A part of the students have several years of shop experience before entering the technical schools.

Aims.-"Between the engineer,' the thought that conceives, and the workman, the hand that executes, a 'technician' is necessary, capable first of grasping easily the projects and plans of the engineer or the architect, and then directing their execution with intelligence.' As the principal aim of these schools is to educate these technicians the principal aim of the mathematics is a purely practical one. It is intended, however, that the mathematics be taught so as to develop logical thought.2 The young technician must know enough mathe matics to organize and solve his problems mathematically, and to be able to read the literature of his subject.

COURSE OF STUDY IN MATHEMATICS.

"The number of hours weekly (spent in the school) is in the neighborhood of 40 throughout the course.3 In the first year 12 or 15 hours are given to pure mathematics, in the second year 5 or 6, and in the third year only 2 or 4. In the second year the course in applied mathematics requires 8 or 10 hours.”

4

The following course in mathematics is given in the section for mechanicians in the Technicum of Berthoud. The length of the course in this school is five semesters.

FIRST SEMESTER. (THE STUDENT'S NINTH SCHOOL YEAR.)

Arithmetic. (Six hours a week.) The principal subjects studied are the fundamental operations, systems of measures, fractions, ratio, and proportion, percentage and its applications, partnership, and alligation.

Algebra. (Four hours a week.) This course includes the fundamental operations with integral expressions, equations of the first degree in one unknown, proportion, and logarithms.

Geometry. (Three hours a week.) The theorems of plane geometry, the constructions, and computations are taken up.

Geometrical drawing. (Five hours a week.)

SECOND SEMESTER. (THE STUDENT'S NINTH SCHOOL YEAR.)

Algebra. (Four hours a week.) The topics studied in this course are equations of the first degree in more than one unknown, equations of the second degree, powers and roots, logarithms, the slide rule, arithmetical and geometrical series, and compound interest.

Geometry. (Four hours a week.) This is a course in solid geometry and includes the most important theorems, and computations of the surfaces and volumes of the elementary solids and also of the prismatoid and of solids of revolution. There is a short introduction to trigonometry which is used in the computations of volumes.

Projective geometry. (Four hours a week.) This course begins with the projection of the point and the straight line and extends through the study of the plane sections of pyramids, prisms, cylinder and cones, and the intersections of bodies bounded by plane and curved surfaces.

1 No. 5, p. 28.

* Ibid., p. 99.

3 Ibid., p. 42.

Ibid., pp. 50-55.

THIRD SEMESTER. (THE STUDENT'S TENTH SCHOOL YEAR.)

Plane trigonometry. (Three hours a week.) This includes the solution of oblique triangles and applications.

Analysis. (Four hours a week.) This course includes the binomial theorem, the notion of a function, and an introduction to differential calculus and its application to the theory of curves and to mechanics, and to the development of functions into power series.

Descriptive geometry. (Three hours a week.) Intersections of solids and applications to numerous problems in the construction of machines. Light and shade and parallel perspective.

Mechanics. (Four hours a week.) This is a course in the elementary theory of mechanics, with many practical applications.

Resistance of materials. (Four hours a week.)

FOURTH SEMESTER. (THE STUDENT'S TENTH SCHOOL YEAR.)

Analysis. (Three hours a week.) Introduction to integral calculus and applications to geometry and mechanics.

Mechanics. (Six hours a week.) This course includes something of the dynamics of rigid bodies, hydrostatics, and hydraulics with applications.

Graphical statics. (Two hours a week.)

FIFTH SEMESTER. (THE STUDENT'S ELEVENTH SCHOOL YEAR.)

Mathematics. (One hour a week.) Review with selected examples. Short introduction to practical geometry.

Graphical statics. (One hour a week.)

Methods of instruction.-What has been said concerning the general aims and methods of instruction in the trade schools applies here. The instruction is expected to be concrete, to involve many practical problems, and in point of view of theory to be limited to what is absolutely necessary. "All the purely theoretical developments are suppressed." 1

The work in descriptive geometry and in geometrical and technical drawing is said to be of great importance, is given much time and attention, and every effort is made to keep the instruction thoroughly practical.

Much importance is attached to the study of mechanics in the middle technical schools. It is sometimes taught by mathematicians and sometimes by engineers. "We will only add that the choice of examples is made in absolute conformity with the technical needs of the students, and that this course is characterized by a great number of carefully chosen graphical and numerical problems." It is urged that in geometry and in mechanics "the professor should prepare the examples according to the necessities of the school. He ought to know the questions of which the technologists have the greatest need; his problems ought to relate the mathematical theory to the other branches of the program.'

Examinations.'-At the end of the course the students take an examination to obtain the diplôme de technicien. In certain schools there is an examination at the end of the second year which includes the examination in pure mathematics.

The examination in mathematics is oral and written, on algebra, geometry, trigonometry, analytics, mechanics, and the resistance of materials. A considerable number of examination questions are given in the report. The most of these examples are theoretical or involve such applications as are found in the books.

Preparation of teachers.3-In general the teachers in the middle technical schools are engineers and architects with diplomas from the higher technical schools, and professors of branches of general culture who have the diplôme de maître de gymnase.

Some of the instructors of technical branches are technologists who after leaving the Technicum have gone into practice, and later have gone into teaching. Some of the teachers of nontechnical subjects have only the diplôme de maîtres secondaires. "We have then no special preparation for the professors of technical instruction. There is found the engineer with no pedagogical knowledge, and the mathematician with no technical knowledge." It generally happens, however, that the engineer has a taste for teaching, and the mathematician is interested in mechanics or architecture; so that the results seem to be good. It is recommended that some special preparation be given instructors in technology, just as is now done in the case of the teachers in the secondary schools. "It is not sufficient for a future professor to be instructed in his subject; it is necessary that he should have practice teaching, show what he can do, and by some well directed trials improve and perfect his individual methods."

UNITED STATES.

The vocational schools of the United States have no general State or national organization, and present great diversity of organization, aims, and methods. As a rule these schools have grown up to meet the needs of particular vocations or of particular localities. The types are not clearly defined, and it is practically impossible to give · detailed descriptions that will apply to a large number of schools.

The data used here are taken almost entirely from the American reports to the International Commission on the Teaching of Mathematics. Only some of the principal facts are given here, and references are made to the reports for details.

1 No. 5, p. 97.

2 Ibid., pp. 91-97.

3

a Ibid., p. 102.

COMMERCIAL SCHOOLS.1

Private commercial schools.-There is generally no definite entrance requirement. The aim is to prepare young persons, in the shortest possible time, to take such positions as that of clerk, bookkeeper, accountant, and stenographer. The courses are from a few months to two years in length. Commercial arithmetic is usually the only work in mathematics.

Commercial departments in high schools.-These are ordinary high schools, and the course offered by the commercial department is only one of several that may be elected by the students. The work in mathematics in the commercial course differs from that of the other courses in including commercial arithmetic and bookkeeping, and less of algebra and geometry.

Commercial high schools. These have the same organization as other high schools, and hence require for entrance the completion of the eighth grade of the elementary school, and the pupils are as a rule 14 years old when they enter. The aim of the commercial high school, as well as of the commercial department in the general high school, is to give a general education and at the same time to give vocational training that may be utilized immediately after leaving the school. Both of these classes of schools prepare for the same kinds of positions as those mentioned under private commercial schools. The courses in mathematics offered are commercial arithmetic, algebra, geometry, bookkeeping, and sometimes trigonometry. The length of the course is four years.

INDUSTRIAL SCHOOLS.

Intermediate industrial, preparatory trade, or vocational schools.2 The aim of these schools is to prepare for a trade. In order to enter these schools the student must be at least 14 years of age and have completed at least the first six years of the elementary school. These schools are as a rule supported by public funds. "In schools of this class the course of study commonly combines bookwork and shopwork in almost equal proportions. The bookwork generally includes English, shop mathematics, industrial history, and civics, together with the elements of physics and chemistry.”

Trade schools with day courses.3 "In these schools the aim is to prepare pupils for actual work in the trades in the shortest possible time." The length of the course varies with the needs of the student. In some schools of this class very little attention is given to mathematics; but other schools "giving longer courses frequently give very

1 International Commission on the Teaching of Mathematics. The American Report: Mathematics in the Technical Secondary Schools of the United States, pp. 21–29.

* Mathematics in the Elementary Schools of the United States, pp. 154, 155.

a Ibid., pp. 155, 156.

thorough instruction in English, mathematics, and science." The work in mathematics sometimes includes the elements of arithmetic, algebra, geometry, and trigonometry.

Some of these schools are supported by private foundations and others by public funds.

Apprenticeship schools.'-"Apprenticeship schools are provided by a number of large industrial corporations for the education of boys who are learning their trade." "Through training in mathematics, drafting, English, and science there are recruited from the ranks of those who work in the shops a group of men much more capable than the ordinary worker." "The fact that in most of these schools shop time is taken for the class work, and that this time is paid for at the regular rate, is an indication of the value attached to such exercises." These schools offer courses in arithmetic, and in some cases courses in algebra, geometry, and trigonometry.

Evening schools.2-These schools are supported by private foundations, by public funds, and by societies such as the Young Men's Christian Association. "They are intended to give a type of education, both theoretical and practical, that will give the worker at the trade a better understanding of his work and of its relationship to the industry toward which his work contributes." Some of these evening schools give free courses. Certain schools give courses extending over a number of years. The free night school of the Cooper Union in New York, for example, offers a five-year course in general science. The mathematics offered in this course includes algebra, plane and solid geometry, trigonometry, analytical geometry, calculus, and mechanics. A student will generally be admitted to a course in an evening school if he gives evidence of sufficient preparation to profit by the course.

Part-time schools."—"The part-time plan, as carried on at Fitchburg, Mass., is an arrangement by which boys in the high school give half of their time to work for wages in the commercial shops in the city and half to school work." Like other high schools, these schools require the completion of the eight grades of the elementary school for admission. "For the first year the boy devotes his entire time to school work, and for the next three years equal groups of boys alternate between shop and school, so that one group is always at work in the shops and one in the school."

Trade schools for the colored races.-"The object of these schools is to teach a skilled trade to Indians and Negroes. In some of these schools there is a course of four years. The entrance requirements vary. In some of these schools the relationship between industrial

1 Mathematics in the Elementary Schools of the United States, pp. 156, 157.