisprojectdemonstratedthattheteachersinterviewedhave
stronginstinctsthatmatch,question,andchallengecurrenteducational
neurosciencefindings,andtheycanbevaluablecontributorsinidentifying
34
importantavenuesforfurtherresearch.AsTokuhama‐Espinosa(2010)states:
“Greatteachershavealwayssensewhatmethodsworked;thankstobrain‐imaging
technologyandbetterresearchtechniques,itisnowpossibletosubstantiatemany
ofthesebeliefswithempiricalscientificresearch.”
ProjectSignificance
Thisprojectapproachestheriftbetweenneuroscienceandeducationfroma
trulyneuroeducationalperspective.AccordingtoKurtFischer,headoftheMind
BrainandEducationinitiativeatHarvard,whateducationneedsisnota“quickfix”
fromneuroscience,butratherthecreationofanewfieldthatintegrates
neuroscienceandothercognitivescienceswitheducation.(FischerandImmordino‐
Yang,2008).Intermsofitsresearchgoals,mechanismsandmethodology,this
projectcanbeseenasatruecollaborationbetweendevelopmentalandcognitive
psychology,educationandneuroscience.
Theprojectdesignrepresentsanewresearchmethod,assembledfromtime‐
honoredmethodsthatsharetheconvictionthatcloseandcarefulobservationofthe
childorstudentiswhatallowsustoenterhismind.(Piaget,1975,Ginsburg,1997,
Kilbane,2008).Thesemethodscannotbeconductedincontrolledlaboratories
wheresubjectsspendafewhourssigningformsandbeingmeasuredwithmachines.
Whilesuchstudiescertainlyhavetheirplaceinscientificstudy,thesuccessful
educationalneuroscienceresearchermustspendtimewithteachers,innatural
settings,havinghonest,respectfulandmeaningfuldiscourse(Goswami,2006).True
andequalcollaborationneedsasharedlanguageandwaysofthinking,andthese
taketime.
ThePrincipalInvestigatorofthecurrentprojecthascompletedboth
undergraduateandgraduatedegreesincognitiveandeducationalneuroscience,and
hasovertwoyearsoftrainingusingthecognitiveclinicalinterviewmethodto
assessbothchildren’smathematicalthinking,andteacher’sknowledgeofstudent
thinking.Shehasalsoco‐developedtheprogramandcurriculaataprogressive
preschool,andhasbeenteachinginthisprogramforfiveyears.Inthissense,the
35
designofthisprojectcanbeseenasanexampleofinnovativeworkbyanew
generationofmulti‐disciplinaryresearcherwhoisdedicatingcombinedexpertisein
bothneuroscienceandeducation“topresentinghighqualityknowledgeonthe
brainindigestibleformandinterpretingneurosciencefromtheperspectiveofandin
thelanguageofeducators,”inorderto“fostersuccessfulexchangeofrelevantdata
betweendisciplines(Goswami,2006).”
Theoverarchinggoaloftheproposedresearchistobeginbuildinga
foundationforcollaborativeexperimentdesignbetweeneducatorsand
neuroscientistsinordertoanswer“bigpicture”questionsthatmattertoteachers.
Mayer(1998)claimsthatdrawingonteachers’experienceandintuitionsabout
students’learningpresentsnewhopeforcollaborativedesignofcognitive
neuroscienceresearchthatwillbetrulymeaningfultoeducators.Withinhisgeneral
skepticismforbuildingabridgebetweenneuroscienceandeducation,Bruer(2006)
holdsouttentativehopefortherareimagingstudythatgoesbeyondsimply
establishinglocalizationclaimsandinsteadexemplifiesbothasufficient
appreciationforthecomplexityandsubtletyofcompetingcognitivemodelsandan
abilityonthepartofresearcherstointerpretimagingdatainlightofalltherelevant
behavioralandneuropsychologicaldataavailable.
Implicationsforfutureresearch
Thisprojectoffersimportantconsiderationsandquestionsforthe
developingfieldofeducationalneuroscience.First,findingawaytoestablisha
commonlanguagebetweeneducatorsandneuroscienceresearchersmayallow
accesstoawealthofintuitiveknowledgeaboutthelearningbrainthatcanresolve
ambiguities,answerquestions,andfine‐tuneinvestigationsforresearchers.What
matcheswillberevealedbetweenteachers’intuitionsaboutstudents’brainsand
neurosciencefindingsoncethebarrierofdifferentlanguageisremoved?
Ontheotherhand,misconceptionsandgapsinteacher’sknowledgewillbe
exposed,allowingforinsightintowhycertainteachingstrategiesandapproaches
maynotbeoptimalforstudentlearning,understandingandachievement,and
36
offeringideasforbetterteaching.Asteacher5expressed,theneuroscience
informationpresentedinthisprojectwas“notenoughtohelpmebeabetter
teacher…“Shefeltthattheinformationshowedherthatsheneededtoknowmore
aboutmathlearninginthebrainordertoguideapplicationtotheclassroom.What
holesinteachers’knowledgeaboutthebrainwillberevealed(independentfrom
unfamiliar/unknownneurosciencevocabularyorterminology)?
AsTeacher3remarked,“Teachingisanintuitivescience.”Insuccessful
classrooms,teachersareconstantlyatwork,applyingapproachesandassessing
theirstudents’behavior,memoryfornewinformationandsubsequent
understanding.Allalongteacher’smodelsmustbeflexibleenoughtoadaptand
changebasedon“whatworks”intheclassroom.Theissuesthataroseinthese
interviews,includingappropriatedegreeoftime,scaffoldingandsensorysupport
foractivities,theinterplayofconcreteandabstractinsupportingconceptual
understanding,andtherolesoflanguage,workingmemoryandspatialskillsin
mathematicalproblemsolving,areissueswithwhichteachersareexperientially
familiar,evenwhentheymaynotbeabletoremembertheterminology.
Manyteachersspendtheircareerstryingandassessingvariousstrategiesin
theirownclassrooms,aswellascommunicatinganddebatingwithotherteachers
abouttheseissues.Sincethe“DecadeoftheBrain,”neurosciencehasbeenhumbled
bytherealization,evenasknowledgehasadvancedmorerapidlythanever,ofhow
verylittleweactuallyknowaboutthethinkingbrain.Thereisadequatereasonto
supposethatteacher’schoicestoredesignorrejectpriorstrategiesthroughouttheir
teachingcareersmayreflectexperienced‐basedintuitionsabouthowchildren’s
brainsatvariousagescanbeengaged,supportedtoprocessandremembernew
information,andtobuildconceptualunderstanding.Howdoteacher’schoicesto
redesignorrejectpriorstrategiesbasedon“whatworks”intheirclassroomreflectan
intuitiveunderstandingofneurosciencebasedrecommendations?
Teachers’intuitionscanalsobeseenasameansofestablishinganotherlevel
ofvaliditybeyondreplicatedfindingsinthebrainlab.Brainresearchistime‐
consuming,expensiveandtedious.TheobstaclespreventingUSstudentsfromthe
37
achievingdeepmathematicalunderstandingneededforacademicsuccessneedto
beremediednow.
Thisethicofcareandrespectforteachers’intuitionsisinitsownrighta
significantaspectoftheproject.Ultimately,theworkoftheeducationalpsychology
researcherandtheteacher,thecognitiveneuroscientistandtheneuroeducatorall
mergewithinthechild'sbrain.Ifneurosciencehopestocrossthebridgeintothe
classroom,itneedstoconsiderthatteachershave,inessence,beenpeeringinto
children'sbrainslongbeforethefirstbrainimagingtechniquesemerged.
Overarchingteachingapproachessuchasstandards‐basedpublicPre‐K
curricula,research‐basedmathcurricula,theMontessorimethod,andvarious
aspectsofprogressivecurriculasuchasconstructivistandemergentapproaches
preserveandpassonteachingmethodsthatshare,tovaryingdegrees,the
convictionthey“work.”TheEarlyChildhoodEducationresearchcommunityhas
cometorealizethatbuildinganunderstandingofwhycertainapproacheswork
whereothersfailisacomplexbutnecessaryendeavorifeducationisgoingto
institutesuccessfulandlonglastingchangesinthequality–andequality–thatwe
provideouryoungeststudents(Clements,2007).
Matheducationisasomberexampleofthecrisisthatoccurswhenmanyof
theacceptedteachingmethodsusedinclassroomsaren’tworkingtobuildsufficient
understandinginstudents.Therearebigquestionsleftunansweredinboththe
educationalandeducationalneuroscienceresearch.Drawingontheanalysisand
comparisonoftwoteacher’sreflectionsonnumbersense(inPreliminaryResults
andDiscussionabove)revealedhowteachers’intuitionscanbebroughttobearon
thesequestions.Forinstance:Howisdevelopingnumbersensebestsupportedinthe
preprimaryyears,andwhatimplicationsaretherefortheclassroomenvironment?
Whatneuroscientificsupportistherefortheimportanceofadequatetimespent
exploringmaterialsinasensorywayandaskingquestions?Whatsupportistherefor
theimportanceofovertinstructionversusactiveconstructionofknowledgeforthe
developmentofthisconcept?
Fromtheothersideoftheclassroomwall,educationalneuroscienceoffersa
setofscientificallysoundrecommendationsforteachingthathavebeencarefully
38
teasedapartfromintelligentspeculation,misinterpretedfindings,andneuromyths.
Forexample,MindBrainandEducationScienceliteratureadvocatesactive
constructionofknowledge,situatedlearningexperiences,andakeyroleofmotor
andsensoryexperiencesinlearning(Hardiman&Denkla,2009;Tokuhama‐
Espinosa,2010).Inwhatwayareteachers’variousphilosophies/approaches
regardingbestpracticesupportedbycurrentneurosciencerecommendationssuchas
thoseprovidedbyMindBrainandEducationScience?Howdothese
philosophies/approachescorrelatewithateacher’shavingintuitivemodelsoflearning
thatalignwithneuroscience?
Inthefaceofthesequestions,thisprojectrepresentsapromisingstart.The
designoffersavaluabletoolforcontinuedinvestigationthroughbi‐directional
collaboration.Themodelisinfinitelyportable,isnotrestrictedbytimeorlocation,
andisinexpensive.Thecollectionandcodingoffurtherdatacanbeginrightawayto
addressthequestionsaboveastheypertaintoallteachersaswellasspecificsub‐
groupsofteachers.Forexample,asthisprojectisbeingsubmitted,upcoming
interviewshavebeenscheduledwithgroupsofearlychildhoodteachersusinga
standarduniversalPre‐Kmathcurriculumatapublicschoolinalow‐incomeareaof
Brooklyn,aswellasatasuccessfulcharterschool,alsoinalow‐income
neighborhoodinHarlem,whereteachersaretrainedinimplementingresearch‐
basedmathcurricula.Newquestions,variablesandobservationcategorieswill
continuetofurtherimproveandrefinetheresearchmethodastheynaturallyarise.
Inconclusion,weareleftwithasenseofonesharedresearchproject
betweenteachersandresearchers.Thelinesdividingknowledgeintotheseparate
spheresofeducation,developmentalandcognitivepsychology,cognitiveand
educationalneurosciencehavecrossedenoughthattheyhaveblurred.Or,asPiaget
putsit,theschemata,orbuildingblocksofknowledge,havebeensufficiently
diversifiedsothatnewinformation,ratherthanproducingannoyanceandcausing
avoidance,“becomesaproblemandinvitessearching(Piaget,1955).“Inthis
picture,thegoalliesnotinwhatcanbeaccomplishedoncethebridgebetween
neuroscienceandeducationhasbeenconstructed,butratherinthedeep
39
connectionsthatarenaturallyformedthroughthesharedprojectofcreative
problemsolving.
PrincipleInvestigator
JuliaNiego
JuliaNiegoreceivedaBAinBehavioralNeurosciencefromColgateUniversity,
whereshedesignedandimplementedERPstudiesexaminingchangesinbrain
activityassociatedwithananti‐stereotypetrainingtask,andco‐authoredanarticle
forDevelopmentalNeuropsychology(Kellyetal,2002)hypothesizingarolefor
gestureinco‐definingspeechduringdevelopment.Beginningin2004shehas
collaborativelydevelopedaprogressiveMontessori‐basedprogramataprivate
preschoolinBrooklyn,NY,wheresheiscurrentlyteaching,researchingand
supportingstaffunderstandingoflearninganddevelopmentthroughregular
meetingsandworkshops.SherecentlyacceptedapositionforFall2011toco‐
developanearlychildhoodcomponentcurriculumforaneuropsychology‐based
tutoringprogramrunbyDr.AnnaWarren‐Levy.JuliaiscurrentlyaMSstudentin
NeuroscienceandEducationatTeacher’sCollege,ColumbiaUniversity,whereshe
hasspenttwoyearslearningtheclinicalinterviewmethodinadditiontocompleting
extensivecourseworkincognitiveneuroscienceandeducation.Sheisgraduatingin
May2011withaMastersofScienceinNeuroscienceandEducation.
ProjectAdvisors
PeterGordon,PhD.Psychology,MassachusettsInstituteofTechnology
PeterGordonistheProgramCoordinatorfortheNeuroscienceandEducation
ProgramatTeacher’sCollege,ColumbiaUniversity.Hisresearchfocusesonthe
developmentalneuroscienceoflanguageandcognition.Hecurrentlyteaches
SpeechandLanguagePathologyandNeuroscienceandEducationatthegraduate
levelatTeacher’sCollegeColumbiaUniversityandadvisesmulti‐disciplinary
doctoralresearch.
40
HerbertGinsburg,PhD.DevelopmentalPsychology,UniversityofNorthCarolina
HerbertGinsburgistheJacobASchiffFoundationprofessorofPsychologyand
HumanDevelopmentatTeacher’sCollege,ColumbiaUniversity.Hehasmade
significantcontributionstoanunderstandingofmentalprocessesinvolvedinthe
developmentofchildren’smathematicalthinkingusingtheclinicalinterview
method.Hecurrentlyteachestheclinicalinterviewmethodasaresearchtoolatthe
graduatelevelatTeacher’sCollegeandadvisesdoctoralresearchintheCognitive
Sciences.
LiteratureCited
Anderson,J.R.(2005).CognitivePsychologyanditsimplications.NewYork,NY:WorthPublishers.
Ansari,D&Coch,D.(2006).Bridgesovertroubledwaters:Educationandcognitive
neruoscience.TrendsinCognitiveSciences,10(4),146‐151.
Baroody,A.J.(2010).Fosteringearlynumeracyinpreschoolandkindergarten.EncyclopediaonEarlyChildhoodDevelopment.PublishedonlineJuly21,2010.www.child‐encyclopedia.com/pages/PDF/BaroodyANGxp.pdf
Berninger,V.W.,&Corina,D.(1998).Makingcognitiveneuroscienceeducationally
relevant:Creatingbi‐directionalcollaborationsbetweeneducationalpsychologyandcognitiveneuroscience.EducationalPsychologyReview,10(3),343‐354.
Beth,E.W.,&Piaget,J.(1966).MathematicEpistemologyandPsychology.Dordrecht:D.
Reidel.Bruer,J.T.(1997).Educationandthebrain:Abridgetoofar.EducationalResearcher,
26(8),4‐16.
Bruer,J.T.(2006).PointsofView:Ontheimplicationsofneuroscienceresearchforscienceteachingandlearning:Arethereany?Askepticalthemeandvariation:Theprimacyofpsychologyinthescienceoflearning.CBELifeScienceEducation,5(2)104‐110.
Campbell,S.R.(2006).EducationalNeuroscience:Newhorizonsforresearchinmathematicseducation.InNovotná,J.,Moraová,H.,Krátká,M.&Stehlíková,N.(Eds.).Proceedings30thConferenceoftheInternationalGroupforthePsychologyofMathematicsEducation,2,257‐264.
41
Clement,J.(2000).Analysisofclinicalinterviews:foundationsandmodelviability.InLesh,R.&Kelly,A.(eds.)HandbookofResearchMethodologiesforScienceandMathematicsEducation.Hillsdale,NJ:LawrenceErlbaum.
Clements,D.H.(2007).Curriculumresearch:Towardaframeworkforresearchbasedcurricula.JournalforResearchinMathematicsEducation.38(1),35‐70.
Clements,D.H.,&SaramaJ.(2007).Effectsofapreschoolmathematicscurriculum:Summativeresearchonthebuildingblockscurriculum.JournalofResearchinMathematicsEducation,38(2),136‐163.
Dehaene,S.,(1997).TheNumberSense.NewYork:OxfordUniversityPress.
Dehaene,S.,Piazza,M.,Pinel,P.,&Cohen,L.(2003).Threeparietalcircuitsfornumberprocessing.CognitiveNeuropsychology,20,487‐506.
deJong,T.,vanGog,T.,Jenks,K.,Manlove,S.,vanHell,J.,Jolles,J.,etal.(2009).Explorationsinlearningandthebrain.Onthepotentialofcognitiveneuroscienceforeducationalscience.TheNetherlands:Springer.
Delazer,M.,&Benke,T.(1997).Arithmeticfactswithoutmeaning.Cortex,33,697–710.
DeSmedt,B.,Ansari,D.,Grabner,R.H.Hannula,M.M.,Scneider,M.,andVerschaffel,L.(2010).CognitiveNeurosciencemeetsmathematicseducation.EducationalResearchReview,5,97‐105.
Eger,F.,Sterzer,P.,Russ,M.O.,Giraud,A.L.,&Kleinschmidt,A.(2003).Asupramodalnumberrepresentationinhumanintraparietalcortex.Neuron,37,719‐725.
Fischer,K.W.andImmordino‐Yang,M.H.(2008).Thefundamentalimportanceofthebrainandlearningforeducation.InTheJosseyBassReaderontheBrainandLearning.SanFrancisco,Jossey‐Bass.
Geake,J.(2004b).Howchildren’sbrainsthink:Notleftorrightbutbothtogether.Education,32,65‐72.
Geake,J.,&Cooper,P.(2003).Cognitiveneuroscience:Implicationsforeducation?WestminsterStudiesinEducation,26(1),7–20.
Gilmore,C.G.,McCarthy,S.E.,Spelke,E.S.,2007.Symbolicarithmeticknowledgewithoutinstruction.Nature,447,589–591.
Ginsburg,H.P.(1997).EnteringtheChild’sMind.TheClinicalInterviewinPsychologicalResearchandPractice.Cambridge,CambridgeUniversityPress.
Ginsburg,H.P.(1989).Children'sArithmetic.(2ndEd.)Austin,TX:Pro‐Ed.
Ginsburg,H.P.,Jacobs,S.F.,&Lopez,L.S.(1998).Theteacher'sguidetoflexibleinterviewingintheclassroom:Learningwhatchildrenknowaboutmath.Boston:AllynandBacon.
42
Ginsburg,H.P.&Opper,S.(1988).Piaget'sTheoryofIntellectualDevelopment.(3rdEd.)EnglewoodCliffs,NJ:Prentice‐Hall.
Ginsburg,HerbP.;Boyd,Judi;SunLee,Joon(2008).Mathematicseducationforyoungchildren:Whatitisandhowtopromoteit.SocialPolicyReport,22(1),1‐23.
Ginsburg,H.P.,Jang,S.,Preston,M.D.,VanEsselstyn,D.,&Appel,A.(2004).Learningtothinkaboutearlychildhoodmathematicseducation:Acourse.InC.Greenes&J.Tsankova(Eds.),Challengingyoungchildrenmathematically(pp.40‐56).Boston:NationalCouncilofSupervisorsofMathematics.
Goswami,U.(2006).Neuroscienceandeducation:fromresearchtopractice?NatureReviewsNeuroscience.AOP,publishedonline,12April,2006.
Goswami,U.(2008a).CognitiveDevelopment:TheLearningBrain.London:TaylorandFrancis.
Greenes,C.,Ginsburg,H.,&Balfanz,R.(2004).BigMathforLittleKids.EarlyChildhoodResearchQuarterly,19,159‐166.
Howard‐Jones,P.(Ed.).(2008).Educationandneuroscience.EducationalResearch,[Specialissue],50(2).
Hardiman,M.andDenckla,M.B.(2009).TheScienceofEducation:Informingteachingandlearningthroughthebrainsciences.TheDANAfoundation:Cerebrum,publishedonlineathttp://dana.org/news/cerebrum.
JoonSunLee,HerbertP.Ginsburg&MichaelD.(2009).Videointeractionsforteachingandlearning(VITAL):Analysingvideosonlinetolearntoteachearlychildhoodmathematics.AustralasianJournalofEarlyChildhood,34(2),19‐23.
Kilbane,C.R.(2008).Preserviceteachers’applicationofaproblem‐solvingapproachonmultimediacases.ActioninTeacherEdcuation,29(4),15‐26.
Lemer,C.,Dehaene,S.,Spelke,E.,&Cohen,L.(2003).Approximatequantitiesandexactnumberwords:Dissociablesystems.Neuropsychologia,41,1942–1958.
Lipton,J.S.,&Spelke,E.S.(2005).Preschoolchildren’smappingofnumberwordstonon‐symbolicnumerosities.ChildDevelopment,76,978‐988.
Mayer,R.E.(1998).Doesthebrainhaveaplaceineducationalpsychology?EducationalPsychologyReview,10,389–396.
Piaget,J.(1955).TheConstructionofRealityintheChild.(MargaretCook,trans.)London:RoutledgeandKeganPaul.
Piaget,J.(1975).TheOriginoftheIdeaofChanceinChildren.London:RoutledgeandKeganPaul.
43
Piazza,M.,Pinel,P.,LeBihan,D.,Dehaene,S.(2007).Amagnitudecodecommontonumerositiesandnumbersymbolsinhumanintraparietalcortex.Neuron,53,293–305.
Rains,J.R.,Kelly,C.A.&Durham,R.L.(2008).Theevolutionoftheimportanceofmulti‐sensoryteachingtechniquesinelementarymathematics:Theoryandpractice.JournalofTheoryandPracticeinEducation,4(2),239‐252.
Rosen,D.(2008).Impactofcase‐basedinstructiononstudentteachers’reflectiononfacilitatingchildrens’learning.Actioninteachereducation,30(1),28‐36.
Rosenfeld,D.(2010).Increasingperceivedefficacyforteachingmathematics:Anexploratorystudy.JournalofMathematicsEducationatTeachersCollege,Spring‐Summer2010,InauguralIssue,2535.
Strauss,S.(1993).Teachers'pedagogicalcontentknowledgeaboutchildren'smindsandlearning:implicationsforteachereducation.EducationalPsychologist,28(3),279‐290.
Strauss,S.andShilony(1994).Teachers’modelsofchildrens’mindsandlearning.InHirschfeld,L.A.andGelman,S.(Eds.)MappingtheMind.DomainSpecificityinCognitionandCulture.Cambridge,CambridgeUniversityPress.
Steen,L.A.(1988)TheScienceofPatterns.Science,240,29April1988,611‐16.Taylor‐Cox,J.(2003).Algebraintheearlyyears?Yes!YoungChildren,January2003,14‐21.Tokuhama‐Espinosa,T.(2008b).Thescientificallysubstantiatedartofteaching:Astudyin
theemergingstandardsinneruoedcuation.Unpublisheddoctoraldissertation,CapellaUniversity.http://www.proquest.com/en‐US/products/dissertations/pdqt.shtml.
Tokuhama‐Espinosa,T.(2010).TheNewScienceofTeachingandLearning.UsingtheBestof
Mind,BrainandEducationScienceintheClassroom.NewYork,Teacher’sCollegePress.
Tomlinson,C.(1999).Thedifferentiatedclassroom:Respondingtotheneedsofalllearners.Alexandria,VA:AssociationforSupervisionandCurriculumDevelopment.
Tomlinson,C.&McTighe,J.(2006).Integratingdifferentiatedinstruction&understandingbydesign:Connectingcontentandkids.Alexandria,VA:AssociationforSupervisionandCurriculumDevelopment.
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Appendix A
Clinical Interview Protocol 1: Teacher background
PreVideo
TeacherInformation
Canyoutellyourname?
Howmanyyearshaveyoubeenteachinginearlychildhood?
Canyoudescribeyourgeneralpathtoteaching?
Can you briefly describe your general philosophy or approach, something that possibly sets you apart from other teachers?
Appendix B
Clinical Interview 2 Protocols: Teacher mental models
(ScaffoldedInterview)
Number sense
Canyoureflectontheteachingexampleyoujustsaw:
1)Didthechildrenseemengagedduringlesson?Howcouldyoutell?
2)Whatmathconceptdoyouthinktheteacherwasattemptingtoteachtothegroup?
3)Wouldyouhavetaughtthisconceptdifferently?Why/whynot?
4)Didthischildperformthetaskcorrectly/incorrectly?Why/whynot?
5)Whatmathabilitieswouldyousaychildrenhadtohaveinordertoperformthistask?
6)Whatwasthischild’sstrategyinperformingthetasksoquickly?
Walksubjectthroughchild’smentalprocess.
(Exploreanymentionofdifferentrepresentationsofmathinformation/connectionsbetweenmathematicalinformation)
Arithmetic
Canyoureflectontheteachingexampleyoujustsaw:
1)Doesthechildseemengagedinthetask?Howcanyoutell?
45
2)Whydoestheteacherchoosetobeginthelessonthisway?
3)Whatmathconceptdoyouthinktheteacherisattemptingtoteach?
Wouldyouhavetaughtthisconceptdifferently?Why/whynot?
4)Whatmathabilitieswouldyousaythechildhadtohaveinordertosolve
thistask?
5)Didthechildsolvethefirsttaskcorrectly/incorrectly?Why/whynot?
Thesecond?Why/whynot?
6))Whatdoyouthinkwasthechild’sstrategyinsolvingtheproblem?
Walksubjectthroughchild’smentalprocess.
(Exploreanymentionofhandlingblocks)
Pattern
Canyoureflectontheteachingexampleyoujustsaw:
1)Didthechildseemengagedduringlesson?Howcouldyoutell?
2)Whatmathconceptdoyouthinktheteacherwasattemptingtoteach?
3)Wouldyouhavetaughtthisconceptdifferently?Why/whynot?
4)Didshesolvethetaskcorrectly/incorrectly?Why/whynot?
5)Whatmathabilitieswouldyousaythechildhadtohaveinordertosolvethistask?
6)Whatdoyouthinkwasthechild’sstrategyinsolvingtheproblem?
Walksubjectthroughchild’smentalprocess.
(Exploreanymentionofchild’sactualconstructionofthepattern,i.e.notfollowingtheexactlanguage)
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Appendix C
Clinical interview protocol 3: Neuroscience follow-up
Protocol2:Followup
(Moreopen‐endedinterview)
Doesthisinformationrelatetotheteachingmomentyoujustsaw?How?/Inwhatways?
Doesthisinformationrelateinanywaytowhatyouobserveinyourownclassroom/students?
Doesthisinformationmatchinanywayswithyourpersonalphilosophy/approachasanearlychildhoodteacher?
Canyouthinkofanywaysthatthisinformationmightimpactyourideas/thoughts/beliefsaboutteachingmath,specificallynumbersense/arithmetic/pattern,toyoungchildren?
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AppendixD
NeuroscienceSupplements
PrintedSupplements
Supplement1:NumberSense
Cognitiveneuroscienceresearcherswanttoknowhowachild’sbraindevelopstoachieveunderstandingofamathematicalconceptsuchas“numbersense.”
Takingpicturesofstudents’brainswhileperformingmathtaskshasrevealedthat:
Theareasinthebrainwheredifferentkindsofmathematicalinformationarerepresentedarelargelyinplacebyage3.
Languageareasatthefrontleftsideofthebrainareinvolvedwithprocessingverbalmathinformationsuchasnumberwords(“five”)andalgorithms(“plus”)
Visionareasattheverybackofthebrainareinvolvedwithprocessingwritten/drawnmathinformationsuchaswrittennumerals(5)orsymbols(+).
Associationareasatthetopbackofthebrainareinvolvedwithprocessingquantitativeinformation(“Howmuch”/”Howmany?”).
Visuo‐spatialareasatthetopbackandfrontofthebrainontherightsideareinvolvedwithprocessingvisualandspatialinformation(shape,structureandspace).
Theseareascanbeactiveornotdependingonthemathtaskapersonisdoing.
Aperson’sbrainlooksdifferentwhenlisteningtonumberwords,orrecognizingawrittennumeral,orcountingdots,orestimatingthenumberofobjectsinagroup.
Aspecificconceptthatachilddevelopsaboutmath,suchasnumbersense,involvesconnectionsbetweenthesedifferentareas
Theseconnectionshappenthroughmathematicalexperienceswheremore thanonekindofmathinformationiscalleduponatthesametime.
Studiessuggestthatthewaymathinformationislearneddirectlyinfluenceshowandwhereitisrepresentedinthebrain.Furthermore,thewaystudentsareguidedtousethismathematicalinformationinfluencesthewayitisintegratedandbuiltintomathematicalconcepts.
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Supplement2:Addition
Cognitiveneuroscienceresearcherswanttoknowhowachild’sbraindevelopstobeabletocompleteatasksuchastheadditiontaskpresentedinthislesson.
Researchonmathlearninghasshownthatchildrenbecomemuchfasteratsolvingarithmetictasksastheygetolder.Takingpicturesofstudents’brainswhileperformingmathtaskshasrevealedthat:
Theareasinthebrainthathandlearithmetictasksarelargelyinplacebyage3
Areasoneithersideofthefrontofthebrainhandlegeneralreasoning
Areasjustbehindthefrontofthebrainserveasamentalworkspacefortemporarilyholdingandmanipulatingfactsandinformationthatachildneedstosolveagivenmathproblem
Areasatthefrontleftsideofthebrain(behindtheleftear)handlemathlanguage(numberwords)
Areasatthebackofthebrainhandlevisualinformation(writtennumerals)
Youngerbrainslookdifferentthanolderbrainswhensolvingarithmetictasks
Youngerchildrenshowmostactivityingeneralreasoningandworkingmemoryareas,whileolderchildrenshowmoreactivityinlanguageandvisualareas
Thisshiftcorrespondswithachangeinchildren’sstrategiesforsolvingarithmeticproblems
Youngchildrenrelyonmanipulationofconcreteobjectstosolveadditionandsubtractionproblems,whereaslaterontheyareabletoperformmath“intheirheads,”usingabstractrepresentationsofnumbersvialanguageandvisualsymbols
Studiessuggestthatthewayachild’sbrainprocessesmathinformationatacertainstageindevelopmentdirectlyinfluencesthestrategytheyusewhensolvinganarithmetictask.Ontheotherhand,recentresearchsuggeststhatteachingchildrennewstrategiesmayactuallyinfluencelanguageandvisualareasto“takeover”duringarithmetictaskssothatgeneralreasoningandworkingmemoryarefreeduptohandlemorecomplexinformation,setgoals,andmonitorprogress.
49
Supplement3:Pattern
Cognitiveneuroscienceresearcherswanttoknowhowachild’sbraindevelopstobeabletocompleteatasksuchasthepatterntaskthatthischildjustdid.Bytakingpicturesofchildandadultbrainsintheprocessofsolvingmathproblems,researchesposethat:
Theareasinthebrainthathandlevariousmathematicaljobsarelargelyinplacebyage3
Areasattheupperbackofthebrainstoreconceptsof“howmuch”
Areasatthefrontleftsideofthebrainhandlelanguage(numberwords)
Areasatthefrontofthebrainholdfactsandinformationthatachildneedstosolveagivenmathproblem
Ontherightsidethisareaholdsneededvisualandspatialinformationlikea“sketchpad”
Children’sbrains,likeadultbrains,seemtohavetwomain“pathways”orconnectionsbetweentheseareas
Theleftsidehandlesquantitativemathinformationlearnedthroughlanguage,likenumbersandcounting
Therightsidehandlesvisuospatialmathinformationlearnedthroughsensoryexperience,likeshape,structureandpattern
BUTchildren’sbrainsdonothandleamathproblemthesamewaythatadults’brainsdo
Bylookingatpicturesofbrainsdoingmath,researcherscanseethatthesemathareasandtheconnectionsbetweenthemlookdifferentinadultsandchildrenwhentheyaresolvingthesamekindsofproblems
Somestudiessuggestthatpreschoolandkindergartenagedchildrenmayrelymoreonvisuospatialknowledgetograspmathconcepts,andthengraduallyshifttoincorporatemorequantitativeknowledgeastheirbrainsdevelopmathematically.
50
PictureSupplements
SupplementA:Numbersense
51
Supplement B: Arithmetic
52
Supplement C: Pattern
53
SupplementReferences
Anderson,J.R.(2005).CognitivePsychologyanditsimplications.NewYork,NY:WorthPublishers.
Anderson,J.R.(1982).Acquisitionofcognitiveskill.PsychologicalReview,89,369‐
406.V.W.Berninger&T.L.Richards.(2002).BrainLiteracyforEducatorsand
Psychologists.SanDiego,CA:AcademicPress.
Ansari,D.(2010).Neurocognitiveapproachestodevelopmentaldisordersofnumericalandmathematicalcognition:Theperilsofneglectingtheroleofdevelopment.LearningandIndividualDifferences,20,123‐129.
Curcio,F.R.,&Shwarts,S.C.(1997).Whatdoesalgebraicthinkinglooklikewith preprimarychildren?TeachingChildrenMathematics,December1997,296‐
300.Baroody,A.J.(2010).Fosteringearlynumeracyinpreschoolandkindergarten.
EncyclopediaonEarlyChildhoodDevelopment.PublishedonlineJuly21,2010.www.childencyclopedia.com/pages/PDF/BaroodyANGxp.pdf
Cantlon,J.F.,Brannon,E.M.,Carter,E.J.,andPelphrey,K.A.(2006).Functional
imagingofnumericalprocessinginadultsand4yroldchildren.PLoSBiology,4(5),844‐854.
Coull,J.T.,&Frith,C.D.(1998).Differentialactivationofrightsuperiorparietal
cortexandintraparietalsulcusbyspatialandnonspatialattention.Neuroimage,8,176–187.
Dehaene,S.(1996).Theorganizationofbrainactivationinnumbercomparison:
Event‐relatedpotentialsandtheadditive‐factorsmethod.JournalofCognitiveNeuroscience,8(1),47‐68.
Dehaene,S.,(1997).TheNumberSense.NewYork:OxfordUniversityPress.Dehaene,S.,&Cohen,L.(1997).Cerebralpathwaysforcalculation:Double
dissociationbetweenroteverbalandquantitativeknowledgeofarithmetic.Cortex,33,219‐250.
54
Dehaene,S.,Dehaene‐Lambertz,G.,&Cohen,L.(1998).Abstractrepresentationofnumbersintheanimalandhumanbrain.TrendsinNeurosciences,21,355‐361.
Dehaene,S.,Piazza,M.,Pinel,P.,&Cohen,L.(2003).Threeparietalcircuitsfor
numberprocessing.CognitiveNeuropsychology,20,487‐506.Delazer,M.,&Benke,T.(1997).Arithmeticfactswithoutmeaning.Cortex,33,697–
710.Eger,F.,Sterzer,P.,Russ,M.O.,Giraud,A.L.,&Kleinschmidt,A.(2003).A
supramodalnumberrepresentationinhumanintraparietalcortex.Neuron,37,719‐725.
Ferrini‐Mundy,J.,Lappan,G.,&Philips,E.(1997).Experienceswithpatterning.
TeachingChildrenMathematics,December1997,282‐288.Finke,K.,Bublak,P.&Zihl,J.(2006).Visualspatialandvisualpatternworking
memory:neuropsychologicalevidenceforadifferentroleofleftandrightdorsalvisualbrain.Neuropsychologia,44(4),649‐661.
Geary,D.C.,Wiley,J.G.(1991).Cognitiveaddition:strategychoiceandspeed‐of
processingdifferencesinyoungandelderlyadults.PsychologyandAging,6,474–483.
Gelman,R.,Meck,E.,1983.Preschoolerscounting:principlesbeforeskill.Cognition, 13,343–359.Gilmore,C.G.,McCarthy,S.E.,Spelke,E.S.,(2007.)Symbolicarithmeticknowledge withoutinstruction.Nature,447,589–591Gruber,O.,Indefrey,P.,Steinmetz,H.,&Kleinschmidt,A.(2001).Dissociatingneural
correlatesofcognitivecomponentsinmentalcalculation.CerebralCortex,11,350–359.
Hanakawa,T.,Honda,M.,Okada,T.,Fukuyama,H.,Shibasaki,H.(2003).Neural correlatesunderlyingmentalcalculationinabacusexperts:afunctional
magneticresonanceimagingstudy.NeuroImage,19,296–307.Ischebeck,A.,Zamarian,L.,Egger,K.,Schocke,M.,Delazer,M.(2007).Imagingearly practiceeffectsinarithmetic.NeuroImage,36,993–1003.Kaufmann,L.,Koppelstaetter,F.,Siedentopf,C.,Haala,I.,Haberlandt,E.,
Zimmerhackl,L.B.,Felber,S.,Ischebeck,A.(2006).Neuralcorrelatesofthenumber‐sizeinterferencetaskinchildren.Neuroreport,17,587–591.
55
Kawashima,R.,Taira,M.,Okita,K.,Inoue,K.,Tajima,N.,Yoshida,H.,Sasaki,T., Sugiura,M.,Watanabe,J.,Fukuda,H.(2004).AfunctionalMRIstudyof
simplearithmetic—acomparisonbetweenchildrenandadults.CognitiveBrainResearch,18,227–233.
Kong,J.,Wang,C.,Kwong,K.,Vangel,M.,Chua,E.,Gollub,R.(2005).Theneural substrateofarithmeticoperationsandprocedurecomplexity.CognitiveBrain
Research,22,397–405.Kucian,K.,vonAster,M.,Loenneker,T.,Dietrich,T.,Martin,E.(2008).Development
ofneuralnetworksforexactandexactandapproximatecalculation:aFMRIstudy.DevelopmentalNeuropsychology,33,447–473.
Lemer,C.,Dehaene,S.,Spelke,E.,&Cohen,L.(2003).Approximatequantitiesand
exactnumberwords:Dissociablesystems.Neuropsychologia,41,1942–1958.Lipton,J.S.,&Spelke,E.S.(2005).Preschoolchildren’smappingofnumberwordsto
non‐symbolicnumerosities.ChildDevelopment,76,978‐988.Martin,T.,Lukong,A.&Reaves,R.(2007).TheRoleofManipulativesinArithmetic andGeometrytasks.JournalofEducationalandHumanDevelopment,1(1),
122‐131.Mayer,R.E.(1998).Doesthebrainhaveaplaceineducationalpsychology?
EducationalPsychologyReview,10(4),389‐396.Menon,V.,Rivera,S.M.,White,C.D.,GloverG.H.&Reiss,A.L.(2000a).Dissociating prefrontalandparietalcortexactivationduringarithmeticprocessing.Neuro
Image12,357–365.Menon,V.,Rivera,S.M.,White,C.D.,Eliez,S.,Glover,G.H.,Reiss,A.L.,2000b. Functionaloptimizationofarithmeticprocessinginperfectperformers.Brain
Res.CognitiveBrainResearch,9,343–345.Piazza,M.&Izard,V.(2009).Howhumanscount:Numerosityandtheparietal
cortex.TheNeuroscientist,15(3),261‐273.Piazza,M.,Pinel,P.,LeBihan,D.,Dehaene,S.(2007).Amagnitudecodecommonto numerositiesandnumbersymbolsinhumanintraparietalcortex.Neuron,53,
293–305.Pinel,P.,Dehaene,S.,Riviere,D.,LeBihan,D.,(2001).Modulationofparietal
activationbysemanticdistanceinanumbercomparisontask.NeuroImage14,1013–1026.
56
Pinel,P.,&Dehaene.S.(2010).Beyondhemisphericdominance:Brainregionsunderlyingthejointlateralizationoflanguageandarithmetictothelefthemisphere.JournalofCognitiveNeuroscience,22(1),48‐66.
Poldrack,R.A.,(2000).Imagingbrainplasticity:conceptualandmethodological issues—atheoreticalreview.NeuroImage,12,1–13.Spelke,E.S.(2002).Developmentalneuroimaging:Adevelopmentalpsychologist
looksahead.DevelopmentalScience,5,592‐396.Spelke,E.S.&Dehaene,S.(1999).Biologicalfoundationsofnumericalthinking.
TrendsinCognitiveScience,3,365‐366.Stanescu‐Cosson,R.,Pinel,P.,vanDeMoortele,P.F.,LeBihan,D.,Cohen,L.,&
Dehaene,S.(2000).Understandingdissociationsindyscalculia:Abrainimagingstudyoftheimpactofnumbersizeonthecerebralnetworksforexactandapproximatecalculation.Brain,123,2240–2255.
Steen,L.A.(1988)TheScienceofPatterns.Science,29,611‐16.Strauss,S.(2003).Teachingasanaturalcognitionanditsimplicationsforteacher
education.InD.Pillemer&S.White(Eds.)DevelopmentalPsychologyandtheSocialChangeofourTime.NewYork:CambridgeUniversityPress.
Taylor‐Cox,J.(2003).Algebraintheearlyyears?Yes!YoungChildren,January2003,
14‐21.Temple,E.&Posner,M.I.(1998).Brainmechanismsofquantityaresimilarin5yr
oldchildrenandadults.ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica,95,7836‐7841.
Terao,A.,Koedinger,K.R.;Sohn,M.‐H.,Qin,Y.,Anderson,J.R.,andCarter,C.S.
(2008).AnfMRIStudyoftheInterplayofSymbolicandVisuo‐SpatialSystemsinMathematicalReasoning.DepartmentofPsychology,Paper59,http://repository.cmu.edu/psychology/59.
Venkatraman,V.,Ansari,D.,&Chee,M.W.L.(2005).Neuralcorrelatesofsymbolic
andnon‐symbolicarithmetic.Neuropsychologia,43,744–753.Xu,F.,Spelke,E.S.,&Goddard,S.(2005).Numbersenseinhumaninfants.
DevelopmentalScience,8,88‐101.Zago,L.,Pesenti,M.Mellet,E.Crivello,F.Mazoyer,B.,&Tzourio‐Mazoyer,N.(2001).
Neuralcorrelatesofsimpleandcomplexmentalcalculation.NeuroImage,13,314‐327.
57