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prominent, are to some extent breaking away from these texts and the corresponding instructional methods. A number of schools have prepared collections of problems taken from shop practice or from matter contained in technical periodical literature. Unfortunately but few of these are accessible in published form.

As to the process of instruction, it would appear that the method of holding recitations upon assigned textbook matter is not extinct. This method, which renders the recitation merely an oral examination, is, however, giving place to the method of development of new matter by questioning on the basis of the pupil's fund of knowledge. The use of problems as an instruction to new matters of theory is apparently more common than the use of problems solely as applications of a didactically presented theory.

It must be said, however, that the evidence on this point is meager and conflicting.

It must be borne in mind, moreover, that even approximate uniformity of method does not exist. The nature of the school, the necessity of preparing students for external examinations, the preparation and personality of the teacher-all have their influence on the method of instruction.

The laboratory method of instruction can not be said to be widely used. The term may be taken as denoting the use of experimental processes devised for the purpose of discovery or emphasis of mathematical truths. The mathematics is the final goal, the physical process the means. The method may be illustrated by the following quotation:

Algebraic problems are developed from the laws of percentage from the sides, angles, and areas of polygons. The laws of the lever and of beams are established by experiments in the classroom and are made the basis for the development of the fundamental processes and the laws of sight. Drawing to scale gives many problems in similarity of triangles and in ratio and proportion.

Allied to the laboratory method, but distinct from it, is that which may be termed the "shop method." Here as in the former physical processes are used, but the physical result is the final goal, the mathematical truth, a thing which is introduced and developed because it becomes necessary to the accomplishment of that end. The method may be illustrated by the following quotation:

WORKSHOP MATHEMATICS.

After a thorough review, which demonstrates to the pupil and the instructor the ability of the former for this important branch of his trade, the apprentice is led, by the solution of practical problems, through the necessary portions of arithmetic, algebra, geometry, and trigonometry. These subjects, when presented to pupils in the abstract, are frequently beyond their mental grasp, but when connected with their trade practice the absolute necessity of this knowledge becomes plain; the student then attacks the problem from a new standpoint and with renewed vigor, and succeeds in mastering the difficulties.

All the problems in this branch of apprenticeship, also, are specially prepared by the instructors and printed by neostyle. Much of this work is required to be done by the students as home study. Lectures and shop talks supplement the workshop mathematics.

It is, of course, scarcely necessary to add that the distinction between the two methods, as they are actually used, is not absolute. It is a difference in the position of the center of gravity of the instruction, which, if great, may amount to a qualitative distinction.

Some of the more distinctively vocational schools and all the commercial schools emphasize to a greater extent than the others the matter of computation, for the evident reason that with them the numerical result is a matter of technical importance.

In the commercial school particularly unremitting drill on the elementary processes of arithmetic is an essential feature of the instruction. Naturally the proportion of time expended on such drill depends on the breadth of the curriculum, and is greatest in the private commercial schools, where the only mathematical subject, commercial arithmetic, is essentially a technical subject. In the "high schools of commerce" the drill is important, but the broader curriculum permits emphasis on the theoretical side of the subject. Naturally, in all the commercial schools much attention is paid to the use of material drawn from commercial practice.

PREPARATION OF CANDIDATES FOR TEACHING.

In the greater number of schools a considerable proportion of the teachers are graduates of "normal schools," or schools of college grade. In many instances they are graduates of engineering schools, and, in a few cases, they possess the doctor's degree (obtained in course). In the trades and industrial schools some of the teachers have had experience in a trade or in one of the engineering professions; in the commercial schools many of the teachers have had experience in business houses.

The question of the preparation of teachers is the gravest which these schools have to face, particularly those in which the trades or industrial element is predominant. "Normal" courses for teachers of "manual training" exist, but there appears to be as yet little provision for the training of men who, with an adequate knowledge of the technique and problems of a trade, also possess a thorough knowledge of the science of mathematics and of the theory and practice of education. Such men exist, but they are few in number and are the result of accidental circumstances, not of organized instruction. A few institutions, notably Teachers College, Columbia University, are now offering courses designed to meet the needs of persons preparing to teach mathematics in the secondary technical schools.

PART II.

MODERN IDEAS CONCERNING SCHOOL ORGANIZATION.

The manual training, the industrial, the trades, the agricultural schools, and the high schools of commerce are, in their present form, essentially new types of school which have been developed as the result of the movement to render instruction more concrete and immediately available.

COEDUCATION.

The schools considered by this committee are fortunate in that the vexed question of coeducation presents itself in so objective a manner as to permit sane discussion, in some respects at least. The question of the simultaneous attendance of the two sexes at the same institution is one which may be regarded as settled in America by custom. This question need not concern us here.

On the other hand, in proportion as the schools are distinctively vocational, the question of segregation rather than coeducation becomes important. Segregation becomes imperative when the vocational element is predominant and differs for the two sexes. Whenever in the schools considered, correlation of the mathematical instruction with that in the technical subjects is regarded as essential, the difference between the technical interests of the two sexes is found to be so great that the successful application of the principle of correlation renders segregation necessary.

On the other hand, in the commercial and agricultural schools the interests of the two sexes are so nearly identical that segregation in the classes in mathematics is neither imperative nor usual. The same considerations hold with reference to the sex of the teacher.

MODERN TENDENCIES CONCERNING THE AIM OF INSTRUCTION AND OF THE BRANCHES OF STUDY.

There is a tendency to omit so-called "useless subjects," but the criteria are variable and often contradictory. The tendency, however, is to omit subjects regarded as involving complex manipulation or difficult theory unless they are of essential and immediate Vocational importance. For example, the extraction of numerical cube root, partial payments, etc., are omitted from the courses in arithmetic given in the agricultural schools. On the other hand, the commercial department of a high school omits as useless the subject of graphs from its course in algebra. In general, however,

the tendency is to retain the traditional content of the courses of the general secondary school, except in those schools whose courses are arranged with reference to immediate technical availability.

The general tendency is, moreover, not to increase the content of the courses or to replace old subjects by new except in so far as the so-called "workshop mathematics" may be regarded as a new subject. Such courses are new in their point of view and their concrete material, but not in their mathematical elements.

In this connection may be mentioned, however, the general tendency to introduce elementary trigonometry into the curriculum because of its numerous applications to shop problems and to surveying.

EXAMINATIONS.

There appears to be no noticeable tendency to abolish examinations but rather to subordinate them to the regular work.

METHODS OF TEACHING.

With the increase in the size of the cities and the centralization of industries has come a decrease in the fund of general knowledge which is available in the child as a basis for mathematical instruction. In the days of the small shop in the small town the artisan's boy frequented the shop and used his father's tools. He learned in a desultory and accidental way, perhaps, but nevertheless he learned to plan, measure, and build, and became acquainted with the materials and methods of the various forms of industrial activity. From all this the city boy of to-day is excluded.

In consequence of this the concrete basis which the boy formerly obtained for himself, and unconsciously, must now be systematically provided by the school. The mathematically ill-equipped teacher of an earlier period had no suitable mathematical superstructure to erect on the excellent foundation provided by the boy; the betterequipped teacher of to-day finds the foundation inadequate to support the structure he is prepared to erect. The consequence of this essentially economic change is that systematic intuitive instruction in the school is becoming increasingly necessary and is supplied with increasing frequency, e. g., by the use of "laboratory" and "shop methods" and by the use of concrete problems.

The technical secondary schools from their very nature tend to adapt their instruction to this need more readily than do the general secondary schools, and with apparent success. One school reports:

The boys take hold of mathematics as if they "needed it in their business." Our boys are not exceptionally bright, and yet the best section of girls in a "regular" high school would not be able to keep pace with them.

Along with this tendency to emphasize the intuitive element in instruction, there is the conservative tendency, partly due to inertia

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and partly to external academic requirements, to emphasize the abstract element.

In view of the lack of central control of American schools, the existence of these two conflicting tendencies, especially the latter, can not be regarded as otherwise than fortunate for the future of mathematical study and research in this country.

The general secondary schools already feel the competition of the technical secondary schools. The danger lies in the possibility that the attractiveness of the intuitional and immediately available element of the instruction in these technical schools, whose ideal approximates to that of the trades school, may so diminish the abstract and logical element in the mathematical curricula of all the secondary schools, general as well as technical, as to hinder the progress of mathematics as a science in this country.

The lack of concrete basis for mathematical instruction is the cause of a tendency, expressed in the curricula of certain schools, to precede the demonstrative work in geometry by work in "constructive or "inventional" geometry. The circular of one school says:

This study (geometry) is, first of all, inventional. With ruler, dividers, compass, and protractor, the pupil is taught to draw geometrical figures and then to study and understand them. After a term's work in industrial geometry, the pupil studies plane and then solid geometry with an interest which would not be possible otherwise.

The age of the pupils in the schools considered is such that, once they have become familiar with the elementary geometric concepts through their work in constructive geometry, the logical element of the subject can be made predominant.

Material aids to mathematical instruction are of course much in evidence in schools which make use of the "laboratory" or the "shop method" of instruction. In geometry and trigonometry, when climatic conditions are favorable, outdoor work is occasionally used. In solid geometry the use of models made by the pupils is not infrequent.

Excepting in the commercial schools, where the matter is one of vital importance, the matter of computation does not receive the general attention one would expect in schools of essentially vocational purpose. A few schools, however, lay much stress on the matter, and one, at least, offers a course on the subject, the outline of which may be quoted:

COMPUTATION, B class. (Applied Mathematics.)

A course in the interpretation and application of standard engineering formulæ, abbreviated methods of calculation, the use of mathematical tables, approximation by graphical methods, and the use of computing devices. The solution of practical problems.

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