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.

occupies a period of not less than three years, with attendance of three evenings a week.

The advanced course includes work of a more advanced and specialized character and normally extends over at least two years.

For purposes of administration it is convenient to speak of a major course and a minor course. A major course will usually extend over not less than four years and will include senior and advanced courses. A minor course will probably be completed in two or three years, so that it will often consist of a senior course only.

1

"The present memorandum deals in detail with the engineering courses which are subsequent to the junior course.'

Nature of the instruction. The minor course is intended primarily to help workers to a more intelligent understanding of their craft. The practical side of the instruction is the important one. It is desirable, but not necessary, that the teachers have technical training.

The major course is intended for persons who expect to become foremen, designers, heads of departments, and the like. "Its curriculum will center around the scientific basis of engineering and will include the training in mathematics and drawing which is necessary for sound progress." "Teachers in the advanced course will be specialists in one or other of the branches into which engineering is divided and will probably confine themselves to their special subjects."

The work of the first three years [subsequent to the junior course] may be planned on lines which are generally similar for each year, while the later treatment will usually be on more specialized lines.

In each of the earlier years the work should be organized to include the treatment oi a number of branches or aspects of the subjects, which may be roughly classified as follows: (i) Mensuration; (ii) manipulation of algebraic expressions, evaluation of formulæ and solution of equations; (iii) the idea of functionality or the dependence of one quantity upon another, and the use of graphs; (iv) trigonometry; (v) variation of functions leading up to the calculus.

The aim of teaching should be to develop a habit of dealing quantitatively with material objects and physical phenomena, and of expressing the results symbolically and in graphical form; also to impart the power to interpret the relationship between quantities when so expressed.

COURSE OF STUDY IN MATHEMATICS.3

SENIOR COURSE. FIRST YEAR. (the student's ELEVENTH SCHOOL YEAR.)

(a) Arithmetic and mensuration. Review of the work of the junior course. areas and volumes. Checks.

(b) Logarithms. Use of common logarithms.

Rules for

(c) Measurement of angles. Notion of an angle. Construction of angles with ruler and compasses. Radian. Sine, cosine, and tangent defined and computed from measurements. Variation and graphs of these functions from 0° to 90°. Solutions of right triangles.

(d) Graphs. Finding the areas of irregular figures by the use of the midordinate rules. Graphs of statistics and of familiar functions.

(e) Algebra. Solution of simple equations in one or two unknowns. Factoring. Simple examples in indices, and much practice in simplifying expressions and in solving formulas for any letter involved.

(f) Slope of a curve. The notion of the slope of a curve and its measurement. (g) Practical work. The use of rulers, scales, micrometers, calipers, planimeter, spring balances, and weighing scales. Much measuring of the usual geometrical solids and of such objects as washers, rings, plates, disks, wires, bars, rods, pipes, and short lengths of standard rolled sections. Drawing to scale.

SENIOR COURSE.

SECOND YEAR. (THE STUDENT'S TWELFTH SCHOOL YEAR.)

(a) Mensuration. More advanced work. Irregular plane and solid figures. (b) Slide rule.

(c) Logarithms.

(d) Trigonometry. Review. Solution of triangles by reducing them to sums or differences of right triangles. Projections of lines and surfaces.

(e) Vectors. Introduction, and simple examples in addition and subtraction. (f) Algebra. Review and extension of the work of the previous year. Quadratic equations. Simple exercises in the graphical solution of equations.

(g) Graphs. "More practice in plotting graphs should be given. This should lead to exercises involving interpolation, the calculation of average rates of increase, the determination of areas, maxima and minima values, the finding of probable laws connecting varying quantities, and the solution of equations."

(h) Calculus. The notion of a derivative with simple examples.

ADVANCED COURSE. FIRST YEAR. (THE STUDENT'S THIRTEENTH SCHOOL YEAR.)

(a) Mensuration. "More advanced work may be taken in mensuration, including the theorems of Guldinus, and a number of exercises on the areas and volumes should be given."

(b) Trigonometry. Trigonometric functions for any angle, the most useful formulas, and the solution of triangles.

(c) Algebra. "The work in algebra may include further exercises in the solution of quadratic equations, and the roots of such equations; further practice in manipulating expressions containing indices; the simplification of fractional expressions; products of binomial expressions, including the consideration of approximations." (d) Graphs. Plotting of certain useful types of curves such as pun=constant, y=aebx, y=a sin (x+c), y=a sin (x+c)+d sin (x+e).

(e) Calculus. The notion of a derivative, and of an indefinite and a definite integral, and simple applications. Much care is to be taken to make the ideas clear.

ADVANCED COURSE.

SECOND YEAR. (THE STUDENT'S FOURTEENTH SCHOOL YEAR.)

(a) General. The greater part of the time may be spent on the calculus and its applications, and some time on exercises in the binomial theorem and with trigonometric formulas.

(b) Differential calculus. "Differential coefficients for standard functions; differentiation of products and quotients of functions and of the functions of a function; partial differentiation; Taylor's and Maclaurin's theorems; exponential, logarithmic, and Fourier's series and their applications; curvature; construction of curves from law of slope."

(c) Integral calculus. "Standard integrals; integration by parts and by substitution; graphical methods of integration; simple differential equations."

(d) Engineering applications. "It is recommended that the teacher of mathematics cooperate with the teacher of engineering and that applications of the calculus be made to such topics as areas, volumes, velocities, accelerations, vibrations, centers of pressure, centers of gravity, moments of inertia, theory of beams, thermodynamics, and problems in electrical engineering."

(e) Plane analytical geometry. If time permits, it is recommended that the equations of the conic sections be studied.

HOLLAND.

COMMERCIAL AND INDUSTRIAL SCHOOLS.

The lower and middle vocational schools discussed in the report from Holland1 are the "Burgeravondscholen," the vocational schools (écoles professionelles), schools of design (écoles de dessin), technical schools, and vocational schools for girls. The first three kinds have the same course of study in mathematics and can be considered together. There is little or no instruction in mathematics in the schools for girls.

BURGERAVONDSCHOLEN, schools of design, vocATIONAL SCHOOLS, AND TECHNICAL SCHOOLS.

Organization. The Burgeravondscholen are evening schools organized by the communes, but in some cases partially supported and supervised by the Government. Their aim is to give theoretical instruction to persons going into commerce and industry. The course is from two to four years in length, with from 6 to 12 or 15 hours of instruction a week.

Schedule of hours per week of a Burgeravondschole of four years.

"For the most part there are no examinations for admission to these schools. A declaration by the head of the primary school that the student has followed the primary instruction with success is sufficient." This statement applies also to the schools of design and to the vocational schools.

3

The schools of design are "in the main the same as the Burgeravondscholen; the course of study is from three to five years in length and from 8 to 15 hours of instruction are given a week." The most of these schools have been founded by private societies, but in some communes there are schools of this kind which receive some support from the State or the Province.

The vocational and technical schools give both theoretical and practical instruction. The most of these schools give a course of three years with from 40 to 44 hours of instruction a week. These

Rapport sur l'Enseignement mathématique dans les Pays-Bas, publié par la Sous-Commission nationale, pp. 11-29. Referred to hereafter as Rapport.

*Ibid., p. 20.

Ibid., p. 14.

schools are in session throughout the year, except five or six weeks. The following schedule of hours a week is given.1

Schedule of hours per week of vocational and technical schools.

The following is the work usually given in the vocational schools and in the Burgeravondscholen:

Arithmetic. (First year.) The subjects studied include the fundamental operations, the metric system, foreign weights and measures, ratio and proportion, percentage, square root, and applications.

Algebra. (First year.) Fundamental operations.

Second year. Review of the fundamental operations and equations of the first degree in one unknown.

Third year. Linear equations in two unknowns and the solution of problems. Geometry. (First year.) Rectilinear figures, areas, similar figures, and problems. Second year. The circle, regular polygons, the ellipse; planes, and regular polyhedrons.

Third year. Prisms, pyramids, cylinders, cones, sphere; calculation of surfaces and volumes.

Mechanics. (Third year.) Elementary mechanics.

Descriptive geometry. "In the teaching of that branch we make as many applications as possible, relating them to the specialty of the student." Many of the purely mathematical examples of the books are discarded and practical applications are substituted.

Methods of instruction.-The above outline of the work in mathematics shows no distinctly vocational material. The examination questions given in the report as examples of those given at the conclusion of the course in the Burgeravondscholen are of the usual textbook type.

In some schools models are used in the instruction. As most of the teachers of mathematics come from the elementary schools, the methods of instruction of those schools are probably used in the vocational schools. There appears to be a tendency to make the instruction more practical, and, in particular, to unify the theoretical development of a topic and its vocational applications.

Preparation of teachers. Some of the teachers are engineers, but the most of them are elementary school teachers.

1 Rapport, p. 15. ⚫

2 Ibid., pp. 18, 19.

3 Ibid., p. 19.