Looking back and ahead — didactical implications for the use of digital technologies in the next decade


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5.Finalremarks TheM -projectwasfinishedinJuly2013.Now,allBavariangrammarschools(Gymnasien)are http://teamat.oxfordjournals.org/ Downloaded from 4.10Visions Visionswillbeimportantinthefuture,inallfieldsofscientificandpubliclife.Withoutvisionsthere arenofurtherdevelopments.Weneed whicharebasedon .12.TheconnectivityofDT.Thisfigureappearsincolourintheonlineversionof TeachingMathematics anditsApplications LOOKINGBACKANDAHEAD http://teamat.oxfordjournals.org/ Downloaded from Thecorrectanswertothisproblemis:For –8or 8,thefunction hasexactlyonezero.Intest examinationsweobtainedalotofstudentanswersalongthelinesof: ‘Sliderc=8.1.Forc=8.1(andmore),thefunctionf (x)hasonlyonezero.’ 8.1thefunctionhasonezero.’ ThisshowsaquiteuncriticalattitudeconcerningSC-resultsandemphasizesagaintheneces- http://teamat.oxfordjournals.org/ Downloaded from .10.(a)Thegraphofthefunction =5;(b)Thegraphofthefunction =8. .9.(a)SolutionwiththeTI-Nspire;(b)SolutionwiththeClassPad400.Thisfigureappearsincolourinthe onlineversionof TeachingMathematicsanditsApplications .11.(a)Thegraphofthefunction =8.1;(b)Azoomedgraphwith =8.1. LOOKINGBACKANDAHEAD http://teamat.oxfordjournals.org/ Downloaded from constructionofitemsandtestswithproblemsofdifferentdifficultylevels.Threelevelsofclassifica- tionaregenerallyaccepted(e.g.inthePISAstudies): Level1:Basicknowledge; Level2:Advancedknowledge; Level3:Complexknowledge. Thisclassificationofaspecificproblemnaturallydependsonthestudents’knowledgeandexperience. Aproblemwhichisnewtothestudentsmaybeclassifiedas transfer ;iftheproblemhasalreadybeen usedintheclassroom(severaltimes),itmaybeclassifiedas reorganisation reproduction The4 http://teamat.oxfordjournals.org/ Downloaded from basicknowledgeconcerningtheimplied mentalrepresentation ofthefunctions.Inall,30%ofthe studentswereabletosolvetheprobleminexample3a)inparagraph4,butlessthan5%wereableto solvetheprobleminexample3b)withthegeneraldomainID= ThesisV :Users(students)ofDTneedtohavestrategiestocontrol,verifyandrevisesolutions obtainedwithDT. .8.(a–c)Screenshotofthefunctionswith f(x)=sin(x)+1 g(x)=2 LOOKINGBACKANDAHEAD http://teamat.oxfordjournals.org/ Downloaded from Youmightbecontentwiththesolution6a,solution6bmightbeacceptedinaSC-environment, butitdemandsspecialknowledgeaboutSC-commands.Butforsureyouwillnotbecontentwith solution6c! ThesisIII :Criteriafortheadequatedocumentationofsolutionsinwrittenexaminationshavetobe WrittenexaminationsofstudentswithDTaskforclearinstructionsforthedocumentationofwritten solutions.Buttherearenoalgorithmicrulesornormshowtodocumentasolutiononpaper.Thisopens thequestionconcerningthe adequatedocumentationform ofthesolutiononpaper. Inourproject,westartedtodevelop for(non-)correct,(non-)accepteddocumentationsof solutions,e.g. Itisnotenoughtoonlywritedownwhatweseeonthescreen! Thesolutionhastobeunderstandable forothers ,andithastobeclearandvisiblewhenandwhere theSCwasused. Thesolutiondescribesthemathematicalactivities;itisnotonlyadescriptionusingaspecial calculatorlanguage .7.Awarningsignappearsinthedisplay:‘Somemoresolutionsmayexist’.Thisfigureappearsincolour intheonlineversionof TeachingMathematicsanditsApplications H.-G.WEIGAND http://teamat.oxfordjournals.org/ Downloaded from traditionalpaperandpencilexaminations.Examinationsinschoollessonsareverysignificanttoboth teachersandstudentsandshouldnotbeunderestimated.Quiteoften,andespeciallyinwrittentests,the solutiontoaproblemhastobedocumentedonpaper.WithaproblembeingsolvedusingtheDT,the documentationofthesolutiononpaperhastocombinetheworkwiththetool(theinputcommands), thescreendisplay(theoutput)andthecalculationsandnotesofthestudentonpaper.Evenafteroneor moreyear(s)ofSC-use,studentsexpressedtheiruncertaintyaboutthecorrectwaytonotedown solutionsonpaperandaskedforaclearinstructionforthedocumentation(seeSchmidt-Thieme& Weigand,2012). Thefollowingexamplesshowsomestudents’solutionsanddifferentdocumentations.Theunder- linedpartsemphasizetheuseofthehandhelddeviceduringtheproblem-solvingprocess. Example2: Giventhefunctionfwithf(x)=(x–2) .6.(a)Astudentsolutionquitesimilartoapaperandpencilsolution;(b)astudentsolutionusinga handheldcommand;(c)astudentsolutionandtheEnglishtranslation.Thisfigureappearsincolourinthe onlineversionof TeachingMathematicsanditsApplications .5.SolutionwiththeTI-NspireNotebookVersion. LOOKINGBACKANDAHEAD http://teamat.oxfordjournals.org/ Downloaded from .4.(a)SolutionusingtheTI-Nspire;(b)SolutionusingtheCasioClassPad.Thisfigureappearsincolourin theonlineversionof TeachingMathematicsanditsApplications .2.ThegraphG andtheparallelline.Thisfigureappearsincolourintheonlineversionof MathematicsanditsApplications .3.Calculationoftheintegral.Thisfigureappearsincolourintheonlineversionof TeachingMathematics anditsApplications H.-G.WEIGAND http://teamat.oxfordjournals.org/ Downloaded from ThefollowingproblemisfromthefinalbaccalaureateexaminationinBavariainMay2012.The texthereisgiveninashortenedformonly. Example1: AceramicartpieceintheCasaBatllo (seeFig.1)hasroughlytheshapeofaparabola. Giveamodeloftheshapeoftheupperrimoftheartworkbyusing q(x)=ax +bx +cx +dx+e (Forcontrol:q(x)=–0.11x –0.81x +5) Thelinegrunsparalleltothex-axisanddividestheworkofartintotwosections.Theareaofthe uppersegmentshouldbe71.5%ofthewholearea. Weconcentrateonlyonproblemc).Thesituationshowsthefollowingscreenshot(Fig.2).Youfirst havetocalculatetheintegralofqfrom 2to2(Fig.3). Ifyousolvetheequation )=cyouget—withthehandheldTI-Nspire—thefollowingresult(Fig.4). Thesmalltrianglesattherightsideofthelastlineofthescreenshotshowthattheformulaisnot finishedattheendofthescreen.Scrollingtotherightshowsasurprisinglongline.Itisimpossible— .1.TheartworkofGaudi LOOKINGBACKANDAHEAD http://teamat.oxfordjournals.org/ Downloaded from ,whicharebasedonstudentandteacherinterviews,onspecialindividualanswersin frameoftheM -project.Andweareespeciallygoingtorefertofinalbaccalaureateexaminations. http://teamat.oxfordjournals.org/ Downloaded from 2.Disillusions http://teamat.oxfordjournals.org/ Downloaded from Lookingbackandaheadçdidactical implicationsfortheuseofdigital technologiesinthenextdecade EORG EIGAND DidaktikderMathematik,Universita tWu rzburg,HublandCampusNord,Emil-Fischer-Str.30,97074Wu *Email:[email protected] [SubmittedNovember2013;acceptedDecember2013] Advantagesanddisadvantagesoftheuseofdigitaltechnologies(DT)andespeciallyof computeralgebrasystemsinmathematicslessonsarediscussedcontroversiallyworld- wide.WhatwillbetheimpactofDTinthenextyearsoreventhenextdecade?Thebasis ofthefollowingconsiderationsisthelong-termempiricalprojectM (Model-project NewMediainMathematicslessons)whichwasstarted10yearsagoin2003totesttheuse ofsymboliccalculators(SC)inBavarianGymnasien(grammarschools)inGermany.In 2013,thereexistsextensiveexperienceregardingtheuseofSCfromgrade10to12inclass- roomactivities,studentandteacherdocumentsaswellastestandexaminationresultsof students.Theimplicationsofthisprojectaregoingtobecollectedin10thesesorhypoth- esesaboutpossible,beneficialdevelopmentsinthefuture.Thesetheseswillbeexplained withexamplestakenfromtheprojectM .Theyareaddressedtomathematicsteachers andmathematicseducators,topeoplewhoareinterestedintheon-goingdevelopmentof http://teamat.oxfordjournals.org/ Downloaded from

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