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- 1911
- Circulaire n. 4 (EM, 13, 1911, 122-137) contains a report on the Commission's achievements so far and sets a date for the next meeting of the ICMI in Milan the following September. This document corrects a mistake made in the Rapport préliminaire: Japan, which was described as an associate member, is here declared to have the right to full membership.
- 18-21 September 1911, Milan - The first plenary meeting of the International Commission on the Teaching of Mathematics
- (Cf. Circulaire n. 5, Compte rendu du Congrès de Milan, 18-21 Septembre 1911, publié par H. Fehr, EM 13, 1911, 437-511). The various member countries give presentations on work being carried out by sub-committees on national level. All appear to have been very active. Two special sub-committees had been appointed previously to prepare for the meeting. These were made up of the following individuals:

A. (Middle-School Teaching): E. Beke, Ch. Bioche, F. Klein, W. Lietzmann, G. Scorza, J.W. Young.

B. (High School Teaching): C. Bourlet, H. Fehr, F. Klein, C. Somigliana, H. Timerding, W. Wirtinger.

The following issues are addressed:- rigour in middle school teaching and the fusion of the various branches of mathematics (sub-committee A)
- the teaching of mathematics to students of physical and natural sciences (sub-committee B).

G. Castelnuovo presents the report on the subject of rigour (EM 13, 1911, 461-68). In order to compare the methods employed in the various countries with which he is concerned, Castelnuovo focuses on just one type of school, high schools with humanistic orientation, and just one area of mathematics, geometry. Taking a number of school textbooks as examples, Castelnuovo divides teaching methods into the following kinds: the purely logical method (Halsted, ...), methods based on empirical principles and logical development (Sannia-D'Ovidio, Veronese, Enriques-Amaldi, ...), methods consisting in alternating and mixing intuitive and deductive considerations (Borel, ...) and finally, the intuitive-experimental method (Perry, ...). During the discussion a number of interesting comments are made, including those by Klein, Veronese, Lietzmann and Enriques.

Ch. Bioche presents the report on the fusions between algebra and geometry, planimetry and trigonometry, and stereometry and descriptive geometry (EM 13, 1911, 468-71). He quotes, amongst other texts, fusionist geometry manuals by C. Meray, G. Lazzeri and A. Bretschneider.

J. W. Young also presents a paper dealing with rigour and fusion. H. Timerding gives a paper discussing mathematics teaching for students of physical and natural sciences (EM 13, 1911, 481-91) This paper gives rise to an interesting discussion, in the course of which C. Bourlet, W. Wirtinger, and H. Fehr outline the situations in France, Austria, and Switzerland, respectively.

The plenary lectures are:

G. Colombo, Sull'insegnamento matematico nelle scuole per gli ingegneri (EM 13, 1911, 499-504)

F. Enriques, Mathématiques et théorie de la connaissance (EM 13, 1911, 505-509) - 21-27 August 1912 - Cambridge Congress on the occasion of the V International Congress of Mathematicians
- (Cf. Publications du Comité central rédigées par H. Fehr, Compte rendu du Congrès de Cambridge, 21-27 août 1912; EM 14, 6, 1912, pp. 441-537)

The Central Committee is composed thus:

President F. Klein,

Vice-Presidents G. Greenhill and D. E. Smith (added on the occasion of the congress),

Secretary General H. Fehr

- After the opening speeches made by C. Godfrey and D. E. Smith,H. Fehr presents a report on the activities of the Commission from 1908 to 1912 with a list of the publications from the various countries (EM 14, 6, 1912, pp. 451-473)

- The representatives of the various countries present work undertaken: in particular, the contributions of A. Gutzmer for Germany, E. Czuber for Austria, J.W.A. Young for the United States, G. Castelnuovo for Italy, and R. Fujisawa for Japan stand out for their wealth of detail.

Results of the inquiries into the following subjects are presented:- The mathematical training of the physicist in the University
- Intuition and experiment in mathematical teaching in the secondary schools.

C. Runge (Göttingen) speaks on the subject The mathematical training of the physicist in the University (EM 14, 6, 1912, pp. 495-507). He stresses the urgent need for a reform of such teaching and states thatlarge portions of the theoretical matter intended for the pure mathematician might be left out and the attention concentrated on those subjects which are of continual application in mathematical physics... Mathematics is suffering from overspecialisation and is cutting itself off from Natural Philosophy and Experimental Science... What is really wanted is that Mathematical teachers should understand the problems and the needs of the Physicist. (p. 500)

Runge's speech is followed by numerous observations from the representatives of the various countries. In particular, F. Enriques (Bologna) notes that Runge has adopted a "utilitarian" approach, wondering, above all, with which mathematical instruments it is necessary to provide physicists, while he himself considers another problem to be of greater importance, namely, how to attract the young to the study of physics. In particular, he is of the opinion that current instruction in mathematical physics is characterised by an excessive development of the algorithmic aspects to the detriment of intuition (p. 503). E. W. Hobson (Cambridge) asserts thatperhaps too large a share of the blame for the undoubted evils of the present divorce of Mathematical and Physical teaching had been put upon the Mathematicians. If the mathematical teacher spent too much time on matters which would be better omitted and in the teaching of merely analytical expertness, it might be said that the Physicist spent too much time on the merely manipulative work of the Laboratory... The main object of the Mathematical teaching ought to be that of developing habits of clear and logical thinking in the pupils... He [the mathematician] would lose his self-respect if he became a mere provider of tools for use in subjects outside his own. (p. 505).

D. E. Smith (New York) presents his paperon the theme of Intuition and experiment in mathematical teaching in the secondary schools (EM 14, 6, 1912, pp. 507-534). After giving some descriptions of the types of schools considered (pupils between the ages of 10 and 19 from secondary schools, excluding vocational ones) he presents a general outline of the situation in the various countries, from which it emerges that in the teaching of mathematics intuition is more highly prized in Austria, Germany and Switzerland, than in Great Britain, France, and the United States, and that the most debated subjects are above all the teaching of geometry, and whether or not to introduce the concept of function. Continuing, he focuses his attention on particular aspects of the teaching of mathematics: "Measuring and Estimating" (geodetic, astronomical measurements, triangulations, ...), "Geometric Drawing and Graphic Representation" (teaching of descriptive geometry), "Graphic Methods" (representation of functions on graph paper, graphic statics, evaluation of surfaces with the aid of graph paper, ...), " Numerical Computation" (use of tables, of slide rule, methods of approximate calculation, ...). Smith's contribution is followed by a broad-ranging debate in which the critique of C.-A Laisant (Paris) emerges:La question me semble mal posée, ou pour

mieux dire posée d'une façon incomplète, ce qui, par cela même, la rende insoluble. Le rôle de l'intuition et de l'expérience dans l'éducation en général, dans l'enseignement mathématique notamment, est l'un des problèmes capitaux de la pédagogie. Mais restreindre l'examen de ce problème à la considération des écoles moyennes (ou secondaires) c'est le rendre insoluble, et méconnaître les conditions psychologiques et physiologiques du développement cérébral. (p. 529)

The final vote of the General Assembly of the Congress testifies to the high appreciation and the recognition for the work of this first international network (cf. Proceedings of the Fifth International Congress of Mathematicians (Cambridge, 22-28 August 1912), Cambridge, at The University Press, 1913, p. 41).

- The United States Bureau of Education offers to publish and distribute - free of charge - the Bibliography of the teaching of mathematics (1900-1912) edited by D. E. Smith and C. Goldziher.