Descriptive geometry. (Ten hours a week, the first half year.)

Calculus. (Two hours a week, the second half year.)

Mechanics. (a) (Six hours a week, the first half year.); (b) (Six hours a week, the second half year); (c) (Six hours a week, the third half year.)

Nature of the instruction.-In discussing the teaching of applied mathematics Ott1 says that there are two methods used, the lecture method, brought from the university, and the heuristic method, which is used in the elementary and secondary schools. The lecture method, pure and simple, is thought not to be a proper method for students with the preparation of those in the middle technical schools. Because of lack of time, and because the greater part of the material in applied mathematics can not be worked out by question and answer, a purely heuristic method can not be used. Hence the method generally used is a combination of lecturing with questions and answers.

Along with the discussion of theory exercises are given, which are solved in the classroom and copied in the notebooks. Not infrequently the class exercise extends through two 45 or 50 minute periods. In many cases the students are in the classroom or the workshop practically all day. Hence there is little opportunity for outside study, and all theory as well as applications must be worked out in the classroom.

The reports recommend that the presentation of theory be made simple and concrete, and be reenforced by many examples. While the students must know and understand the fundamental theorems, it is not thought necessary that all the formal proofs be given. It is recommended that, in applied mathematics at least, use be made of models and other similar aids.

"We have pointed out that there must be developed first of all in the technical school skill and ability to do." This aim determines the choice of subject matter and of method. It is said that in the choice of subject matter the only question to be raised is, "What parts of mathematics are necessary or important for the study of technology, and moreover can be taught in an elementary way suitable to the development of the student?" "What the technologist can not use will be unhesitatingly omitted."4

Methods of approximation form an important topic in mathematics, as these often give the best methods for the solution of problems, both because of brevity and because the limited mathematical knowledge of the students makes it impossible to develop a complete theoretical treatment.

1 Die angewandte Mathematik an den deutschen mittleren Fachschulen der Maschinenindustrie, by Dipl.-Ing. Karl Ott, p. 6. This is Band IV, Heft 2 of the German Report.

2 Band IV, Heft 1, p. 36.

* Ibid., p. 41.

" Ibid., p. 35.

Grünbaum' criticizes the teaching of algebra for not doing more with the notion of a function, which is of great importance in applied mathematics.

Graphical methods have at present, and have had for a long time, an important place in the teaching of mathematics in the middle technical schools.

The slide rule is much used in some institutions, but apparently not much used in others.

Preparation of teachers.-In speaking of the instruction in applied mathematics, Ott says: "At present the instruction in this field is with increasingly few exceptions in the hands of engineers, and there appears to be no prospect that this condition will be essentially changed."

The proportion of engineers among the teachers of pure mathematics is probably not so great, as Grünbaum says that the question of the course of preparation for the teachers in the technical schools is not settled. At present the instruction in pure mathematics is given by engineers and mathematicians. The first may lack theoretical preparation and good methods of instruction, and the second may be deficient in technical knowledge and experience. There has been some discussion of special instruction in the technical high schools for teachers in the middle technical schools, and also of requiring a year of practice teaching; but present opinion is not in favor of these movements.4

Books. There is no lack of texts in mathematics for the middle technical schools, and they are of various grades of excellence as to presentation of theory and choice of problems. Many of these books show the influence of the secondary schools. In not a few cases the books prepared for the technical schools are merely revisions of books written for the secondary schools, and contain more than is necessary of the purely formal exercises of the old books, as well as large numbers of conventional problems that have no technical applications.

NAVIGATION SCHOOLS.

Organization.-The report on the navigation schools has to do with the schools which prepare pilots and seamen for ocean navigation.

1 Band IV, Heft 1, p. 68.

2 Band IV, Heft 2, p. 24.

3 Band IV, Heft 1, p. 93.

• Ibid., p. 99.

Ibid., pp. 50-65.

• Der mathematische Unterricht an den deutschen Navigationsschulen, by Dr. C. Schilling and Dr. H. Meldau. This is Band IV, Heft 4, of the German Reports. Referred to hereafter as Band IV, Heft 4.

90987-15-4

When the report was written (1911) there were 18 such schools 1 in Germany, of which 12 were in Prussia, 1 in Oldenburg, 2 in Mecklenburg, and 1 in each of the Hanseatic cities of Hamburg, Bremen, and Lubeck. All of these are State schools, except one in Mecklenburg, which is under the control of the city in which it is located. There are, furthermore, 6 Prussian preparatory navigation schools. There are no private navigation schools, properly speaking, but some candidates do get their preparation for taking the seaman's examination from private sources.

Aim of the schools. The chief aim of the schools seems to be to prepare for examinations the pilots and seamen who will engage in ocean navigation. The following outline of the requirements for these examinations shows what is, apparently, the extent of the mathematical instruction.

COURSE OF STUDY IN MATHEMATICS.2

Algebra. The examination covers the fundamental operations in arithmetic with common and decimal fractions, equations of the first degree, powers, roots, and logarithms.

Plane geometry. This includes the simpler theorems about angles, congruence, similarity, and equality of polygons, the simpler theorems about the circle, simple constructions, and easy computation of areas.

Solid geometry. This includes the simpler theorems about lines and planes in space, and about the sphere and spherical triangles; and the computation of the volumes of prisms, cylinders, and casks, and of the ship's hold by Simpson's rule.

Plane trigonometry. This covers the simplest relations of the trigonometrical functions and the solutions of triangles.

Spherical trigonometry. (Required only of seamen auf grosser Fahrt.) This includes the solution of spherical triangles.

Methods of teaching.-The authors of the report would have the geometry made intuitional as far as possible, proving only those important theorems which do not appear at once evident. There should be many applications to nautical problems. The rules for the computation of areas should be taught in the simplest concrete way.

The trigonometry seems to be taught much as in other schools, except that the development is simpler, and special emphasis is placed on the parts which are of the greatest importance to navigation.

"In contrast to the technical schools, drawing and methods of graphical solution are in the background in the navigation schools.” ♦ The reasons given are that the students in the navigation schools

lack previous practice in drawing, and that conditions on board in small, poorly lighted cabins are not favorable to accurate drawing. It seems that little is done with the notion of a function except in trigonometry. The authors of the report favor a more extended use of this idea, and give some examples the solution of which would be simplified by its use.

1

Preparation of teachers.—“To-day 1 the question of the preparation of the teachers for the navigation schools has not found a thoroughly satisfactory answer. The problem is to get teachers with the proper academic and practical training; and because of the small number of teachers in navigation schools, there is no prospect that a university or technical school will take up the problem of preparing these teachers.

In the schools outside of Prussia part of the teachers are academically prepared and part are seamen. In general, the first class have had experience of a year or more in navigation and the second have received their mathematical preparation by private study or by attending a university or technical high school.

"In the Prussian navigation schools there are employed at present, besides 3 directors, 29 navigation-school teachers, 17 preparatoryschool teachers, and 5 candidates. It is a general principle here that there will be given positions as teachers only former seamen who have passed the examination for pilots and sailors auf grosser Fahrt. There are no further requirements as to school preparation.'

An examination is generally necessary before a position as teacher in a navigation school can be obtained.

Examinations. As is mentioned above, the students attend the navigation schools to prepare for the nautical examinations. Since 1870 these examinations have been uniform in the German Empire. They have also, according to the report, become stereotyped, and are criticized by many teachers in the navigation schools, who say that before 1870 the schools were interested in teaching the students something, but now are forced to prepare the students to pass the examinations. "If, however, this criticism can not be truly denied, yet there is no doubt that on the whole the introduction of the new system of examinations has a good influence on the preparation of the German seamen.” 5

Books.-In many schools no books are used. In some schools Aufgabensammlungen or Leitfaden that are specially prepared for the navigation schools are used.

Modern tendencies.-There is some discussion of the question of requiring attendance of the navigation schools from those who take the nautical examinations.

1 Band IV, Heft 4, p. 49.

Ibid., p. 55.

3 Ibid., p. 17.

The authors of the report wish a change in the examinations, so that they will not be so stereotyped, and so that they will require more thorough preparation on the part of the candidates.

Modern improvements in ship construction and equipment, and the use of modern instruments also, require changes in the school instruction.

GREAT BRITAIN.

Some of the most noteworthy work that has been done in recent years in the improvement of the teaching of mathematics, especially in technical schools, has been done in England. The effect of this work has been felt in many other countries. The reports of the British Subcommission do not, however, give a sufficiently detailed account of the teaching of mathematics in the vocational schools to make it possible to discuss the subject here for Great Britain as is done for other countries. But it has been thought useful to give such an account of present tendencies as is furnished by the "Memorandum on the Teaching of Engineering in Evening Technical Schools," published by the board of education.

This memorandum does not attempt to give an account of present conditions. Its aim may be seen in the following extract from the prefatory note.1

The following memorandum has been drawn up with the object of furnishing suggestions to teachers and organizers of schools which provide evening classes in mechanical and electrical engineering. It is not in the least intended to lay down a scheme of instruction suitable for universal application; it is obviously necessary and desirable that there should be great variety both in methods of teaching and in organization to meet the needs of different types of students and the varying industrial conditions of different areas. Further, the last thing which the board desires in making these or any other suggestions is to fetter the liberty of the teachers or discourage individuality in teaching. The object of the memorandum is simply to assist teachers and organizers to work out for themselves the schemes of instruction best suited to the conditions of their classes.

The memorandum has been prepared by a number of the board's inspectors, many of whom have had recent experience as teachers.

Organization.-A complete curriculum 2 of evening instruction falls naturally into three stages, which may for administrative purposes be classified as follows: (a) The junior course; (b) the senior course; (c) the advanced course.

The junior course usually occupies two years and is intended for boys who leave the public elementary schools at the age of 14. It gives a general preparation for technical courses of all types, in particular for the senior course, and comprises instruction in mathematics, drawing, science, and English.

It is assumed that all students entering a senior course have had the equivalent of the junior course. The senior course normally

1 P. 4. All references are to the above memorandum.

2 P. 6.