MATHEMATICS IN THE TECHNICAL SECONDARY SCHOOLS IN THE UNITED STATES.

PART I.

INTRODUCTORY.

The secondary technical schools of the United States, because of their heterogeneity, present peculiar difficulties to an investigation along the lines laid down by the International Commission. While such schools have existed for many years, it is particularly within the last decade that a great increase in their numbers has taken place, for it is within that period that the tendency to break away from the traditions of the general secondary schools and to bring the schools into close contact with industrial and commercial life, rather than to raise to a maximum their efficiency in furnishing preparation for higher education, has become a movement of sufficient strength to alter essentially the character of existing schools and to determine that of those newly established.

The "manual training high school " is the oldest of the important types of public secondary technical schools in the country. As a type, moreover, it is the most conservative of the schools to be considered in this report, in that to a large extent the traditions of the general secondary school have been retained and the function of the school as an instrument of preparation for higher education emphasized. The ideals of this type of school are essentially scientific rather than classical or technical.

At the other extreme stands the "trades school," a type which is in its infancy as a public institution but examples of which have existed for many years as private or endowed institutions. Here the aim is primarily that of training for immediate entrance into a definite trade or trades and for efficiency in the work thereof. The school is not, or is not primarily at least, an intermediate step in the student's progress toward higher education.

These types represent the extremes. It must, however, be remembered that the lines of demarcation are by no means sharply drawn and that rigid classification is, at the present time at least, scarcely possible.

The schools just mentioned are in the province of subcommittee 1. The schools considered by subcommittee 2 fall into three classeshigh schools of commerce, commercial departments of general secondary schools, and private commercial schools (the so-called "business colleges"). On these schools, too, though to a less extent, the influence of the traditions of the general secondary school is in evidence at one end of the series, while at the other end the ideal of the trades school, i. e., training for immediate business activity, is dominant.

The secondary agricultural schools studied by subcommittee 3 are of recent origin. More than schools of the other two classes they are supported in whole or in part by State rather than by municipal appropriations, and consequently are to a greater extent under State supervision. Their object is to provide such an education for the youth of the agricultural community as will tend to retain him in that community as an efficient member thereof. In consequence, except in so far as they lead to the agricultural colleges, their tendency is toward producing immediate vocational efficiency rather than to serve as a step toward higher education.

In view of this diversity it may well be asked what there is in common among these schools that justifies their inclusion in the same category. The obvious answer is that the schools in question are the most recent result of that movement which has led to the establishment of the technical colleges and the broadening of the curricula of the classical colleges and secondary schools, namely, the movement toward bringing the instruction within the school into closer contact with the phenomena and problems of life outside the school, and toward making the knowledge gained in the school more immediately useful to the pupil when he leaves.

AIM OF THE MATHEMATICAL INSTRUCTION.

With the wide differences in general object which exist in the technical secondary schools are naturally associated similar differences in the aim of the mathematical instruction therein.

The schools show in different degrees the common tendency to emphasize the utilitarian side of the subject. Large, well-organized schools which form an integral part of a municipal school system or are controlled by university authorities, while recognizing the utilitarian side and providing for it by suitable selection of problems and correlation with the work of the technical departments, also emphasize the logical element of the subject and the importance of mathematics as an independent science.

There are schools in which, if the character of the text used is any indication, the formal element is predominant; and on the other

hand, as in the trades schools and the private commercial schools, the aim is mainly to produce accuracy and speed in the application of a limited range of mathematical principles to the problems of a definite trade or occupation.

The situation may be illustrated by the following quotations from information furnished by various schools:

(1) The ends to be attained are the knowledge of a body of geometrical truths to be used in the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest in the science of mathematics.

(2) In mathematics two ends are constantly kept in view: First, stimulation of the inventive faculty, exercise of judgment, development of logical reasoning, and the habit of concise statement; second, the association of the branches of pure mathematics with each other and with applied science, that the pupil may see clearly the true relations of principles and things.

(3) It is the aim to give that knowledge and training to the students that shall make them capable men, ready to meet successfully the practical questions of everyday life, and to solve intelligently the problems constantly arising in office, factory, and field; hence, the practical side of mathematics is emphasized rather than the purely theoretical. Abstract mathematical discussions, as such, are avoided except as they are necessary to a better comprehension of results, and then they are made as direct and clear as possible. Stress is placed upon the application to mechanical, physical, and electrical problems, but it is intended that the instruction shall be of such a character as to give the student power and incentive to perform ordinary mathematical work with confidence, precision, and success.

(4) In the courses in mathematics the main purpose is to train the students, not to prove propositions and formulæ, but to make intelligent use of these propositions and formulæ in the solution of original problems.

(5) They must know enough of mathematics, drawing, and science to insure intelligent, progressive workmanship, as contrasted with rule of thumb methods. (6) We aim to give them some idea of the subject of elementary mathematics with special reference to its application in technical studies.

(7) The aim of the courses is twofold: First, to teach the methods of computation necessary for the solution of common problems arising in shop practice; second, to present in condensed form the essentials of algebra, geometry, and trigonometry for the benefit of those who have not had a high-school training, and to show the applications of these subjects to the more advanced types of shop problems.

(8) The practical results of the method are usually the acquisition of certain "rules of thumb" which are immediately available in the trade work of the student. There is probably no very great increase in mathematical ability.

(9) The course in mathematics is designed:

First, to develop in the pupil the power of independent thought, to cultivate the inventive faculty, and to inculcate the habit of clear, concise, logical statement. To this end the course is so arranged that the graphic, concrete branches of the subject precede those that are abstract and analytic.

Second, to teach the student the importance of mathematics in relation to the applied sciences, the mechanic arts, and to business life. For this purpose he is required to apply the formulas of algebra and trigonometry to physics, mechanics, chemistry, and engineering; and the short methods of arithmetic and mensuration to the practical work of bookkeeping and architecture.

The course in mathematics as taught in this school is both preparatory and complete. Those boys who finish their studies here possess a good working knowledge of the subject; and those who continue their studies in colleges or in technical schools possess an adequate preparation for higher work.

(10) The aim of the instruction is to inculcate habits of accuracy, rapidity, and neatness in the manipulation of algebraic operations, and to inspire a thorough knowledge of the fundamental principles and laws of the subject. To aid in securing these results the pupils are required to solve a large number of carefully selected problems.

During the first half of the sophomore year algebra is again taken up. A thorough review of such portions of the elementary algebra as are deemed necessary by the instructor is followed by a course in advanced algebra. This course covers topics usually studied in the freshman year in the colleges and higher technical schools.

Five recitations a week during the second half of the sophomore year and the first half of the junior year are devoted to plane and solid geometry. The instruction aims primarily to use the subject as an instrument of education. Geometry contains a system of knowledge that is indispensable to success in many of the pursuits of life, but the presentation of this system of knowledge can never be other than a secondary object in a course of proper instruction in the subject. In reality the pupil ordinarily comes to the subject with many of its leading facts already in his possession. The real objects kept constantly in view in teaching the subjects are training in logical reasoning, an object of increased importance, as it is the only course in strict reasoning with which a large number of young people ever become closely acquainted; training in clear and accurate expression, an object not wisely neglected in any department of instruction; training in imagination and invention. To aid in these objects, extensive practice in original exercises is given, in which the pupil is required to devise his own proof, under the guidance and suggestions of the instructor.

The last half of the junior year is given to plane and analytical trigonometry. The textbook is supplemented by frequent familiar talks pointing out the best methods of procedure and illustrating the applications of the subject to surveying, navigation, etc. Special stress is placed upon the use of logarithms in computations and also upon analytical work to insure familiarity on the part of the pupil with the transformations and definitions necessary to success in future mathematical and engineering courses.

The schools considered are "secondary" schools, whatever their specific name, and consequently the aim of their instruction, except in the case of the private commercial schools and certain trades schools, is to a greater or less degree influenced by the requirements for admission to higher institutions. In fact, this element is an important consideration in all of the activities of these schools, though of course it is not necessarily predominant.

The mathematical curricula include arithmetic, commercial arithmetic, algebra, geometry, trigonometry, analytic geometry, the calculus, history of mathematics, and so-called "applied" or "shop" mathematics.