One Hundred Years of L'Enseignement Mathematique Moments of Mathematics Education in the Twentieth Century Proceedings of the EM±ICMI Symposium Geneva, 20±22 October 2000 edited by Daniel CORAY, Fulvia FURINGHETTI, Hel eneÁ GISPERT, Bernard R. HODGSON and Gert SCHUBRING L'ENSEIGNEMENT MATHEMA TIQUE GenevÁ e Ð 2003 c 2003 L'Enseignement Mathematique , GenevÁ e Tous droits de traduction et de reproduction reserv es pour tous pays ISSN : 0425-0818 Ð ISBN : 2-940264-06-6 TABLE OF CONTENTS List of participants . 5 The Authors . 6 INTRODUCTION : DANIEL CORAY ± BERNARD R. HODGSON . 9 L'ENSEIGNEMENT MATHEMA TIQUE : BIRTH AND STAKES FULVIA FURINGHETTI Mathematical instruction in an international perspective : the con- tribution of the journal L'Enseignement Mathematique . 19 GERT SCHUBRING L'Enseignement Mathematique and the ®rst International Commis- sion (IMUK) : the emergence of international communication and cooperation . 47 GILA HANNA Journals of mathematics education, 1900±2000 . 67 REACTION : JEAN-PIERRE BOURGUIGNON . 85 GENERAL DISCUSSION (reported by CHRIS WEEKS) . 89 GEOMETRY RUDOLPH BKOUCHE La geom etrie dans les premieresÁ annees de la revue L'Enseignement Mathematique . 95 GEOFFREY HOWSON Geometry : 1950±70 . 113 COLETTE LABORDE Geom etrie ± Periode 2000 et apresÁ . 133 REACTION : NICOLAS ROUCHE . 155 GENERAL DISCUSSION (reported by MARTA MENGHINI) . 160 4 EM±ICMI SYMPOSIUM ANALYSIS JEAN-PIERRE KAHANE L'enseignement du calcul differentiel et integral au deb ut du vingtiemeÁ siecleÁ . 167 MAN-KEUNG SIU Learning and teaching of analysis in the mid twentieth century : a semi-personal observation . 179 LYNN STEEN Analysis 2000 : challenges and opportunities . 191 REACTION : MICHELEÁ ARTIGUE . 211 GENERAL DISCUSSION (reported by MARTA MENGHINI) . 224 APPLICATIONS OF MATHEMATICS : MATHEMATICS AS A SERVICE SUBJECT PHILIPPE NABONNAND Les debats autour des applications des mathematiques dans les reformes de l'enseignement secondaire au deb ut du vingtiemeÁ siecleÁ 229 HEL ENEÁ GISPERT Applications : les mathematiques comme discipline de service dans les annees 1950±1960 . 251 MOGENS NISS Applications of mathematics `2000' . 271 REACTION : GERHARD WANNER . 285 GENERAL DISCUSSION (reported by CHRIS WEEKS) . 293 PERSPECTIVES FOR MATHEMATICS EDUCATION UBIRATAN D'AMBROSIO Stakes in mathematics education for the societies of today and tomorrow . 301 JEREMY KILPATRICK Scienti®c solidarity today and tomorrow . 317 REACTION : HYMAN BASS . 331 LIST OF PARTICIPANTS Nestor AGUILERA (Argentina) Djordje KADIJEVIC (Yugoslavia) MicheleÁ ARTIGUE (France) Jean-Pierre KAHANE (France) Nataliya BANTSUR (Ukraine) Jeremy KILPATRICK (USA) Hyman BASS (USA) Colette LABORDE (France) Rudolph BKOUCHE (France) Thomas LINGEFJARDÈ (Sweden) Jean-Pierre BOURGUIGNON (France) Marta MENGHINI (Italy) Aldo BRIGAGLIA (Italy) Alain MERCIER (France) Daniel CORAY (Switzerland) Philippe NABONNAND (France) Ubiratan D'AMBROSIO (Brazil) Mogens NISS (Denmark) Viviane DURAND-GUERRIER (France) JoaoÄ PITOMBEIRA DE CARVALHO (Brazil) Trevor FLETCHER (UK) LuÂõs REIS (Portugal) Ruhal FLORIS (Switzerland) Andre REVUZ (France) Fulvia FURINGHETTI (Italy) Nicolas ROUCHE (Belgium) Claude GAULIN (Canada) Lee RUDOLPH (USA) Livia GIACARDI (Italy) Jean RUFFET (Switzerland) Hel eneÁ GISPERT (France) Gert SCHUBRING (Germany) Alexandra GOMES (Portugal) Mansa C. SINGH (Canada) Francine GRANDSARD (Belgium) Man-Keung SIU (China) Pierre-Paul GRIVEL (Switzerland) Per SIVERTSEN (Norway) Andre HAEFLIGER (Switzerland) Lynn STEEN (USA) Ernst HAIRER (Switzerland) John STEINIG (Switzerland) Gila HANNA (Canada) Alain VALETTE (Switzerland) Jean-Claude HAUSMANN (Switzerland) Gerhard WANNER (Switzerland) Bernard R. HODGSON (Canada) Claude WEBER (Switzerland) Geoffrey HOWSON (UK) Chris WEEKS (UK) Jean-Pierre IMHOF (Switzerland) Jochen ZIEGENBALG (Germany) THE AUTHORS MicheleÁ ARTIGUE : IREM, Case 7018, Universite Paris 7 Denis Diderot, 2, place Jussieu, F-75251 Paris Cedex 05, France [[email protected]] Hyman BASS : 4204C School of Education, 610 E. University, University of Michigan, Ann Arbor, MI 48109-1259, U. S. A. [[email protected]] Rudolf BKOUCHE : 64, rue Negrier , F-59800 Lille, France [[email protected]] Jean-Pierre BOURGUIGNON : Institut des Hautes EÂtudes Scienti®ques, 35, route de Chartres, F-91440 Bures-sur- Yvette, France [[email protected]] Daniel CORAY : Section de mathematiques, Universite de GeneveÁ , 2±4, rue du LievrÁ e, Case postale 240, CH-1211 GeneveÁ 24, Switzerland [[email protected]] Ubiratan D'AMBROSIO : R. Peixoto Gomide, 1772 ± Apt 83, 01409-002 ± SaoÄ Paulo ± SP, Brazil [[email protected]] Fulvia FURINGHETTI : Dipartimento di Matematica, UniversitaÁ di Genova, Via Dodecaneso 35, I-16146 Genova, Italy [[email protected]] Hel eneÁ GISPERT : Groupe d'Histoire et de Diffusion des Sciences d'Orsay (GHDSO), Batimentà 407, Centre universitaire, F-91405 Orsay Cedex, France [[email protected]] Gila HANNA : Ontario Institute for Studies in Education, University of Toronto, 252 Bloor St. West, Toronto, Ontario M5S 1V6, Canada [[email protected]] Bernard R. HODGSON : Departement de mathematiques et de statistique, Universite Laval, Quebec G1K 7P4, Canada [[email protected]] Albert Geoffrey HOWSON : 1 Chidden Holt, Chandlers Ford, GB-Eastleigh SO53 4RJ, United Kingdom [[email protected]] Jean-Pierre KAHANE : 11, rue du Val-de-Graceà , F-75005 Paris, France [[email protected]] EM±ICMI Symposium 7 Jeremy KILPATRICK : 105 Aderhold Hall, University of Georgia, Athens GA 30602-7124, U. S. A. [[email protected]] Colette LABORDE : Laboratoire IMAG-Leibniz, Universite Joseph Fourier ± CNRS, 46, avenue Felix Viallet, F-38000 Grenoble, France [[email protected]] Marta MENGHINI : Dipartimento di Matematica, UniversitaÁ di Roma ªLa Sapienzaº, Piazzale Aldo Moro, I-00185 Roma, Italy [[email protected]] Philippe NABONNAND : Archives Henri Poincare, Universite de Nancy 2, 23, Boulevard Albert 1er, BP 3397, F-54015 Nancy Cedex, France [[email protected]] Mogens NISS : IMFUFA, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark [[email protected]] Nicolas ROUCHE : CREM, 5, rue Emile Vandervelde, B-1400 Nivelles, Belgium [[email protected]] Gert SCHUBRING : Institut furÈ Didaktik der Mathematik, UniversitatÈ Bielefeld, Postfach 100 131, D-33501 Bielefeld, Germany [[email protected]] Man-Keung SIU : Department of Mathematics, University of Hong Kong, Hong Kong SAR, China [[email protected]] Lynn Arthur STEEN : Department of Mathematics, St. Olaf College, North®eld MN 55057, U. S. A. [[email protected]] Gerhard WANNER : Section de mathematiques, Universite de GeneveÁ , 2±4, rue du LievrÁ e, Case postale 240, CH-1211 GeneveÁ 24, Switzerland [[email protected]] Chris WEEKS : Downeycroft, Virginstow Beaworthy, Devon EX21 5EA, United Kingdom [[email protected]] Charles-Ange LAISANT (1841±1920) INTRODUCTION Organized jointly by the University of Geneva and the International Commission on Mathematical Instruction, a symposium on mathematics education was held in Geneva in October 2000 under the theme ªONE HUNDRED YEARS OF L'ENSEIGNEMENT MATHEMA TIQUE : MOMENTS OF MATHEMATICS EDUCATION IN THE 20TH CENTURYº It was an occasion for celebrating the 100th anniversary of the international journal L'Enseignement Mathematique , founded in 1899 by Henri Fehr (Geneva) and Charles-Ange Laisant (Paris). Among periodicals devoted to mathematics education, the new journal was the ®rst to seek explicitly an international audience; in fact, an original characteristic of the beginnings was a series of articles on the teaching of mathematics in different countries. The fact that such a journal was launched in Geneva is not surprising if we consider the local context in which it emerged. Apart from obvious references to internationalism as seen from Henry Dunant's birth-place, it is also true that, for centuries, Geneva had been haunted by the myth of its pedagogic vocation. To describe this strong historical background, it is useful to recall here that the city of Geneva was not a part of the Swiss Confederation until 1815. It had been an episcopal principality for several hundred years Ð with a relatively large degree of autonomy granted to the people Ð until it became an independent protestant republic, by a decree of 21st May 1536, in which the Conseil gen er al promulgated the Reformation 1 ). It is quite remarkable that the same decree set the principle of compulsory education, which was also declared free for the poor : [¼] que chescung soit tenu envoyer ses enfans aÁ l'escholle et les faire apprendre. 2 ) [Encyclopedie de GenevÁ e 1986, 127] 1 ) To avoid a common confusion, it is important to note that this decision was reached under the determining in¯uence of Guillaume Farel, a few months before Jean Calvin arrived in Geneva. 2 ) [¼] that everyone be required to send their children to school and have them learn. 10 INTRODUCTION In 1559 Jean Calvin inaugurated two new institutions : the CollegÁ e (with seven grades) and the Academie . Led by Mathurin Cordier, Theodore de Beze,Á and others, both schools attracted numerous students from France, the German States, the Low Countries and the whole of Europe, right from the beginning 3 ) but even more after the Saint-Barthelemy massacre (1572). Mathematics and physics were also taught. However, a formal chair of mathematics was introduced in the Academie only in 1724, with Jean-Louis Calandrini and Gabriel