The aims of the course are:

1. To give the student some adequate acquaintance with computing methods. 2. To develop in him at the same time accuracy and speed.

3. To cultivate the ability to estimate results with a reasonably close degree of approximation.

4. To minimize labor in his calculations.

The experience of the vocational schools indicates that the solution of the problem of giving mathematics a better place in popular instruction and of reacting against the prejudices against the science lies in bringing the subject into such close relation with the activities of daily life, especially those of an industrial nature, that the necessity of a knowledge of the subject is felt. From the standpoint of the progress of the science this is the valuable element in "workshop mathematics."

The dangerous element in "workshop mathematics" is not fundamentally distinct from that which is so often the bane of abstract instruction and against which the representatives of the secondary technical schools so emphatically protest, namely, excessive formalism. By formalism is here meant the unthinking and mechanical execution of mathematical processes without regard to the significance of the data, the operations, or the results.

This formalism may be illustrated by the following problem taken from a published collection that is somewhat widely used: The dimensions of the parts of a rather complex combination of crank, screw, and gears are given, and it is required to find the weight which can be raised by a force of 60 pounds applied to the crank. The mathematical work involved is merely the numerical evaluation of a rational fraction the factors of whose terms are given. The published answer is 203,575.68 pounds; that is, a weight of a hundred tons is given to the sixth part of an ounce. It is evident that such a result can be obtained only by a mechanical and unthinking use of the mathematical processes involved, and without the slightest consideration of the significance of the concrete elements of the problem. This is formalism pure and simple, and it is the more pernicious in that it masquerades under the guise of "shop mathematics" and claims to be an example of "how to apply mathematical principles, rules, and formulas to the solution of such (i. e., shop) problems.”

There is a desire among some of the teachers of mathematics in the secondary technical schools to break down the conventional barriers between the several branches of mathematics, and in a few cases this desire has been realized. The principal difficulty mentioned concerning this movement appears to be the lack of textbooks designed for such combined courses, a difficulty which is not inconsiderable by reason of the dominant position of the textbook in American mathematical instruction. A few such texts, however, have been published.

There are administrative difficulties, particularly in the case of schools which articulate closely with the grades on the one hand and the colleges on the other, and these difficulties, while not mentioned by the schools, have proved a serious barrier to the movement in the colleges. The conservatism of teachers also retards the establishment of these combined courses.

It would appear that the technical and industrial schools offer unusual advantages for the development of combined courses, because of the fact that, in the newer ones at least, the articulation with established schools is less close and the force of tradition less strong than in the general secondary schools. It must be remembered, however, that these combined courses are of comparatively recent origin, while the traditional courses are the result of a long period of evolution. The combined course is in the experimental stage, and for this stage of its evolution the secondary technical schools offer, for the reasons just mentioned, a favorable culture medium.

Wherever the courses are separated it appears that the study of the elements of algebra precedes that of plane geometry. The relative position of the second course in algebra and that in solid geometry is variable.

RELATION BETWEEN MATHEMATICS AND OTHER BRANCHES.

While systematic coadaptation of the courses in mathematics and those in other subjects is not general, there is a strong tendency to make such adaptation to a greater or less degree. The tendency may, and sometimes does, take the form of emphasis on the application of mathematical results at the expense of the logical and demonstrative element of mathematics and of its dignity as an independent science. The "pocketbook engineer" has his counterpart in the secondary schools. More frequently, however, the tendency finds a more rational expression in the form of emphasis, in the mathematical courses, upon problems derived from other branches. These problems may serve as an introduction to the demonstrative work or as an application of its results.

The difficulties and dangers in the working out of this tendency are precisely the same as those which arise in a similar situation in the higher schools. Unfortunately, neither the teacher of mathematics nor the teacher of the technical subjects is omniscient. The one lacks technical training, the other a thoroughly grounded knowledge of the science of mathematics.

A study of published collections of problems used by some of the schools indicates that the mathematical principles involved in the technical problems considered in their courses are, for the most part, of a very elementary character. In geometry, the propositions

of congruence and similarity, the theorem of Pythagoras, and the mensurational theorems; in trigonometry, the definitions and elementary properties of the functions with their use in the composition and resolution of vectors; and in algebra the fundamental operations and the solution of linear equations and binominal equations of lower degree form the theoretical basis for the greater part of the problems in question. The problems arising in surveying, of course, require more extended knowledge of trigonometry, and the varied problems of the machine shop involve algebraic principles of a more advanced character.

For example, problems on the efficiency of hoisting devices (friction considered) and in the design of cone pulleys involve geometric progressions; problems of gearing and screw cutting involve indeterminate equations, Euclid's algorithm of the greatest common divisor and continued fractions.

REPORT OF SUBCOMMITTEE ON SECONDARY COMMERCIAL SCHOOLS.

Sources of information.-This report is based on statistical information obtained by means of questionnaires and on other data available to the members of the committee, especially the chairman, as members of the instructing staffs of commercial schools.

The report was prepared by the chairman of the subcommittee in consultation with the other members and with the chairman of the committee on secondary technical schools.

Aim of the report.-As some of the work of the schools considered does not greatly differ from that of general secondary schools the subcommittee has confined itself largely in this report to the consideration of the points of difference between the work of the commercial and of the general secondary schools-the nature, cause, and results of these differences.

Classification of schools.-The schools considered by the subcommittee fall into three classes, viz, high schools of commerce, commercial departments of general secondary schools, and private commercial schools (the so-called business colleges).

The private schools were first in the field; their primary aim was and is preparation for immediate vocational activity. Though they are, therefore, essentially of the same nature as "trades schools" and are largely conducted for profit, the committee is impressed with their educational value and the professional spirit of their instructing staffs.

There are many of these schools throughout the country. The fact of their existence is proof of the demand for the kind of education they offer. It is a further testimony of the work done by these schools when it is cited that some cities, as Berkeley, Cal., already offer a two-years' course in commercial subjects. Boston, Mass., has just voted to establish a central clerical high school, to be in session from 9 to 5; its scope of work will be that usually offered by the best business colleges.

The commercial department in the general high school is the natural outgrowth of the success of the private commercial school, just

as earlier in the history of our country the general high school was the outcome of the success of the academy. Moreover, just as the private academies began to go out of existence with the success of the general high school, so that now but a comparatively few of those in existence from 1850 to 1860 still remain, in the same way it is probable that the public schools will do more and more of the work now being done by the private commercial schools. And just as there are now some academies still in existence, and in a most healthy condition, so is it likely that we shall always have our private commercial school; but it is improbable that we shall have all that we have at present.

The same observation which is made here with regard to the pioneer work of the "commercial college" is manifest in the history of many features of our present educational system; the need is first shown by experiment carried by private enterprise, either philanthropic or commercial; then the public-school department, hitherto passive, becomes eager to incorporate the private success into its own field of activities.

Since the commercial department of the general high school aims to do the work of the private commercial school, there will be but little to report in regard to them in addition to what will be reported for the private school beyond the fundamental differences between the two kinds of schools in all respects.

The commercial high school is but one step beyond the commercial department and must be from the nature of affairs restricted to the larger cities. However, this step to the separate high school is a long one. The "high school of commerce" expects to give a better fitting for business life than either the private commercial school or the commercial department can; better than the one because the course is longer and its scope is broader; better than the other because the work of the four years in the high school is much more specialized.

COURSES OF THE SEVERAL TYPES OF SCHOOLS.

In the private commercial school the length of course varies from three months to two years, depending on the preparation and wish of the pupils. The usual course, however, is for one year. Certificates or diplomas are given for work covered. The same work is given for both boys and girls, as would be expected since they are found in the same class. The entrance requirements are not as clearly defined as for the public secondary school; the scheme is rather to put the pupils into those classes where they can take the work to best advantage. The age of the pupils varies from 14 to 20, as a general rule; both younger and older pupils, however, will be found in attendance.

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In the commercial department of the high school, the course is of four years' length; the studies in the commercial department are but a part of the student's studies; the others are taken from the general course of the school. The diploma given at the end of the course is the general school diploma and does not usually specify that the pupil is a graduate of the commercial department; it states rather that he is a graduate of the school; the diploma is the same as that given to the graduates of all the other departments of the schoolthis statement holds true so far as facts have come to the observation of the members of this committee. The work of the first year for pupils in this course does not vary materially from that of the other pupils of the school; accordingly, the same entrance requirement holds for all-that they satisfactorily complete the grammar-school course. The average age is from 14 to 20 years from entrance to graduation.

In the commercial high school, usually called "high school of commerce," the same general consideration for age and entrance holds true as in the case of the commercial department. The curriculum, however, is more specialized. The aim of the school is to prepare for a commercial life in a broad sense. The commercial departments prepare more for secretarial and clerkship positions and make bookkeeping, stenography, and typewriting the courses around which the work of the school centers. The schools of commerce make the economic sciences and courses the subjects around which the work of the school centers. The other subjects are studied, but are given but comparatively small emphasis in working out the aims of the school. Business men's organizations connected with such schools are of great value to them.

THE AIM AND SCOPE OF THE MATHEMATICAL INSTRUCTION.

The only branch of mathematics that is taught in the private commercial school is commercial arithmetic-sometimes called business arithmetic. The placing of commercial arithmetic as one of the branches of mathematics is one of the points of difference between the American and the German practice, for in Germany it is considered as one of the branches of the commercial studies. The aim of the work is to give drill, constant drill, in the ordinary operations of business, so as to secure habits of accuracy, speed of computation, and skill in mental operations.

There are usually five recitations per week, and the usual length of the recitation is 45 minutes. Some schools have recitations, however, that are one hour in length. The work in commercial arithmetic in these schools is made to correlate more with the work in bookkeeping than with any other subject. The teachers in these schools frequently make the complaint about the previous prepara