In schools where curricula are arranged with more or less reference to the entrance requirements of the colleges, algebra, geometry, and trigonometry receive about the same amount of time as in general secondary schools, viz, one year each of elementary algebra and plane geometry, and one-half year each of advanced algebra, solid geometry, and trigonometry. A year's work ordinarily represents five exercises per week for 33 to 40 weeks, i. e., a total of 165 to 200 exercises. The length of the exercises is from 40 to 50 minutes.

The data accessible render a precise statement of the division of time between the two algebraic subjects and between the two geometric subjects impracticable. The more advanced subjects are occasionally omitted, and in this case the tendency appears to be toward the retention of solid geometry and trigonometry rather than advanced algebra.

The schools to which the above statements apply are ordinarily, though not necessarily, of the type known as "manual-training schools, as distinguished from the "technical," "industrial," or "trades" schools.

These latter schools, as well as some of the manual-training schools, prefer to offer a course in arithmetic of from 50 to 200 exercises.

A few schools present courses in so-called "shop mathematics," the nature of which appears from the following outline taken from the circular of a trades school:

COURSE I, ELEMENTARY.-SHOP ARITHMETIC.

This course comprises work with common and decimal fractions, measurements, percentage, ratio and proportion, square and cube root; applying these principles to such shop problems as gearing-simple and compound; how to select gears to cut screws and spirals; computations on the lever, including the lathe indicator, lever safety valve, the Prony brake; pulleys and hoists; simple, compound, and differential indexing with the milling machine; problems connected with the speed lathe and engine lathe; computing the horsepower of steam engines, electric dynamos, and motors.

COURSE II, ADVANCED.-ALGEBRA, GEOMETRY, AND TRIGONOMETRY, WITH APPLICATIONS TO SHOPWORK.

This course is open to those who have completed Course I or who have had a preparation equivalent to a good grammar-school education. It treats of the most important principles of algebra, especially of the equation as a means of solving problems and of the derivation and use of formulas. The practical side of geometry is next taken, emphasizing the methods of finding areas and volumes, weights of bars of various shapes and materials, heating surface of boilers, etc. The last half of this course is spent on trigonometry, including the use of logarithms and logarithmic tables and emphasizing the applications of trigonometry to the more advanced types of shop problems.

The private commercial schools give no mathematics but arithmetic and commercial arithmetic; the commercial departments of several secondary schools give commercial arithmetic and sometimes algebra and geometry; while the "high schools of commerce" always give

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commercial arithmetic and a course in algebra and offer geometry and trigonometry as elective subjects.

The agricultural schools generally give one-half to one year's work in arithmetic, usually with reference to the problems of farm life, e. g., farm accounts, mensuration, bookkeeping, etc. They ordinarily give the same amount of time to algebra and geometry as do other secondary schools. Advanced algebra is occasionally given, trigonometry in about 25 per cent of the schools, frequently with reference to its use in surveying.

Analytic geometry and the calculus are seldom given except in schools which properly belong in the province of Committee IX in that their work, while not leading to a degree, nevertheless covers the first two years of the work of the higher technical schools. Except for these schools, the subjects mentioned are offered only in courses preparatory to the colleges.

The history of mathematics is specifically mentioned by only one school (and that a school for girls) and is given in connection with the regular work in algebra and geometry.

As to the matter of correlation of the mathematical subjects among themselves or with other subjects, it would appear that it is necessary to distinguish between the actual state of affairs and the tendencies at work. Taking the schools as a whole, it may fairly be said that systematic correlation is not widespread. The principle of correlation, however, is generally regarded with favor, but as a rule it is not systematically applied, except where the relation of the subjects (e. g., commercial arithmetic and bookkeeping) is so obvious that the necessity is apparent. In so far as the term "correlation " indicates use of problems taken from the applied sciences or from daily life, there is fairly general application of the principle, but in the sense of systematic coadaptation of mathematical and other courses it is not generally applied.

The situation may be illustrated by the following quotations:

(1) On the technical side the pupil articulates the mathematics with the work of the drafting room, shop, domestic science, and domestic art. Teachers of technical subjects are in constant touch with the mathematics department, anticipating problems which will arise and reporting immediately to that department any weakness shown by a pupil in problem or principle.

(2) In this work a great deal of time will be spent in laboratory study, so that the pupil will obtain such a first-hand knowledge of the subject that he can afterwards readily and efficiently apply it in the shops and laboratories. No sharp distinction will be drawn between algebra, geometry, etc., but the different methods will be treated merely as various ways of getting at the same thing, of which one way may be the more useful in one case and another method that best adapted to deal with another situation. At all times the work in mathematics will be kept in close touch with the shopwork; the aim will be to so train the pupil that he can use his mathematics in the shop readily and efficiently.

These statements are made by schools which strongly emphasize the principle of correlation. Both are recently organized schools. On the other hand, we have in answer to the question, "Are any systematic attempts made to correlate mathematics with other subjects?" the following:

(3) Some, but more later.

(4) In past, 25 cents on the dollar; will aim to do better in future.

In many cases the answer was a flat "No."

The chairman of the committee is of the opinion, based on internal contradictions in the evidence submitted by the schools, that no satisfactory conclusion as to the nature and results of the application of the principle of correlation can be obtained except on the basis of a study of a considerable number of schools, this study to be made by a single individual or a small committee and on the spot.

EXAMINATIONS.

There is no evidence to indicate that examinations are to any extent used as the sole means of determining the proficiency of the pupil. They are used as auxiliaries for that purpose, but the results are combined with those of daily work. They are ordinarily written, may be from 40 minutes to 4 hours in length, and occur from 30 to 2 times per year. The tendency is toward relatively frequent examinations, not exceeding 2 hours in length.

METHODS OF TEACHING.

The movement which has led to the establishment of the secondary technical schools finds its principal expression in the emphasis laid on the concrete. Consequently one should expect to find, and does find, that increasing attention is paid to the concrete element of the instruction, both in material and method. Nothing, however, could be further from the truth than an assertion that as a class those schools have developed and are using methods of instruction widely different from those of general secondary schools. Certain schools, it is true, have developed such methods; the majority have not.

The information derived from the questionnaires is not extensive or detailed enough to warrant detailed statements. The basis of the above assertions lies rather in the direct and indirect evidence contained in the catalogues of the schools.

As illustrations of such evidence the following statements are quoted:

(1) Pupils in mathematics are given acquaintance with the language of mathematical symbols, called formulæ, in which problems and laws involving weight, size, time, force, and the like are frequently stated. They are taught to understand these formulæ, to solve problems so stated, and to use the

mathematical symbols in the statement and solution of new problems. Pupils are taught also to state and solve problems by graphical methods, i. e., by scale drawing or by the graph, and immediately to solve the same by the algebraic methods of the equation or the proportion. The pupil becomes familiar with the standard geometrical forms, the laws of their structure, measurement, and relation to other forms, and acquires the power to state these laws algebraically, together with some ability to make a clear and logical proof of the truth of geometrical theorems. Geometry and algebra are carried along together for two years and a part of the third. In the first year the geometrical laws and concepts furnish much material for developing algebraic problems and processes. In the second year algebra is used to develop geometrical theorems, and to fix them in mind through use. The school offers a continuous four years' course of elementary and advanced algebra, plane and solid geometry, and plane trigonometry.

In the classroom a combination of laboratory, recitation, and examination methods is employed. The theory of a new subject, especially in the earlier years, is usually developed by the instructor; and home work is assigned to clarify and impress it, and to enlarge its application. The method of approach to new subject matter is, in general, that of induction, the particular leading to the general, the concrete to the abstract. Deductive work becomes more prominent in the late years.

(2) Throughout the entire course this study (mathematics) will be pursued as a means to quantitative determination in the workshop, laboratory, office, and countingroom. Much of the educational value lies in the grasp which is gives the students of quantitative relations.

Objective work will introduce new subjects, so that there may be a rational basis for intelligent use of symbols and a thorough conception of the power of the equation. Formulæ should be deduced from relations actually seen, so that the pupil may discriminate between the abstract formula and its concrete practical relations to real things.

The boys of our school will have several weeks of constructional geometry work at the beginning of their mechanical drawing. This helps to lay a good foundation for demonstrative geometry, as well as to be of great practical value in their future use of drawing.

Supplementary exercises are given to show some of the uses of algebra in the natural sciences. Correlation between algebra, geometry, and the sciences is shown wherever possible. The graph and some of its uses are taught in linear equations and in easy quadratic equations.

In geometry the pupil will to a considerable extent originate his demonstrations instead of simply memorizing those of the author. Model proofs will be given when necessary to teach good form and logical arguments, but as a rule such demonstrations will be given only when the pupils would otherwise be at a loss to know how to proceed.

When a class in trigonometry has developed the working formulas it does most of its problem work in the field with the transit, leveling rod, and tapeline. Much of the work is plotted to scale. This work is found to be interesting and practical.

FIRST YEAR.

First term. Algebra to simple equations, including the application of factoring in simplifying fractions, and in solving easy quadratic equations of one unknown quantity.

Second term. Algebra to ratio, including easy exercises taken from the physical and chemical laboratories. The simpler uses of the graph will be taught during this term.

SECOND YEAR.

First term. The first two books of plane geometry. Special attention wil! be given as to what constitutes a rigid proof. Suggestions are given on methods of attacking propositions and problems. Neat, accurate form work will receive special attention, and pupils will be required to bisect lines, angles, erect perpendiculars, and draw parallel lines by actually using compasses and ruler.

Second term. Plane geometry completed. We expect most pupils by this time to be able to do considerable work on their own initiative; to be able to have some determination to master a proposition set before them.

The practical applications of the subject are shown whenever possible to do so.

First term. Solid geometry.

THIRD YEAR.

Second term. Elementary algebra completed. This includes ratio, proportion, variation, imaginaries, series, partial treatment of binomial theorem, logarithms, review.

FOURTH YEAR.

First term. (1) Plane trigonometry. Development of formulas. Fieldwork with transit, leveling rod, and tapeline.

(2) Higher algebra.

Second term. (1) Descriptive astronomy. A brief, simple, and accurate account of the heavens as they are known to-day. It is intended only for high-school pupils, and to give some information on the subject that is needed for the person of ordinary culture. Interesting facts will be studied, but no attempt can be made to gain any clear conception of the processes by which the fundamental truths of astronomy have been established. The methods of discovering the wonderful truths of this subject belong to the advanced student. (2) Higher arithmetic for those who expect to become teachers.

NOTE. The formation of these classes depends on the number of pupils who can arrange their programs so that it would justify the taking of a teacher's time from the crowded classes in first, second, and third years.

These quotations indicate what is done in certain schools which have paid much attention to the development of methods.

On the other hand, there are two sources of evidence for the statement that the schools as a class have not developed special methods of instruction. The one is the negative answer in the answers to the questionnaire; the other is in the lists of textbooks. The use of texts dominates the instruction in the majority of American schools, and hence the nature of the text is to some extent an index of the character of the teaching. When one finds, as is actually the case, a widespread use of a few texts in which the treatment is essentially abstract and the problems constructed with small reference to other than formal requirements, it is a justifiable inference in the absence of evidence to the contrary that the instruction in the schools using these texts is not markedly different from that in general secondary schools using the same texts, and this inference is strengthened by detailed statements of matter covered, which not infrequently are extracts from the tables of contents of these texts. While the use of texts, and, indeed, of traditional texts, is general, the newer schools, especially those in which the industrial idea is