European Traditions in Didactics of Mathematics PDF
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ICME-13 Monographs Werner Blum · Michèle Artigue · Maria Alessandra Mariotti · Rudolf Sträßer · Marja Van den Heuvel-Panhuizen Editors European Traditions in Didactics of Mathematics ICME-13 Monographs Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, Hamburg, Germany Each volume in the series presents state-of-the art research on a particular topic in mathematics education and reflects the international debate as broadly as possible, while also incorporating insights into lesser-known areas of the discussion. Each volumeisbasedonthediscussionsandpresentationsduringtheICME-13conference and includes the best papers from one of the ICME-13 Topical Study Groups, DiscussionGroupsorpresentationsfromthethematicafternoon. More information about this series at http://www.springer.com/series/15585 è Werner Blum Mich le Artigue (cid:129) (cid:129) Maria Alessandra Mariotti (cid:129) äß Rudolf Str er Marja Van den Heuvel-Panhuizen (cid:129) Editors European Traditions in Didactics of Mathematics Editors WernerBlum Michèle Artigue Department ofMathematics University Paris Diderot—Paris 7 University of Kassel Paris, France Kassel, Hesse,Germany RudolfSträßer Maria Alessandra Mariotti Institut für Didaktik derMathematik Department ofInformation Engineering University of Giessen andMathematics Giessen, Hesse,Germany University of Siena Siena, Italy Marja Van denHeuvel-Panhuizen Freudenthal Institute/Freudenthal Goup UtrechtUniversity Utrecht, Netherlands and NordUniversity Bodø,Norway ISSN 2520-8322 ISSN 2520-8330 (electronic) ICME-13 Monographs ISBN978-3-030-05513-4 ISBN978-3-030-05514-1 (eBook) https://doi.org/10.1007/978-3-030-05514-1 LibraryofCongressControlNumber:2018964682 ©TheEditor(s)(ifapplicable)andTheAuthor(s)2019.Thisbookisanopenaccesspublication. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents 1 European Didactic Traditions in Mathematics: Introduction and Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Werner Blum, Michèle Artigue, Maria Alessandra Mariotti, Rudolf Sträßer and Marja Van den Heuvel-Panhuizen 2 The French Didactic Tradition in Mathematics . . . . . . . . . . . . . . . . 11 Michèle Artigue, Marianna Bosch, Hamid Chaachoua, Faïza Chellougui, Aurélie Chesnais, Viviane Durand-Guerrier, Christine Knipping, Michela Maschietto, Avenilde Romo-Vázquez and Luc Trouche 3 Didactics of Mathematics in the Netherlands . . . . . . . . . . . . . . . . . . 57 Marja Van den Heuvel-Panhuizen 4 The Italian Didactic Tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Maria Alessandra Mariotti, Maria G. Bartolini Bussi, Paolo Boero, Nadia Douek, Bettina Pedemonte and Xu Hua Sun 5 The German Speaking Didactic Tradition . . . . . . . . . . . . . . . . . . . . 123 Rudolf Sträßer 6 Didactics of Mathematics as a Research Field in Scandinavia . . . . . 153 Frode Rønning 7 Czech and Slovak Research in Didactics of Mathematics . . . . . . . . . 187 Jarmila Novotná, Marie Tichá and Na(cid:1)da Vondrová v Chapter 1 European Didactic Traditions in Mathematics: Introduction and Overview WernerBlum,MichèleArtigue,MariaAlessandraMariotti,RudolfSträßer andMarjaVandenHeuvel-Panhuizen Abstract Europeantraditionsinthedidacticsofmathematicssharesomecommon featuressuchasastrongconnectionwithmathematicsandmathematicians,thekey roleoftheory,thekeyroleofdesignactivitiesforlearningandteachingenvironments, andafirmbasisinempiricalresearch.Inthisfirstchapter,thesefeaturesareelaborated byreferringtofourcases:France,theNetherlands,ItalyandGermany.Inaddition, thischaptergivesanoverviewontheotherchaptersofthebook. · Keywords Europeandidactictraditions Overview 1.1 Introduction AcrossEurope,therehavebeenavarietyoftraditionsinmathematicseducation,both inthepracticeoflearningandteachingatschoolandinresearchanddevelopment, which have resulted from different cultural, historical and political backgrounds. Despite these varying backgrounds, most of these traditions share some common B W.Blum( ) DepartmentofMathematics,UniversityofKassel,Kassel,Germany e-mail:[email protected] M.Artigue UniversityParisDiderot—Paris7,Paris,France M.A.Mariotti DepartmentofInformationEngineeringandMathematicsScience, UniversityofSiena,Siena,Italy R.Sträßer InstitutfürDidaktikderMathematik,UniversityofGiessen,Giessen,Germany M.VandenHeuvel-Panhuizen UtrechtUniversity,Utrecht,TheNetherlands M.VandenHeuvel-Panhuizen NordUniversity,Bodø,Norway ©TheAuthor(s)2019 1 W.Blumetal.(eds.),EuropeanTraditions inDidacticsofMathematics,ICME-13Monographs, https://doi.org/10.1007/978-3-030-05514-1_1 2 W.Blumetal. features,onefeaturebeingtheuseinmanylanguagesoftheworddidactic(derived from the Greek didáskein, which means teaching) to denote the art and science of teachingandlearning(didactiek inDutch,didactiqueinFrench,didácticainSpan- ish,didattica inItalian,didaktika inCzech,dydaktyka inPolishanddidaktik(k)in Swedish,Danish,NorwegianandGerman)ratherthaneducation,whichispreferred inAnglo-Saxontraditions.TheseEuropeandidactictraditionscanbetracedbackas farasComenius’DidacticaMagnainthe17thcentury,thefirstcomprehensiveopus onaims,contentsandmethodsofteaching.Thesetraditionsshareinparticularthe followingcommonfeatures:astrongconnectionwithmathematicsandmathemati- cians,thekeyroleoftheory,thekeyroleofdesignactivitiesforlearningandteaching environments,andafirmbasisinempiricalresearch.Othercommonfeatures(such astheimportantroleofproofsandprovingoroflinkingmathematicswiththereal world)canbeconsideredpartofthosefourfeatures. In the following sections, we will elaborate a bit more on these four common features.1Theywillbemademoreconcretebyreferringbrieflytofourselectedcases ofEuropeantraditionsinthedidacticsofmathematics:France,theNetherlands,Italy andGermany.Inthefollowingfourchapters(Chaps.2–5)ofthisvolume,thesefour traditionsarepresentedinconsiderabledetail.Inparticular,theroleofthefourkey featuresinthosetraditionswillbecomemoretransparent.Thelasttwochaptersare devotedtoanothertwoEuropeantraditions,theScandinavian(Denmark,Norwayand Sweden;Chap.6)andtheCzech/Slovak(Chap.7).Althoughthischapteriswritten inEnglish,wewillspeak,throughoutthechapter,ofthe‘didacticsofmathematics’ instead of ‘mathematics education’ when we refer to the discipline dealing with allaspectsofteachingandlearningmathematicsintheabove-mentionedEuropean traditions. 1.2 TheRoleofMathematicsandMathematicians Herewewillhighlighttherolethatsomeoutstandingmathematicianshaveplayed inthedidacticsofmathematicsinthesefourcountriesbytheirinvolvementinedu- cationalissuessuchasdesigningcurriculaforschoolandforteachereducationand writingtextbooksandbytheirfosteringofthedevelopmentofdidacticsofmathe- maticsasaresearchfield.Inthisrespect,aprominentexemplarisFelixKlein(see Tobies,1981),whoalsohadagreatinfluenceonothermathematicianswhohadthe opportunityofgettingtoknowhisworkduringtheirvisitstoGermanyasresearchers. Animportantoccasionforinternationalcomparisonofdifferentexperiencesinthe didacticsofmathematicswastheFourthInternationalCongressofMathematicians, whichtookplaceinRomefrom6to11April1908.Duringthiscongress,theInter- nationalCommissionontheTeachingofMathematics(CommissionInternationale de l’Enseignement Mathématique, Internationale Mathematische Unterrichtskom- 1ThesesectionsrefertothecorrespondingsectionsinBlum,Artigue,Mariotti,Sträßer,andVan denHeuvel-Panhuizen(2017). 1 EuropeanDidacticTraditionsinMathematics:Introduction… 3 missionandCommissioneInternazionaledell’InsegnamentoMatematicoinFrance, GermanyandItaly,respectively)wasfounded(detailsofthehistoryofthisinstitution canberetrievedathttp://www.icmihistory.unito.it/timeline.php). AfterthetraumaticinterruptionsfortheFirstandSecondWorldWars,mathemati- cianswereagaininvolvedinvariousreformmovements.Inmanycountries,theideas andprinciplesoftheso-calledNewMathweresharedinthe1960sand1970s.We canrecogniseacommoninterestinreformingcurricula,whichiscertainlyrelatedto theimpactanewgenerationofmathematicianshadonthereorganisationofmathe- maticsthatwasinitiatedbytheBourbakiGroup.Thus,althoughtheconcreteresults oftheNewMathmovementwereverydifferentinvariouscountries,acommonfea- turewasthatsubstantialinnovationenteredintoschoolpracticethroughtheactive involvementofeminentfiguressuchasGustaveChoquet,JeanDieudonnéandAndré Lichnerowicz inFrance; EmmaCastelnuovo inItaly;and HansFreudenthal inthe Netherlands. Inthecontextofthisreform,newperspectivesdeveloped,beginninginthelate 1970s,thatmovedthefocusofreflectionfromissuesconcerningmathematicalcon- tentanditsorganisationinanappropriatecurriculumtoissuesconcerningthedescrip- tionandexplanationofthelearningandteachingofmathematics,givingbirthtoa newscientificdiscipline,thedidacticsofmathematics,thatrapidlydevelopedthrough activeinternationalinteraction.Insomecases,forinstanceinFranceandItaly,itis possibletorecogniseagainthestronginfluenceofthemathematicians’community, sincethefirstgenerationofresearchersinthedidacticsofmathematicsconsistedin these countries for the most part of academics affiliated with mathematics depart- ments.Thisobservationdoesnotignoretheexistenceofarecurrenttensionbetween mathematiciansandresearchersindidacticsofmathematics. Insummary,somecommonfeaturesthatcanbeconsideredthecoreoftheEuro- pean tradition of didactics of mathematics can be directly related to the fruitful commitmentofmathematicianstoeducationalissuesandtheirintenttoimprovethe teaching and learning of mathematics. One example is the strong role that proofs andprovinghaveintheseEuropeantraditions.Itcanbesaidthatinallfourcases, mathematicshasbeenandstillisthemostimportantrelateddisciplineforthedidac- ticsofmathematics,andthereisstillalivelydialoguebetweenmathematiciansand didacticians(researchersinthedidacticsofmathematics)oneducationalissues. 1.3 TheRoleofTheory Thewordtheoryinthedidacticsofmathematicshasabroadmeaning,rangingfrom very local constructs to structured systems of concepts; some are ‘home-grown’ whileothersare‘borrowed’withsomeadaptationfromotherfields,andsomehave developedoverdecadeswhileothershaveemergedonlyrecently.Thisdiversitycan alsobeobservedinthefourEuropeantraditionsunderconsideration. TheFrenchtraditioniscertainlythemosttheoreticalofthese.Ithasthreemain pillars: Vergnaud’s theory of conceptual fields (see Vergnaud, 1991), Brousseau’s 4 W.Blumetal. theoryofdidacticalsituations(TDS;seeBrousseau,1997)andtheanthropological theoryofthedidactic(ATD)thatemergedfromChevallard’stheoryofdidactictrans- position(seeChevallard&Sensevy,2014).Thesedevelopedoverdecadeswiththe convictionthatthedidacticsofmathematicsshouldbeascientificfieldofresearch with fundamental and applied dimensions supported by genuine theoretical con- structionsandappropriatemethodologies,givinganessentialroletotheobservation andanalysisofdidacticsystemsandtodidacticalengineering.Thesetheorieswere first conceived as tools for the understanding of mathematics teaching and learn- ingpracticesandprocesses,takingintoconsiderationthediversityoftheconditions andconstraintsthatshapethem,andfortheidentificationofassociatedphenomena, suchasthe‘didacticcontract’.Thethreetheoriesarealsocharacterisedbyastrong epistemologicalsensitivity.Overtheyears,thistheoreticallandscapehasbeencon- tinuously enriched by new constructions and approaches, but efforts have always beenmadetomaintainitsglobalcoherence. TheDutchtraditionislessdiversified,asithasdevelopedaroundasingleapproach known today as Realistic Mathematics Education (RME; see Van den Heuvel- Panhuizen&Drijvers,2014).Italsoemergedinthe1970swithFreudenthal’sinten- tiontogivethedidacticsofmathematicsascientificbasis.SimilartotheFrenchcase, thisconstructionwassupportedbyadeepepistemologicalreflection:Freudenthal’s didactical phenomenology of mathematical structures (see Freudenthal, 1983). In thistradition,theoreticaldevelopmentanddesignarehighlyinterdependent.Thisis visible in the RME structure, which is made of six principles clearly connected to design: activity, reality, level, intertwinement, interactivity and guidance. Through designresearchinlinewiththeseprinciples,manylocalinstructiontheoriesfocus- ing on specific mathematical topics have been produced. RME is still in concep- tualdevelopment,benefitingfrominteractionswithotherapproachessuchassocio- constructivism,instrumentationtheoryandembodiedcognitiontheory. In the Italian tradition, it is not equally possible to identify major theories that wouldhavesimilarlyemergedanddeveloped,despitealong-termtraditionofaction researchcollaborativelycarriedoutbymathematiciansinterestedineducationand byteachers.Progressively,however,aspecificresearchtrendhasemergedfromthis actionresearchandconsolidatedwithinaparadigmofresearchforinnovation,lead- ingtothedevelopmentofspecifictheoreticalframesandconstructs(foranoverview, seeArzarello&BartoliniBussi,1998).Boero’sconstructoffieldofexperience,Bar- toliniBussiandMariotti’stheoryofsemioticmediation,andArzarello’sconstructs ofsemioticbundleandaction,productionandcommunication(APC)spacerepresent thistrendwell. In Germany, scholars since the early 1970s have aimed to create the field of didactics of mathematics as a scientific discipline, as shown by articles published in ZDM in 1974–75 (see Griesel, 1974; Winter, 1975; Wittmann, 1974) and the effortsmadebyHans-GeorgSteinertoestablishaninternationaldebateonthetheory of mathematics education and the underlying philosophies and epistemologies of mathematicswithinaninternationalTheoryofMathematicsEducation(TME)group he founded in 1984. However, it would be difficult to identify a specific German wayofapproachingtheoreticalissuesinthedidacticsofmathematicseventhough,
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