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