Challenging Mathematics In and Beyond the Classroom: The 16th ICMI Study PDF
Preview Challenging Mathematics In and Beyond the Classroom: The 16th ICMI Study
Challenging Mathematics In and Beyond the Classroom New ICMI Study Series VOLUME 12 PublishedundertheauspicesoftheInternationalCommissionon MathematicalInstructionunderthegeneraleditorshipof Miche`leArtigue,President BernardR.Hodgson,Secretary-General Thetitlespublishedinthisseriesarelistedattheendofthisvolume. Edward J. Barbeau Peter J. Taylor l Editors Challenging Mathematics In and Beyond the Classroom The 16th ICMI Study 1 3 Editors EdwardJ.Barbeau PeterJ.Taylor UniversityofToronto UniversityofCanberra ON,Canada ACT,Australia [email protected] [email protected] ISSN:1387-6872 ISBN:978-0-387-09602-5 e-ISBN:978-0-387-09603-2 DOI10.1007/978-0-387-09603-2 LibraryofCongressControlNumber:2008931140 #SpringerScienceþBusinessMedia,LLC2009 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScienceþBusinessMedia,LLC,233SpringStreet,NewYork, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. Printedonacid-freepaper springer.com Preface Inthemid1980s,theInternationalCommissiononMathematicalInstruction (ICMI) inaugurated a series of studies in mathematics education by commis- sioningoneontheinfluenceoftechnologyandinformaticsonmathematicsand its teaching. These studies are designed to thoroughly explore topics of con- temporary interest, by gathering together a group of experts who prepare a StudyVolumethatprovidesaconsideredassessmentofthecurrentstateanda guidetofurtherdevelopments. Studieshaveembracedarangeofissues,somecentral,suchastheteachingof algebra,somecloselyrelated,suchastheimpactofhistoryandpsychology,and somelookingatmathematicseducationfromaparticularperspective,suchas culturaldifferencesbetweenEastandWest. These studies have been commissioned at the rate of about one per year. Once the ICMI Executive decides on the topic, one or two chairs are selected and then, in consultation with them, an International Program Committee (IPC) of about 12 experts is formed. The IPC then meets and prepares a Discussion Document that sets forth the issues and invites interested parties tosubmitpapers. ThesepapersarethebasisforinvitationstoaStudyConference,atwhichthe variousdimensionsofthetopicareexploredandabook,theStudyVolume,is sketched out. The book is then put together in collaboration, mainly using electroniccommunication.Theentireprocesstypicallytakesaboutsixyears. Thetopicofthe16thStudywaschosenin2002,andwewereappointedas JointChairs.Soonafter,theIPCwasselectedanditmetinModena,Italy,in November2003todraftaDiscussionDocument.ThiswasfinalizedatanIPC meetinginCopenhagenatICME-10inJuly2004.Thecallforpaperswasthen issued.AboutfiftypeoplewereinvitedtoattendtheStudyConference,which washeldinTrondheim,Norway,atthebeginningofJuly2006. Initially, the participants in the Study Conference were divided into two groups, examining challenges beyond the classroom, and in the classroom. About two-thirds wished to work on the second topic, so the second group wassplitintwo,onefocusingontheviewpointofthestudentandtheotheron thatoftheteacher.EachgroupwasconvenedbymembersoftheInternational ProgramCommittee. v vi Preface Peter Kenderov convened Group 1: ‘‘Beyond the classroom’’. Maria Bartolini Bussi, with assistance from Mark Saul, convened Group 2: ‘‘In the classroom:studentperspective’’,andDerekHoltonconvenedGroup3:‘‘Inthe classroom:teacherperspective’’. The groups refined the organization of the material, and then broke into teamstoworkonindividualchapters.Chapters1–3aretheresultoftheworkof Group1,Chapters4and5camefromGroup2andChapters6–8fromGroup3. Eachteamselectedacoordinatingauthortoassemblethecontributionsand develop a coherent chapter. Communicating by email, each team was able to discussthedocumentasawholebeforesubmittingittotheeditors. Asaresult,eachchapterwaswrittenasajointpaper,eventhoughitmaybe obviousattimesthatsomesectionsoriginatedfromaparticularauthor.Inmost cases,thefirstnamedauthorcoordinatedthewriting. Duringtheeditingphaseparticipantswereabletoaccessageneralworking websitewiththelatestversionofeachchapter.Thusallwereabletocomment onanypartofthevolumethroughouttheeditingprocess. On very few occasions, writing teams co-opted work from outside the original group when it was felt to be particularly appropriate. The author list wasthenextended. EachICMIStudyhasitsownevolutionandcharacter.Inourcase,theStudy Volumeconsistsofeightpapers,eachwithitsownreferencelist. Inaseparatesection,weacknowledgespecialindividualcontributionstothe volume andsomebenefactors whosesupporthasenabled authors towork on theStudy. References Asdocumentsevolveinamoderntechnologicalagethisvolumecontainsmany instantreferencestowebsites.Wehavechosentoleavetheseinthetexttoavoid cross-referencing(ingeneral).Inmanycases,normalreferencescanbefoundon theInternet.Theyareshownwiththelastdayofknownaccess,inthenormal way,attheendofachapter.Onotheroccasionsareferenceisnotgiven,dueto it being a relatively common term, which can easily be accessed by a direct Internetsearch. School years There is no international consistency in naming school years. In order to standardizeourdiscussionwehavegenerallyadoptedtheKto12systemused inanumberofcountries.InthissystemYearKisforstudentswhohaveturned 5,Year1forstudentswhohaveturned6,uptoYear12forstudentswhohave turned 17. In many countries this is the last year before normal university Preface vii enrolment.Wehavegenerallyadjustedallyearlevelreferencestothissystem, but in places, to keep the original flavor we kept some French terminology. Whereused,thismeans6eisapproximatelyYear6,5eisapproximatelyYear7, 2eisapproximatelyYear10,andTerminaleisapproximatelyYear12. June2008 EdwardJ.Barbeau PeterJ.Taylor Contents 0 Introduction ............................................ 1 EdBarbeau 1 ChallengingProblems:MathematicalContentsandSources ....... 11 VladimirProtasov,MarkApplebaum,AlexanderKarp, RomualdasKasˇuba,AlexeySossinsky,EdBarbeau, andPeterTaylor 2 ChallengesBeyondtheClassroom—Sources andOrganizationalIssues ................................. 53 PetarKenderov,AliRejali,MariaG.BartoliniBussi, ValeriaPandelieva,KarinRichter,MichelaMaschietto, DjordjeKadijevich,andPeterTaylor 3 TechnologicalEnvironmentsbeyondtheClassroom.............. 97 ViktorFreiman,DjordjeKadijevich,GerardKuntz, SergeyPozdnyakov,andIngvillStedøy 4 ChallengingTasksandMathematicsLearning.................. 133 ArthurB.Powell,IngerChristinBorge,GemaIne´sFioriti, MargoKondratieva,ElenaKoublanova,andNeelaSukthankar 5 MathematicsinContext:FocusingonStudents ................. 171 MariaG.BartoliniBussi,SharadaGade,MartineJanvier, Jean-PierreKahane,VincentJ.Matsko,MichelaMaschietto, Ce´cileOuvrier-Buffet,andMarkSaul 6 TeacherDevelopmentandMathematicalChallenge.............. 205 DerekHolton,Kwok-cheungCheung,SesuthoKesianye, MariaFalkdeLosada,RozaLeikin,GregoryMakrides, HartwigMeissner,LindaSheffield,andBan-HarYeap 7 ChallengingMathematics:ClassroomPractices................. 243 GloriaStillman,Kwok-cheungCheung,RalphMason, LindaSheffield,BharathSriraman,andKenjiUeno ix x Contents 8 CurriculumandAssessmentthatProvideChallenge inMathematics.......................................... 285 MariaFalkdeLosada,Ban-HarYeap,GunnarGjone, andMohammadHosseinPourkazemi ConcludingRemarks ......................................... 317 Acknowledgements........................................... 319 AuthorIndex............................................... 325 SubjectIndex............................................... 331
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