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UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina1

WIMS attheUniversityofMilano-Bicocca MarinaCazzola DipartimentodiMatematicaeApplicazioni Facolt`adiScienzedellaFormazione 29may2010

W IM S

September2000:firstItaliantranslationofWIMS Inashort,inthewholesystem,youshouldtranslateeach fileorsubdirectoryterminatingwithan.enora.frintoan additionalfile/subdirectoryterminatingwithan.it.[...] Toknowwhatinthefileislanguage-dependent(soshould betranslated)fromwhatisnot(forexamplethe commands),youcancomparevisitor.phtml.enwith visitor.phtml.fr,tolocatetheplaceswherethetwofilesare different.

PetithistoriqueIT-WIMS WIMS •Petithistorique IT-WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina4

December2000:firstusewith“real”students LinearAlgebraforstudentsincomputerscience. Useofthereadyavailableexercisesonthetopic (mostlybyXiaoGang),inEnglish(veryfewtranslation intoItalian). Usedbothforselfstudyand(“selectively”)forexams; 145registeredusers. PetithistoriqueIT-WIMS WIMS •Petithistorique IT-WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina5

October2002:wims.matapp.unimib.itlistedasaWIMS mirrorsite Sporadicworkonthetranslation,asteachingtono classsuitablefortheuseofWIMS(nomorelinear algebracourse)andnotime/notenoughtexperienceto writedownmyownexercises.

PetithistoriqueIT-WIMS WIMS •Petithistorique IT-WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina6

Novembre2004:colleaguesinFlorencestartusing WIMS. Milano-Bicoccatakescareofthetranslationofthe core/admmodules,whileFlorencestartstotranslate theexercisemodules.

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WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina7

PetithistoriqueIT-WIMS WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina8

October2003:IwasenrolledbytheFacolt`adiScienze dellaFormazione Teachingactivitymainlyinthepre-serviceprimary schoolteachertrainingprogramme. CoursesnotsuitablefortheuseofWIMS •nocomputerlabsavailableforthestudents •pedagogicalconcerns

PetithistoriqueIT-WIMS WIMS Teachertraining Testingstudents’ proficiency E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina9

Futureteachersneedstoacquireabilitiesthatgo beyondcomputationalskills:theyneedtobeableto formulateabstractmathematicaldefinitionsor statementsandtowritesimpleproofs. Intheacademicyear2005/06computerlabswere madeavailable,sowecouldreallystartusingWIMS withourstudents.TheuseofWIMSwasalso stimulatedbythewillofthefacultytostart experimenting“e-learning”.

T e s ti n g s tu d e n ts ’ p ro fi c ie n c y

WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina10 Mathsforprimaryschoolteachers WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina11

Thepre-serviceteachertraininguniversitydegree courseisafouryearsprogram.Futureteacherhave thefollowingmathcourses •“IstituzionidimatematicheI”(Elementsof mathematics,firstpart):arithmetic(firstyear) •“IstituzionidimatematicheII”:geometry(second year) •“Didatticadellamatematica1B”(mathematics education):probabilityandexampleofgood teachingpractice(thirdyear) •“Didatticadellamatematica2”:moreteaching practice(fourthyear) Basiccomputationalskills WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina12

TheeasiestthingtodoistouseWIMScomputational capabilitiesinordertoteststudents’basic computationalskillsattheverybeginningoftheir courseofstudy. Allthefirstyearstudents(about400/450everyyear) hastotaketheWIMStest. 10%ofthestudentsfailthetest (Academicyears:2006/07,2007/08,2008/09and 2009/10). Butwithstudentsintheirsecondandthirdyear,we needtogobeyondtestingbasiccomputationalskills. Manipulatives WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina13

Manipulatives WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina14 ...contrattodidattico UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina15

Whenmathematicsisconcerned •afewthingsareamustknow ◦beingabletodothecomputations(withintegersandwith fractions), ◦beingabletoconvertbetweensquaremetersandsquare centimeters,cubemetersandcubecentimeters, ◦beingabletotellasquarefromanhexagon •butfutureteachersalsoneedtobeableto“talk”aboutmaths

mustknow WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina16

youdonotreallyknowifyoucannottell UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina17

Inmy“Didatticadellamatematica1B”course(thirdyear),theexams consistinaWIMStest,awrittentestandanoralexamination. Thewrittentestpartlyconsistofgivingafullexplanationofoneof thequestionalreadyansweredintheWIMStest. RisponderealseguentequesitoWIMS,dandounapiena giustificazione(NOTABENE:inquestoesamesar`a valutatalacapacit`adiargomentareedareuna giustificazioneenonsemplicementeilraggiungimentodi unrisultatocorrettoinquantoquest’ultimoaspetto`egi`a statoverificatoconlaprovapreliminare informatizzata)

youdonotreallyknowifyoucannottell WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina18 Geometry WIMS Teachertraining Testingstudents’ proficiency •Mathsforprimary schoolteachers •Basiccomputational skills •Manipulatives •...contrattodidattico •mustknow •youdonotreallyknow ifyoucannottell •Geometry E-learning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina19

Duringthisacademicyearforthefirsttimeweare experimentingWIMSactivitiesforthe“Istituzionidi matematicheII”course(secondyear). •bigeffortincollecting(translating+creating) suitableexercises •stillthinkpartofthefinalexamshouldconsistof exercisesinwhichthestudenshaveto“give explanations”(i.e.stillneedforawrittenexam)

E-l e a rn in g

WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca•Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina20 E-learningprojectatBicocca UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina21

Theuniversityfundsaproject withtheaimofexperimenting e-learningonavastescale. InparticulartheFacolt`adi ScienzedellaFormazione (Facultyofeducation) (e.g.inthelastacademicyear thefacultyhas37coursestaught ine-learning)

Howtoteachinane-learningcourse? UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina22

Whichtoolsandwhichactivitiesaresuitablefore-learning? •Didatticadellamatematica(master) •Didatticadellamatematica1B(2005/06) •Didatticadellamatematica1B(rec-2006/07) •Didatticadellamatematica1B(2008/09) Manipulatives WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca•Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina23

Activelearning WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca•Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina24

Whattheteacherssaysintheclassroomis notunimportant,butwhatthestudentsthinkis athousandtimesmoreimportant.Theidea shouldbeborninthestudents’mindandthe teachershouldactonlyasamidwife.1 1?

Problemsandmathematics WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca •Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina25

...thereisagrainofdiscoveryinthesolution ofanyproblem.Yourproblemmaybemodest; butifitchallengesyourcuriosityandbrings intoplayyourinventivefaculties,andifyou solveitbyyourownmeans,youmay experiencethetensionandenjoythetriumph ofdiscovery.Suchexperienceatasusceptible agemaycreateatasteformentalworkand leavetheirimprintonmindandcharacterfora lifetime.2 2?

Problem-basedlearning WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca •Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina26

Problem-basedlearning(PBL)isaconstructivist learner-centredinstructionalapproachbasedonthe analysis,resolutionanddiscussionofagivenproblem. Itcanbeappliedtoanysubject,indeeditisespecially usefulfortheteachingofmathematics. PBL“isaninstructional(andcurricular) learner-centeredapproachthatempowerslearnersto conductresearch,integratetheoryandpractice,and applyknowledgeandskillstodevelopaviablesolution toadefinedproblem”.3 3? Problem-basedlearning UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina27

TypicallyaPBLsessionfollowsthesesteps4: •pupilsaregivenaproblemtheyhaveneverseenbefore; •theydiscusstheproblemand/orworkontheprobleminsmall groups,collectinginformationusefultosolvetheproblem; •allthepupilsgathertogethertocomparefindingsand/ordiscuss conclusions;newproblemscouldarisefromthisdiscussion,in thiscase •pupilsgobacktoworkonthenewproblems,andthecyclestarts again. 4?

The“problem” UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina28

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Avendocompletatolaproceduradiiscrizioni,siamopronti acominciare. Chiedoatuttivoiunapartecipazioneattivafindaora(“non siimparamatematicasenonsifamatematica”). te Ilcorsocominciaconunarevisionedelprimocapitolodel `etesto“LalotteriaaBabilonia”diGiulianoSpirito,cheil testofondamentaleperlapartedicalcolodellaprobabilit`a. Comeprimo“compito”vichiedodiiniziarelaletturain `eparticolaredeiparagrafi“Checosaunacorrispondenza” e“Unadefinizioneelegantedicorrispondenza”. The“problem” UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina29

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Nellarevisionedeltestovichiedoinparticolarmodotener contodeiseguentiaspetti: •checosa`eunacorrispondenza?te •saperdareesempidicorrispondenze(oltreaquelli contenutineltesto) •dataunacorrispondenza,saperlarappresentarein diversimodi

The“problem” WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca •Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina30

Do cen

Infinevichiedodileggereconattenzionei te“Quesiti”,checostituisconounariccafontedi esempi,ediscriverequidiseguitoglieventuali dubbichevisonosortiechenonsieteriuscitia risolverenellavorodigruppo. The“problem” WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca•Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina31

•Readthetextbook •trytounderstanddefinition,explanations, examples&exercisesgiveninthetextbook •discussaboutthemwithyourcolleguesinyour groupusingtheforum •practicewithWIMSexercisescompletingthe assignedworksheets •writedownanexplanationofyourfindingsforthe teacher

The“problem” WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca•Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina32

Theassignementemphasizedtheneedof“active participation”byallofthestudents:onlyaftergoing throughthestepsabovethestudentscouldaskforhelp fromtheteacher. Oneofthemainthemesofthecoursewas combinatorics,withrespectto“counting corresondance,permutationsandfunctions”,soitwas easytoimplementWIMSexercisescorrespondingto theexercisesgiveninthetextbook Note:inthespiritof“discovery”thestudentswere askedtofindouttheformulasbythemselves.

Roleoftheteacher WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca •Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina33

Astheaimofthecoursewasindeed“teachtoteach”, theactionsoftheteacherwherementtostimulate metacognitivereflections

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´eteSecondovoiperchgliesercizi[deltestdi autovalutazione]vicostringevanoalavorarecon numerianchepiuttostoalti? (WhydoyouthinkWIMSposesexerciseswithquitebig numbers?)

Roleoftheteacher WIMS Teachertraining Testingstudents’ proficiency E-learning •E-learningprojectat Bicocca •Howtoteachinan e-learningcourse? •Activelearning •Problemsand mathematics •Problem-based learning •The“problem” •Roleoftheteacher UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniWIMSatMilano-Bicocca–pagina34

Itwasgoodtodiscoverthatsomestudentsdidgetthe point mivienedapensarecheusareinumerialti`e stataunastrategiaper“costringerci”atrovare unageneralizzazioneeforseancheuna giustificazione,che,apensarcibene,con numeripiccolinon`enecessaria.

Stu de nte

(Isortofthinkthatbignumbersforceustofindouta generalizationadajustificationofourprocedure, justificationwedonotfeeltheneedforwithlower numbers)