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ICME-13 Topical Surveys Paul Ernest · Ole Skovsmose Jean Paul van Bendegem · Maria Bicudo Roger Miarka · Ladislav Kvasz Regina Moeller The Philosophy of Mathematics Education ICME-13 Topical Surveys Series editor GabrieleKaiser,FacultyofEducation,UniversityofHamburg,Hamburg,Germany More information about this series at http://www.springer.com/series/14352 Paul Ernest Ole Skovsmose (cid:129) Jean Paul van Bendegem Maria Bicudo Roger Miarka (cid:129) Ladislav Kvasz Regina Moeller (cid:129) The Philosophy of Mathematics Education PaulErnest RogerMiarka GraduateSchoolofEducation DepartamentodeEducaçãoMatemática ExeterUniversity UniversidadeEstadualPaulista Exeter “JúliodeMesquitaFilho” UK BelaVista,RioClaro,SãoPaulo Brazil OleSkovsmose DepartmentofLearningandPhilosophy LadislavKvasz AalborgUniversity DepartmentofMathematicsEducation, Aalborg PedagogicalFaculty Denmark CharlesUniversity Prague JeanPaulvanBendegem CzechRepublic CenterforLogicandPhilosophyofScience VrijeUniversiteitBrussel ReginaMoeller Brussels ErziehungwissenschaftlicheFakultät Belgium UniversitätErfurt Erfurt MariaBicudo Germany DepartamentodeMatemática UniversidadeEstadualPaulista “JúliodeMesquitaFilho” RioClaro Brazil ISSN 2366-5947 ISSN 2366-5955 (electronic) ICME-13TopicalSurveys ISBN978-3-319-40568-1 ISBN978-3-319-40569-8 (eBook) DOI10.1007/978-3-319-40569-8 LibraryofCongressControlNumber:2016943630 ©TheEditor(s)(ifapplicable)andTheAuthor(s)2016.Thisbookispublishedopenaccess. OpenAccessThisbookisdistributedunderthetermsoftheCreativeCommonsAttribution4.0International License(http://creativecommons.org/licenses/by/4.0/),whichpermitsuse,duplication,adaptation,distribution, andreproductioninanymediumorformat,aslongasyougiveappropriatecredittotheoriginalauthor(s)and thesource,alinkisprovidedtotheCreativeCommonslicense,andanychangesmadeareindicated. Theimagesorotherthirdpartymaterialinthisbookareincludedinthework’sCreativeCommonslicense, unlessindicatedotherwiseinthecreditline;ifsuchmaterialisnotincludedinthework’sCreativeCommons licenseandtherespectiveactionisnotpermittedbystatutoryregulation,userswillneedtoobtainpermission fromthelicenseholdertoduplicate,adaptorreproducethematerial. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditors giveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissions thatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland The Main Topics You Can Find in This ICME-13 Topical Survey (cid:129) Brief analysis of the philosophy of mathematics education including multiple ways of interpreting it and some of its central questions and problems. (cid:129) The varying roles of philosophy in mathematics education research including applications of concepts from branches of philosophy and philosophical styles of analysis and thinking. (cid:129) Introduction to critical mathematics education, an area of research concerned withcriticalandsocialjusticeissuesthatisacentralstrandinthephilosophyof mathematics education. (cid:129) An update on the philosophy of mathematical practice, a vital, emerging movement in contemporary philosophy of mathematics. (cid:129) A case study is included of how one strand of research in the philosophy in mathematicseducationhasdevelopedinonecountryrecently. Wehavechosen Brazil because it is very active in this area of research, drawing on the special knowledge of several of our authors. v Contents The Philosophy of Mathematics Education. . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 An Overview of the Philosophy of Mathematics Education . . . . . . . . . 3 2.1 Introduction: What Is the Philosophy of Mathematics Education? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A ‘Top Down’ Analysis of the Philosophy of Mathematics Education. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Critical Mathematics Education: Concerns, Notions, and Future . . . . . . 9 3.1 Some Concerns in Critical Mathematics Education. . . . . . . . . . . . 9 3.2 Some Notions in Critical Mathematics Education. . . . . . . . . . . . . 10 3.3 Critical Mathematics Education for the Future. . . . . . . . . . . . . . . 12 4 The Philosophy of Mathematical Practice: What Is It All About? . . . . . 13 4.1 Lakatos as the Starting Point. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Kitcher as the Next Step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 A Tension Is Introduced to Stay . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 Enter the Sociologists, Educationalists and Ethnomathematicians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.5 Brain and Cognition Complete the Picture . . . . . . . . . . . . . . . . . 17 4.6 Conclusion: Working in Different “Registers”. . . . . . . . . . . . . . . 17 5 The Philosophy of Mathematics Education in Brazil . . . . . . . . . . . . . . 18 6 Summary and Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 vii The Philosophy of Mathematics Education 1 Introduction ThisICME-13TopicalSurveyisdesignedtoprovideanoverviewofcontemporary research in the philosophy of mathematics education. This is a broad cluster of overlapping but at times disparate themes. In the first instance, this publication exposes some of the problems and questions in mathematics education that the philosophy of mathematics education clarifies, illuminates and sometimes helps to solve. A metaphor for what is offered is a three tier pyramid. At the apex is this publication, presenting an abbreviated ‘problematique’ of the subfield, that is, the cluster of problems and issues at the heart of the area. At the next level, is the breadthofproblems,issuesandresearchresultsthatwillbesharedwhenthegroup of authors of this publication meet with the ‘public’ at the ICME 13 conference in Hamburg,July2016.Finally, atthebaselevelisthefullspreadofresearchandits results, books, journals, papers, conference presentations and other activities that make up the subfield, the philosophy of mathematics education, which is beyond thescope ofthispublicationandconference.Thispublicationthusbrieflysketches some of the topics, problems and areas of active research (the apex) and through this point to some of what will be offered at the conference (the middle level). In doing so it serves as an introduction to the extent of the sub-field overall, through references to current publications and classic literature (the base of the pyramid). Why the philosophy of mathematics education? What does it offer? The phi- losophy of any activity comprises its aims or rationale. Given our shared com- mitment to the teaching and learning of mathematics it is vital ask: What is the purpose of teaching and learning mathematics? What do we value in mathematics anditsteachingandlearning?Whydoweengageinthesepracticesandwhatdowe ©TheAuthor(s)2016 1 P.Ernestetal.,ThePhilosophyofMathematicsEducation, ICME-13TopicalSurveys,DOI10.1007/978-3-319-40569-8_1 2 ThePhilosophyofMathematicsEducation hope will be achieved? The sub-field can also help us uncover whatever implicit assumptions and priorities underlie mathematics education. These can including paradigmatic assumptions of which we may be unaware, but that can be identified through, let us say, a philosophical archaeology. The philosophy of mathematics education applies philosophical methods to a critical examinationoftheassumptions,reasoning andconclusionsofmathematics education, systematically enquiring into fundamental questions: (cid:129) What is mathematics? (cid:129) How does mathematics relate to society? (cid:129) Why teach mathematics? (cid:129) What is the nature of learning (mathematics)? (cid:129) What is the nature of mathematics teaching? (cid:129) What is the significance of information and communication technology in the teaching and learning of mathematics? (cid:129) What values underlie these activities, overt and covert? (cid:129) Howandtowhatextentissocialjusticepromulgatedbytheseactivitiesandthis field of study? (cid:129) What is the status of mathematics education as knowledge field? (cid:129) What deep and often unacknowledged assumptions underlie mathematics edu- cation research and practice? The philosophy of mathematics education matters because it gives people new ‘glasses’ through which to see the world. It enables people to see beyond official storiesaboutthesociety,mathematics,andeducation.Itprovidesthinkingtoolsfor questioningthestatusquo,for seeing‘whatis’isnotwhat‘hastobe’;enablingus to imagine alternatives possibilities. In addition to laying out the conceptual traditions and ‘problematique’ of the philosophyofmathematicseducation,thistopicalsurveyalsopresentsacasestudy of research in the topic conducted in one country. As will emerge in all of the sections, Brazil has played a strong role in the formation and development of philosophy of mathematics education research, especially in the connected and well-knownareasofcriticalmathematicseducationandethnomathematics(see,for example, D’Ambrosio 1985, 1998, 2006). In addition to these special interests Brazilhasanactiveresearchcommunityinthetheoreticalaspectsofthephilosophy of mathematics education and their applications to policy and practice. Recent developmentsintermsoforganisation,basisandresearchorientationofonestrand in the area are sketched in Sect. 5. 2 AnOverviewofthePhilosophyofMathematicsEducation 3 2 An Overview of the Philosophy of Mathematics Education Paul Ernest 2.1 Introduction: What Is the Philosophy of Mathematics Education? The philosophy of mathematics education can be interpreted both narrowly and more widely. Understood narrowly the philosophy of some activity is its aim or rationale.Sointhenarrowsensethephilosophyofmathematicseducationconcerns the aims or rationale behind the practice of teaching mathematics. However, the aims, goals, purposes, rationales, etc., for teaching mathematics do not exist in a vacuum, belonging to people, whether individuals or social groups (Ernest 1991). Since the teaching of mathematics is a widespread and highly organised social activity,itsaims,goals,purposes,rationales,andsoon,needtoberelatedtosocial groups and society in general, while acknowledging that there are multiple and divergentaimsandgoalsamongdifferentpersonsandgroups.Aimsareexpressions ofvalues,andthustheeducationalandsocialvaluesofsocietyorsomepartofitare implicated in this enquiry. Therearebroaderinterpretationsofthatgobeyondtheaims,rationaleandbasis for teaching mathematics. Expanded senses of the philosophy of mathematics education include: 1. Philosophy applied to or of mathematics education 2. Philosophyofmathematicsappliedtomathematicseducationortoeducationin general 3. Philosophy of education applied to mathematics education (Brown 1995). Each of these possible applications of philosophy to mathematics education represents a different focus, and might very well foreground different issues and problems.However,theseofapplicationsofphilosophyinvolvesubstantivebodies of knowledge. In fact, philosophy, mathematics education and other domains of knowledgeencompass processesofenquiryandpractice, personalknowledge,and as well as published knowledge representations. They are not simply substantial entities inthemselves,butcomplexrelationshipsandinteractions betweenpersons, society, social structures, knowledge representations and communicative and other practices. Thus a further expanded sense of the philosophy of mathematics edu- cation includes the applications of philosophical processes, methods and critical modes of thought.

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