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
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4.10Visions
Visionswillbeimportantinthefuture,inallfieldsofscientificandpubliclife.Withoutvisionsthere
arenofurtherdevelopments.Weneed
whicharebasedon
.12.TheconnectivityofDT.Thisfigureappearsincolourintheonlineversionof
TeachingMathematics
anditsApplications
LOOKINGBACKANDAHEAD
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Thecorrectanswertothisproblemis:For
–8or
8,thefunction
hasexactlyonezero.Intest
examinationsweobtainedalotofstudentanswersalongthelinesof:
‘Sliderc=8.1.Forc=8.1(andmore),thefunctionf
(x)hasonlyonezero.’
8.1thefunctionhasonezero.’
ThisshowsaquiteuncriticalattitudeconcerningSC-resultsandemphasizesagaintheneces-
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.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.
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constructionofitemsandtestswithproblemsofdifferentdifficultylevels.Threelevelsofclassifica-
tionaregenerallyaccepted(e.g.inthePISAstudies):
Level1:Basicknowledge;
Level2:Advancedknowledge;
Level3:Complexknowledge.
Thisclassificationofaspecificproblemnaturallydependsonthestudents’knowledgeandexperience.
Aproblemwhichisnewtothestudentsmaybeclassifiedas
transfer
;iftheproblemhasalreadybeen
usedintheclassroom(severaltimes),itmaybeclassifiedas
reorganisation
reproduction
The4
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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
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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
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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.
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.4.(a)SolutionusingtheTI-Nspire;(b)SolutionusingtheCasioClassPad.Thisfigureappearsincolourin
theonlineversionof
TeachingMathematicsanditsApplications
.2.ThegraphG
andtheparallelline.Thisfigureappearsincolourintheonlineversionof
MathematicsanditsApplications
.3.Calculationoftheintegral.Thisfigureappearsincolourintheonlineversionof
TeachingMathematics
anditsApplications
H.-G.WEIGAND
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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
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,whicharebasedonstudentandteacherinterviews,onspecialindividualanswersin
frameoftheM
-project.Andweareespeciallygoingtorefertofinalbaccalaureateexaminations.
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2.Disillusions
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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
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