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method, if there is sufficient time available to both the instructor and the students. I have followed it to a considerable extent. In regard to Professor Brook's suggestion, that is simply a detail as to assignments of work. I remember a time when some students of mine wanted to do work similar to that which Professor Brooks outlined, and they put in two and a half days during a vacation in making up the connection for one single experiment. When they had finished they had most of the apparatus in the laboratory in operation.

PROFESSOR RADTKE: Perhaps I did not emphasize sufficiently the statement that the electrical engineering course, or any engineering course, is not a final one. The student has then about completed his theoretical training, and he is going to make an application of it. His success depends on whether he can connect up what he has learned with what he will find in engineering practice. Special attention should be paid to ensure to the student having developed his power and ability to correlate his theoretical training to its application.



Professor of Applied Mathematics, University of Cincinnati.

The most promising feature of modern scientific and technical education is the widespread and growing interest in the improvement of present methods of instruction. This agitation for improved pedagogic methods is not confined to any one locality, but is general throughout England, Europe and America. It is, in fact, a manifestation of the modern spirit which tends increasingly to specialization, and which has classified teaching, among other professions, as a science rather than an art.

In common with all great truths, the fundamental principles underlying scientific pedagogy are characterized by two distinctive features, simplicity and naturalness. In the engineering profession it has long been recognized that the sole function of the engineer is the utilization of natural forces in the simplest and most efficient manner. By reason of the fact that mental processes are more elusive than physical, this idea has been longer in penetrating the teaching profession. Its acceptance, however, lies at the basis of scientific pedagogy, and it is now generally recognized that the chief aim of the teacher, whether in the class-room or through the textbook, is not merely to impart information, but to direct natural thought processes to a particular end.

To accomplish this purpose there are at least three essential requisites; the stimulation of interest in a particular subject, the utilization of this interest for the mastery of the fundamental principles underlying the subject, and the acquirement of proficiency in their practical application.

The relation of interest to instruction is one of the most important problems of pedagogy, and as such has received careful study, notably by Herbart and his school. To arouse interest is practically to put the mind in a receptive attitude, and such a preparation of the soil for the seed is obviously of fundamental importance. Without going into a scientific discussion of interest, it may be said that interest depends on the completeness of the apperception. In other words, an idea is interesting in proportion as it is related to other ideas already familiar. Instruction must therefore proceed through a chain of related ideas, the more fully each idea is related to previous experience the more rapid being its assimilation and the greater the interest aroused. In short, education is a process of growth rather than of accretion, and for growth continuity is essential.

This fact is often overlooked, especially in mathematical instruction, where the mistake is frequently made of setting up a series of definitions as the basis of presentation, and teaching the subject as an exercise in formal logic. A definition, however, is an isolated idea and can be of no vital interest until it is related to experience. To be able to mechanically follow out the results of a definition is not the aim of instruction, and teachers who rely upon their weight of authority to attain this end are unscientific as well as unpractical. To produce the continuity essential to growth, elementary instruction should therefore begin and continue with familiar things, following inductive methods, and waiting until thorough familiarity with the subject in hand is acquired before attempting to summarize knowledge in rules and formal definitions.

Closely connected with the question of the stimulation of interest is that of the sequence of development most favorable to the mastery of fundamental principles. This is also a psychological problem, but its solution is found at once in the simple fact, long since established, that the mental development of the individual follows the same course as that of the race, or, as Herbert Spencer expressed it, that the mental development of the individual is but a repetition of civilization in miniature. Since education means normal mental growth, it is apparent that the historical sequence of development is that which should be followed by scientific instruction. This so-called historical principle has been almost entirely overlooked by mathematicians and engineers, probably by reason of its extreme simplicity. It is, however, of fundamental importance to scientific pedagogy, especially in such subjects as mathematics and mechanics, which by reason of their long history afford an unusual opportunity for its application. The need of a universal criterion of instruction seems to be widely felt, and that furnished by the historical principle satisfies all demands, being simple and practical in its application as well as powerful in its results.

To illustrate the practical application of the historical method, a few instances may be cited from mathematics and mechanics. In elementary mathematics, the history of the subject indicates that instruction should begin by teaching addition as a method of counting, multiplication as an abbreviation of addition, involution as an abbreviation of multiplication, and subtraction, division and evolution as their inverses respectively. Logarithms should then be immediately introduced, and may be simply illustrated by the construction of paper slide rules. Logarithms were invented to supplement involution, and their introduction at this point adds greatly to clearness, especially if the word logarithm is dropped and they are called simply exponents. Calculation with five and seven place tables may be conveniently deferred until the student is farther advanced and greater accuracy is required than is possible with the ordinary slide rule. The remainder of elementary algebra should consist chiefly in an inductive proof of the binomial theorem, followed by the elementary theory of equations, including graphs. Ratio and proportion should be omitted as irrelevant, and the treatment of series reserved until demanded by the development of the calculus, where the history of the subject shows that it properly belongs. The introduction of graphs in the theory of equations is in strict accordance with the historical principle and the law of continuity. Not only does such correlation elucidate both subjects, but it also establishes a new series of connections between two branches of mathematics which students usually have difficulty in associating,

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