ICME-13 Monographs Hans-Georg Weigand · William McCallum Marta Menghini · Michael Neubrand Editors Gert Schubring The Legacy of Felix Klein ICME-13 Monographs Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, 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 Hans-Georg Weigand William McCallum (cid:129) Marta Menghini Michael Neubrand (cid:129) Gert Schubring Editors The Legacy of Felix Klein Editors Hans-Georg Weigand Michael Neubrand UniversitätWürzburg University of Oldenburg Würzburg, Bayern,Germany Oldenburg,Germany William McCallum GertSchubring University of Arizona UniversitätBielefeld Tucson,AZ, USA Bielefeld, Germany Marta Menghini and Department ofMathematics Sapienza University of Rome Universidade FederaldoRiodeJaneiro Rome, Italy RiodeJaneiro, Brazil ISSN 2520-8322 ISSN 2520-8330 (electronic) ICME-13 Monographs ISBN978-3-319-99385-0 ISBN978-3-319-99386-7 (eBook) https://doi.org/10.1007/978-3-319-99386-7 LibraryofCongressControlNumber:2018952593 ©TheEditor(s)(ifapplicable)andTheAuthor(s)2019.Thisbookisanopenaccesspublication. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents Part I Introduction 1 Felix Klein—Mathematician, Academic Organizer, Educational Reformer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Renate Tobies 2 What Is or What Might Be the Legacy of Felix Klein? . . . . . . . . . 23 Hans-Georg Weigand Part II Functional Thinking 3 Functional Thinking: The History of a Didactical Principle . . . . . . 35 Katja Krüger 4 Teachers’ Meanings for Function and Function Notation in South Korea and the United States. . . . . . . . . . . . . . . . . . . . . . . 55 Patrick W. Thompson and Fabio Milner 5 Is the Real Number Line Something to Be Built, or Occupied?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Hyman Bass 6 Coherence and Fidelity of the Function Concept in School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 William McCallum Part III Intuitive Thinking and Visualization 7 Aspects of “Anschauung” in the Work of Felix Klein. . . . . . . . . . . 93 Martin Mattheis 8 Introducing History of Mathematics Education Through Its Actors: Peter Treutlein’s Intuitive Geometry. . . . . . . . . . . . . . . 107 Ysette Weiss v vi Contents 9 TheRoadoftheGermanBookPraktischeAnalysisintoJapanese Secondary School Mathematics Textbooks (1943–1944): An Influence of the Felix Klein Movement on the Far East . . . . . . 117 Masami Isoda 10 Felix Klein’s Mathematical Heritage Seen Through 3D Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Stefan Halverscheid and Oliver Labs 11 The Modernity of the Meraner Lehrplan for Teaching Geometry Today in Grades 10–11: Exploiting the Power of Dynamic Geometry Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Maria Flavia Mammana Part IV Elementary Mathematics from a Higher Standpoint—Conception, Realization, and Impact on Teacher Education 12 Klein’s Conception of ‘Elementary Mathematics from a Higher Standpoint’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Gert Schubring 13 Precision Mathematics and Approximation Mathematics: The Conceptual and Educational Role of Their Comparison . . . . . 181 Marta Menghini 14 Examples of Klein’s Practice Elementary Mathematics from a Higher Standpoint: Volume I . . . . . . . . . . . . . . . . . . . . . . . . 203 Henrike Allmendinger 15 A Double Discontinuity and a Triple Approach: Felix Klein’s Perspective on Mathematics Teacher Education . . . . . . . . . . . . . . . 215 Jeremy Kilpatrick Part I Introduction Hans-Georg Weigand, William McCallum, Marta Menghini, Michael Neubrand and Gert Schubring Throughout his professional life, Felix Klein emphasised the importance of reflectinguponmathematicsteachingandlearningfrombothamathematicalanda psychological or educational point of view, and he strongly promoted the mod- ernisation of mathematics in the classroom. Already in his inaugural speech of 1872, the Erlanger Antrittsrede (not to be mistaken with the Erlanger Programm which isa scientific classification ofdifferent geometries) for his first position as a full professor atthe University ofErlangen—atthe age of 23—he voiced his view on mathematics education: Wewantthefutureteachertostandabovehissubject,thathehaveaconceptionofthe presentstateofknowledgeinhisfield,andthathegenerallybecapableoffollowingits furtherdevelopment.(Rowe1985,p.128) FelixKleindevelopedideasonuniversitylecturesforstudentteachers,whichhe laterconsolidatedatthebeginningofthelastcenturyinthethreebooksElementary Mathematics from ahigher standpoint.1 Inpart IV ofthis book,the three volumes areanalysedinmoredetail:Klein’sviewofelementary;hismathematical,historical and didactical perspective; and his ability to relate mathematical problems to problemsofschoolmathematics.Intheintroductionofthefirstvolume,FelixKlein also faced a central problem in the preparation of mathematics teachers and expressed it in the quite frequently quoted double discontinuity: The young university student finds himself, at the outset, confronted with problems, whichdonotremember,inanyparticular,thethingswithwhichhehadbeenconcernedat school.Naturallyheforgetsallthesethingsquicklyandthoroughly.When,afterfinishing hiscourseofstudy,hebecomesateacher,hesuddenlyfindshimselfexpectedtoteachthe traditional elementary mathematics according to school practice; and, since he will be 1ThepreviousEnglishtranslationofthefirsttwovolumesbyEarleRaymondHedrickandCharles AlbertNoble,publishedin1931and1939,hadtranslated“höheren”erroneouslyby“advanced”; seethecommentbySchubringin:Klein2016,p.v–vi,andtheregularlectureofJeremyKilpatrick (2008)atICME11inMexico. 2 H.-G.Weigandetal. scarcely able, unaided, to discern any connection between this task and his university mathematics, he will soon fell in with the time honoured way of teaching, and his university studies remain only a more or less pleasant memory which has no influence uponhisteaching.(Klein2016[1908],Introduction,Volume1,p.1) Atthe13thInternationalCongressonMathematicalEducation(ICME-13)2016 in Hamburg, the “Thematic Afternoon” with the The Legacy of Felix Klein as one major theme, provided an overview of Felix Klein’s ideas. It highlighted some developments in university teaching and school mathematics related to Felix Klein’s thoughts stemming from the last century. Moreover, it discussed the meaning,theimportanceandthelegacyofKlein’sideasnowadaysandinthefuture in an international, global context. Threestrandswereofferedonthis“ThematicAfternoon”,eachconcentratingon oneimportantaspectofFelixKlein’swork:FunctionalThinking,IntuitiveThinking and Visualisation, and Elementary Mathematics from a Higher Standpoint— Conception, Realisation, and Impact on Teacher Education. This book provides extendedversionsofthetalks,workshopsandpresentationsheldatthis“Thematic Afternoon” at ICME 13. Felix Klein was a sensitised scientist who recognised problems, thought in a visionary manner, and acted effectively. In part I, we give an account of some biographical notes about Felix Klein and an introduction to his comprehensive programme. He had gained international recognition through his significant achievementsinthefieldsofgeometry,algebra,andthetheoryoffunctions.Based in this, he was able to create a centre for mathematical and scientific research in Göttingen.Besideshisscientificmathematicsresearch,Kleindistinguishedhimself through establishing the field of mathematics education by having such high regardsforthehistoryofmathematicsasakeystoneofhighereducation.Hewasfar ahead of his time in supporting all avenues of mathematics, its applications, and mathematicalpedagogy.Heneverpursuedtheunilateralinterestsofhissubjectbut rather kept an eye on the latest developments in science and technology (see the article by Renate Tobies in this book). Klein investigated functions from many points of view, from functions defined bypowerseriesandFourierseries,tofunctionsdefined(intuitively)bytheirgraphs, to functions defined abstractly as mappings from one set to another. Part II examines the development of the concept of function and its role in mathematics education from Klein’s time—especially referring to the “Meraner Lehrplan” (1905)—totoday. Itincludesstudents’andteachers’thinking about theconcept of function,thecommunication(problems)andtheobstaclesthisconceptfacesinthe classroom. Klein made an important distinction between functions arising out of applications of mathematics and functions as abstractions in their own right. This distinction reverberates in mathematics education even today. Alongside the concept of function or functional thinking, the idea of intuition andvisualisationissurelyanothercentralaspect toKlein’smathematicalthinking. ThearticlesinpartIIIhighlightFelixKlein’sideas.Thecontributionslookforthe origins of visualisation in Felix Klein’s work. They show the influences of Felix Klein’sideas,bothinthenationalandintheinternationalcontext.Theythengoon PartI:Introduction 3 to confront these ideas with the recent possibilities of modern technological tools and dynamic geometry systems. PartIVpresentsthenewlytranslatedversionsofthethreebookson“Elementary Mathematics from a Higher Standpoint”. At ICME-13, the third volume Precision andApproximationMathematicsappearedinEnglishforthefirsttime.Referringto these three famous volumes, this chapter presents a mathematical, historical and didactical perspective on Klein’s thinking. The whole book intends to show that many ideas of Felix Klein can be rein- terpretedinthecontextofthecurrentsituation,andgivesomehintsandadvicefor dealingtodaywithcurrentproblemsinteachereducationandteachingmathematics in secondary schools. In this spirit, old ideas stay young, but it needs competent, committed and assertive people to bring these ideas to life. References Kilpatrick, J. (2008). A higher standpoint. In ICMI proceedings: Regular lectures (pp. 26–43). https://www.mathunion.org/fileadmin/ICMI/files/About_ICMI/Publications_about_ICMI/ ICME_11/Kilpatrick.pdf.Accessed6Oct2018. Klein, F. (1908). Elementarmathematik vom höheren Standpunkte aus: Arithmetik, algebra, analysis(Vol.1).Leipzig,Germany:Teubner. Klein,F.(2016).ElementaryMathematicsfromaHigherStandpoint.Vol.I.Arithmetic,algebra, analysis(Translated2016byGertSchubring).Berlin&Heidelberg:Springer. Rowe,D.(1985).FelixKlein’s“ErlangerAntrittsrede”.HistoriaMathematica,12,123–141.