Bulletin

The International Commission on the Teaching of Mathematics, created by the International Congress of Mathematicians at Rome, Italy, in 1908, submitted a large body of reports to the congress at Cambridge, England, in 1912. Those for the United States have been published as bulletins of the Bureau of Education (Bulletin, 1911, Numbers 6, 7, 8, 9, 12, 13, 16; 1912, Numbers 2, 4, 13, 14). The congress directed the commission to continue its activities and to submit further reports to the congress to be held at Stockholm, Sweden, in 1916. The following bulletin has been prepared as a part of the report of this country for the second period of the commission's labors. The work of this period will naturally consist, to a considerable extent, of a preparation by each nation of a digest of the results of the earlier work of the commission, as seen from the point of view of that nation.

No one believes in the bodily transplantation of a course of study or of methods of work from one nation to another; yet the very existence of the commission, and the phenomenal success of its work, are evidences of the widespread conviction that every nation can profit from a careful comparative study of the work of other nations. The material that follows shows that the types of school organization and the curricula in mathematics of the leading nations of Europe have much more in common with each other than with those of the United States. European nations will therefore sometimes find in the practice of their sister nations encouraging confirmation of their own customs, whereas the United States will be confronted by the question as to whether she alone has found the path that is best, at least for herself. Some lines of divergence between the general practice in Europe and that of the United States are mentioned in the concluding remarks of this report (p. 87). Historical and other reasons for the existence of the divergencies are easily found. Reasons, sometimes of a theoretical, sometimes of a practical, nature, may doubtless also be found sufficient to warrant the present continuance of some or all of these divergencies. At the same time the possession of authoritative statements of the practice of the world's leading nations, as contained in the reports from which the material that follows is taken, can but prove helpful and stimulating to the educators of the United States. The efficient supervisor or the alert teacher will surely profit by a serious study of the curricula in other countries as set forth in this bulletin. He may find little or nothing

that is directly usable in his own environment, but he can not help being aided in his effort to improve his own work, according to local needs and conditions, by a wider knowledge of what is done in corresponding cases by the world at large.

All of the statements of fact in the bulletin are based upon reports of the international commission to the congress at Cambridge. A complete bibliography of the reports used is given on page 88. The data for the various nations have perforce been given with varying degrees of completeness, depending upon the information available in the reports. As the age at which pupils enter school varies somewhat in different countries, the age of the pupil, rather than the school year, was chosen as the basis of comparison.1

Unless otherwise stated, it is to be understood that the European schools are for boys only. Relatively little is as yet done in most of the European countries for the mathematical education of girls beyond the fundamentals of arithmetic. The scanty information that is available is given in appropriate connections.

1 The public schools of the United States comprise, in most cases, an eight-year elementary course, followed by a four-year course in the secondary school. In some cases the course in the elementary school is seven years and in the secondary school five years in length. In rare instances both the elementary and the secondary school courses are six years in length.

The age of compulsory school attendance varies somewhat in the different States, but it is usually from the age of 6 or 7 to 14. Most pupils enter the elementary school at the age of 6 or 7. Both the elementary and the secondary schools are free to all pupils of the school district.

The courses in the elementary schools of the country are more nearly uniform than those of the secondary schools. In the elementary schools the attempt is made to lay the foundation for a good general education. Reading, writing, arithmetic, history, geography, elementary science, and spelling are the important subjects of the curriculum. There is but little opportunity for choice of courses in these schools. In the secondary school a pupil is sometimes permitted to select those subjects that he especially wishes. Certain subjects are often required, and the pupil is permitted to select a prescribed number of additional subjects. In many of the larger cities there are secondary schools of various types, such as commercial, manual training, and industrial.

A pupil who has completed the course in a good elementary school may enter a secondary school without examination. Many of the colleges and universities admit graduates of secondary schools upon the presentation of a diploma or a certificate, but some colleges and universities admit only by examination. In this report the New York State course of study is rather closely followed in outlining the subject matter of the elementary school. This course is fairly typical of the best courses of the country.

CURRICULA IN MATHEMATICS.

I. GENERAL ARRANGEMENT OF THE COURSES IN TYPICAL SCHOOLS OF THE VARIOUS COUNTRIES.

The minister of education has general supervision of education in all of the 14 Provinces into which Austria is divided. In each Province there is a superintendent of education, who stands midway between the minister and lower-school authorities. Each Province is divided into districts and each district has its school council. The members of this school council are nominated by the teachers of this district and appointed by the higher-school council. Each district is divided into smaller subdivisions and each of these has its local board. Compulsory education extends from the sixth to the fourteenth year. Coeducation is the rule in the rural districts and the exception in the cities.

The administration of secondary education is vested in the minister of instruction. Each Province has an inspector and a local board. All schools, whether public or private, are subject to the same regulations as the State schools. Most of the secondary schools are maintained by the State.

There are two kinds of elementary schools: (a) The common primary, usually called the Volksschule, and (b) the superior elementary school, called the Bürgerschule. Many of the Bürgerschulen in the rural communities are ungraded. In the cities the course in these schools comprises from two to eight grades. The Bürgerschulen usually consist of three classes, which follow the fifth year of the primary school. Pupils may enter directly from the primary school. Different courses are offered for boys and girls, and as far as possible the sexes are instructed in different schools. Sometimes an intermediate course of one year follows the Bürgerschule. The nature of the course which a pupil follows during this year depends largely upon the

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