“Dig where you stand” 3 “Dig where you stand” 3 Proceedings of the Third International Conference on the History of Mathematics Education September 25–28, 2013, at Department of Education, Uppsala University, Sweden Editors: Kristín Bjarnadóttir Fulvia Furinghetti Johan Prytz Gert Schubring Uppsala University Department of Education “Dig where you stand” 3 Proceedings of the Third International Conference on the History of Mathematics Education September 25–28, 2013, at the Department of Education, Uppsala University, Sweden Editors: Kristín Bjarnadóttir University of Iceland, School of Education, Reykjavík, Iceland Fulvia Furinghetti Dipartimento di Matematica dell’Università di Genova, Italy Johan Prytz Uppsala University, Department of Education, Sweden Gert Schubring Universidade Federal do Rio de Janeiro, Instituto de Matemática, Brazil Institut für Didaktik der Mathematik, Universität Bielefeld, Germany Cover design: Graphic Services, Uppsala University Printed by: DanagårdLITHO AB, Ödeshög 2015 The production of this volume is funded by the Faculty of Education at Uppsala University, the Department of Education at Uppsala University and the Swedish Research Council. All rights reserved © 2015 Authors 1st edition, March 2015 ISBN 978-91-506-2444-1 Published by Uppsala University – Department of Education Uppsala Contents Introduction 7 Klein’s Elementary Mathematics from a Higher Standpoint – An analysis from a historical and didactic point of view 9 Henrike Allmendinger The influence of New School ideas in the preparation of mathematics teachers for liceus in Portugal from 1930 to 1969 21 Mária Cristina Almeida Euclides Roxo’s deductive geometry 33 Regina de Cassia Manso de Almeida Top-down: the Role of the Classes Préparatoires aux Grandes Écoles in the French teaching of descriptive geometry (1840–1910) 49 Evelyne Barbin Mathematics education in twentieth century Iceland – Ólafur Daníelsson’s impact 65 Kristín Bjarnadóttir Teaching traditions in Swedish school algebra – a project description 81 Kajsa Bråting Lucienne Félix and Osvaldo Sangiorgi: Interchanges between two Bourbakist militants in the 1960s 93 Elisabete Zardo Búrigo From Lancaster to Pestalozzi – changing views of mathematics education in Latin America during the nineteenth century 109 João Bosco Pitombeira Fernandes de Carvalho The understanding of parallel lines in early nineteenth century textbooks: A comparison between two Norwegian geometry books from 1827 and 1835 123 Andreas Christiansen Contents Teaching the mathematical sciences in France during the eighteenth century: a few examples from some of the most used textbooks 137 Sara Confalonieri Modern mathematics at the 1959 OEEC Seminar at Royaumont 151 Dirk De Bock, Geert Vanpaemel Decimal fractions in school mathematics in Great Britain and North America, 1667–1887 169 Nerida F. Ellerton, M. A. (Ken) Clements Francesco Severi and mathematics teaching in secondary schools. Science, politics and schools in the first half of the twentieth century 187 Livia Giacardi, Alice Tealdi Problems in old Russian textbooks: How they were selected 203 Alexander Karp Warren Colburn and the inductions of reason 219 Jeremy Kilpatrick Abraham Gotthelf Kästner and his “Mathematische Anfangsgründe” 233 Desirée Kröger Change and stability: Dutch mathematics education, 1600–1900 245 Jenneke Krüger Emma Castelnuovo’s commitment to creating a new generation of mathematics teachers 263 Marta Menghini Fortification teaching in seventeenth century French Jesuit colleges 279 Frédéric Métin American mathematical journals and the transmission of French textbooks to the USA 293 Thomas Preveraud Swedish mathematics curricula, 1850–2014. An overview 309 Johan Prytz Academics, textbooks and reform of mathematics education in secondary French schools (1890–1905) 327 Hervé Renaud Contents Epistemology, methodology, and the building of meaning in a new community of mathematics educators in England, 1950–1980 345 Leo Rogers Analytical geometry in An elementary treatise on plane and spherical trigonometry, and on the application of algebra to geometry by S. F. Lacroix 361 Isabel María Sánchez, Maria Teresa González Astudillo Maximum and minimum: Approaches to these concepts in Portuguese textbooks 377 Ana Santiago, Maria Teresa González Astudillo The emergence of the profession of mathematics teachers – an international analysis of characteristic patterns 389 Gert Schubring “Sickened by set theory?” – New Maths at German primary schools (abstract) 405 Tanja Hamann The autonomy of secondary school mathematics culture (abstract) 406 José Matos Arithmetic textbooks – on the origins of an European tradition (abstract) 407 Barbara Schmidt-Thieme Bento de Jesus Caraça: criticism and proposals for the mathematics teaching in Portugal in the 1940s (abstract) 408 Jaime Carvalho e Silva New geometries for old schools? (abstract) 409 Klaus Volkert Contributors 411 Index 419 Introduction From 25 to 28 September 2013 the third International Conference on the History of Mathematics Education (ICHME-3) was held at the Department of Education, Uppsala University, Sweden. The department also sponsored the conference financially. The local organizer was Johan Prytz. The Scientific Program Committee was composed by Kristín Bjarnadóttir (University of Iceland), Fulvia Furinghetti (University of Genoa, Italy), Johan Prytz (Uppsala University), Gert Schubring (Universität Bielefeld, Germany/ Universidade Federal do Rio de Janeiro, Brazil). Altogether there were 35 participants from 13 countries, 31 contributions were presented. After processing by peer reviews, 26 papers are published in these Proceedings. They may be categorized according to the following thematic dimensions: Ideas, people and movements Kristín Bjarnadóttir, Elisabete Búrigo, João Bosco Carvalho Pitombeira, Dirk De Bock and Geert Vanpaemel, Livia Giacardi and Alice Tealdi, Jeremy Kilpatrick Transmission of ideas Nerida F. Ellerton and McKenzie A. (Ken) Clements, Thomas Preveraud Teacher education Henrike Allmendinger, Mária Almeida, Marta Menghini, Gert Schubring. Geometry and textbooks Andreas Christiansen, Regina Manso De Almeida, Frédéric Métin, Isabel María Sánchez and Maria Teresa González Astudillo Textbooks – changes and origins Sara Confalonieri, Alexander Karp, Desirée Kröger. Curriculum and reforms Evelyne Barbin, Kajsa Bråting, Jenneke Krüger, Johan Prytz, Hervé Renaud, Leo Rogers, Ana Santiago and María Teresa González Astudillo. 7 Introduction The abstracts of five more papers presented at the Conference are included by the end of this volume. To emphasize the continuity of the project behind the conference on research in the History of Mathematics Education held in Uppsala, the volume containing the proceedings keeps the original title of the first conference, i.e. “Dig where you stand” (followed by 3, which is the number of the conference). This sentence is the English title of the book Gräv där du står (1978) by the Swedish author Sven Lindqvist. Hansen (2009) uses it to explain what he did when took up the position as teacher in mathematics. His “Dig where you stand” approach is based on the idea that “there was important and interesting history in every workplace, and that the professional historians had neglected this local part of history writing, so you had to do it by yourself.” (p. 66) We deem that “Dig where you stand” may be a suitable motto for those (historians, educators, teachers, educationalists) who wish to sensitively and deeply understanding the teaching and learning of mathematics. References Hansen, Hans Christian (2009). From descriptive history to interpretation and explanation – a wave model for the development of mathematics education in Denmark. In K. Bjarnadóttir, F. Furinghetti, & G. Schubring (Eds.), “Dig where you stand”. Proceedings of the conference on “On-going research in the History of Mathematics Education (pp. 65–78). Reykjavik: University of Iceland – School of Education. The editors: Kristín Bjarnadóttir, Fulvia Furinghetti, Johan Prytz, Gert Schubring 8 Klein’s Elementary Mathematics from a Higher Standpoint – An analysis from a historical and didactic point of view Henrike Allmendinger University of Siegen, Department of Mathematics, Germany Abstract In the early 20th century, a demand arose for a course of university studies considering the special needs of future teachers. One of the well-known representatives of this movement is Felix Klein. Inter alia, he held lectures on Elementary Mathematics from a Higher Standpoint. In the work at hand, the lecture notes are analyzed concerning the underlying intention and inner structure. The results show that Klein adheres closely to several principles, such as the principle of mathematical interconnectedness, the principle of intuition, the principle of application-orientation and the genetic method of teaching. Those principles contribute greatly to the development of Klein's higher standpoint. In addition, Klein conveys a multitude of perspectives that widen this higher standpoint. As a result, in the lectures two different orientations can be declared: Klein regards elementary mathematics from a higher standpoint and higher mathematics from an elementary standpoint. Introduction Felix Klein pinpointed the main problem of teachers' education: The young university student [is] confronted with problems that did not suggest [...] the things with which he had been concerned at school. When, after finishing his course of study, he became a teacher [...] he was scarcely able to discern