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7/31/2019 Technology in Learning Atricle(1)
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CalculatorsintheClassroom:HelporHindrance?
ChristinaL.SheetsWallace,NE
MathintheMiddleInstitutePartnershipActionResearchProjectReport
inpartialfulfillmentoftheMATDegreeDepartmentofMathematicsUniversityofNebraska-LincolnJuly2007
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CalculatorsintheClassroom:HelporHindrance?
Abstract
Inaworldwheretechnologyiseverpresentandeverchanging,istoomuchtechnologyattooyoungofanagedetrimentaltoachildseducationalsuccess?Thepurposeofthispaperistosharetheresultsofafour-monthstudythatfocusedontheuseofcalculatorsingradeeight.Thisstudywasconductedinaneighthgradeclass,inasmallkindergartenthroughtwelfthgradeschool.Thispaperwillsharethefindingsofastudyofaclassroominwhichcalculatorusewaslimitedandmentalcomputationwasemphasized.Themainfocusofthisstudywaswhetherornottherewouldbeanyimprovementinthecomputationskillsofmystudentsandhow,orif,theirproblemsolvingwouldbeaffected.Asaresultofthisresearchproject,Iplantopermanentlylimitcalculatoruseingradessevenandeight,aswellastoimplementacomputationalreviewthatwillbeconductedyearlywithallofmyclasses.
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Themostelaboratebuildingwillneverstandifitsbaseandsupportstructurearentstrongenoughtosupportit.Thesamecanbesaidwhenitcomestoachildsmathematicseducation.Ifastudentdoesnthaveastrongbase,itisverydifficult,ifnotimpossible,tounderstandthecomplexitiesofupperlevelmathematics.Thatbeingsaid,therearesometoolsthatcanhelpthestudentwhounderstandstheconceptsbutstrugglestorememberhisorhertimestablesorwhomakescarelesserrorsinthedivisionalgorithm.Ifthesetoolsexist,shouldnttheybeutilized?
PriortothisresearchprojectIpermittedallofmystudentsingradesseventhroughtwelvetousescientificand/orgraphingcalculators.Ibelieved,frompersonalexperience,thatthestudentswouldonlyusecalculatorstohelpthemspeeduptheirlearningprocess,andIdidntfeeladependencywouldbeformed.Ihopeditwouldgivethestrugglingstudentsaboostsothat
theycouldcatchupwiththeirpeersoratleasthaveachancetounderstandthemoreprofoundtopics.IneverbelievedIwasdoinganyharm,untilaparentapproachedmeataconference.ShewasconcernedthatherstudentwaslosingallofhispriorskillsinmathematicsbecauseIpermittedunrestrictedcalculatoruseinmyclassroom.Iassuredherthatthiswasnotthecase,andbelievedthiswasthetruth.
InMayoflastyear,whenmyschoolreceivedachievementtestscores,Ididnoticethatourscoreswerelowerfortheportionofthetestwherecalculatorswerenotper
mitted.Iwassurprisedtosaytheleast.Inseventhandeighthgrade,Icoveragreatdealoftopicsdesignedtohelpstudentsmakegooddecisions,suchasdivisibilityrules,reducingbyfactors,andthereasonablenessofananswer.Iteachthemtherules,and,perhapsnaively,believethattheywillrelyonthemanduseacalculatoronlywhentimeanddifficultynecessitateit.IalsoseldomuseacalculatorwhenIamworkingthroughaprobleminaclassdiscussion.Iassumedtheywouldfollowmyexample.
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WhatdoIwanttohappen?Iwantmystudentstoretaintheirpriormathematicalknowledge.Iwantmystudentstochoosetothinkforthemselves,andnotmindlesslypushbuttonsontheircalculator.Iwantstudentstousetechnologyasatooltobroadentheirknowledgebase,notacrutchtostabilizeaweakfoundation.Ialsowantstudentstobecomeconfidentintheirabilitytosolveaproblemwithouttechnology.
Whatistheintervention?Iplantolimittheuseofcalculatorsinmylowerlevelclasses(7th,8th,andAlgebraI).Ialsoplantoemphasizetheneedtoshowalloftheworknecessarytoobtainananswer.Inthismannerevenifcalculatorsareused,thethoughtprocesswillbeapparent.
Whyisthisimportant?Ifstudentsdonottrulycomprehendthebasics,itwillbedifficult,ifnotimpossible,tobuildtheabstractconceptsthatarenecessaryinupperlevelmathclasses.Studentsneedtobuildconfidenceintheirmathability,andiftheyareentirel
ydependentontechnology,Ifearthiswillneverhappen.Thisstudywillhopefullyrevealiftechnologyisabenefitorabarriertomiddlelevelmathstudents.
ProblemStatement
Thepurposeofmyprojectistouncovertheeffectthattechnologyhasonstudentlearning,specificallytheeffectofpermittingcalculatorsingradessevenandeight.Iwillbeexaminingthevariabilityofthenumberofstudentsthatshowimprovementonthestandardized
test(TerraNova),thenumberofstudentsthatbecomeconfidentintheirabilitytosolveaproblemwithouttechnology,andthenumberofstudentswhoretainpriormathematicalknowledge.
Ibelievethisprojectisworthyofconsiderationbecauseitwillhelptoresolvetheage-olddebateaboutcalculatorsintheclassroom.Isacalculatorahelpfultoolthatfreesthestudenttospendmoreofhisorhertimeproblemsolving,orisitmerelyanexcuseforastudenttoforget
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everythingheorshewastaught?Everyonehasanopiniononthismatter,andIhopetogivethereaderenoughinformationtomakeaninformeddecisionastohowthispertainstomiddleschoolstudents.
LiteratureReview
Iamatechnologyaficionado.Iloveallofthelatesttoys,andIenjoyutilizingtheminmyclassroom.Besidesthescientificcalculator,mystudentshavebeenexposedtographingcalculatorsandProbes,GeometerSketchpad,PowerPoint,theWebandanythingelseonwhichIcangetmyhands.Thestudentsreallyenjoythebreakfrompaperandpencil,areengagedinthelearningprocess,andseemtolearnagreatdealfromtheexposure.
So,whatstheproblem?Lastyearatparent-teachersconference,Ihadaparentaccusemeofusingtoomuchtechnology.SheinsistedthatIshouldbefocusingonthebasi
cs,assertingthatherson(aneighthgrader)nolongerknewhismultiplicationfacts,orthedivisionalgorithm.AdmittedlythesearenottopicsonwhichIspendtime,becausetheyarethoroughlycoveredattheelementarylevel.WhenIgotachievementtestresultsbacklastspring,however,Iwassurprisedathowlowourscoreswereforthesectionofthetestwherethestudentswerenotpermittedtousecalculators.Perhapsthisparentwasright.Bygivingstudentspermissiontousecalculators,amIgivingthempermissiontoforgetwhattheyhavelearned?Itisforthisreason
thatIchosecalculatorusageasmyproblemofpractice.
AsIbeganresearchingthisproblem,IturnedfirsttotheNationalCouncilofTeachersofMathematics(NCTM)(2000)PrinciplesandStandardsforSchoolMathematics.NCTMsupportstheadvancementoftechnologyintheclassroom,andassertsthattheuseoftechnologyoftenallowsforproblemsolvingtooccuringreaterdepth.Italsoacknowledgesthattheoverarchingprincipleoflearningrequiresstudentstoactivelylearnwithunderstandingandbuild
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newknowledgefromexperienceandpriorknowledge.Althoughmystudentsareactivelyengagedinthelearningprocess,Iamconcernedthattheyarelosingtheirgriponvitalpriorknowledge.Newknowledgeis,afterall,builtonabasisofpre-existingideas,accordingtoBazzini(2002)inherstudyofteachingexperimentsbasedonbody-relatedmetaphors.InmystudyIhopetouncoverwhetherornotmystudentsareretainingthefoundationthatwasbuiltinelementaryschool.
Mynextstepwastoturntoresearch.ThefirstquestionItriedtoanswerwhilesiftingthrougheducationaljournalswas,Whateffectwilllimitingcalculatorusagehaveonstudentachievement?Thiswasadauntingtask.BecausetheNCTMsupportstechnologyusage,mostofthejournalsfoundcalculatorstobeapositiveforceintheclassroom.Eveninveryyoungstudents,calculatorusagecanbeseenasapositive.InanarticlebyThomand
Pirie(2002),agroupofthirdgradestudentsweregivenaninvolvedvolumeproblem.Thesestudentswererequiredtoconstructabox(withoutatop)fromapieceofcardboardthatwouldholdthemostlinkingcubes.Thestudentswouldcreateanumberofboxesandeventuallyappearedtograsptheconceptofvolume.Thetaskrequiredagreatdealofproblemsolvingonthestudentspart,andatonepointintheprocessonesmallgroupbecamefrustratedafterattemptingalongstringofrepeatedaddition.Theauthorscommentonthefactthatthestudentsfrustrationandimpatience
couldhavebeenalleviatedifacalculatorwouldhavebeenavailable(Thom&Pirie,2002,p.16).Clearlytheintegrityoftheproblemwouldnthavebeenchallengedbythepresenceofacalculator,andmaybetheuniquenessofthetaskisthekey.Nowhereinthearticledoesitsuggestthatthesestudentsusedcalculatorsonaregularbasis.
Inanotherarticle.Bridgeman,Harvey,andBraswell(1995)wantedtoseeifusingacalculatoronastandardizedtesthadaneffectonmathematicalreasoning.Bridgeman,Harvey,
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andBraswellstudiedasampleofcollege-boundjuniorsfromtwohundredandseventy-fivehighschoolswhotookatestwhichconsistedof70mathquestionsfromtheSAT.Halfofthestudentswerepermittedtousecalculators,andtheotherhalfwerenotbasedonrandomselection.Theirstudyappearedtodemonstratethattheuseofcalculatorsonastandardizedtestresultedinamodestincreaseintestscores.Ialsofounditinterestingthatthestudentswhousedcalculatorsinthisstudyrequiredthesameamountoftimetocompletethetestasthestudentswhowerenotpermittedtousethemonthetest(Bridgeman,Harvey&Braswell,1995,p.323-340).TimeisoneofthereasonsthatIhavealwaysallowedmystudentstousecalculators.Perhapsthisisamisconceptiononmypart,orperhapsitdependsontheproblemsgiven.Theseproblemsmayhaverequiredmoredepthofthought,andmaybeIgivetoomanyproblemsthatfocusoncalculation.
Iwasabletofindonearticlethatdealttosomedegreewiththeideaoflimitingcalculatorusage.ThearticlebyKilpatrick(2001)triedtosynthesizetheliteratureonmathematicslearningandprovideresearch-basedrecommendations.Inthearticle,inanattempttodefinethetermfuzzymath,LynneCheney,wifeofthecurrentvice-presidentoftheUnitedStates,isquotedascriticizingNCTMforintroducingintotheschoolsfuzzymath.Bythisshemeantinstructionthatdown-playscomputationalskills,advocatescooperativelearningandtechnology,and
promotestheviewthatgettingtherightanswertoamathproblemcanbemuchlessimportantthanhavingagoodrationaleforawrongone(Kilpatrick,2001,p.102).Hercommentsreallyseemedunsettlingtome,primarilybecauseIdontunderstandwhereshebelievestheproblemlies.AlthoughIaminterestedinwhethertechnologyishurtingmystudents,herstatementsseemuninformedandpoliticallybiased.
AfterreadingcountlessarticlesIbegantowonderaboutadifferentquestion,What
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effectwilllimitingcalculatorusagehaveonstudentsproblemsolvingskills?Addingproblemsolvingtomysearchcriteria,IencounteredanarticlebyWheatley(1980).Wheatleyconductedapilotstudywithtwenty-fivestudentsinafifthgradeclassinordertoseewhateffectcalculatorswouldhaveonthestudentsproblemsolvingabilities.Basedonherfindings,shebelievesthatcalculatorscanfacilitatemathematicsperformancewithlittleriskoflossofcomputationalproficiency.Thissurprisedme.Wheatleywantedtoknowifastudentwaspermittedtouseacalculatoronaregularbasis,thenwouldthestudentbemorefocusedonstrategyneededtosolveaproblem.BasedonthisstudyWheatleyconcludedthatstudentswhoarepermittedtousecalculatorsspendapproximatelythesameamountoftimeontaskasthosewhoarenot.ThiswasinagreementwiththearticlebyBridgeman,Harvey,andBraswell.Thecalculatorgrouphadfewercomputationalerrorsandspentmoretimeontheprocessofproblemsolving
asopposedtocomputation.WheatleysfindingsappeartosupportthebeliefsofNCTMthattechnologypromotesproblemsolving(Wheatley,1980,p.323-334).However,Istillwonderaboutcomputationalproficiency.Huntley,Rasmussen,Villarubi,Sangtong,andFey(2000)conductedacomparativestudyoftheeffectsoftheCore-PlusMathematicsProjectcurriculumandamoreconventionalcurriculumwithregardstogrowthofstudentunderstanding,skill,andproblem-solvingabilityinalgebra.Intheirarticle,theyclaimthatalthoughtheuseofacalculatoroffersa
studentavarietyofpowerfulnewlearningandproblemsolvingstrategies,itdiminishestheneedforthatstudenttoacquireahighdegreeofskillinsymbolmanipulation(Huntley,Rasmussen,Villarubi,Sangtong,&Fey,2000,p.331).IsthiswhatIwantformystudents?
ForsterandTaylor(2000)studiedaneleventhgradeGeometryandTrigonometryclassofeighteenstudentsinaprivatecollegeforgirlsinWesternAustralia.Inthisclassroom,graphingcalculatorswereusedonaregularbasis,andthelearningconditionswereconsideredtobe
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exemplary.Basedontheirfindings,ForsterandTaylorconsidertechnologytobeatoolorapartnerforthestudent.Theyseethestudentsnotonlyusingthetooltoimproveproblemsolving,butalsotakingwhattheyhavelearnedfromthetoolandapplyingitelsewhere(Forster&Taylor,2000,p.38).
Similarly,Ifoundseveralarticlesthatsupportedtheuseofcalculators,specificallygraphingcalculators,astoolsthatdeepenastudentsunderstandingofabstractormerelydifficultmathematicalconcepts.OnesuchconceptistheideaofavariableinAlgebra.GrahamandThomas(2000)conductedapilotstudyinfiveUnitedKingdomschoolsusingtheTI-80graphingcalculator.Thisstudyfocusedonvariables,oneofthemoredifficulttopicsinAlgebra.Theyimpliedthatthestudentswhousedgraphingcalculatorscouldgainabetterunderstandingoftheuseoflettersasvariables,withoutanadverseeffectontheirbasicabilityto
manipulatethesymbolstosolveanequationorinequality(Graham&Thomas,2000,p.271).GrahamandThomasbelievethestudentsinvolvedinthisstudyallsawanincreaseinunderstandingbutstatedthatthestudentswiththelowestabilityappearedtoshowthemostimprovement(Graham&Thomas,2000,p.279).Theauthors,however,didnotethatoneteacherinvolvedwiththeirstudystatedthatherstudentswereveryreluctanttogiveupthecalculators.Herstudentshadbecomedependentonthetechnologyandfearedfailurewithoutit(Graham&Thomas,2000,p.
277).Thissoundslikemystudents,manyofwhomwanttopushthebuttonsonthecalculatortodothesimplestoftaskssuchasdividingbynegativeone.
Thisledmetomyfinalquestion,Whateffectwilllimitingcalculatorusagehaveonstudentsabilitiestojustifythereasonablenessofananswer?Ihavedeterminedthatastudentwithacalculatorinhandoftenloseshisorhercommonsense.Oneofmyfreshmenwasconvincedthathersolutiontoadistanceproblemwascorrect.Inherdefense,theproblemwasa
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littletricky.Thestudentsweregivenaproblemthatwouldgeneratethestoppingdistanceacarwouldneedbasedonthespeeditwastraveling.Thetrickypartofthisproblemwasthatthestudentswereasked,afterseveralreasonablesituations,whatthestoppingdistanceofaparkedcarwouldbe.Thisstudentpluggedsomenumbersintohercalculatorandcameupwithanegativedistance.WhenIaskedherifthismadeanysense,sheinsisteditwasrightbecausethatwasthenumberthecalculatorhadgivenher.Iaskedherthespeedofaparkedcar,andsheknewthatitwaszero,butittookalotoftalkingonmypartbeforeshewouldadmitthatheranswerdidntmakeanysense.Dostudentsreallythinkthatcalculatorsareintelligent?Donttheyunderstandthatacalculatormerelyperformstheactionsaskedofit?
DoerrandZangor(2000)conductedaclassroom-basedstudyinordertobetterunderstandthemeaningofatoolsuchasthegraphingcalculatorasitwasconstructedbyboth
teacherandstudent.Theyalsohopetounderstandhowthestudentusedthetooltoconstructmathematicalmeaningoutofparticulartasks.DoerrandZangorsingthepraisesofthegraphingcalculatorasacomputationaltool,transformationaltool,datacollectionandanalysistool,visualizingtool,andacheckingtoolforstudents(Doerr&Zangor,2000,p.161).Theseauthorsdid,however,notethelimitationsofthistool.Thefirstlimitationisthestudentsideathatthecalculatoristhemagicblackbox.Theyimpliedthatoccasionallythestudentsdidnthaveameaningfulstrategyforusingthecalculator,butratherpushedbuttonsrathert
hanattendingtothemeaningoftheproblempresentedtothem.Thesecondlimitationwastheeffectthatcalculatorshadongroupwork.Studentstendedtousetheircalculatorsasprivatedeviceswhichoftenledtothebreakdownofgroupinteractions(Doerr&Zangor,2000,p.158).Thisseemsreasonablebecausestudentscouldbeatdifferentstagesoftheproblem,basedontheirability.Studentsseldomsecondguesswhatthecalculatorsaysandfeelfreetocontinuewithout
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verificationfromthegroup.
Althoughthesestudieshavegivenmeagreatdealtoconsider,notonestudyevaluatedcalculatoruseinthemiddlegrades.Middleschoolstudentsareuniqueinthemannerinwhichtheyapproachproblems.Theyhavetheknowledgefoundationfromelementaryschool,buttheyarenotalwayscertainhow,when,orwheretoapplyit.Givenapowerfultoollikeascientificcalculator,willtheyuseitwiselyormerelymindlesslypushbuttons?Middleschoolstudentsaretheheartofmystudy,andIfeeltheyarewhatsetsthispaperapartfromtheothersbasedonthistopic.
PurposeStatement
Thepurposeofmyproblemofpracticeresearchistoattempttouncovertheeffectthattechnologyhasonstudentlearning,specificallytheeffectofpermittingcalcul
atorsingradesseven,eightandnine.AlotoftheresearchthatIhavereadforthisprojectsingsthepraisesoftechnologyintheclassroom,andpriortolastyearIwouldhaveagreedwithouthesitation.Iamnowconcernedabouttheeffectthatunlimitedcalculatorusehasonstudents.WhileIwouldneverwanttoteachinatechnology-freeclassroom,Iwanttotrytodetermineifandwhereboundariesneedtobeset.Mygreatestconcernisformymiddleschoolstudents,soIamgoingtolimitmyseventhandeighthgradestudentscalculatorusage.Iplantogiveapre-andpost-test
andseveralhabitsofmindproblemstodeterminetheeffectthelimitationhasoncomputationalskills,problemsolving,andreasoningskills.Ifeelthisstudyisnecessaryinaworldthatisbecomingmoretechnologicallyadvanced.Inwantingtogiveourstudentseveryadvantage,areweunderminingtheireducation,oristheremerelyaneedtoclingtothefamiliar?
ThequestionsIconsideredwere:
Whateffectwilllimitingcalculatorusagehaveonstudentachievement?
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Whateffectwilllimitingcalculatorusagehaveonstudentsproblemsolvingskills?Whateffectwilllimitingcalculatorusagehaveonstudentsabilitiestojustifythereasonablenessofananswer?Method
Thisstudywasconductedinmyeightgrademathematicsclassroomduringthespringsemester,2007.Becausemyinitialconcernwasthescoresthestudentsreceivedonthecomputationsectionoftheirachievementtests,thisremainedamajorfocusofthisstudy.Idecidedtocollectachievementtestscoresfromthesamesetofstudentsoverathree-yearperiod.Iwantedtoseethechangeintheircomputationalscoresfromsixthgradewherenocalculatorswereused,seventhgradewhichallowedunlimitedcalculatoruse,andeighthgradewhich
allowedlimitedcalculatoruse.TheschoolsguidancecounselorgavemeacopyofallthreesetsofscoresonMay1,2007.IthenputthedataintoaspreadsheetsothatIcouldvisualizethechange.Ialsocalculatedthemeanandstandarddeviationofthedifferencesfromsixthtoseventhgrade,seventhtoeighthgrade,andsixthtoeighthgrade.
Similarly,everyyearIgiveasemestertestinDecemberandafinalinMayforeveryclassthatIteach.BecauseIwantedtocomparethisyearsMath8teststolastyearstests,IdecidedtogivethesameformIhadgivenlastyear.Onthesemesterteststuden
tswerepermittedtousecalculators.Onthefinalteststudentswerenotpermittedtousecalculators.Ikeepcopiesofthesetests,sothisdatawasalsoeasytocollect.IthenputthedataintoaspreadsheetsothatIcouldseethechangeandcalculatedthemeanandstandarddeviationofthedifferenceslastyeartothisyear.
Toseehoworifcalculatorusageaffectedthestudentsproblemsolvingskills,Ihadthestudentsdohabitsofmindproblems.Theseproblemsweregiveneveryotherweek,startingthe
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weekofMarch19th,2007.Theywerecomplicatedproblemsthatinvolvedmorethansimplecomputation.Ipermittedcalculatoruseonsomeoftheproblemsandforbidthemonothers.WhilegradingtheseproblemsIlookedforalogicalthoughtprocess,soundmathematics,areasonableexplanation,andthecorrectanswer.Iusedarubrictogradetheseproblems(seeAppendixA).Fromthisdata,Ichosefourofthebestsamplestocompare.Twosamplesallowedtheuseofcalculators,andtwodidnot.Iwantedtoseeiftherewasanydifferenceintheproblemsolvingprocess,andiftheanswerstotheproblemswerereasonableandwelljustified.
Finally,IaskedmystudentstogivedailyhomeworkpresentationsstartingtheweekofMarch19th,2007.Iwouldchoosethreetofivehomeworkproblemseachday,andstudentstookturnspresentingtheseproblemsontheboardtotheirclassmates.Studentswereexpectedtoshow
alloftheirworkandwerenotpermittedtouseacalculator.Thesepresentationswerealsogradedwitharubric(seeAppendixB).Fromthisdata,Ilookedateachindividualstudenttoseeiftheirproblemsolvingskillswereimprovingordecliningandiftheiranswersmadesenseandwerejustifiable.
Findings
Whateffectwilllimitingcalculatorusagehaveonstudentachievement?
AsIhavementionedinprevioussections,oneofthemainreasonsIchosetodo
thistopicformyactionresearchprojectwasbecauseInoticedthecomputationsectionofmystudentsachievementtestscoreswaslow.Inordertodetermineiftherewasanyimprovement,Idecidedtocomparethecomputationscoresofmycurrenteighthgradestudentstoscoresforthesesamestudentswhentheywereinbothsixthandseventhgrade.ThedataIgatheredfromtheachievementtestscanbeseeninAppendixC.
BasedonthedatathatIwasabletocompileandcompare,Ibelievethatmystudentsdid
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showaslightimprovementinthecomputationsectionoftheachievementtestthisyear.Ilookedatthedifferencesforeachstudentfromsixthgradetoseventhgrade,fromseventhgradetoeighthgrade,andfromsixthgradetoseventhgrade.Themeanofthedifferencesfromsixthtoseventhgradewas-3.43withastandarddeviationof22.46.Thismeansmanyofthestudentssawadropintheircomputationscoresinseventhgrade,eightoffourteentobeexact.Thismeansunlimitedcalculatorusagehadanegative,thoughminimal,effectontheirscores.Themeanofthedifferencesfromseventhtoeighthgradewas4.57withastandarddeviationof
14.98.Thismeanstherewasanincrease,thoughsmall,inthestudentscomputationscoreswhenIchosetolimitcalculatorsintheclassroom.Themeanofthedifferencesfromsixthtoeighthwas1.14withastandarddeviationof27.01.Iwassurprisedtodiscoverthatsixofmyfourteenstudentsachievedtheirhigh
estscoreontheireighthgradetest.Iassumedthatthesixthgradescorewouldbethehighestforeveryone,becausemostofthemathematicsinelementaryhastodowithcomputation.Tenofmyfourteenstudentsscoredhigherthisyearthantheydidintheirseventhgradeyear,soapproximatelyseventy-onepercentofmystudentsseemedtobenefitfromlimitingcalculatoruse.
EveryyearIgiveasemestertestandfinalforeveryclassIteach.Thisyear,asIhadmentionedintheprevioussection,IchosetogivethesametestsothatIcould
comparethisyearsscorestolastyearsscores.Themeansofthesemestertestswerequiteclose.Themeanwas90forlastyearand91forthisyear.However,thestandarddeviationforlastyearwaslarger;thestandarddeviationforlastyearwas8.49,anditwas4.83thisyear.Ibelievethedifferencesinthestandarddeviationcouldbeduetolastyearsclasssizeandmakeup.Therewere12studentslastyearand16studentsthisyear.Themeansofthefinalexamswerealsofairlyclose;themeanwas88.4lastyearand91.7thisyear.However,therewasaslightincrease
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withthisyearsclass.Thestandarddeviationsof10lastyearand6.3thisyearmayhavehadagreaterdifference,onceagainbasedonclasssizeandmakeup.
Wantingtoknowmore,Idecidedtogenerateaboxandwhiskerplot(seeAppendixD).Themedianforthesemestertestwashigherlastyear,butthevalueshadagreaterrange.Thefirstquartilethisyearwashigher,butthethirdquartilewashigherlastyear.Overall,basedontheplots,Ibelievethesemestertestswerefairlyeven.Lookingattheplotsforthefinalexam,however,theredoesseemtobeanincreaseinthescoresthisyear.Thisyearsmedianisslightlylowerthanlastyears,butboththefirstandthirdquartilesarehigher,andtherangeissmaller.Thisdatasupportsmybeliefthatmystudentsdidshowaslightimprovementwhencalculatorswerelimited.
TheproblemIhavewithansweringthisquestionsolelyonthebasisoftwotests
ismydefinitionofachievement.Somepeopleonlychoosetolookattestscoreswhentheymakethedecisionastowhetheroranotastudentisachieving.I,ontheotherhand,prefertolookatthestudentsasawhole.Someofthesestudentshaveshownsuchgreatprogress,yetthiscanbedifficulttoseeonanystandardizedtest.Oneofthedynamicsofthisclassisthatassixthgradersahandfulofthesestudentswereplacedbackinthefifthgradetextbecausetheirteacherdidntbelievethattheywerecapableofsixthgrademath.Ifeelthatthisisoneofthemostdetrimental
thingsthatcanbedonetoachild,especiallythesekids.WhenIstartedseventhgradewiththemtherewassuchadivisioninabilitybasedprimarilyonalackofconfidence.WhenIgavethesestudentspermissiontousecalculators,IfeltlikeIevenedtheplayingfield.Thesestudents,whowereconvincedthattheywouldneverbesuccessfulinmath,begantoachievebyleapsandbounds.Onestudent,whoclaimedthathehadhatedmathduringhisseventhgradeyear,actuallyaskedattheendofthisyearifhecouldtrytotakealgebra,becauseIthinkIfinallyunderstand
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howtousetheletters.Thisstudenthadfailedsixthgrademath.Thistomeisachievement,and
thishappenedwiththeaidoftechnology.
Whateffectwilllimitingcalculatorusagehaveonstudentsproblemsolvingskills?
Ibelieve,basedonevidencegatheredfromHabitsofMindproblemsanddailyhomeworkpresentations,thatmystudentaremoresuccessfulproblemsolverswhentechnologyisavailable.Myfocalpointfortheseproblemswasproblemsolvingandreasoning.First,Icomparedsolutiontohabitsofmindproblemswhencalculatorswereandwerenotpermitted.WhilegradingtheseproblemsIlookedforalogicalthoughtprocess,soundmathematics,areasonableexplanation,andthecorrectanswer.Iusedarubrictogradetheseproblems(seeAppendixA).
Idiscoveredthatwithoutacalculatormystudentsseemedtofocusoncomputation.OneexampleofthisoccurredwhenIhadmystudentsdotheMATHproblem.Themathproblemconsistsofsixteenlettersarrangedinadiamondshape(seeAppendixE).ThestudentsareaskedtofigureouthowmanywaystherearetogetfromthetopMtothebottomH,giventhattheycanonlygodiagonallydown.Oneofmystudentsstartedmultiplyingthenumberoflettersineachrow(1*2*3*4*3*2*1=144),andthenproceededtoaddthenumbersineachrow(1+2+3+4+3+2+1=16).Shebroughtherpaperuptome,andsaid,Here!Iknowi
thastobeoneofthesesincewecantuseourcalculators.Iaskedhertothinkabouttheproblem,towhichshereplied,IknewImultipliedwrong.Ineedacalculator!Itoldherthathermultiplicationwasright,butherapproachtotheproblemwasincorrect.Shetookherpaperandsatbackdown,butshedidntchangeheranswer;thiswasoneofmybetterstudents.
OnthecontrarywhenIpermittedcalculators,thefocusseemedtoshifttoproblemsolving.Oneexampleofthisoccurredwhenmystudentsworkedonthechessboardproblem.We
hadjustcompletedasectioninourtextthatdealtwithscientificnotationandlargenumbers,soI
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decidedthisproblemwouldbeappropriate.Thisproblemtellsthestudentsaboutakingwhowassoexcitedaboutlearningtoplaythegameofchessthatheofferedhisteacheranything.Theteacheraskedforrice;heaskedforonegrainonthefirstsquare,twoonthesecond,fouronthethird,doublingoneachnewsquareforeverysquareonthechessboard.Becauseoftheiryouth,Iassumedmostwouldcontinuedoublingandthenaddthenumberstogether.Iwasexcitedtohaveastudentwrite,Becausethenumbersgottoolarge,Idecidedtotryasmallerboardtotryandfindapattern.Hecontinued,Inoticed,byusingatwobytwoandathreebythree,thatwhenyouaddupallthericeitisthesameasmultiplyingthelastsquarebytwoandthensubtractingone.Hedidntgetthefinalanswerbuthisthoughtprocesswaswonderful.Anotherstudentknewthatthelastsquarewouldhave263grainsofricebutdidntknowwhattodobecausehercalculatorhadgivenheranoverflowerror.
Ialsoaskedmystudentstogivedailyhomeworkpresentations.Iwouldchoosethreetofivehomeworkproblemseachday,andstudentstookturnspresentingtheseproblemsontheboardtotheirclassmates.Everystudenthadtopresentatleastonceaweek.Studentswereexpectedtoshowalloftheirworkandwerenotpermittedtouseacalculatorinclass.Thesepresentationswerealsogradedwitharubric(seeAppendixB).Aftereachstudenthadpresented,fellowclassmateswereencouragedtoaskquestions.Thisreallyseemedtoboostmystudents
problemsolvingskills.Theyhadtothinkquicklyandusuallydidntneedacalculatortogivetheiropinionorsolution.Inthisrespect,nothavingcalculatordidntseemtoaffecttheirskills.Ibelievethiswasbecausetheywerealreadyfamiliarenoughwiththeproblem,andtheirconfidencedidntwaiverwhenpeersaskedthemquestions.
Whateffectwilllimitingcalculatorusagehaveonstudentsabilitiestojustifythe
reasonablenessofananswer?
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Ibelievethatwhenmystudentsdidnthaveacalculator,providedthatnobasicmatherrorsweremade,theywereactuallyalittlebitbetterpreparedtoanswerthequestionIsthatreasonable?AgainIwillturntothedatagatheredfromtheHabitsofMindproblemsanddailyhomeworkpresentations.OneofthethingsthatIwasastickleraboutwasshowingeverybitoftheworkneededtoobtainanyanswer.ThiswastrueforboththeHabitsofMindproblemsandthehomeworkpresentations,soevenwhencalculatorswerepermittedthestudentshadtoshoweverystep.WiththeHabitsofMind,Ialsotriedtoencouragethemtowriteincompletelogicalsentences,mathematicalorotherwise.Ibelievethatthishelpedthestudentsthinkthroughtheiranswerandtheirthoughtprocess.TheHabitsofMindproblemsweredoneinclass,andIdecidedbasedontheproblemwhetherornotcalculatorswouldbeused.Ibelieveoverallthatstudentswerebetterabletojustifytheiranswer,becausetheyhadtoshowall
oftheirworkregardlessiftechnologywaspresent.
Forthemostpart,studentsdidverywellwiththedailyhomeworkpresentationproblems,butIquestionmydatabasedonthefactthatthestudentsdidmostoftherequiredproblemsathomewherethestudentsdidnotknowiftheyweresupposedtouseacalculatorornot.Becausethestudentsknewthattheyhadtoshowalloftheirworkontheboard,theydidseemtoworkthroughtheproblemsbyhand,butIamfairlycertainthatmostcheckedtheiranswerswitha
calculator,perhapsmotivatedbyfearofridiculebytheirpeers.Ibelieve,onceagain,thatthestep-by-stepbreakdownoftheproblemlentitselftoimprovedreasoning.
Conclusions
Whateffectwilllimitingcalculatorusagehaveonstudentachievement?
Basedonmyfindings,Iwouldhavetosaythattheanswertothisquestiondependslargelyonhowonechoosestodefineachievement.Ididseeimprovementinmystudents
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achievementtestcomputationscores,soclearlytherewasgoodreasontoquestioncalculatoruse.ThestudybyBridgeman,Harvey,andBraswellseemedtoshowthattheuseofcalculatorsonastandardizedtestresultedinamodestincreaseintestscores(Bridgeman,Harvey&Braswell,1995,p.323-340).Mydataappearedtoshowtheoppositeistrueforthecomputationsectionofthetestwhichdoesnotpermitcalculatoruseatall.IalsosawaminimalimprovementinmyfinalexamthatIgavethisyear.Iamnotcertain,however,ifeitherofthesecomparetotheresultIwasgettingwithsomeofmystudentswhencalculatorswerepermitted.
Whateffectwilllimitingcalculatorusagehaveonstudentsproblemsolvingskills?
Iwouldhavetosaybasedonmyfindingsthatwithoutacalculatortherewassomedeclineinmystudentsproblemsolvingability.ItriedtouseHabitsofMindproblemsthat
forcedthestudenttothink,butwhenIdidntletthemusecalculatorsitseemedlikesomeofthemnolongercouldthink.Itwasntthattheyusedthecalculatorsthatmuch;itwasmorethattheyneededtoknowthatthecalculatorwasthereiftheyneededit.Thisprobablyoccurredinpartbecausetheywerepermittedtousecalculatorswhenevertheyfeltlikeitlastyearinmyclassroom.IsharethefearofoneoftheteachersinvolvedwiththestudyconductedbyGrahamandThomas;mystudentshavebecomedependentonthetechnologyandfearfailurewithoutit(Graham&Thomas,2000,p.277).
Whateffectwilllimitingcalculatorusagehaveonstudentsabilitiestojustifythe
reasonablenessofananswer?
Mystudentsreasoningskillsdidimprove,butIamnotcertainofwhythisoccurred.Ibelievethatstudentsweregettingbetteratcheckingtheirworkbecausetheyhadtogivetheiroralpresentationstotheclass.Theygotusedtomeaskingthemifsomethingmadesense,andtheyknewthattheyneededtobesurethatitdid.Ihavenoideaifcalculators
wereusedathome,
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butIdoknowthathavingtoexplainandteachsomethingtosomeoneelsemakesyoumorethoughtfulofyourwordsandwork.Ibelievethatpriortothisstudymystudentssometimesthoughtofthecalculatorasthemagicblackbox.SimilartothestudybyDoerrandZangor,sometimesmystudentsdidnthaveameaningfulstrategyforusingthecalculatorbutpushedbuttonsratherthanattendingtothemeaningoftheproblempresentedtothem(Doerr&Zangor,2000,p.158).Whenforcedtoshowalloftheirstepsandwritetheirthoughtsdown,mystudentshavebecomemorethoughtfulwhichinturnhasimprovedtheirreasoningskills.
Implications
AlthoughnotallofmyresultswereasclearasIwantedthemtobe,Idothinkthatteachingastudenttouseatoolandallowingastudenttobecomedependentonatoolaretwoverydifferentresults.Iwillalwayspermitmystudentstousecalculatorsift
hetaskdeemsthemnecessary,butIwillnowsetrestrictions.Idontthinkithurtstoremindthestudentshowtomultiplyordividebyhand,especiallysinceIdontthinkthecomputationsectionofourachievementtestswillbechangedanytimesoon.Mostadultsuseacalculatoronaregularbasis,butstruggletoseethecalculatorsuseintheclassroom.Theywanttheirstudentstobetaughtinthemannerthattheyweretaught.Afterall,theyturnedoutallright.Ifirmlybelievethatmatheducationneedstofocuslessoncomputationandmoreonproblemsolving.
IplantokeeptheHabitsofMindproblemsaswellasthedailypresentationsapermanentfixtureinmyclasses.Ibelievethatbothofthesepractices,whichwereborrowedfromtheMathintheMiddleProject,allowmystudentstogrowacademicallyandareveryworthyofclasstime.IalsohaveinsistedonmoreworkbeingshownandhavelessenedthenumberofproblemsIrequiremystudentstodowiththeemphasisonqualityoverquantity.
Myrecommendationstoteacherswhofaceasimilarsituationistonotbeafraidtolet
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yourstudentsgainwhattheycanfromtechnology,butalsodontallowthemtobecomedependent.Requireyourstudentstoshowalloftheirwork,andchallengethemtosupporttheiranswers.Encourageyourstudentstothink,andteachthemhowtoproperlyusethetoolsprovidedbytodaystechnology.
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References
Bazzini,L.(2001).Fromgroundingmetaphorstotechnologicaldevices:Acallforlegitimacyinschoolmathematics.EducationalStudiesinMathematics,47(3),259-271.
Bridgeman,B.,Harvey,A.,Braswell(1995).Effectsofcalculatoruseonscoresonatestofmathematicalreasoning.JournalofEducationalMeasurement,32(4),323-340.
Doerr,H.M.,Zangor,R.(2000).Creatingmeaningforandwiththegraphingcalculator.EducationalStudiesinMathematics,41(2),143-163.
Forster,P.A.,Taylor,P.C.(2000).Amultiple-perspectiveanalysisoflearninginthepresenceoftechnology.EducationalStudiesinMathematics,42(1),35-59.
Graham,A.T.,Thomas,M.O.J.(2000).Buildingaversatileunderstandingofalgebraicvariableswithagraphiccalculator.EducationalStudiesinMathematics,41(3),
265-282.
Huntley,Rasmussen,Villarubi,Sangtong,Fey(2000).Effectsofstandards-basedmathematics
education:Astudyofthecore-plusmathematicsprojectalgebraandfunctionsstrand.
JournalforResearchinMathematicsEducation,31(3),328-361.
Kilpatrick,J.(2001).Understandingmathematicalliteracy:Thecontributionofresearch.EducationalStudiesinMathematics,47(1),101-116
Thom,J.S.,Pirie,S.E.B.(2002).Problems,perseverance,andmathematicalresidue.EducationalStudiesinMathematics,50(1),1-28.
Wheatley,C.L.(1980).Calculatoruseandproblem-solvingperformance.JournalforResearchinMathematicsEducation,11(5),323-334.
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AppendixA
HabitsofMindRubric
Logicalthoughtprocess:12345678910
Didthestudentsworkappeartofollowalogicalsequence?YesNoIfnot,isthereareasonforthedeviation?Soundmathematics12345678910
Isthestudentsworkmathematicallysound?YesNoIfnot,explain:Reasonableexplanation12345678910
Doesthestudentsexplanationmakesense?YesNo
Ifnot,explain:Correctanswer12345678910
Istheanswercorrect?YesNoIfnot,where/whydidtheerroroccur?Calculator:12345678910
Didthestudentuseacalculator?YesNoIfyes,howdiditaffecttheirthoughtprocess?
Comments:
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AppendixB
StudentPresentationScoringGuide
Rating(1-5)Comments:LogicalthoughtprocessSoundmathematicsReasonableexplanationCorrectanswerTotal(4-20)Calculator:YesN
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AppendixC
AchievementTestScores
120100806040200
StudentNumber
6thgradescore7thgradescore8thgradescore6of14achievedtheirhighestscoreduringtheir8th
gradeyear.
10of14didbettertheir8thgradeyearthantheydidtheir7thgradeyear.
Themeanofthedifferencesfrom6thgradeto7thgradeis-3.43withastandarddeviationof22.46.
Themeanofthedifferencesfrom7thgradeto8thgradeis4.57withastandarddeviationof14.98.
Themeanofthedifferencesfrom6thgradeto8thgradeis1.14withastandarddeviation
of27.01.
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AppendixD
SemesterTestandFinalComparison
LastYear'sSemesterTest
ThisYear'sSemesterTest
LastYear'sFinalExam
ThisYear'sFinalExam
0102030405060708090100110
SemesterTest-2005SemesterTest-2006Mean9091Variance72.1818181823.36666667StandardDeviation8.4959883584.833908012Median93901stQuartile84.587.875
3rdQuartile96.595Maximum100100Minimum7683Range2413SampleSize1216
FinalExam-2005FinalExam-2006Mean88.4166666791.6875Variance99.9015151540.09583333StandardDeviation9.9950745456.332127078Median92.5911stQuartile79.2587.53rdQuartile96.2597
Maximum100103Minimum7180Range2923SampleSize1216
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AppendixE
MMAMATMATHATHTHH
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