[PDF] History and Epistemology in Mathematics Education

View - Download History and Epistemology in Mathematics Education

Previous PDF

Next PDF

HistoryandEpistemology

inMathema ticsEducation

Proceedingsofthe

5 th

EuropeanSummerUniver sity

Prague

July19-24,2007

Publisher:Vydavatelsk!servis,Plze"

Editors:EvelyneBarbi n,Na

aSt ehlíková,ConstantinosTzanakis,2008

ISBN978-8 0-86843-19-3

3

Contents

Preface13

1.History andEpistemologyasto olsforanin terd isciplinarya pproachin

theteac hingandlearningofMathemati csandtheSci ences15

PlenaryLecture

1.1LeoCO RRY

AxiomaticsBetweenHilbertandthe NewMath:DivergingViewson MathematicalResearchandt heirConsequencesonEducation....................17 Workshopsbasedonhi storicaland epistemological material

1.2Didi erBESSOT

CalculusbyAugustin Fres nel(1788-1827)toImprovetheE ciencyofParabolic

1.3Franti ˇsekKU

RINA,Chr istianSIEBENEICHER

AlgebraandGeometry inElement aryandSecondarySchool.....................29

1.4Fr´ ed´ericMETIN

AdamFrita ch'sNewFortification...............................................39

1.5Yanni sTHOMAIDIS,Const antinosTZANAKIS

OriginalTextsintheCl assroom.................................................49 Workshopsbasedonpe dagogicalanddi dacticalmateria l

1.6Mic helROELENS

TheVolu meofaPyramidThr oughtheA ges: Tosliceornottoslice,that's the

Oralpresentations

1.7Chri stianGERINI

LesAnna lesdeMath´ematiquesdeGergonne:Un journaldu 19 `eme si`ecle

num´eris´eetm´ediatis´eaub´en´eficed'unei nterdisciplinarit´eentremath´ematiques,

histoire,didactiqueetphilosophie ...............................................71

1.8RogerGOD ARD

LinearProgrammi ngandItsMathematicalRoots................................83

1.9Ladis lavKVASZ

HistoricalandEpistemological Aspects ofTeachingAlgebra......................91

1.10Konst antinosNIKOLANTONAKIS

Didweha ve"Rev olutions"inMa thematics?ExamplesfromtheHistoryo f MathematicsinthelightofT. S.Kuhn 'shisto ri calphilosophyofsc ience.........97 4

1.11Syl viePROVOST

Pourquoi"fairehistoire» dansl 'industrie,larecher cheetl'enseignement ,selon C.Combes (1867),W.H.B ragg(1912),P .L angevin(1926)?....................107

1.12Man-K eungSIU

HarmoniesinNature:ADialo guebet weenMathematicsand Phy sics.............115

1.13Atha nassiosSTRANTZALOS

GeometricTransformationsas aMeansfortheIntroductionofInterdisciplinarit y andofEdu cationalElemen tsinHighSchool.....................................125

Shortor alpresentations

1.14Ji

´ıCIHL

A

R,Pe trEISENMANN,Magda lenaKR

ATK

A,Pe trVOP

ENKA

1.15Mirand aISAIAS

LinearMovementofO bjects:D i

erencesbetweenAr istotle'sand highsc hoolstudents 'analysis(abstract).........................................143

1.16Anna MariaMERCURIO,N iclaPALL ADINO

Onth eResoluti onofAlgebraicEquations(abstract).............................145

1.17Ra

aelePISANO EvaristeGalois'GroupThe ory:Epis temologicalnotesonitslogical structure

1.18Sevim SEVG

I Harmonographapplicationintoourle ssons(abstract)...........................149

1.19Domi niqueTOURN

ES

GeometricalApproachtoDi

erentialEquations:From historytomathematics

2.Intro ducingahistoricaldimensioni nthe teachingandlearningof

Mathematics153

PlenaryLecture

2.1Luis PUIG

ResearchingtheHistoryofAlgebra icIdeasfr omanEducationalPoint ofView

Paneldiscussion

2.2Evely neBARBIN,FritzSCHWEI GER,LuisRADFORD,FrankSW ETZ

MathematicsofYesterdayan dTeaching ofToday................................157 5 Workshopsbasedonh istoricala ndepistemologi calmaterial

2.3Rit aBASTOS,Eduardo VELOSO

Episodesofthe History ofGeom etry:TheirInterpretationThrough Models inDy namicGeometry..........................................................171

2.4RobertB URN

TowardsaDefinitionof Lim it...................................................181

2.5Renau dCHORLAY,Philipp eBRIN

UsingHistor icalTextsintheClassroom:Exa mplesinStatisti csandPr obability

2.6CarlosC ORREIADESÁ

ConqCoutbes AvecLeurH istoire:laQuadrat rice,laSpirale,laConc ho¨ıde,la Cisso¨ıdeet laCyclo¨ıde..........................................................195

2.7Ale jandroS.GONZ

ALEZ-MART

IN,C arlosCORREIADES

A NewTea chingPractices.........................................................211

2.8Odil eKOUTEYNIKOFF

AboutFibonac ci'sBookofSquares:Howelementaryt oolscansolve quite elaborateproblems ..............................................................225 Workshopsbasedonpe dagogicalanddi dacticalmateria l

2.9Mic helBALLIEU,Marie-France GUISSARD

PourUneCu ltureMath ´ematiqueAccessible`aTous..............................237

2.10Adri anoDEMATT

E HistoricalDocumentsinEverydayC lassroomWork.............................251

2.11Isabel CristinaDIAS

Fromtheori ginaltext sofPedroNunestothemathe maticsclassroo mactiv ities

2.12Vict orKATZ

HistoricalModulesfortheTeac hingandLearningofMathema tics...............261

2.13Mich aelKOURKOULOS,Constan tinosTZANAKIS

EnhancingStudents 'UnderstandingontheMethodofLeastSqu ares:an InterpretativeModelInspiredByHistorical andEpistemologi calConsiderations..271

2.14Mich elaMASCHIETTO,FrancescaMARTIGNONE

ActivitieswithMathematicalM achines:Pant ographsandCurveDrawers........285

2.15Theo dorusPASCHOS,VassilikiF ARMAKI

TheIn tegrationofGeneticMomentsintheHi stor yofMathematicsandPhysics inth eDesigningofD idacticActivitiesAimingtoInt roduce First-Year 6

2.16Chri sWEEKS

UsingHistor icalMaterialintheMathematic sClassroom:Condorcet'sPa radox..311

2.17Robin WILSON

SolvingDottyProble ms(abstract)...............................................319

2.18Greis yWINICKI-LANDMA N

PlayingwithF ractionsaLaL eibniz.............................................321

Oralpresentations

2.19EvaC AIANIEL LO

LePr oblemeD'Oiseaux:Procede sdeResolutionDansL'Histoirede s

2.20Kathleen CLARK

ReflectionandRevision:Firs tExperien ceswithaUsingHistoryCou rse.........343

2.21JohanH DEKLERK

HistoryandEpistemolo gyAsToo lsinTeachingMathematics....................353

2.22EleniD IMITRIADOU

DidacticalandEpistemological Issues RelatedtotheConceptofProof:Some mathematicsteachers'ideasabo uttheroleofproofinGreeksec ondarycurriculu m363

2.23Mich aelR.GLAUBITZ

TheU seofOriginalSo urces intheClassroom:The oreticalPerspectivesand EmpiricalEviden ce.............................................................373

2.24HansN ielsJAHNK E

StudentsWorkingonThe irOwnIdeas:Bernoulli 'sLec turesontheDi erential

2.25Po-Hu ngLIU

InvestigationofStudents'Perceptionsofth eInfinite:A HistoricalDime nsion....385

2.26Andr easPOULOS

2.27Mariad oC´euSILV A

BentoFernan des'TratadodaArtedeArismetica(P orto,1555)(abstract).......405

2.28Frank J.SWETZ

HistoricalProblems:AValuable ResourceforMathematicsClas sro omInstruction

Shortor alpresentations

2.29OscarJo˜aoAB DOUNUR

AnEx hibitionAsaTooltoApproachDida ctical and HistoricalA spects ofthe RelationshipBetweenMathematicsand Music(abstract).........................409 7

2.30Fun daGONULATES

MathematicsTheather:Mathematician sonStage(abstract).....................411

2.31Gavi nHITCHCOCK

ApproachingMathematicalAnalys iswiththeHelpofItsHistory(abstract).......413 2.32U eTh omasJANKVIST

2.33JimJ .TATTE RSALL

BringingRamanujanInto theClassroom(abstract)..............................417

2.34PaolaG ALLOPIN,LucianaZ UCCHERI

MathematicalMethodsinApproachingthe ConceptsofA rea andVolume

3.Histor yandEpistemologyin Mathematicstea cherseducation 421

PlenaryLectur e

3.1Frit zSCHWEIGER

TheGr ammarofMathematicalSymb olis m......................................423 Workshopsbasedonhi storicaland epistemological material

3.2Giorgio T.BAGNI,Cateri naVICEN TINI

HistoryandEpistemol ogyofC alculusandAlgebra,CelebratingLeonhardE uler's

Tercentenary:CooperativeLearningand E

ectivenessofPerspectiveTeacher

3.3Jind ˇrichBE

CV A

R,Vlas timilDLAB,DagHRUB

Y,Fr antiˇsekKU

RINA EducationofMathematicsT eachers(InAlgebra andGeometry,inParticular)...441

3.4AnneBOY

E WhenHighSc hoolStudents AreTaughtChasles':" G´eom´etrieSup´ erieure" Lorsquel'onenseigna itla"g´eom´etriesup´ erieure"deChasles`alafinducursus

3.5Mich aelN.FRIED,AlainBERNAR D

ReadingandDoingMat hematicsi ntheHumanistTr adition:AncientandModern

3.6Jean- PaulGUICHARD

Vi`eteandt heAdventofLit eralCalculus........................................475

3.7Martin eB

UHLER,AnneMICHEL-P AJUS

AboutDi

erentKindofPro ofsEncounter edSpe cificallyinArithmeti c(Fermat's LittleTheorem):Fe rmat'sLittleTheorem.......................................489 8

3.8Ste liosNEGREPONTIS,DionysiosLAM PRINIDIS

ThePlat onicAnthyphaireticInterpret ationofPappus'AccountofAnalysisand

3.9Irene POLO-BLANCO

RegularandSemi-Regu larPolyto pes:AdidacticapproachusingBoole Stott's

3.10LuisR ADFORD

GeneralityandMathematicalIndetermin acy:Variables,U nknownsand Parameters,andtheirSymbolizatio ninHisto ryandintheClass room(abstract).521 Workshopsbasedonpe dagogicalanddi dacticalmateria l

3.11Mil anHEJN

Y,Na aSTEH LÍKOVÁ DidacticSimulationofHist oricalDiscoveriesinMathemat ics....................523

3.12Nits aMOVSHOVITZ-HAD AR

Today'sMathematical NewsAreTomorrow'sHistory:Interweavi ngMathNews SnapshotsintheTeachingo fHighSch oolMath..................................535

3.13LeoROGER S

EarlyMethodsforSolvin gRealProblems(abstract).............................547

3.14BjørnSM ESTAD

VariousMaterialsfor PrimarySchoolTeacherTraini ng: Or:Cany oudosomet hingevenif youca n'tdomuch?...........................549

Oralpresentations

3.15Marti naBE

VÁ

OVÁ

HistoryofMathematic sasaPa rtofMathematicalEducation....................561

3.16Alain BERNARD

HistoryofScienceandTec hnology intheFrenchSyste mforTe acherTraining: AboutaRecent Ini tiative........................................................569

3.17Anton ´ınJAN &A'

IK TheIn fluenceofITontheDevelopmentofMathemati csandont heEducation ofFu tureTeachers..............................................................577

3.18James F.KIERNAN

HowMuch HistoryofMath ematicsShouldanEle mentary MathematicsTeacher

3.19Bernadete MOREY

NavigationInstrumentsandTeacher Training...................................587

3.20Sifis PETRAKIS

TheRole oftheFifth Postulate intheEu clideanCons tructionofParallels.......595 9

3.21Klaus VOLKERT

ThePr oblemoftheDimensi onso fSpaceintheHi storyofGeometry.............605

Shorto ralpresentations

3.22Kamila

MEJRKOV

A SolvingLogicalPr oblemsandCommunicativeSk illsBytheGroupsofPupils

3.23E.Meh met OZKAN,HasanUNAL

InfluenceofMathematiciansinH istory onPre-ServiceMathematicsTea chers' BeliefsA bouttheNatur eofMathemati cs(abstract)..............................613

3.24David STEIN

SomeQuestio nsRegardingtheHistorica lRoleofConstructivisminMa thematic s

3.25Lucian aZUCCHERI,Veren aZUDINI

TheQu estionofChangingMathematicsSecon darySchoolCu rriculainVen ezia GiuliaAfter theFirstWorldWar(1918-1923)(abstract).......................617

4.Cu lturesandMathematics619

PlenaryLecture

4.1Ulric hREBSTOCK

MathematicsintheService oftheIs lamicComm unity...........................621 Workshopsbasedonhi storicaland epistemological material

4.2Chris tinePROUST

4.3Marie-N o¨elleRACINE

FemmesMathemati ciennesDansL'Histoire.....................................641 Workshopsbasedonp edagogicalandd idacticalmateri al

4.4GailE .FITZSIMON S

MathematicsandthePersonalCul tures ofStuden ts..............................651

Oralpresentations

4.5Adri anaCESARDEMATTOS

TheP rocessofRecognitionint hehis toryofMathematics.......................659

4.6Harald GROPP

TheR elationsBetweenMathema ticsandMusicinDi

erentRegionsandPeriods ofWor ldHistory(abstract).....................................................667

4.7LeoRO GERS

LeonardandThomasDig ges:16

th

CenturyMathematicalPract itioners..........669

10

4.8Edu ardoSebastianiFERREIRA

UsingEthno-M athematicsintheGraduationoftheIndianTeacher..............679

4.9Robin WILSON

LewisCarrolli nNumberland....................................................685

Shortor alpresentations

4.10Cec´ ıliaCOST A,PaulaCATARINO, MariaManuelSILVA NASCIMENTO

CouldMathemati csTransformMyLandintheCapital ofUniverse?(abstract)..697

4.11GoharM ARIKYAN

AnaniaShirakats i's7

th CenturyMethodologyofTeaching Arithmet icAcross the CenturiesandDiverseCultu res(abstract).......................................699

4.12Elfri daRALHA,

AngelaLOPES

Onth ePortugue seMathematicalReadingsaboutthe GregorianCalendarReform

4.13Carlos A.S.VILAR

LaD´ epressionduSoleil,AuD´ebutduC r´epusculeMatinalet `aLaFindu Vesp´eral,D'apr`esPedr oNunes,DansS onOuvrageDesCrepuscu lis(abstract)...703

5.History ofMathemati csEducationin Europe705

PlenaryLecture

5.1H´el`eneGISPE RT,GertSCHUBRING

TheHis toryofMathematicsEducationand Its Contextsin20 th

CenturyFrance

andGerma ny...................................................................707

Paneldiscussion

5.2GertS CHUBRING, H´el`eneGISP ERT,NikosKASTANIS,LiviaGIACARDI

TheEm ergenceofMathematicsasaMajorTeachin gS ubjectinSecondarySchools719 Workshopsbasedonhi storicaland epistemological material

5.3MariaEl fridaRALHA ,MariaFernandaESTRA DA

ReflectionUpona"Methodfor StudyingMath s", byJos´eMonteiroDaRocha

5.4Jos´ eManuelMATOS

VariationsinMathematicalKno wledge OccurringintheModernMathematics

5.5MartaM ENGHINI

TheEl ´ementsDeG´eom´etrieofA.M.L egendre:An analysisofso meproofsfr om yesterdayandtoday'spointofview ..............................................743 11 Workshopsbasedonp edagogicalandd idacticalmateri al

5.6HarmJan SMID

HeuristicGeometryTeaching :PreparingtheGroundOraDead End?...........755

Oralpresentations

5.7Eric BARBAZO

LeRˆ oled eL'Ass ociationDesPr ofesseursdeMath´ematiquesdeL'E nseignement Public(A.P.M.E.P .)DansLaCr´eationDesInstitu tsdeRe chercheSur L'En seignementDesMath´ematiques(I.R.E.M.)(abstract)......................765

5.8M´onicaBLANCO

Di erentialCalculusinMil itarySchoolsinLateEightee nth-Cent uryFranceand Germany:Acomparativete xtb ookanalysis......................................767

5.9TonyC RILLY

MathematicalEducationatCambridge UniversityintheNinet eenthC entury.....775

5.10Renau dD'ENFERT

DuCa lculAuxMathematiq ues?:L'int roductiondes"math´ematiquemodernes» dansl'enseign ementprimairefran¸cais,1960-1970...............................781

5.11JavierD OCAMPO

ManuscriptsandTeachersofCommercialAr ithmeti cinCatalonia(1400-1521)..791

5.12LiviaG IACARDI

TheIt alianSchoolofAlgebraicG eometryan dth eFormativeRoleof Mathematics inSe condaryTeaching(abstract)................................................803

5.13Snezana LAWRENCE

DescriptiveGeometryinEngland - aHis toricalSketch.........................805

5.14Pave l

I MA TeachingattheTechnicalU niver sitiesinR etrospect(abstract)...................813

Shorto ralpresentations

5.15J´erˆomeAUVI NET

C.-A.Laisant ThroughHisBookLaMathemat ique,Philosophie - Enseign ement

5.16Krist´ ınBJ ARNAD

OTTIR TheNu mberConceptandtheR oleofZero inNorthern-Eur opeanArithmetic

5.17Cec´ ıliaCO STA

IntroducingaHistoricalDimension Into Teaching:APortugueseExample - J.Vic enteGon¸calves(abstract).................................................819

5.18Alexand raGOMES,ElfridaRALHA

MathematicalTrainingandP rimarySchoolTeachers:Wherear ewecom ingfrom andwherear ewegoingtoin Portugal?(abstract)...............................821 12

5.19G´erardHAMON,LoocLE CORRE

EvidenceandC ulture ,RigorandPedagogy:EuclidandArnauld(abstract)......823

5.20MarioH .RAM

IREZD IAZ TheEvolut ionintheIntroduct ionof LearningS tyleintheTeachingofCalculus inMexi co(abstract)............................................................825

6.Mathema ticsinCentralEurope827

PlenaryLecture

6.1Magdalen aHYK

OVÁ

ContributionofCzechMathematiciansToP ro babilityTheory....................829 Workshopsbasedonhi storicaland epistemological material

6.2EwaLA KOMA

FromVitell onis'sGeometrytoUnravellingtheSecre tof"Enigma"Millenniumof thePol ishMathematicalThough tandItsImpactonToday'sMathematic s Education:Millenniumofth ePolishMathematicalThoughtandItsI mpacto n Today'sMathematics Education(abstract)......................................841 Workshopsbasedonp edagogicalandd idacticalmateri al

6.3JanVA NMAAN EN

TheWor kofEulerand theCurren tDiscussionAbou tSkills(abstract)...........843

Oralpresentations

6.4Mic haelaCHOCHOLOVÁ

WilhelmMat zka(1798-1891)andHisA lgebraicWorks..........................845

6.5KarelLEPKA

E.and K.Mathem aticalOlym pics...............................................855

6.6Frédéri cMETIN,PatrickGUYOT

ErasmusHabermel'sGe ometricalSquare(abstract)..............................861

6.7Rein hardSIEGMUND-SCHULTZE

NaziRuleand TeachingofMat hemat icsintheThirdReich,Pa rticul arlySchool

6.8DanaT RKOVSKÁ

TheIn fluenceoftheErlangerandtheMeranerPr ogrammon Mathemat ics EducationinCzechCountrie s...................................................877

6.9Leonardo VENEGAS

The(int ernational)ScientificProgramCommittee(SPC)891

TheLo calOrganizingCommittee(LOC)892

AuthorIndex893

Participants/countries897

13

Preface

Thisvo lumecontainsthet extsofthecontributionstothescientificpr ogr ammeofthe5 th EuropeanSummerUn iversity(ESU5)onthe HistoryandEpistemolo gyinMa thematics ofth iskindsinceJul y1993,when,ont heinitiative oftheFrench IRE Ms 1 thefir stEuropean SummerUniversityo ntheHistoryandEpistemology inMath ematicsE ducationtookplace inMont pellier,France.ThenextESUto okplaceinBraga,Portugalin1996,conjointlywith theHPM 2

SatelliteMeetingofI CME8), the3

rd inLouv ain-la-NeuveandLeuven,Belgium in19 99andthe4 th inUp psala,Swedenin2004,con jointlywiththeHPMSatellitemeeting ofIC ME10. Thepur poseofESUisnotonlytos tresst heuseofh istory andepiste mologyinthe teach- ingand learningof mathematics,inthesenseo fatechnical toolforinstruction, butalso to revealthatmat hematicssho uldbeconceivedasalivingscience, asciencewithal onghistory , notonlyt hecoreofthe teaching ofmathematic s,butals otheima geofmathematicsspre ad totheo utsidew orld.Inthisco nnection,theemphasisputonhistorica lan de pistemological issuesof mathematicsma yleadtoabetterunderstanding ofmathematicsitselfand toa deeperawarenesso fthefactthatmathematicsisnotonlyasystemofwell-organizedfinal- izedan dpolishedmentalp roducts,butalsoahuma nactivity,inwhichthe processesthat leadtothesep roducts are equallyimportantwith theproductsthemselves.I nthisway, asani nte rnationalactivity,ESUmainlyaimstop rovideaforumforpresentingresearc hin mathematicseducationandinnov ativeteachingmethodsbasedon ahistoric al,epistemologi- calandcultura lapproac htomathematicsa ndtheirteaching.So,itgivestheo pportunity to mathematicsteachers,educat orsandresearchers,toshare theirteachingideasandclassroom experience,andto graduatestuden tstobenefit fromth is.Inthisway,i tmotivat esfurther collaborationinthisperspectiveamongmember sof thema thematicseducationcommunity inEu ropeandbeyond,andstimu latesandencourage sgraduatestudentsint hisareato pursuefurther theirresearchin terestsbyestablishing newcollab orations. Thisismo stim- portantespeciallyto daythatmanycountriesarecon cernedaboutthele velofmathem atics theirstuden tslearnandabouttheirdecreasi nginter estinmathematicsatatimewhenthe needfor bothtec hnicalskillsandabettere ducationisrising. ThesePro ceedingscollect120papersorabstracts correspondi ngtoall types ofactivities includedin thescientificp rogramme ofESU5:S ixplenarylectures,twopaneldiscussions,

19wo rkshopsbasedondidactical andpedagogical materi aland25workshopsbasedon

historicalandepistemologicalm ateria l,44oralpresentationsandanother26shortoralcom- munications.Thisvolumeisdividedin tosixsection s,correspondingto thesix mainthemes ofESU 5:

1.Hi storyandEpistemo logyastoolsfor aninterdisciplinaryapproachinthete achi ng

andlearni ngofMathematicsandthe Sci ences

2.In troducingahistoricaldimensioninthe teac hingandlearningofMathemati cs

3.Hi storyandEpistemo logyinMathem aticsteachers'education

1 InstitutdeRecherchesur l'Ens eignementdesMathématiques. 2 TheIn ternationalStudyGrouponthe Relati onsbetweentheHistoryandPedag ogyofMathematics, a liatedtoICMI. 14

4.Cu lturesandMathematics

5.Hi storyofMathemat icsEducat ioninEurope

6.Ma thematicsinCentralEurope

Foreac hmaintheme,o neplenarylec turewasdelivereda nditstext appearsinthecorre- spondingsection.Therea realsopa perscoming fromanother twoplenarysession s,namely, thepane ldiscussions.T hesubjectofthesetwopanelswerecomplem entary;on" Mathe- maticsofYe sterdayand TeachingofToday",ando n"TheEm ergenceofMathematicsasa MajorTeaching SubjectinSecondarySchools".Finally ,workshops we reatypeofactivity ofsp ecialinterest.Theym adefocusonstudyingaspecifi csubjectandha vingafollo w-up discussion.Theroleo ftheworksho porganizerw astopr epar e,presentanddistributethe his- torical/epistemologicalorpedagogical/didacticalmaterial,whichmotiv atedand orientedthe exchangeofideasandt hed iscussionamong thepartici pants.Partici pantsreadandworked onth ebasisofth ismaterial(e.g.ori ginal historicaltext s,didacticalmaterial ,students' worksheetsetc).Thereaderoft heseProceedingswillfind heremany historicalresources, likeabst ractsoforiginaltexts,andmany pedago gicalresourcesfora lllevelsofmathematicsquotesdbs_dbs42.pdfusesText_42

[PDF] a first course in abstract algebra rotman solutions PDF Cours,Exercices ,Examens

[PDF] a first course in abstract algebra with applications 3rd edition pdf PDF Cours,Exercices ,Examens

[PDF] a g c t PDF Cours,Exercices ,Examens

[PDF] A Gringotts, 5ème Mathématiques

[PDF] a hombros de gigante clonación terapéutica PDF Cours,Exercices ,Examens

[PDF] a intersect b complement PDF Cours,Exercices ,Examens

[PDF] a intersection b formula PDF Cours,Exercices ,Examens

[PDF] a intersection b intersection c formula PDF Cours,Exercices ,Examens

[PDF] A justifier 3ème Mathématiques

[PDF] a l aide de vos connaissances, montrer comment on est passé de la dérive des continents ? la tectonique 1ère SVT

[PDF] À l ouest rien de nouveau 4ème Français

[PDF] A LAIDE 3ème Autre

[PDF] A l'aide 3ème Mathématiques

[PDF] A l'aide 4ème Chimie

[PDF] a l'AIDE 4ème Mathématiques