Geometry. There is usually a course of one year, with one or two double periods a week. Six books of Euclid are studied, particular attention being given to problems of construction. Practice is given in the computation of areas and volumes. In some industrial schools trigonometry and calculus are taught.

Methods of instruction.-There are short lectures by the teacher, followed by the solution of problems by the students. The same books are used as in the secondary schools.

MIDDLE TECHNICAL SCHOOLS.

Organization.-Students who have finished the Realschule enter without examination. These schools are said to belong to group C in the plan of the International Commission. The average age of entrance is about 18 years. The length of the course is three years.

COURSE OF STUDY IN MATHEMATICS.

There is much freedom given the teacher as to the nature of the instruction. The official plan requires only that certain subjects be taught. These are arithmetic, plane and solid geometry, algebra including computations with series and logarithms, plane trigonometry, and the elements of analytical, descriptive, and practical geometry. Calculus is taught in some schools.

Aims and methods of instruction.-The official instructions recommend that the teacher keep constantly in mind the needs of the industries. Mathematics is to be regarded as a tool. Hence the parts that have only a theoretical value will be omitted, such as certain formal parts of algebra. Geometry is of great importance in design. Therefore it should be developed so as to contribute as much as possible to practical applications. The principal aim of the calculus is to prepare the student to read technical literature.

Examinations.-Examinations are held at the option of the teacher to determine the progress and promotion of the students. There is no final examination. A diploma is granted at the completion of the

course.

SWITZERLAND.

COMMERCIAL SCHOOLS.2

COMPLEMENTARY COMMERCIAL INSTRUCTION.

Organization. The complementary commercial instruction is both public and private. Courses are organized by commercial socities, by communes, and by one Canton (Fribourg). The students are

1 Der mathematische Unterricht in Schweden: Die Mathematik an den technischen Lehranstalten in Schweden, II: Der mathematische Unterricht an den technischen Mittelschulen, by O. Gallander, pp. 14-21. L'Enseignement mathématique en Suisse, No. 6: Les mathématiques dans l'enseignement commercial suisse, by L. Morf. Referred to hereafter as No. 6.

apprentices and adult employees. The programs given in the report are for a course of three years.

Aim of the instruction.-The aim of the mathematical instruction is to give the students accuracy and speed in the calculations involved in commercial affairs.

COURSE OF STUDY IN MATHEMATICS.2

In the courses of study given in the report, from one to two hours a week for three years are given to elementary and commercial arithmetic, and the same to bookkeeping. Arithmetic. The following are the principal topics taken up in the different years: First year. Computing by the hundred and by the thousand, English money and measures, and a review of fractions.

Second year. Interest, discount, exchange, ratio and proportion, alligation, and partnership.

Third year. Accounts current, accounts involving public funds, oral calculation, and a review of the work of the three years.

Bookkeeping. Some of the topics given in this course are simple and double entry by American and Italian methods, accounts of a society, loss and gain accounts, balances, partnership accounts, accounts of an industry and of a hotel.

Methods of instruction.3-The method used is that of exposition by the teacher, followed by the solution of numerous problems in and outside of the recitation.

Books.-A list is given of the books most widely used. For arithmetic these books are mainly collections of problems.

Examinations." At the end of the last year of the course an examination is given to which any person who has served two years' apprenticeship is eligible, whether or not he has attended the cours complementaires. This is an examination for a diploma and is required of apprentices in 10 Cantons. The tendency is to make it compulsory everywhere. This examination requires, in arithmetic, a written examination lasting from 2 to 24 hours, and a test in oral calculation lasting from 6 to 7 minutes; in bookkeeping, a written examination lasting from 3 to 34 hours, and an oral one lasting from 6 to 7 minutes.

The examinations are set by the Cantons or by the commercial societies, according to the school. They are under the supervision of the Federal department of commerce and industry, and in certain Cantons under that of the cantonal department of commerce and industry.

The examples of examination questions given in the report 5 are in the main practical ones, such as would occur in business practice.

1 L'Enseignement mathématique en Suisse, No. 6: Les mathématiques dans l'enseignement commercial suisse, by L. Morf. Referred to hereafter as No. 6, p. 3.

Ibid., p. 5.

Ibid., p. 11.

4 Ibid., p. 6.

Ibid., pp. 6-11.

Preparation of teachers.'—The teachers belong to three groups—elementary school teachers, secondary school teachers, and persons engaged in business. Apparently no special preparation is required. These teachers are usually paid by the hour, at the rate of from 3 to 5 francs. In the larger places there are permanent positions which may pay as much as 4,800 or 6,000 francs for work of 28 hours a week.

SECONDARY COMMERCIAL INSTRUCTION.

1. THE LOWER SECONDARY SCHOOLS OF COMMERCE.2

There are few of these schools. They have had support from the Federation since 1909. They have a large attendance of girls. The length of the course is one or two years. From five to six hours a week are given to arithmetic and bookkeeping in each year in a course of study outlined in the report. This is a two-year course and is used in a school in Basel. The subject matter in this course is similar to that already given on page 74.

2. HIGHER SCHOOLS OF COMMERCE.3

Organization. These schools are partly supported by the Federal Government. In some of the Cantons students of the ages of 14 or 15 who are normal in their studies are admitted without examination, and some schools require an examination of all students. Of 31 official schools of commerce, 6 are attended by boys only, 11 by girls only, and 14 are coeducational.

Aims of the mathematical instruction. The aims of the mathematical instruction are said to be to develop ability to reason, to give precision in language, to give ability in abstract and applied calculation, and to prepare the students for advanced commercial work in the university.

COURSE OF STUDY IN MATHEMATICS.5

The department of commerce of the Federal Government has issued a type program for courses of three and of four years. In the former, 10 hours a week are prescribed for arithmetic and 11 for bookkeeping and commercial work; and in the latter, 13 hours for arithmetic and 16 hours for bookkeeping and commercial work. Besides these, courses in algebra, geometry, and trigonometry are usually given.

Arithmetic. (a) Elementary arithmetic. "Review of arithmetic from the point of view, especially, of rapid calculation." This includes a review of the fundamental operations with integers and fractions, the metric system, square root, ratio and proportion, percentage, interest, discount, proportional division, alligation, and partnership, and oral calculation.

I No. 6, p. 12.

Ibid., p. 13.

3 Ibid., p. 14.

4 Ibid., p. 18.

Ibid., pp. 19-21.

(b) Commercial arithmetic. This includes a study of notes, bills, and accounts; the moneys and measures of the principal countries; methods of exchange and values of the precious metals; and buying and selling of stocks and bonds and other securities. Algebra. (a) Elementary. This course begins with the fundamental operations and extends through square and cube root, equations of the second degree in one or more unknowns, arithmetical and geometrical progressions, logarithms, and permutations and combinations.

(b) Financial. This includes a study of compound interest, annuities, sinking funds, and public loans.

(c) Insurance. A study is made of probabilities and of mortality tables and their

uses.

Elementary geometry. "Study of plane surfaces. Surfaces and volumes of the principal bodies."

Plane trigonometry. This course includes the solution of triangles.

Bookkeeping. This is a fairly comprehensive course in bookkeeping, including sets of books for banks and for mercantile and industrial establishments.

The courses in mathematics in certain schools include a few topics not mentioned above.

Methods of instruction.'-"The method employed is the method of exposition; it is reduced, generally, to a strict minimum; it is possible to devote much time to the solution of numerous and varied problems drawn, usually, from the domain of practical affairs." Little use is made of models. There is a list given of the textbooks in general use.

Examinations."-"The examinations for promotion at the end of the year are giving way more and more to quarterly examinations." The grade for the quarter is obtained by combining the examination grade with the grade made in recitation. The average of the three quarterly grades is the grade for the year. A diploma is given students who finish the course and pass an examination. This examination is both oral and written and is based mainly upon the work of the last year of the course. The question is being considered of creating a Federal diploma for these schools.

About 20 pages3 of the report are taken up with questions used in the examinations. These questions are in the main in arithmetic and bookkeeping, some in algebra and geometry. They are almost entirely applied problems, those in arithmetic dealing with somewhat complicated questions of accounts current, exchange, stocks and bonds, insurance, and profit and loss.

Preparation of teachers.-The teachers prepare themselves either in a university, by private study, or, in a few cases, by commercial practice. It is recommended that some practical experience be made a part of the preparation of those who expect to teach in the commercial schools. A part of the teachers of mathematics in the commercial schools are selected from those who have prepared to teach in the secondary schools. It appears, then, that no definite require

1 No. 6, p. 47.

a Ibid., pp. 25, 26.

Ibid., pp. 26-46.

4 Ibid., p. 48.

ments are made of teachers of mathematics except the passing of a special examination.

Present tendencies.-"The actual experiences are too recent for it to be possible to indicate a definite tendency in the teaching of mathematics in the higher schools of commerce." 1 It is of interest, however, to note the growth of these schools. In 1891, when the Swiss Federation first began to give them aid, there were 8 schools of commerce, with about 700 students. In 1910 there were 31 official schools of commerce, with 3,477 students-2,249 male and 1,228 female.

SCHOOLS OF ADMINISTRATION AND FOR THE RAILWAY SERVICE.2

Organization.-"By a school of administration is meant a school that prepares the youth to enter the Federal post office, telegraph, telephone, or customs service." 3

The schools of administration and for the railway service have courses of two or three years. Students enter at the age of 14 or 15 years. "Students who have completed their previous studies normally are admitted without examination; each school reserves the right, however, to examine all students." The entrance examination includes the mother tongue, arithmetic, geography, and national history.

COURSE OF STUDY IN MATHEMATICS.4

The time given to arithmetic varies from four hours a week for one year to three hours a week for three years; to algebra, from one hour a week for six months to two hours a week for two years; to geometry, from nothing to one hour a week for three years; to bookkeeping, from nothing to two hours a week the first year and one hour a week the second year.

The following outline of the course is given:

Arithmetic. (a) Elementary. "Review and extension of the four fundamental operations with integers and fractions. Common fractions. Ratio and proportion. Simple and compound rule of three. Interest and discount. Proportional division. Partnership and alligation. Metric system. Square root." ·

(b) Commercial. "Calculation of interest by different methods. Stocks and bonds. Accounts current. Bills and notes."

Elementary geometry. "Calculation of surfaces and volumes."

Elementary algebra. "The four fundamental operations with positive and negative numbers. Fractions. Square and cube root. Equations of the first and second degree. Progressions and logarithms."

Elementary bookkeeping. This is an elementary course.

Methods of instruction."-The methods used are similar to those used in the commercial schools.

Books.-A list of the books in general use is given.5

1 No. 6, p. 53.

Ibid., pp. 64-69. a Ibid., p. 64.

4 Ibid., pp. 65, 66.

5 Ibid., p. 68.