During the deliberations of the International Congress of Mathematicians at Rome in 1908 steps were taken to organize an International Commission on the Teaching of Mathematics, the members of which were to prepare or procure reports on the teaching of mathematics in different countries. Most of these reports were ready for the Cambridge congress in 1912, but since then several more have appeared. At this writing 18 countries have published 178 reports, containing over 12,000 pages. Germany has already issued 50 reports, with a total of 5,393 pages. About a fifth of this space is required by the United States for its 14 reports (the present report being the fifteenth), and about a sixth of the same space by each of the following countries: Austria, with 13 reports; Great Britain, with 34 reports; Switzerland, with 9 reports; and Japan, with 2 volumes. The reports of France cover some 700 pages. Of more modest dimensions are, in order of size, the reports from Belgium, Russia (including Finland), Italy, Sweden, Spain, Netherlands, Hungary, Denmark, Australia, and Roumania (1 report of 16 pages).

From this statement it will be observed that much greater detail is given in the case of some countries than in others. Moreover, even in reports of about the same length different subjects are emphasized. As this bulletin is based very largely upon facts drawn from the reports to the International Commission, the treatment of its sections varies with the extent of data at hand, and lack of uniformity is a necessity. No claim is made for originality of presentation.

For the most part only those schools which are under the immediate direction of the Government have been considered. And even here discussion is limited to the best schools for boys and to the teachers in such schools. As a rule the schools for girls are not as completely organized nor of so high a standard.

It has seemed to me desirable to include in this bulletin, when possible, very brief independent sketches of the educational conditions in the various countries, so that the reader may receive here in connected form condensed but definite accounts of the following phases of educational work in the country under discussion, in so far as they bear on the preparation of teachers of secondary mathematics: (1) The general educational scheme; (2) secondary

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1 For greater detail along this line the reader may be referred to Monroe's Cyclopedia of Education (5 volumes, New York, 1911-1913). As to mathematical instruction the Bibliography of the Teaching of Mathematics, 1900–1912, by D. E. Smith and K. Goldziher (Bu. of Educ., Bul., 1912, No. 29), Washington, Government Printing Office, 1912, contains titles supplementary to the bibliographies in the following pages.

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schools and their relation to that scheme; (3) the mathematics taught in the secondary schools and the pupils to whom it is taught; (4) the inducements (such as salary, pensions, social position) to young men to take up secondary-school teaching as a profession; (5) the universities of the country, the courses of mathematics and allied subjects they offer, and the diplomas or certificates they confer. With this in mind, one may get an intelligent idea both of the preparation of the secondary-school teacher for his duties and of the type and caliber of men who take up such work. An endeavor has been made to picture conditions of the present day. Only occasional brief historical comments have been introduced.

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At a meeting of the International Commission on the Teaching of Mathematics held at Paris in April, 1914, the commission decided to study the theoretical and practical preparation of professors of mathematics in the secondary schools of different countries. It was considered that such a work "would constitute in a certain sense the crown of the labors of the commission." Early in 1915 a questionnaire was published in order to acquaint those who might consent to prepare the special reports for different countries with the questions which the commission wished to have them answer. As far as I have been able to learn, only two reports based upon this questionnaire have been published at this writing; these are the brief report (14 pages) by W. Lietzmann, concerning Germany, and the longer report on Belgium by J. Rose. It is hoped that the general report submitted herewith may be considered a worthy contribution to the commission's special inquiry.

At the present time superintendents, inspectors, and principals in many parts of the United States have been forced by public opinion to consider numerous radical changes in methods of secondaryschool education. If a high minimum standard of preparation were required on the part of each teacher, and the position of the teacher were made such as to attract in sufficient numbers the best talent in the country, other difficulties would disappear. Most countries considered in this bulletin have far higher standards than we with respect to teachers of mathematics in secondary schools. The degree of this superiority is exhibited throughout the following pages, and some of the chief points are summarized in the last chapter.

In what follows, statements concerning countries now at war refer, for the most part, to ante bellum days.

December, 1917.

R. C. A.

For fuller account of the work in mathematics see Curricula in Mathematics, a comparison of courses in the countries represented in the International Commission on th Teaching of Mathematics, by J. C. Brown (Bu. of Educ., Bul., 1914, No. 45), Washington, 1915.

L'Enseignement Mathématique, tome 17, January, 1915, pp. 61-65. See also pp. 129-145.

TRAINING OF TEACHERS OF MATHEMATICS FOR

THE SECONDARY SCHOOLS.

I. AUSTRALIA.

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Australia is politically divided into five States (New South Wales, Victoria, Queensland, South Australia, Western Australia), which with the island of Tasmania form what has been known since 1901 as the Commonwealth of Australia. At each of the six capitals (Sydney, Melbourne, Brisbane, Adelaide, Perth, and Hobart) there exists a university supported in part by public funds and in part by private endowments and fees paid by students. And while the educational conditions vary greatly in the different States and not a little in the same State, the universities, whatever their status, are the crown of the educational system of which they form a part. The conditions in New South Wales and Victoria, each with a population of more than one and one-half millions and with universities founded well over half a century ago, are greatly superior to those in the enormous State of Western Australia, with its scattered population of less than 300,000 people (in 1911) and a university which has been in operation but a short time.

The peculiarity of Australian education is that the State not only controls, but completely and absolutely supports and regulates the system of public education, without support from or interference by the localities in which the schools may lie. Australian education tends therefore to be centralized, systematic, and homogeneous; but since local interest is naturally fitful, the external equipment of the schools is usually of an inferior character, while the qualifications of the teachers are distinctly superior. Primary education throughout Australia is free, but secondary is not. The State secondary schools are fewer and somewhat less important than those of a semipublic endowed or denominational character.

The organization of secondary education in Australia is passing through a period of active development. But until very recently the chief influence upon the work of the secondary schools has been exerted by the universities, not only through their requirements at matriculation, but also through a system of public examinations taken by pupils of the schools whether they proposed to enter the univer

"The universities, though not State universities in the usual sense of the term, are in most cases largely supported by the State. In some instances the proportion of their revenues derived from the treasury is so large that, except for the freedom of their administration, it would be difficult to distinguish them from State institutions."!

sities or not. These examinations are similar to the Cambridge and Oxford local examinations. The machinery in connection with them has differed from State to State. Since 1912 they are gradually being displaced by a much more satisfactory system of State intermediate and leaving certificates. The requirements for these examinations and certificates give a fair idea of the ground the teacher must cover in his instruction. For the States in order these requirements are:

JUNIOR AND SENIOR EXAMINATIONS.

UNIVERSITY OF SYDNEY.

Till very recently the junior examination was held in June of each year. Candidates averaged 15 years of age. The examination was intended to cover the first two or three years' work of the secondary school. Every student matriculated at the university had to pass the mathematical papers of this or an equivalent examination. Almost all of those taking mathematical classes in their university course also passed the senior examination in mathematics before entrance. The senior examination was usually taken from one and a half to two and a half years after the junior; additional papers or separate questions were set for honor candidates.

In New South Wales the above junior and senior examinations are now replaced by the system of State intermediate and leaving certifiThe former is awarded after the successful completion of two years' work in the high school. The courses of study in the mathematical subjects are, in outline, as follows:

Arithmetic: This forms a part of the first and second years' work for all pupils. It includes mensuration, the plane figures named in the syllabus being the rectangle, triangle, parallelogram, quadrilateral, and circle. The solids are the rectangular box, prism, pyramid, cylinder, cone, and sphere. The simple numerical trigonometry of the right-angled triangle is also introduced. This is not taken in the arithmetic course till after a simple geometric treatment has made possible a satisfactory discussion of the points involved.

Algebra: The work in algebra of these two years goes up to simple case of simultaneous quadratics. The variation and change of sign of the expressions ax+b and ax2+bx+c are studied graphically and algebraically.

Geometry: This course covers the subject matter of Euclid's Elements, Books I-III, with some freedom from his methods and sequence. Prop. 4, Book I, is accepted as an axiom. Preliminary practical work in geometry will in most cases have been done before entrance upon the course of the secondary school.

The third and fourth years' work in the high schools is divided into pass and honor sections. Practically all pupils have to do some mathematical work in these two years, but only those who have shown special aptitude for this study attempt the full course. Indeed, some take only part of the pass course, but all who desire leaving certificates to count as equivalent to the matriculation examination