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Trends in the History of Science Brendan Larvor Editor
Mathematical Cultures The London Meetings 2012–2014
TrendsintheHistoryofScience TrendsintheHistoryofScience isaseriesdevotedtothepublicationofvolumes arisingfromworkshopsandconferencesinallareasofcurrentresearchinthe historyofscience,primarilywithafocusonthehistoryofmathematics,physics, andtheirapplications.Itsaimistomakecurrentdevelopmentsavailabletothe communityasrapidlyaspossiblewithoutcompromisingquality,andtoarchive thosedevelopmentsforreferencepurposes.Proposalsforvolumescanbesubmitted usingtheonlinebookprojectsubmissionformatourwebsitewww.birkhauserscience.com.
Moreinformationaboutthisseriesathttp://www.springer.com/series/11668
BrendanLarvor Editor
MathematicalCultures TheLondonMeetings2012 Editor BrendanLarvor SchoolofHumanities
UniversityofHertfordshire
Hatfield,Hertfordshire
UK
ISSN2297-2951ISSN2297-296X(electronic)
TrendsintheHistoryofScience
ISBN978-3-319-28580-1ISBN978-3-319-28582-5(eBook) DOI10.1007/978-3-319-28582-5
LibraryofCongressControlNumber:2016934201
© SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernorthe authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade.
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TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland (www.birkhauser-science.com)
EditorialIntroduction ....................................1
BrendanLarvor
PartIFirstMeeting:Varieties
UnderstandingtheCulturalConstructionofSchoolMathematics .....9 PaulAndrews
EnvisioningTransformations ThePracticeofTopology ...........25
SilviaDeToffoliandValeriaGiardino
CreativeDiscomfort:TheCultureoftheGelfandSeminar atMoscowUniversity ....................................51 SlavaGerovitch
MathematicalCultureandMathematicsEducationinHungary intheXXthCentury .....................................71 KatalinGosztonyi
OntheEmergenceofaNewMathematicalObject: AnEthnographyofaDualityTransform ......................91 StavKaufman
WhatAreWeLike … ....................................111 SnezanaLawrence
MathematicsasaSocialDifferentiatingFactor:MenofLetters, PoliticiansandEngineersinBrazilThroughtheNineteenth Century ..............................................127
RogérioMonteirodeSiqueira
“TheEndofProof”?TheIntegrationofDifferentMathematical CulturesasExperimentalMathematicsComesofAge
HenrikKraghSørensen
PartIISecondMeeting:Values
MatthewInglisandAndrewAberdein
WhatWouldtheMathematicsCurriculumLookLikeifInstead ofConceptsandTechniques,ValuesWeretheFocus? .............181
AlanJ.Bishop
MathematicsandValues ..................................189
PaulErnest
PurityasaValueintheGerman-SpeakingArea .................215
José Ferreirós
ValuesinCaringforProof .................................235
JohnMasonandGilaHanna
AnEmpiricalApproachtotheMathematicalValuesofProblem ChoiceandArgumentation ................................259
MikkelWillumJohansenandMortenMisfeldt
TheNotionofFitasaMathematicalValue .....................271
ManyaRaman-Sundström MathematicalPull .......................................287
ColinJ.Rittberg
PartIIIThirdMeeting:Interfaces
MathematicsandFirstNationsinWesternCanada: FromCulturalDestructiontoaRe-Awakening ofMathematicalReflections ................................305
TomArchibaldandVeselinJungic
RemunerativeCombinatorics:MathematiciansandTheirSponsors intheMid-TwentiethCentury ..............................329
MichaelJ.Barany
CallingaSpadeaSpade:MathematicsintheNewPattern ofDivisionofLabour.....................................347
AlexandreV.Borovik
MathematicsandMathematicalCulturesinFiction: TheCaseofCatherineShaw ...............................375
TonyMann
MoralityandMathematics .................................387
MadelineMuntersbjorn
TheGreatGibberish MathematicsinWesternPopularCulture
MarkusPantsar
IsMathematicsanIssueofGeneralEducation?
EmilSimeonov
Contributors AndrewAberdein SchoolofArtsandCommunication,FloridaInstituteof Technology,Melbourne,USA
PaulAndrews DepartmentofMathematicsandScienceEducation,Stockholm University,Stockholm,Sweden
TomArchibald DepartmentofMathematics,SimonFraserUniversity,Burnaby, Canada
MichaelJ.Barany DepartmentofHistory,PrincetonUniversity,Princeton,USA
AlanJ.Bishop FacultyofEducation,MonashUniversity,Melbourne,Australia
AlexandreV.Borovik DepartmentofMathematics,TheUniversityofManchester,Manchester,UK
RogérioMonteirodeSiqueira SchoolofArts,SciencesandHumanitiesofthe UniversityofSãoPaulo,SãoPaulo,Brazil
SilviaDeToffoli PhilosophyDepartment,StanfordUniversity,Stanford,CA, USA
PaulErnest SchoolofEducation,UniversityofExeter,Exeter,UK
José Ferreirós FacultaddeFiloso fia,UniversidaddeSevilla,Sevilla,Spain
SlavaGerovitch DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA,USA
ValeriaGiardino Laboratoired’HistoiredesSciencesetdePhilosophie ArchivesHenri-Poincaré,UMR7117CNRS Université deLorraine,Nancy, France
KatalinGosztonyi BolyaiInstitute,UniversityofSzeged,Szeged,Hungary; LaboratoiredeDidactiqueAndré Revuz,UniversityParisDiderot-Paris7,Paris, France
GilaHanna DepartmentofCurriculum,Teaching,andLearning,Universityof Toronto,Toronto,Canada vii
MatthewInglis MathematicsEducationCentre,LoughboroughUniversity, Leicestershire,UK
MikkelWillumJohansen DepartmentofScienceEducation,Universityof Copenhagen,Copenhagen,Denmark
VeselinJungic DepartmentofMathematics,SimonFraserUniversity,Burnaby, Canada
StavKaufman CohnInstitute,TelAvivUniversity,TelAviv,Israel
BrendanLarvor SchoolofHumanities,UniversityofHertfordshire,Hatfield, Hertfordshire,UK
SnezanaLawrence InstituteforEducation,BathSpaUniversity,Bath,UK
TonyMann DepartmentofMathematicalSciences,UniversityofGreenwich, Greenwich,UK
JohnMason DepartmentofMathematicsandStatistics,OpenUniversity,Milton Keynes,UK;UniversityofOxford,Oxford,UK
MortenMisfeldt DepartmentofEducation,LearningandPhilosophy,Aalborg University,Copenhagen,Denmark
MadelineMuntersbjorn DepartmentofPhilosophyandReligiousStudies, UniversityofToledo,Toledo,OH,USA
MarkusPantsar DepartmentofPhilosophy,History,CultureandArtStudies, UniversityofHelsinki,Helsinki,Finland
ManyaRaman-Sundström DepartmentofScienceandMathematicsEducation, Umeå University,Umeå,Sweden
ColinJ.Rittberg SchoolofHumanities,UniversityofHertfordshire,Hertfordshire,UK;VrijeUniversiteitBrussel,Brussels,Belgium
EmilSimeonov DepartmentAppliedMathematics&Science,Universityof AppliedSciencesTechnikumWien,Wien,Austria
HenrikKraghSørensen DepartmentofMathematics,CentreforScienceStudies, AarhusUniversity,Aarhus,Denmark
EditorialIntroduction BrendanLarvor
1WhyMathematicalCultures? Mathematicshasinternationallysharedstandardsofvalidity.Nevertheless,thereare localmathematicalcultures,whichcanaffectthedirectionandcharacterofmathematicalresearch.Therefore,philosophersofmathematicsandotherswhostudythe growthandepistemologyofmathematicsshouldhaveaninterestinthem.Theyalso matterbecauseofthewiderculturalimportanceofmathematics.Mathematics enjoysenormousintellectualprestige,andrecentyearshaveseenagrowthof popularexpositorypublishing, fi lmsaboutmathematicians,novelsandplays. However,thissameintellectualprestigeencouragesdisengagementfrommathematics.Ignoranceofevenrudimentarymathematicsremainssociallyacceptable. Policyinitiativestoencouragethestudyofmathematicsusuallyemphasisethe economicutilityofmathematics.Appealsofthissortdonotseemtobeeffective, perhapsbecausetheyspeaktotheneedsofthenationratherthantheinterestsof individualstudents,manyofwhomnoticethatitispossibletodoverywellinour economywithoutknowingverymuchmathematics.Moreover,policyresponses rarelyaddressanunhelpfulperceptionofmathematicsasremoteandforbidding. There-presentationofmathematicsascultureoffersthepossibilityofanalternative approach,inwhichmathematicsmightenjoythesameappealasthestudyof literatureorhistory.Thatistosay,wemightre-framethestudyofmathematicsas trainingoftheintellect,thefurnishingofawell-stockedmindandtheappreciation oflocal,nationalorglobalculturalriches.
B.Larvor(&)
SchoolofHumanities,UniversityofHertfordshire,HatfieldAL109AB,UK e-mail:phlqbpl@herts.ac.uk
© SpringerInternationalPublishingSwitzerland2016
B.Larvor(ed.), MathematicalCultures,TrendsintheHistoryofScience, DOI10.1007/978-3-319-28582-5_1
2HistoryandContext Thisbookisoneoftheoutcomesofaseriesofthreemeetingsonmathematical cultures,1 principallyfundedbytheUKArtsandHumanitiesResearchCouncil (AHRC),asanetworkunderits ScienceinCulture theme.2 Thisgeneraltheme presentsawelcomeopportunityforthestudyofmathematicalcultures,andasthe convenerofthemathematicalculturesnetwork,Iamgladofthisopportunityto recordthegratitudeoftheparticipantsforthefundingwereceivedfromtheAHRC.
This ScienceinCulture initiativebytheAHRCintheearly2000swastimely, becauserecentdecadeshaveseentheemergenceoftwoscholarlymovements seekingtounderstandmathematicalcultures.Oneisthephilosophyofmathematicalpractice,alooseassociationofphilosophers,historiansofmathematics,psychologistsandresearchersinotherhumansciencesinvestigatingmathematicsand mathematicians;theotherisa ‘culturalturn’ inmathematicseducationresearch.
Thephilosophyofmathematicalpracticehasnowacquiredascholarlyliterature,3 regularconferences4 andaninternationalsociety,theAssociationforthe PhilosophyofMathematicalPractice.5 However,culturalapproachestomathematicspredatethismovementbysomedecades.Oneoftheearliest(beginningin the1950s)wastheworkofthetopologistRaymondL.Wilder,whoproposedthat mathematicsis,essentially,culture.Wilder ’sworkdidnotdirectlystimulatethe formationofanewsub-discipline.Thismayhaveresultedfromhisrelatively under-theorisednotionofculture,whichdidnotofferpromisingnewtoolstoothers whomighthavefollowedhimandunitedtheireffortsunderacommontheoretical outlook.6 ThemanyinsightsinWilder ’smainexpositionofhisview, Mathematics asaCulturalSystem,7 seemtooriginateinhisexperienceasacreativemathematician,ratherthaninthedeploymentofanthropologicaltheory.Thecultural approachseemstohaveliberatedhimtoexpressgeneralthoughtsaboutmathematicsthathebelievedonothergrounds.Thistestimonyisvaluable,butitisnota programmeofresearchthatotherscantakeupandcontinue unlesstheytooare 1https://sites.google.com/site/mathematicalcultures/ .
2http://www.sciculture.ac.uk/ .
3SeeLarvorreviewof ThePhilosophyofMathematicalPractice PaoloMancosu(ed.)OUP2008 in PhilosophiaMathematica (2010)18(3):350–360forarepresentativelist.
4SeeLarvor “WhatareCultures?” in CulturesofMathematicsandLogicSelectedpapersfromthe conferenceinGuangzhou,China,9–12November2012.ShierJu,BenediktLoewe,Thomas Mueller,YunXie(eds.)Birkhäuser,Basel(2016)forapartiallist. 5http://www.philmathpractice.org/
6Wilderofferedthisdefinitionofculture: “Weuse[theterm ‘culture’]inthegeneral anthropologicalsense Inthissense,acultureisthecollectionofcustoms,beliefs,rituals, tools,traditions,etc.,ofagroupofpeople…” IntroductiontotheFoundationsofMathematics (JohnWiley;seconded. 1965 (firstpublished1952)p.282).
7PergamonPress, 1981.
professionalmathematicianswithinsightsgroundedintheirpersonalexperienceof mathematicalresearch.Nevertheless,Wildermayhaveinfluencedsomeothers, suchasPhillipJ.Davis,ReubenHersh,DavidBloorandperhapsAlvinWhite,who foundedandformanyyearsedited(whatisnowcalled)the JournalofHumanistic Mathematics. LikeWilderbeforethem,Davis,HershandWhitecoulddrawon theirexperiencesasprofessionalmathematicians,andBloorfoundhistheoretical frameinWittgensteinratherthanWilder.Hershincludedafewculturalor anthropologicalapproachesinhiscollection 18UnconventionalEssaysonthe NatureofMathematics (Springer, 2006).OneisbyLeslieWhite,whointroduced Wildertoanthropology;othersarebyAndrewPickering,EduardGlasandHersh himself.AsHershnotesinhisintroduction(p.xv),mostofthearticlesinthis collectionareonthecognitiveaspectsofmathematicalpracticeratherthanthe culturalorsocialaspects,andinthisitreflectsthestateofthe field.Thesameistrue ofthemostimportantrecentbookinthisarea, ThePhilosophyofMathematical Practice,editedbyPaoloMancosu.8
Whilethephilosophyofmathematicalpracticecommunityhasbeenrelatively slowtotakeupculturalapproachestomathematics,therehasbeenaturntowards cultureinmathematicseducationresearch.9 Beforethis ‘culturalturn’,mathematics educationresearchwasinasimilarconditiontothestateofphilosophyofmathematicalpracticeatthemomentofpublicationof 18UnconventionalEssays,thatis, mostlyfocusedonthecognitivepsychologyinindividuals.Now,followingthis ‘culturalturn’,researchersinmathematicseducationincreasinglystudythelearning andteachingofmathematicsasculturalactivities.Thisisnotquitethesameas treatingmathematicalresearchasaculturalactivity,anditremainstobeseen whethernotionsdrawnfromeducationcouldhelpphilosophersofmathematical practice.Educationisanaturalcontextinwhichtoemployaculturalapproach, becausecultureischaracterisedby(andinsomewriters,definedas)non-biological reproduction.Philosophersofmathematicalpracticetendtofocusmoreonthe productionofmathematicalknowledgethanonitsreproduction.Moreover, researchinmathematicseducationtendstobemoreovertlypoliticisedthaninthe philosophyofmathematicalpractice,becauseoneofthemotivesforworkingon educationistoimprovemattersforpeoplewhomthecurrenteducational arrangementsservepoorly.
8Andtherefore,thisobservationisnocriticismofMancosu’seditorialwork.OxfordUniversity Press, 2008.
9Fortheoriginofthisterm,seeLerman,S.(2000).Thesocialturninmathematicseducation research.InJ.Boaler(ed.), Multipleperspectivesonmathematicsteachingandlearning (pp.19–44).Westport,CT:Ablex.Forarecentoverview,seeKarenFrancois’ videopresentationtothe culturesoflogicandmathematicsconferenceinGuangzhou https://youtu.be/umuKvJFR_7U (2012).SeealsoFrançois,K.&Stathopoulou,C.(2012). ‘In-BetweenCriticalMathematics EducationandEthnomathematics.APhilosophicalReflectionandanEmpiricalCaseofaRomany Students’ groupMathematicsEducation.’ JournalforCriticalEducationPolicyStudies,10(1), 234-247ISSN1740-2743.
3TheMathematicalCulturesResearchNetwork Thedevelopmentofthesetworesearchcommunitieswithinterestsinmathematics asculturesuggestedthepossibilityofmultidisciplinaryengagement,fundedunder theAHRCscienceinculturetheme.TheMathematicalCulturesResearchnetwork hadthefollowingobjectives(adaptedfromthefundingapplication):
(1)Createaninterdisciplinary,internationalnetworkofresearcherswithinterests inmathematicsaswellasinculture,withasupportinginternetnodethatwill outlivethisfundedproject.
(2)Facilitatediscussionofthemethodologicalchallengesfacingthestudyof mathematicsasculture.
Mathematicsissimultaneouslycultureandknowledge.Scholarswhotreatitas culturemustrespectitsstatusasknowledge;thosewhoengagewithitas knowledgemustacknowledgethatitisacollectionofhumanpractices.
(3)Exploreandmapsomeofthevariouscontemporarymathematicalcultures
Thesecanbetheculturesofprofessionalresearchmathematicians,butalso usergroupssuchasengineersoractuaries,andcultureswithineducation, amongteachersandstudents.Ofparticularinterestaretheimagesofmathematicsamongreluctantusersofmathematics.
(4)Exploretherationalstructureofmathematicalvalue-judgments
Whenmathematiciansawardorwithholdprizes,scholarships,PhDsand grants,correctnessisalmostneverthedecisivecriterion.Rather,thequestion iswhethertheworkisworthwhile,interesting,elegant,promising,insightful, etc..Ifthesejudgmentsarenotarbitrary,theyshouldrefertosomestandards orvalues.Arethesecommonacrossallthemathematicalculturesexploredin (3)?Howaretheytaught?Howdotheyevolve?
(5)Articulatetheculturalandeducationalvalueofmathematicsinaformuseful foreducationalistsandpolicy-makers
Thevalueofmathematicsisusuallyarguedeitherineconomicterms,orin termsoftheexcitementofmakingrarebreakthroughs.10 Thereisaneglected middleground:mathematicsasaproperpartoftheculturaldietofaneducated person.
(6)Publishasabookandontheinternethigh-qualityscholarshiprelatingto mathematicsasculture
Thisprojecthostedthreeconferences.The first (September2012)hadtheaimof exploringandbeginningtomapthevarietyofandconnectionsamongcontemporarymathematicalcultures.The second (September2013)aimedtoarticulateand classifymathematicalvalues.The third (Easter2014)discussedmathematicsin publicculture.
10MichaelHarris,oneofthecontributorstotheseries,haswrittenanentirebookonthistension: MathematicswithoutApologies:PortraitofaProblematicVocation (PrincetonUniversityPress, 2015).HepresentedChaptereightofthisbookatthethirdmeetingofthemathematicalcultures network.
Howwelldiditsucceedinthesesixobjectives?Takingtheminorder:participantsdidengageacrossdisciplinaryboundariesandsomeconversationsandcollaborationshavedevelopedamongscholarswhodidnotknoweachotherbeforethis project.Ontheotherhand,the ‘supportinginternetnode’ wasalwaysmoreofa repositorythanahubofactivity theparticipantsinthisprojectrevealedthemselves tobelargelyindifferenttosocialmediaandfewtookupopportunitiestoblogor tweet.Thereweresomenotableexceptionsfollowingthethirdand finalmeeting.11 Programmes,participationlists,abstractsofallthecontributionsandvideosofmany ofthetalksareavailableontheprojectsite.Wediddiscussmethodology,butthis waslesssalientatthemeetingsthanweoriginallyhoped.Perhapsthiswasbecause thecentralterm, ‘culture’,wasnotsufficientlywellarticulatedtostimulatedebate. Fromthepresentationsinthisseries,itisclearthattheconceptofcultureremains relativelyuntheorisedamongwritersonmathematicalpractice.Onlyafewofthe philosophersandhistoriansatthemeetingsdeployedanotionofcultureexplicitly, andtheseweremostlylackinginconceptualarticulation.Thereweresomeexceptionswhohonouredtheprojectthemeandmadeadeliberateefforttotheorise mathematicalculturesassuch.12 Regardingthethirdaim,thepresentationsexplored asatisfyingvarietyofmathematicalcultures,includingresearchcommunitiesanda valuablediscussionoftheculturalsignificanceofmathematicsintheworldof finance.Sincemanyofthecontributorsarehistorians,wehavetotakeagenerous readingof ‘contemporary’ toincludemostofthetwentiethcentury,butgiventhatit iscommontoregardmathematicsashavingassumeditscurrentformintheinter-war years,thisisnotunreasonable.Thechiefdisappointmentisthatwedidnothavea presentationthatvividlyarticulatedtheperspectivesofreluctantmathematicspupils. Ourprojectwouldhavebenefittedfromanexplorationofthepossibilitythatsomeof thereluctanceofreluctantlearnersiscultural thatmathematicshasculturalassociationsthatsomechildren findoff-putting.Wedidhearsomecontributionsinthis directionfromteachers(seeSnezanaLawrence’schapterinthisvolume).Wealso hadareportonthereceptionofEuropeanmathematicsamongtheindigenouspeople ofWesternCanada,andoneffortstoovercometheresistancetoEuropeanmathematicsresultingfromitsassociationsoverdecadeswitheducationdesignedto eradicateindigenouscultures(seeArchibaldandJungicinthisvolume).Perhaps moreobviouslyethnographicstudiesofthissortmightsupplyamodelforreading theculturalpositionofmathematicsinotherclassrooms.13
Weweremoresuccessfulwithrespecttothefourthobjective,inthatwehad severalexcellentpapersonwhatitmeansforaprooftobebeautiful thoughthe resultswerenotencouragingforanyoneseekingarational,objectiveaestheticsof
11https://sites.google.com/site/mathematicalcultures/blog
12SeeSørenseninthisvolumeandLarvor(2016)inShierJu,BenediktLoewe,ThomasMueller, YunXie(eds.)forfurtherdetail.
13Thereisplentyofresearchonpupils’ feelingsaboutmathematics;lessontheircultural perceptionofmathematics.Thereissomeinterestingmaterialonlearningandteachingculturesin Hersh&John-Steiner(2011) LovingandHatingMathematics,especiallypp.273–300& 312–315.
proof.The fifthobjective,toarticulatetheculturalandeducationalvalueof mathematics,was(unsurprisinglyinretrospect)problematic.Somecontributors tookmathematicstobealivingembodimentofvaluesnormallyassociatedwiththe EuropeanEnlightenmentphilosophersoftheeighteenthcentury:universality, objectivity,rationality.Thisstandsintensionwiththepluralistspiritofour enterprise:wetookcaretospeakofmathematicalcultures,inplural.Ifmathematicalculturesreallyaredistinctandhaveimportantdifferences,thenonecannot assumequitesoeasilythattheyareallequallyvaluableandvirtuous orvaluable andvirtuousinthesameways.Nevertheless,thereisstillaworthwhilejobtodoin articulatingthevaluethatsomemathematicshashadforsomepeopleatsometimes andplaces thismaystillprovideeducatorswithbettermotivationalresourcesthan vaguepromisesofimprovedcareerprospects.
Theprincipalrealisationofobjectivesixisthisbook.Allofthecontributions havebeenthroughaprocessofreviewbyanonymousreferees.Iamimmensely gratefultothesereferees,whogavetheirtimeandexpertise probono,andwhoalso helpedwiththeselectionoftalksfortheconferences.Imustalsorecordour gratitudetotheLondonMathematicsSocietyandthestaffatDeMorganHousefor theirhelpwithandsupportforthisproject.
References Artsandhumanitiesresearchcouncilscienceinculturethemewebsite. http://www.sciculture.ac. uk/ Associationforthephilosophyofmathematicalpracticewebsite. http://www.philmathpractice.org/ Francois,K.(2012).Videopresentationtotheculturesoflogicandmathematicsconferencein Guangzhou. https://youtu.be/umuKvJFR_7U
François,K.&Stathopoulou,C.(2012).In-Betweencriticalmathematicseducationand ethnomathematics.AphilosophicalreflectionandanempiricalcaseofaRomanystudents’ groupmathematicseducation. JournalforCriticalEducationPolicyStudies,10(1),234–247. ISSN1740-2743.
Harris,M.(2015). Mathematicswithoutapologies:Portraitofaproblematicvocation.Princeton UniversityPress. Hersh,R.(2006). 18Unconventionalessaysonthenatureofmathematics. Springer. Hersh,R.&John-Steiner,V.(2011). Lovingandhatingmathematics. PrincetonUniversityPress. Larvor,B.(2010).Thephilosophyofmathematicalpractice.P.Mancosu(Ed.)OUP2008, PhilosophiaMathematica, 18(3),350–360.
Larvor,B.(2016).WhatareCultures?InS.Ju,B.Loewe,T.Mueller&Y.Xie(Eds.), Culturesof mathematicsandlogicselectedpapersfromtheconferenceinGuangzhou, China 9–12 November2012.Basel:Birkhäuser.
Larvor,B.Mathematicalcultures. https://sites.google.com/site/mathematicalcultures/ Lerman,S.(2000).Thesocialturninmathematicseducationresearch.InJ.Boaler(Ed.) Multiple perspectivesonmathematicsteachingandlearning (pp.19–44).Westport,CT:Ablex. Mancosu,P.(2008). Thephilosophyofmathematicalpractice. OxfordUniversityPress. WilderRaymondL.(1965). Introductiontothefoundationsofmathematics (2nded.).JohnWiley. WilderRaymondL.(1981). Mathematicsasaculturalsystem. PergamonPress.
PartI FirstMeeting:Varieties UnderstandingtheCultural ConstructionofSchoolMathematics PaulAndrews
1Introduction
FormorethantwentyyearsIhavebeenobserving,video-recordingandanalysing mathematicsteachinginvariousEuropeancountries.Inaddition,duetothegenerosityofmygraduatestudentsandothercolleagues,Ihaveacquiredvideo recordingsofmathematicslessonsfromseveralotherEuropeancountries.My interpretationofallthismaterialhasledmetotwoconclusions.The fi rst,asfound throughouttheliterature(Schmidtetal. 1996;Hiebertetal. 2003),isthatmathematicsteacherstendtobehaveinwaysthatalignthemmorecloselywiththeir compatriotsthanwithteacherselsewhere.Thesecond,whichmayinpartexplain the first,isthattheculturesinwhichteachersoperatehaveasmuch,ifnotmore, influenceonstudentachievementasthewaysinwhichmathematicsistaught.For example,successiveiterationsoftheprogrammeofinternationalstudentassessment (PISA)(OECD 2001; 2004; 2007; 2010a; 2013)havehighlightedFinnishstudents’ linguisticandmathematicalcompetence.However,recentanalysesindicatethat suchsuccessesmaybeduemoretomattersculturalthandidactical(Andrews 2011, 2013;Andrewsetal. 2014).IreturnlatertotheissueofFinland,whichisparticularlysigni ficantinlightofcontinualmediaandotherattentionpaidthatcountry. But firstitasimportanttoexplainwhatthischapterwill,andwillnot,attemptto achieve.Theprincipalaimistoshowhowculture,themeaningofwhichwillbe discussedbelow,underpinsallaspectsofschoolmathematics,whetheritbethe curriculumspecifiedbythesystem,thedevelopmentofthetextbooksthatteachers mayormaynotbecompelledtouse,thewaysteachersteach,theclassroom
P.Andrews(&) DepartmentofMathematicsandScienceEducation,
StockholmUniversity,Stockholm,Sweden e-mail:paul.andrews@mnd.su.se
interactionsprivilegedbythesystemorthebeliefs,attitudesandaspirationsof teachers,studentsandparents.Todothis,however,requiressomeunderstandingof thenatureofcultureanditseducationalmanifestation.
2TheNatureofCulture Ofcourse,suchambitionsmaybemoredifficulttoachievethanmightbeexpected, ascultureisnotonly “oneofthetwoorthreemostcomplicatedwordsinthe Englishlanguage” (Williams 1976,p.76)butalso “aprofoundlycongestedconcept” (Lewis 2007,p.867).Inpart,thisisduetothefactthat;
Cultureisordinary:thatisthe firstfact.Everyhumansocietyhasitsownshape,itsown purposes,itsownmeanings.Everyhumansocietyexpressesthese,ininstitutions,and learning.Themakingofasocietyisthe findingofcommonmeaningsanddirections,andits growthisanactivedebateandamendment,underpressuresofexperience,contact,and discovery,writingthemselvesintotheland.Thegrowingsocietyisthere,yetitisalsomade andremadeineveryindividualmind.Themakingofamindis, first,theslowlearningof shapes,purposes,andmeanings,sothatwork,observationandcommunicationarepossible. Then,second,butequalinimportance,isthetestingoftheseinexperience,themakingof newobservations,comparisons,andmeanings(Williams 1958,p.75).
FromWilliams ’ defi nitioncanbeinferredseveralkeycharacteristics;cultureisa groupconstructionbasedoncommonexperiences.Itisnot fi xedbutchangeable. Anindividual’sculturalaffiliationistheresultofanimplicitnegotiativeprocess, indicatingthattheinfluenceofcultureasaguidingforceinthelifeofindividualsis largelyhidden.Culture,inthissense,canbeconstruedasa wayoflife (Lewis 2007; Williams 1976).
This wayoflife interpretationimpliesaconsensusthatculturecomprisesa collectivepsychologicalconditioning(TriandisandSuh 2002).Itembodiesthe “implicitlyorexplicitlysharedabstractideasaboutwhatisgood,right,and desirableinasociety” (Schwartz 1999,p.25),andincludesthosebeliefs,artefacts, practicesandinstitutionsthathistoryhasshowntobeeffectiveforthemaintenance ofasocietyanditsfuturegenerations(Fiske 2002;Hofstede 1980;Triandisand Suh 2002).Culturesaresocial,historicalandbehaviouralconstructions(Fiske 2002)thatreflectthe “collectivementalprogramming” oftheirpeople(Hofstede 1980,p.43).Throughthetransmissionoftheirembeddedvalues,beliefs,knowledgeandskills,theyensurereplication.Hence,people’spsychologicalprocesses “arelikelytobecon figuredindifferentwaysacrossdifferentsocio-culturalgroups” (ErezandGati 2004,p.568).Summarisingtheabove,Triandis(2007,p.64)writes thatculturecomprisesthreemaincharacteristics;it “emergesinadaptiveinteractionsbetweenhumansandenvironments”,itcomprisessharedelementsand “is transmittedacrosstimeperiodsandgenerations” and,therefore,largelystable (Soaresetal. 2007).Inthefollowing,Iconsiderseveraleducationally-salientfeaturesofcultureandshow,invariousways,howtheyimpactonstudents’
opportunitiestolearn.Insodoing,Iacknowledgethat, “schoolscanrisenohigher thanthecommunitiesthatsupportthem” (Boyer 1983,p.6)and,intryingto understandtheprocessesofeducation,that “thethingsoutsidetheschoolsmatter evenmorethanthethingsinsidetheschools,andgovernandinterpretthethings inside” (Sadler(1900)inBereday 1964,p.310).
3ModellingCulture Anumberofresearchers,typicallyalignedwithananthropologicaltradition(Soares etal. 2007)haveattemptedtoidentifythosedimensionsthatdistinguishoneculture fromanother,anditistothemthatIturn first.Hofstede’swellknownstudyinitially identi fiedfourdimensionsofculture.The first, powerdistance,concernstheextent towhichfollowersacceptbeingled. “Asociety’spowerdistancelevelisbredinits familiesthroughtheextenttowhichitschildrenaresocializedtowardobedienceor towardinitiative” (Hofstede&McCrae 2004,p.62).Thesecond, uncertainty avoidance,relatestotheextenttowhich “acultureprogramsitsmemberstofeel eitheruncomfortableorcomfortableinunstructuredsituations”.Thethirdis individualism,orthe “degreetowhichindividualsareintegratedintogroups”.Lastly, thereare masculine asopposedto feminine cultures. Masculine cultures “strivefor maximaldistinctionbetweenwhatmenareexpectedtodoandwhatwomenare expectedtodo”.Fromtheperspectiveofeducation,Hofstede(1986)hasusedthese dimensionstopredictculturallydetermineddifferencesinthesocialpositionsof teachersandstudents,participants’ cognitiveandaffectiveexpectations,andpatternsofparticipantinteractions.
Othershaveproposeddifferentbutrelatedmodelsofculture.Forexample, Triandis(2001)offeredelevendimensionsandSchwartz(1999)seven.Like Hofstede’sdimensions,bothcategorisationshelpustounderstandtheroleof cultureinframinghumans’ decisionmakingingeneralandtheprocessesofeducationinparticular.Signifi cantly,Schwartzfocusedonelementaryteachers’ education-relatedvaluesinmorethan40countries,arguingthatteachers “playan explicitroleinvaluesocialisation”,arelikelytobe “keycarriersofculture,and… reflectthemid-rangeofprevailingvalueprioritiesinmostsocieties ” (Schwartz 1999,p.34).Hefound,forexample,thatconservativeculturesthatemphasisethe “maintenanceofthestatusquo orthetraditionalorder” wouldstructureeducationalopportunitiesverydifferentlyfromanautonomouscultureinwhichan individual finds “meaninginhisorherownuniqueness andisencouragedtodo so” (Schwartz 1999,p.34).Allsuchmodelshelpustounderstandthatforces, possiblybeyondparticipants’ consciousness,acttoshapewhathappensinclassrooms.They “determinewhatistobetaught,towhomitistaught,howitistobe taughtandwhereitistaught” (Andrews 2010,p.5).
4CultureandCurricula Thecurriculum,accordingtothesecondinternationalmathematicsstudy,comprisesintended,implementedandattainedforms.However,whatisfrequently missingfromsuchdiscussionsistheextenttowhichculturalforcesshapecurricula developmentsindifferentcountries.Inthisrespectacurriculumcanbeconstruedas “bothmaterialartefact”,thesphereoftheculturalanthropologist,and “symbolic system”,thesphereoftheculturalsociologist.Thatis,thecurriculumreflectsbotha wayoflife,includingthe “sharedvaluesandmeaningscommontomembersofthe group” andthepracticesbywhichmeaningisconstructedandsharedwithinthe group(Mason 2007,p.172).Inmuchthesamewayaswithcultureabove,various analystshavemodelledcurricula.Forexample,HolmesandMcLean(1989)have presentedfourcurriculummodels: essentialism,derivedfromtheEnglishpublic1 school, encyclopaedism,linkedtopost-revolutionaryFranceanditsEnlightenment principles, polytechnicalism,tiedtoSovietexpectationsofvocationalismunderpinnedbyanencyclopaedicmodelofknowledge,and pragmatism,derivedfrom theUnitedStatesandaneedfortheknowledgeandskillsnecessaryfortacklingthe realworldproblemsofaliberaleconomy.Inasimilarvein,Kamensetal.(1996) categoriseuppersecondarycurriculainfourways; classicalcurricula addressthe maintenanceofthenaturalsocialorderthroughthetrainingofapoliticalandsocial elite, comprehensivecurricula,reflectingegalitarianprinciples,aimtoproduce competentandproductivecitizens, mathematicsandsciencecurricula reflect economicneedsforatechnologicallycompetentwork-force,and arts,humanities, andmodernlanguagescurricula, reflectingamodernisationoftheclassical Europeancurriculum,focusonthemaintenanceofanelitehighculture.Finally,it hasbeenassertedthatfewoftheworld’scurriculaarenotadaptationsofoneofsix core curricula thePrussian,Russian,French,English,JapaneseandUnitedStates (Cummings 1999).
Ofcourse,suchcharacterisationsarecrudeandmaynotreflecttheparticularities ofindividualsystems.Forexample,socialscientistshavetendedto “conceptualize individualismastheoppositeofcollectivism,especiallywhencontrastingEuropean AmericanandEastAsianculturalframes(Oysermanetal. 2002,p.3).However, suchconceptionsare,insomeways,confoundedbyanalysesoftheEuropean context.Weber(1930)observedthatProtestantcultures,focusedonthepromotion ofself-reliance,tendedtowardstheindividualistwhileCatholic,duetoemphases onthemaintenanceofestablishedhierarchicalrelationships,tendedtothecollectivist.Admittedly,increasingmobilityasaresultoftheEuropeanUnionandother postcolonialimmigrationmayhavecompromisedtheobviousrelevanceofsuch characterisations,butfewnationalcurriculadonotattempttoinstilintheirstudents asenseofthenationalcharacter,ifonlyinthewaysthatliteratureismanaged.
1Theadjective public inthiscontextismisleading;anEnglishpublicschoolisaneliteindependent school.Suchschoolssustainthehigherranksofthecivilserviceandthejudiciary,halfofall studentsatCambridgeandOxforduniversities,andadisproportionatelyhighnumberofmembers ofparliament.
Inotherwords,whilemodelssuchasthosepresentedabovehighlighthowcurricula mayvarysubstantiallyinbothprincipleandmanifestation,fewmoderncurricula aresotightlylocatedinasingleculturalcontextthattheyremainuntouchedby developmentselsewhere(Kamensetal. 1996),particularlyinthelightofrecent internationaltestsofstudentachievement,especiallyTIMSSandPISA.Indeed, suchtestsare,essentially,underpinnedbyassumptionsthatmathematicsisinthe vanguardofcurricularconvergence;afterall,anequationisanequationirrespective ofwhereitisfound,isn’tit?Inthefollowing,thereforeIshow,withreferenceto variousEuropeaneducationalsystems,howcurricularmathematicsanditsclassroompresentationvariesaccordingtoculturallyestablishednorms.
5CultureandMathematicsCurricula Inthefollowing,usingthemasaplaceholderformathematics,Isummarisefour Europeansystem’scurricularperspectivesonlinearequationsbeforediscussing theminrelationtothevariouscharacteristicsofculture.Insodoing,Itrytoshow howcurriculaareconstructed.Choice,inrespectofthecountriesunderscrutiny, wasconstrainedbytheavailabilityofcurriculainEnglish,whilethetopicwas determinedbyotherworkonwhichIamengaged.Eachispresentedalphabetically bycountry,withdetailsreferringonlytocontentrelatedexplicitlytotheformulationandsolutionoflinearequations.Thus,referencestosolvingsimultaneous linearequations,forexample,havebeenomittedintheinterestsofnarrative simplicity.
TheEnglishnationalcurriculum 2 forstudentsintheagerange11–14assertsthat pupils,undertheumbrellaofdeveloping fluency,should,interalia, “solveequations”.Theyshould “movefreelybetweendifferentnumerical,algebraic,graphical anddiagrammaticrepresentations[forexample,… equationsandgraphs]”.More particularly,pupilsshouldbetaughtto “understandandusetheconceptsand vocabularyof… equations” , “usealgebraicmethodstosolvelinearequationsinone variable(includingallformsthatrequirerearrangement)” .
TheFinnishnationalcurriculum 3 forgrades6–9assertsthatstudents,bytheend ofgrade8, “willknowhowto… solvea fi rstdegreeequation”
TheFlemishmathematicscurriculum,4 expectsstudentsinthe fi rstgradeof secondaryeducationto “solveequationsofthe fi rstgradewithoneunknownand simpleproblemswhichcanbeconvertedtosuchequations”.Duringthesecond gradetheywill “solveequationsofthe fi rstandseconddegreeinoneunknownand problemswhichcanbeconvertedintosuchequations”
2See https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/ SECONDARY_national_curriculum_-_Mathematics.pdf
3See http://www.oph. fi/english/publications/2009/national_core_curricula
4See http://www.ond.vlaanderen.be/dvo/english/ .
TheHungariancurriculum 5 forgrades5–8(upperprimary)writesthatinyear5 studentsshould “solvesimpleequationsofthe firstdegreebydeduction,breaking down,checkingbysubstitutionalongwithsimpleproblemsexpressedverbally”.In year6theyshould “solvesimpleequationsofthe firstdegreeandonevariablewith freelyselectedmethod”.Byyear7theyshould “solvesimpleequationsofthe fi rst degreebydeductionandthebalanceprinciple.Interprettextsandsolveverbally expressedproblemsandequationsofthe firstdegreeandonevariablebythe graphicalmethod”.Lastly,byyear8studentsshould “solvedeductivelyequations ofthe firstdegreeinrelationtothebasesetandsolutionset.Analysetextsand translatethemintothelanguageofmathematics.Solveverballyexpressedmathematicalproblems”
Fromtheseexamplescanbediscerneddifferentperspectivesonacoretopicof thelowersecondarycurriculum.UnliketheFlemishandHungariancurricula, neithertheEnglishnortheFinnishdocumentsofferyear-on-yearprogression,with bothspecifyingoutcomestobeachievedbytheendofaparticularphase.Indeed,it isdifficulttoimaginealessdetailedspeci ficationthantheFinnish.Neithercurriculumspeci fiesmethods,althoughtheEnglishalludestoarecommendedalgebraicrearrangementmethod.Neitherdocumentmakesexplicitmentionofproblem solvingorwordproblemsandthederivationofequationsfromtext.TheFlemish document,whichalsosaysnothingaboutapproaches,appearssuper ficiallymore tightlyspecifiedalthoughtheshiftfromoneyeartothenextisvague.However, thereisaclearsensethatstudentsareexpectedtoderiveandsolveequationsfrom, byimplication,wordproblems.Finally,theHungariancurriculumoffersatightly speci fiedprogressionoverafouryearperiodwithmethodsandproblemsolving, includingwordproblems,increasinglyspeci fiedandexploited.
So,howdothevariousculturalnormsplayoutinthecurricularpresentationof mathematics?Thetwocountrieswiththeloosestcurriculaexpectations,England andFinland,reflecthorizontalcultures(Triandis 2001)wherepowerdistanceand uncertaintyavoidancearelow(Hofstede 1980).Incountrieswithlowpowerdistance, “peopleatvariouspowerlevelsfeel preparedtotrustpeople” (Hofstede 1980,p.46).Inotherwords,curriculumauthoritiesseenoneedtoover-specify, say,teachingapproachesbecauseteachersaretrustedtomanagetheirresponsibilitiesappropriately.Indeed,Finnishteachersoperatewithinacultureoftrust (Välijärvi 2004)thatextendsfromthetoptothebottom(Sahlberg 2007);theyenjoy highpublicesteem(Simola 2005)andareviewedasprofessionalswhoknowwhat isbestfortheirstudents(Ahoetal. 2006).Insimilarvein,looselystructured curriculafoundincountrieswithlowuncertaintyavoidancereflectsocietalnormsin whichnotonlyis “theuncertaintyinherentinlife … moreeasilyacceptedandeach day… takenasitcomes” butalsoacultureinwhichdissentanddeviationare toleratedandpeoplearewillingtotakerisks(Hofstede 1980,p.47).TheFlemish situationisdifferent,althoughitisimportantforthereadertounderstandthat Hofstede’sstudiesfocusedonBelgiumasawholeandnotjustFlanders.Relatively highlevelsofpowerdistanceanduncertaintyavoidancemayhelpexplainwhythe
5See http://www.okm.gov.hu/letolt/nemzet/kerettanterv36.doc .
FlemishauthoritiesproducedamoretightlystructuredcurriculumthantheEnglish ortheFinns.Further,theverytightlyspeci fiedHungariancurriculumwouldbea productofacultureinwhichpowerdistanceanduncertaintyavoidancearehigher thantheotherthreecountriesunderscrutiny.Inotherwords,thereappearstobea relationshipbetweenthedegreetowhichacurriculumisspecifiedandthelevelsof bothpowerdistanceanduncertaintyavoidance thelowertheculturaldimensions thelessspecifi edthecurricula.Alternatively,thelooselystructuredcurriculaofthe EnglishandtheFinnsmaybeartefactsofautonomous(Schwartz 1999)protestant (Weber 1930)cultures,whilethemorestructuredcurriculaoftheFlemish,and particularlytheHungarians,couldbeconstruedasreflectingmoreconservative (Schwartz 1999)Catholic(Weber 1930)cultures.Inshort,itisnotinconceivable thatdifferencesinthepresentationofmathematicscurriculaareconsequencesof signifi cantdifferencesintheunderlyingstructuresoftheculturesthemselves.Inthe followingIconsiderhowsuchdifferencesplayoutinmathematicsclassrooms.
6CultureandMathematicsTeaching If,assuggestedabove,teachersareproxiesforasystem’svaluesthenitwould surprisingifteachers’ actionswerenotculturallydeterminedmanifestationsofthose values.Inthisrespect,itiswidelythoughtthatmathematicsteachingintheWest differsfromthatintheEast.Forexample,anumberofresearchershaveemphasised theinfluenceofSocraticandConfucianphilosophiesontheculturallyWestand culturallyChineseeducationaltraditions(Leung 2001;TweedandLehman 2002; Watkins 2000).Indeed,inrespectofmathematics,Leung(2001)proposedsix dichotomiesthatdistinguishbetweenEastAsianandWesternmathematicsclassrooms;product(content)versusprocess,rotelearningversusmeaningfullearning, studyinghardversuspleasurablelearning,extrinsicversusintrinsicmotivations, wholeclassteachingversusindividualisedlearning,andteachercompetencein relationtothesubjectmatterversuspedagogydebate.However,suchdistinctionsare notonlycrudeand,attimes,inaccurate(Clarke 2006)butfrequentlyslideinto unconsciousstereotypingorevenracism(Mason 2007).Inthefollowing,mindfulof suchproblems,IfocusonEuropeansystemsandthemoresubtlewaysinwhich culturesinfluenceeducationalpracticesingeneralandmathematicsteachingin particular.Forexample,atthegenerallevel,Osborn(2004)hasshownhowsocietal privilegingoftheindividual,thecommunityandthenation emphasesexplicable byreferencetothevariousculturalmodelsdiscussedabove underpineducational expectationsandpracticesin,respectively,England,DenmarkandFrance.With respecttotheparticularsofmathematics,theanti-scientifictraditionsoftheEnglish curriculumandtherationalencyclopaedictraditionsoftheFrenchcurriculum (HolmesandMcLean 1989;Cummings 1999)mayexplain findingsthatEnglish mathematicsteachersworktoreducethecomplexityofmathematicsfortheirstudents,whileFrenchteachersworktowardsan “inductionintothatcomplexity”
(JenningsandDunne 1996,p.51).Inasimilarvein,Kaiser(2002)foundsubstantial differencesbetweenthetraditionsofEnglishandGermanmathematicsteaching explicablebythesocio-historicalunderpinningsofthetwosystems.Forexample,in Germanclassrooms “theoreticalmathematicalconsiderationsareofgreatimportance” whileEnglishmathematicsteachingprivilegesa “pragmaticunderstandingof theory” (p.249).Germanmathematicsisdefinedbythestructureofthesubjectwith largethematic fieldstaughtindependentlyofeachother,whileinEnglandaspiral curriculumallowstopicstobeintroduced,revisitedlaterandtaughtindependentlyof anyobvioussenseofstructure.InGermanynewtopicsormethodsareintroducedby meansofclassdiscussionsandoftenillustratedbyreal-worldexamples,whilein Englandtheyhavealowimportanceandareoftengiven “asinformationorinthe styleofarecipe” (p.250).Germanlessonsareprogressedbyhighexpectationsof students’ correctandconfidentexecutionofalgorithms,withclearexpectationsthat eachisundertakeninwell-definedandpredeterminedways.InEnglishlessons “rulesandstandardalgorithms(are)ofminorimportance” (p.252),withpriority giventostudents’ ownsolutions.Germanteachersplacegreatimportanceonprecise mathematicallanguageatalllevelsofdiscourse,whileEnglishteachersviewsuch mattersasofminorimportance.
Thecontentandstructureofmathematicstextbooks,too,showevidenceof beingculturallydetermined.HaggartyandPepin(2002)analysedcommonlyused English,FrenchandGermantextbooksandfoundevidencesupportiveofthedifferencesdiscussedinthepreviousparagraph.Forexample,Frenchtexts,which werecognitivelychallenging,incorporatedextensiveexplanatorytextandtechnical vocabularysufficientforstudentsto “dothequestions … withoutadditionalsupport” (p.576).Germantexts,incorporatingdetailedexplanatorytextandmuch technicalvocabulary,attemptedtoestablishlinks “betweeneverydaysituationsand whatpupilsaretolearn” (p.578),althoughtypicallyquestionsrequiredonly low-levelapplicationsoftheskillsbeingpractised.Englishtextswerelessdense andcontainedfewerexamples.Therewaslittleemphasisontechnicalvocabulary andnoexplanatorytext.Exerciseswerelowlevelwithlittlescopeforextension (p.582).Finally,withrespecttotextbooks,culturesdifferintheirproductionand deployment.Forexample,allGreekteachersaremandatedtousecentrallyproducedtextbooks(Hatzinikitaetal. 2008),whereasthesupplyoftextbooksin Englandisunregulated.Suchmatterscanalsobeexplainedbyculturalanalyses. Greece,despitebeinganOrthodoxcommunity,wouldbeconstruedascollectively CatholicunderaWeberiananalysis,anargumentsupportedbyitshavingoneofthe largestpowerdistancesofallEuropeancountriesandthelargestuncertainty avoidanceintheworld(Hofstede 1986).Suchculturalcharacteristicsdiffergreatly fromtheEnglishdescribedabove.
TofurtherillustratethediversityofEuropeanmathematicsdidacticstraditions,I returntothetopicoflinearequationsandaqualitativeanalysisofasequenceof five lessonstaughtineachofFinland,FlandersandHungary(AndrewsandSayers 2012).Theyfound,atthelevelofdidacticalstructure,thateachsequencepassed throughfourphases,whichtheydescribedasdefi nition,activation,expositionand
consolidation.Thedefinitionphase,alwaysawhole-classactivity,introducedstudentstothenotionofanequationand,eitherimplicitlyorexplicitly,presenteda definition.Theactivationphase,predominatelywhole-class,involvedstudents solvingequationswiththeunknownononesideoftheequalssign.Suchequations allowforintuitiveapproaches,typicallybasedoninverseoperations.Theexpositionphase,whichwasalwayswholeclassactivity,focusedonsolvinganequation withtheunknownonbothsidesoftheequalsign,exposedtheinadequaciesof intuitiveapproachesandwarrantedtheintroductionofthebalanceasadidactictool. Finally,theconsolidationphase,whichincorporatedbothwhole-classandindividualworking,enabledstudentstopracticeandexploittheirnewlyacquiredskills. However,whentheanalyseswentbeyondsuchstraightforwardsimilarities,substantialdifferenceswereobserved.TheHungarianteacher,focusedmuchofher timeonencouragingherstudentstoconstructequationsfromwordproblems.These equations,whichwerealternatedwithequationsderivedfromproblemssetina worldofmathematics,werealwayscognitivelychallengingbut,afterperiodsof individualwork,collectivelysolvedinwaysthatprivilegedthestudents’ voice focusedonacollectivelyagreedsolution.TheFlemishteacherusedawordproblem tointroducethetopicbutthenspentmuchofhertimeworkingoncognitively challengingproblemslocatedinaworldofmathematics.However,thesolutionsto suchproblemsweremanagedinwaysthatminimisedopportunitiesforstudentsto engageinindividualproblemsolving,withheralwaysbeingthe finalarbiterof whatconstitutedanappropriatesolution.Finally,theFinnishteacherexploitedonly problemssetinaworldofmathematics.Thesewerealwaysroutine,nevercognitivelychallenging,andalwayssolvedinwaysthatminimisedorignoredthestudent voice.Suchteacherbehaviours,whichresonatedcloselywiththeavailableliteratureoneachteacher ’scountry’sdidacticalpractices,furtherhighlighthowteachers areconditionedtoactinparticularwaysbythecultureinwhichtheyliveandwork (AndrewsandSayers 2013)andleadus,aspromisedearlier,backtoFinland.
Finland,acountrymuchadmiredforitsrepeatedsuccessesonthe fi veiterations ofPISA,hasattractedmanythousandsofforeignenvoyskeentouncover curriculum-relatedinsights(Laukkanen 2008, 2013).Indeed,suchactivityhasbeen encouragedbytheOECD’sextraordinaryassertionthat “bringingallcountriesup totheaverageperformanceofFinland,OECD’sbestperformingeducationsystem inPISA,wouldresultingainsintheorderofUSD260trillion” overthe “lifetimeof thegenerationbornin2010” (OECD 2010b,p.6).However,asIshowbelow,the assumptionthatemulatingFinnishpracticewillraiseachievementelsewheremay benaïve.Forexample,andtypicallydownplayedbytheFinnishauthorities,has beenFinnishstudents’ modestachievementonthetwoTIMSSinwhichFinland hasparticipated.Indeed,inboth1999and2011,Finnishstudents’ algebraicand geometriccompetencebarelyreachedtheleveloftheinternationalmean(Mullis etal. 2000, 2012).ThisdisparitybetweenhighPISAand,inrealterms,lowTIMSS hasnotonlypromptedFinnishmathematicianstosuggestthatthemathematical knowledgeandskillsnecessaryforfurtherstudyofthesubjecthavebeensacri ficed inthecontinuingpursuitofPISAsuccess(Astalaetal. 2006;Martio 2009;
TarvainenandKivelä 2006)butleadsonetoask,whatishappeninginFinnish classroomsthatmightexplainsuccessononeformoftestandfailureontheother?
EarlyanalysesofFinnishclassroomshavebeenparticularlycritical.Research undertakenmorethanaquarterofacenturyagofoundteachingpracticesthathad notonlychangedlittlein fi ftyyearsbutcreatedanintelligenceandemotional wasteland(Carlgrenetal. 2006).Morerecently,butprecedingFinland’sPISA successes,Norrisetal.(1996,p.29)found “rowsandrowsofchildrenalldoingthe samethinginthesamewaywhetheritbeart,mathematicsorgeography”,adding thattheyhad “movedfromschooltoschoolandseenalmostidenticallessons,you couldhaveswappedtheteachersoverandthechildrenwouldneverhavenoticed thedifference”.DuringthePISAyears,Andrewsandhiscolleagues,inanattempt tounderstandtherelationshipbetweenFinnishmathematics-relateddidacticsand PISAsuccessundertookthree,differentlyframedqualitativeanalysesofvideo sequencesofFinnishmathematicslessons.The first,basedonKilpatricketal.’s (2001) fi vestrandsofmathematicalproficiency,foundnoevidenceofFinnish teachersencouragingstudentstoacquirethehigherorderskillsconduciveto problemsolving;theprimaryobjectiveseemedtobeproceduralcompetence locatedonlimitedconceptualknowledge(Andrews 2013).Thesecond,exploiting PISA’sownassessmentframework,foundnoevidenceofstudentsbeingencouragedtoacquirethehigherlevelcompetencestheydemonstratedinrepeatedPISA assessments(Andrewsetal. 2014).Thethird,basedonaconstantcomparison analysisinwhichdatawerenotsubjectedtopredeterminedframeworks,founda didacticaltraditioninwhichteachersposedmanyclosedquestionsbutnever evaluatedorsoughtclari ficationrelatingtoastudent’sresponse.Thus,correct, partiallycorrect,incompleteorincorrectstudentresponsesreceivedthesame, noncommittalteacherfeedback.This implicit tradition,wherestudentsinfer meaningfromanygivensetofteacher-studentinteractions,dominatedallanalysed lessons(Andrews 2011).Thus,whileFinnishdidacticaltraditionsmaybeunableto explainFinnishstudents’ continuingPISAsuccesses,theymayhelpustounderstandtheirrepeatedTIMSSfailures,whichleadsnaturallytothe finalquestion, whatdoesexplainFinnishPISAsuccess?Thistakesustoanotherkeyrelationship betweencultureandeducation.
Anumberoffactors,assumedcontributorytostudents’ continuingPISAsuccesses,havebeenproposedbytheFinnsthemselves.Theseconcernthequalityof thecomprehensiveschoolsystemanditscompulsorycommoncurriculum(Aho etal. 2006),exceptionalspecialeducationalneedsarrangements(Hausstätterand Takala 2011;KiviraumaandRuoho 2007)andthehighqualityofteachereducation (Antikainen 2006;Laukkanen 2008;NiemiandJakku-Sihvonen 2006;Tuovinen 2008;Välijärvi 2004).However,suchstudieshave,asisincreasinglytypicalin Finland,failedtoacknowledgetheproblemofTIMSS.Afew,lesstriumphal, authorshavesuggestedthatFinnishPISAsuccessmaybeaconsequenceofother factors,largelyindependentofschool.Firstly,culturalhomogeneity(Välijärvietal. 2002)hasmadeit “comparativelyeasyinFinlandtoreachmutualunderstandingon nationaleducationpolicyandthemeansfordevelopingtheeducationsystem” . (Välijärvietal. 2002,p.45).Moreover,Finnishculturalhomogeneity,bornof
frequentlyviolentstrugglesbetweentheFinnishpeopleandoppressorsfromboth eastandwest,hascreatedacollectivemindsetnotdissimilartothoseofJapanand Korea(Simola 2005).Inparticular, “thereissomethingarchaic,something authoritarian,possiblyevensomethingeastern,intheFinnishcultureandmentality” (Simola 2005,p.458).Secondly,thissenseoftheauthoritariancollective finds resonanceinthetraditionthatparticipationinFinnishculturallifehas,since post-reformationtimes,beendependentonapublicdemonstrationofreading competence,whichwasapreconditionnotonlyforreceivingthesacramentsbutalso forcontractingaChristianmarriage(Linnakylä 2002).Consequently,asMason (2007,p.167)notes, “givenwhatwenowknowoftherelationshipbetweenlevelsof parentaleducationandtheeducationalachievementsoftheirchildren,itdoesnot takeasocialDarwinianperspective torealizetheeffectovercenturiesofacultural practicethathasmeantthatalmostallchildreninFinlandhavebeenraisedin familieswherebothparentsareliterate”.Consequently,theFinnshaveacquireda collectiveappreciationforeducationingeneralandFinnishliteratureinparticular (HalinenandJärvinen 2008),totheextentthattheFinnishlibrarynetworkisamong theworld’sdensest,withFinnsborrowingmorebooksthananyoneelse(Sahlberg 2007).Suchtraditions,basedonthecreationsofoneoftheworld’smostliterary cultures,explainwhyFinnishstudentsachievewellonPISAandnotonTIMSS; theyreadwithcompetence,theycaninterprettextandextractrelevantmaterial beforeundertakingthesimplemathematicsexpectedofatypicalPISAitem. Mathematically,TIMSSaskssomuchmoreofastudent,and,inthisrespect,Finnish studentsarenomorecompetentthananyotherstudentsintheWest.Inshort,the availableevidenceshowsthatculturemayplayamoresignificantrolethanpedagogyindeterminingtheeducationalachievementsofcountry,a findingthatshould beofgreatconcerntoanyonewithaninterestinimprovingbothmathematics teachingandstudentachievement.
7Conclusions Inthischapter,Ihavetriedtodemonstratetheextenttowhichteachingingeneral andmathematicsteachinginparticularareculturallydeterminedactivities.Thereis atemptation,particularlyamongacademicmathematiciansandpolicymakers,to assumethatschoolmathematicsisthesamewhereveritisexperienced.Thisis simplynottrue.Researchshowsconsistentlythatteachers’ practicesreflect “culturallydeterminedpatternsofbeliefandbehaviour,frequentlybeneatharticulation, thatdistinguishonesetofteachersfromtheirculturallydifferentcolleagues” (AndrewsandSayers 2013,p.133).Wherevertheyarelocated,lessons “havea routinenessaboutthemthatensuresadegreeofconsistencyandpredictability” (Kawanakaetal. 1999,p.91).Thissenseofroutinepredictabilityhasbeenvariouslydescribedasthe traditionsofclassroommathematics (Cobbetal. 1992),the characteristicpedagogical flow ofalesson(Schmidtetal. 1996),the culturalscript (StiglerandHiebert 1999)andthe lessonsignature (Hiebertetal. 2003).Such
descriptionsalludetorepeatedlyenactedpedagogicalstrategiestypicalofacountry’slesson(CoganandSchmidt 1999).Inthismannerculture “shapestheclassroomprocessesandteachingpracticeswithincountries,aswellashowstudents, parentsandteachersperceivethem” (Knipping 2003,p.282).Indeed,sosignificant isthishiddenroleofculturethatmanyoftheprocessesofteachingareso “deepin thebackgroundoftheschoolingprocess… sotaken-for-granted… astobebeneath mention” (HuftonandElliott 2000,p.117).Thatsaid,Ihopetheabovepageshave highlightedthecomplexrelationshipbetweenculture,howeveritisdefined,andthe processesandpracticesofeducationingeneralandmathematicseducationin particular.
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