technology in learning atricle(1)

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


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


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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technology in learning atricle(1)

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