teaching mathemacs with reasoning tools: learning, teaching and … misfeld... · 2016-10-27 ·...
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Teachingmathema,cswithreasoningtools:learning,teaching
andcurriculumplanningMortenMisfeldt
Professoroftechnologyandmathema,cseduca,on,researchmangerforITandlearningdesign.
AalborgUniversityCopenhagen
WhereamIfrom…ResearchLab:ITandlearningdesign
• Researchthemes– Studentsasproducersanddesigners– GameBasedLearning– ICTandcoretopicsinprimaryandsecondaryschool– ICTandter,aryeduca,on– Organiza,onalaspectsofICTandlearning
The,tle?
• Teachingmathema,cswithreasoningtools:learning,teachingandcurriculumplanning
• Reasoningtools:=CAS,DGS,Toolsforspecifictasks• Learning,Teaching,Curriculumplanning:=– Whatkindoflearningprocessesandleaningdifficul,escanweobserve
– Whatkindofvaluesaboutourdisciplinearewetryingtoteachandwhatconflictsareweexperiencingasteachers
– Whattopicsshouldbeinthecurriculumandhowdoweorganizecurriculummaterialssuchatextbooks
Overviewoftalk
• ExplainaboutwhatIdoinordertoconveyfromwhereIspeak
• Thesitua,oninDenmark• Threeexamplesofreasoningtoolproblems– instrumentalunderstandingandlearningdifficul,es
– Trianglesandtrigonometry– ProofsandCAS
• Zoomout,discussionandconclusions
Myresearchportefolio
• Iamyourfriend(seriously)– ButIwilltrytochallengeyouhere.
• Researchinmathema,calprac,ce• Youngstudentscrea,vemathema,calICTwork
• Teachersuseoftechnology• …andtheinfluenceofCAS/DGS/ectonsecondaryeduca,on.
FromwhereIspeak1:Mathema,calPrac,ces
• Philosophicalandeduca,onalproject• Objec,ve:
– Gainempiricalknowledgeaboutmathema,calresearchprac,ce• Problemsolving,posingandchoosing• theuseofmateriality/representa,oninmathema,calac,vi,es
– Developknowledgethatinformsmathema,cseduca,on• Mean:
– Interviewresearchersofmathema,csaboutworkprac,ces,howtheya\ackandsolvemathema,calproblems,howtheyusevisualrepresenta,onsandcomputers,andhowtheycollaboratewithpeers.
Misfeldt,M.,&Johansen,M.W.(2015).Researchmathema,cians’prac,cesinselec,ngmathema,calproblems.Educa,onalStudiesinMathema,cs,89(3),357-373.DOI:10.1007/s10649-015-9605-3
Workingwithmathema,calproblems
• “Trytodescribehowyoustarttoworkonamathema4calproblem.”– Well,infactIbelieveitallstartsabitearlier,becauseyoualsoneedto
findtheproblemyouwishtoa?ack.Therearesomanyproblems,soyouhavetomakesomekindofselec4on.
– Well,themostimportantisofcourse…Themostessen4al,whichcomesearlier,isthatyouhavetoasksomeques4ons,right,andthenyoucanaskwherethoseques4onscomefrom.
• “Howdoyouchosewhatproblemtoworkon”:– “Asama\eroffact,Ibelieve…infact,itisthehardestpart.”– “Itisnoteasy.Thegreatartinmathema4csistofindaproblemthat
youhaveideasabouthowtosolve.Andinfacthaveachanceofsolving.[…]Thegreatestartisnottosolveproblems,thegreatestartistofindtheproblemswecansolve.”
Mathema,calproblems
• Isacontextformathema,calinves,ga,on• Leadstonewmathema,calproblems• Isnotathingtobesolvedandlem• Thechoiceofwhatproblemstoworkoniscrucialforamathema,cian,he/shebalances:– Interest– Headstart– Audience(community)
FromwhereIspeak2:Crea,veDigitalMathema,cs
• Teachersasdesignersoflearningenvironments/scenarios
• StudentsasproducersofboardgamesandvisuallayoutswithGeoGebra.
• 2011-2014(approx70teacherstotal)• Goalwastopush– Thetypicalstudentac,vityinprimarymathed.towardsproducingandexploringwithandwithinmathema,cs
– Teachercollabora,onandteacherroles,towardsproducingandexploringnewlearningsitua,onsincollabora,on.
Misfeldt,M.,&Zacho,L.(2016).Suppor,ngprimary-levelmathema,csteachers’collabora,onindesigningandusingtechnology-basedscenarios.JournalofMathema,csTeacherEduca,on,19(2-3),227-241.DOI:10.1007/s10857-015-9336-5
Teachersasscenariodesigners
Currently36scenariosonthewebpage
Aprototypicalscenario–thegamefactory
1.Welcomegamebuilder2.TryGeoGebra3.Tasks4.Gameproject5.Test6.Applyforwork
12
Exampleofgame
ThepurposeofSPGameisthatplayernumberonethrowsadicethree,mes,andifyouthrowa“4“youcanmovetothenearestnextfield.Whengettoafieldwithamul,plica,oncalcula,on,youshoulddothecalcula,onright,andthenyoucanmovealloverthegameboardbutnotintothetargetzone.Ifyouarenexttothetargetzoneyouhavetothrowa5withthediceinordertogetintothetarget.
Sorryforthelongintro• Thegoalofmytalkistozoomintosomeoftheproblems
thatIseewithreasoningtoolsinsecondaryeduca,oninDenmark
• Theanecdotesandrumors– Allstudentlackskillsinmathema,cs(anditisallbecauseofCAS!J)
– Cleverstudentsareboredinmathema,cs– Newtypesoflearningdifficul,esemerges
• Thefacts– Prominentmatheduca,onscholarsarguestronglyagainsttomuchuseofCASandsimilartoolsinDanishuppersecondary
– Teachersdisagreeaboutthevalueofthesetools– PolicydocumentpromoteICT
TheStateofthe:tousetoolsornottousetool
• Toolsareanessen,alpartofbothdoingmathema,csandteachingmathema,cs
• Theuseofefficientcogni,vetoolscharacterizemathema,cs
• Toolsinfluencesmathema,calac,vi,esoutsideschool• AndCASworks–ametaanalysisfrom2008showsitis
efficient(Tokpah2008)
• Iamnotheretoadvocateagainsttheuseoftools– Butwedoneedatdeeperunderstandingoftheproblemsthatcomesfromusingthesetools
Tokpah,C.(2008).TheEffectsofComputerAlgebraSystemsonStudents'AchievementinMathema4cs.(ElectronicThesisorDisserta,on).Retrievedfromh\ps://etd.ohiolink.edu/
TheStateofTheArt:mainconcerns
• Understandingtoolsandtechnologiesinmathema,cseduca,onisredocean
• Thehistory– Fromchangeindisciplinetounderstandingnewlearningprocesses,toteachingsitua,ons
• Theprac,calaspects– Cabri,GeoGebra,Geomatech…
• Thetheore,calaspects– Understandinghumanswithtools,blackboxing,instrumentalgenesis,cogni,vear,facts
• Thecoreproblems– Assessment,teachingwithtools
• Andthevalues– IsCASmathasgoodaschalkmath…whataboutcomputa,onalthinking.
Myapproachtoday• Examplesoftensionsandproblems• Goal:friendlycri,que• Levelsinunderstandingthesitua,on– Learningandreasoningtools– Teachingandreasoningtools– Curriculumdesignandreasoningtools
• Threeexamples– CASandalgebraicwork– Trianglesandtrigonometry– CASandproofs
TheDanishsitua,on
Successfulimplementa,on–problemsandcri,que
Technologyinloweranduppersecondarymathema,cseduca,onin
Denmark• Lowersecondaryeduca,on
– majorleapintechnologyuseinthelast2-3years– MaintechnologiesareGeoGebraandExcel– AlotofCAItoolsareused– Partofthecurriculumfrom2015– Seemssuccessful–buthardtogetallteachersonboard
• Uppersecondary– Atleasttenyearsofheavytechnologyuse– CASsystems(maple,TIinspire,mathcad)aredominant– Partofthecurriculumfrom2005– Inten,onsisthatCASshouldnotonlyservetheroleofatoolfor
solvingproblems,etc.butalsobeseenasaninstrumentforconceptualunderstanding.
– Thedetailsoftheimplementa,onislemtotheschools,theteachers,andthetextbookauthors.Anditisratherdiverse.
TheDanishdiscussionProorconict• IT-alis4cversusmathema4calapproachtomathema,cseduca,on(M.Nissinvarioustalksandinterviews)– Wemightnotagree–butweneedtotakeno,cewhenverystrongscholarsinthefieldpointtoproblemsrelatedtotechnologyuse
• Informalreportsonproblemsinuppersecondary,butdefinitelynotconsistent.
• Cri,queoftheuseofCAItoolsinlowersecondary/primary
Learningdifficul,es
InducedbyCASuse
Asimpletaskandaninnocentmistanke
• GivendN/dt=–16N+32withtheboundarycondi,onthatN(10)=1,findanexpressionforN(t)
• Casestudybasedoninterviews,documentsandobserva,on
Jankvist,U.,&Misfeldt,M.(2015).CAS-InducedDifficul,esinLearningMathema,cs?FortheLearningofMathema,cs,35(1),15-20.
Problem• WeseealargenumberofsuchverybasicmistakesinDanishuppersecondaryschool
• Mistakesthatwecanunderstandintermsoflackofrela4onalunderstanding(Skemp1976)aswellasintermsofthedevelopmentofinstrumentedtechniquesthatchangesthelandscapeofmathema,calrela,ons.
• Exampleofinstrumentedtechniques– Seeanequa,on->solve(“theequa,on”,x)– Seeadifferen,alequa,on->desolve(“theD-equa,on,x,y)
Thevisciouscircleofreifica,on(Sfard1991)
CAS-Relatedlearningdifficoul,esinmathema,cs
• Proceduralunderstanding• Studentscanactwithoutofrela,onalunderstandingands,llobtainrequestedresults
• InthatsenseCAScanhelp”overcome”SfardsViciouscirclewhichthenbecomesaviciousspiralandinsteadofkeepingstudentsawayfromperformingmathema,cally–itallowsthemtoperforminacompletelysuperficialway.
Teachingtrigonometry
Salientinfluenceoftechnology
TrigonometryinsecondaryschoolinDenmark
• Lowersecondary– Simpletriangleproblems(since2009)
• Uppersecondary– Complextriangleproblems(includingthevarioustrigonometricrea,ons)
– Theunitcircle– Thetrigonometricfunc,ons(thelateryears/higherlevels)
Misfeldt,M.(2014).Trekantsberegningerogteknologi:eteksempelpåhvordanteknologihar(ellerbørhave)indflydelsepåudviklingafMatema,kcurriculum.Mona,2014(1),27-43.
Thetechnologicalsitua,on
• DGSwithprecisemeasuringtools(givesstudentstheabilitytocreate“exact”drawings
• Trianglecalculatorsautomatesthealgebraic/trigonometricsolu,onstrategy
Example
• ”InatriagleABCthesidesarea=10,b=15ogc=21,findtheangleA”
Euclidian–onsteroids
Algebraic
Automa,c
!
Lookingatthesitua,on• Wehavethreestrategiestowardstheproblem– Euclidian– Algebraic– Automa,c
• Twooftheseare“new”• Thischangesthedynamicofteachingtrigonometry– cos,sinandtanareobsoletefordealingwithtrianglesitua,ons
– Thereisastrategyforstudentstoavoidworkingwithtrigonometry(automatedsolu,on)
Influenceoncurriculumlevel• Triangletrigonometryloosesrelevance• S,llrelevantasmeantoteachaboutthetrigonometricfunc,ons
• Butiscurrentlytaught3yearsbeforethetrigonometricfunc,ons
• Thissitua,onisnotunusual– Longdivision– Analyzingafunc,on(findzeros,f’(x)=0..Ameralotofalgebraicworkyoucandrawthegraph)
Onthepedagogicallevel– Howdoweanswerstudentwhoalwaystaketheeasyandautomatedapproach?
– Whatdowesaytostudentsthatasks“whydoweneedtolearnthetrigonometricapproach”• Itismorecorrect• Itpreparesyouforfuturestudy• Itispre\yandhistoricallyinteres,ng
– Whatimageofmathdoweconveybyansweringlikethis• Authoritarian• Inefficient• Textoriented• Prepara,ontolife–notlifeitself
Proofswithcomputeralgebrasystems
Implemen,ngreasoningtoolsintextbooks
CAS-Proofs
• AsdescribedCASismassivelypresentinuppersecondarymathema,csinDenmark(sincea2005reform)
• Ano,onandprac,ceof“CASproofs”or“CAS-assistedproofs“haveemergedinthetextbooks.
• ACASassistedproofisatextbookproofofastatementortheorem,whereoneorseveralstepsareperformedbyaCAStool.
WhatkindofCASProofsdoweseeintetbooks
• AlmostonlyCAS• CASforaspecificaspect• CheckingProofswithCAS
• Theremightbeotherforms
Example1:Completeoutsourcing
Technoauthoritarianproofscheme
• Noa\emptismadeatexplainingorvisualizingwhycos'(x)=-sin(x)andsin'(x)=cos(x)
• The“CASproof”isbothconvincingandtrustworthyifwetrusttheCAS,butisdoesnotexplainitisonlyaproofthatconvince(Hanna1989),
• anditreferstoasexternaltechnologicalauthority(theCAS),thatcanbemanipulatedbythestudent!Leadingtoinstrumentalgenesis(ifIdon’tknowthetruthvalue–IaskCAS)
• “techno-authoritarian”externalconvic,onproofscheme.
Example2:Outsourcingandcheckingofspecifictechnicalaspects
Provingthatthederiva,veofsin(x)iscon(x)byalgebraicmanipula,onsandthelimit.
Allisdonealgebraically(byhand)exceptthelimitsin(k)/kwhenkapproaches0,whichishandledinthreedifferentways,twooftheminvolvingCAS.1. Byavisualargumentusingtheunitcircle2. Byexperimen,ngwithsmallnumbersfork3. ByaskingtheCAStocalculate
Augmen,ngandcheckingtheproofbyuseofCAS
• Wehavealmostafullalgebraicproof,andCASaugmentsintwoways– TheuseofCASforcalcula,onofexampleswithverysmallvaluesofkisexperimentalanddoesnotplayaformalroleintheproof.
– TheuseofCAStoverifythelimitcalcula,onplaytheroleofcheckinganalgebraicandvisualargument
Ques,onsaboutCASandthatshouldbeaddressed
• DoestheCASuseestablishtruth?• DoestheCASuseallowinterac4onandexperimenta4on?
• Istheargumenta,oninduc4ve,deduc4veorauthoritarian?
• Doestheargumenthighlightimportantaspectoftheprooforthemathema,calrela,onships?Misfeldt,M.,&ThomasJankvist,U.(2016).Instrumentalgenesisandproof:understandingtherolesofcomputeralgebrasystemsintextbookproofs.PaperpresentedatICME,13Interna,onalCongressonMathema,csEduca,on,Hamburg,Germany.
Recab
• Reasoningtoolscanremovestudentsgroundingandcreateanewtypeoflearningdifficul,es
• Changeintechnologicalsitua,onisachangeinthelearningecologyaroundspecificmathema,caltopics
• Toolsaffecttheformats/genresthatareusedinmathema,csitua,on,leadingtocomplex/paradoxicalphenomenon'ssuchasautomatedsolu,onsanddistributedproofs
TheLevels• Learningandreasoningtools– Wedohavestrongtheore,caltoolse.g.instrumentedtechniquesbutIwouldargueformixingwithothertheoriesaboutlearningmathema,cs.
• Teachingandreasoningtools– Howshouldteachersreacttoscriptssuchastheautomatedtrianglecalculators,orheavyCASuseinalgebraicexercises
• Curriculumdesignandreasoningtools– Whentoteachwhat– Howtorefertotoolsintextbooks
Possibleac,ons(1):Onthelevelofschoolsystems
• Regulatetools(inteaching,inexams)• Inservicetraininganddiscussionamongteachers(normsandprac,cesdiffersgreatly)
• Con,nuouslyconsiderthetechnologicalsitua,onandchangesyllabusaccordingly
Possibleac,ons(2):Onthelevelofresearchandmatheduca,on
community(lastslide…)• Focusontheseparadoxes–unfoldandunderstandthem–Iwouldagueforaba\eryofproblemsitua,ons
• Ac,vatelearningtheoriesandtheoriesabouttheteachingandlearningofmathema,cs– Nottoprescribe–buttounderstand
• Wearemathema,cs!– We–matheducators–shouldseriouslyconsiderwhatkindofimageofthedisciplineweconveytostudent• Pushingbu\ons,learningoldredundantstrategiestoproblemsolving,anddistribu,ngmathema,calinsightstomachinesisnotwhatwewantstudentstolearn
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