Table Of ContentChallenging 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
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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
