The aims of college mathematics can perhaps be most clearly understood by recalling the fact that mathematics constitutes a kind of intellectual shorthand and that many of the newer developments in a large number of the sciences tend toward pure mathematics. In particular, "there is a constant tendency for mathematical physics to be absorbed in pure mathematics."[[8]] As sciences grow, they tend to require more and more the strong methods of intellectual penetration provided by pure mathematics.
The principal modern aim of college mathematics is not the training of the mind, but the providing of information which is absolutely necessary to those who seek to work most efficiently along various scientific lines. Mathematical knowledge rather than mathematical discipline is the main modern objective in the college courses in mathematics. As this knowledge must be in a usable form, its acquisition is naturally attended by mental discipline, but the knowledge is absolutely needed and would have to be acquired even if the process of acquisition were not attended by a development of intellectual power.
The fact that practically all of the college mathematics of the eighteenth century has been gradually taken over by the secondary schools of today might lead some to question the wisdom of replacing this earlier mathematics by more advanced subjects. In particular, the question might arise whether the college mathematics of today is not superfluous. This question has been partially answered by the preceding general observations. The rapid scientific advances of the past century have increased the mathematical needs very rapidly. The advances in college mathematics which have been made possible by the improvements of the secondary schools have scarcely kept up with the growth of these needs, so that the current mathematical needs cannot be as fully provided for by the modern college as the recognized mathematical needs of the eighteenth century were provided for by the colleges of those days.
There appears to be no upper limit to the amount of useful mathematics, and hence the aim of the college must be to supply the mathematical needs of the students to the greatest possible extent under the circumstances. In order to supply these needs in the most economical manner, it seems necessary that some of them should be supplied before they are fully appreciated on the part of the student. The first steps in many scientific subjects do not call for mathematical considerations and the student frequently does not go beyond these first steps in his college days, but he needs to go much further later in life. College mathematics should prepare for life rather than for college days only, and hence arises the desirability of deeper mathematical penetration than appears directly necessary for college work.
Advanced work in college mathematics
Another reason for more advanced mathematics than seems to be directly needed by the student is that the more advanced subjects in mathematics are a kind of applied mathematics relative to the more elementary ones, and the former subjects serve to throw much light on the latter. In other words, the student who desires to understand an elementary subject completely should study more advanced subjects which are connected therewith, since such a study is usually more effective than the repeated review of the elementary subject. In particular, many students secure a better understanding of algebra during their course in calculus than during the course in algebra itself, and a course in differential equations will throw new light on the course in calculus. Hence college mathematics usually aims to cover a rather wide range of subjects in a comparatively short time.
Since mathematics is largely the language of advanced science, especially of astronomy, physics, and engineering, one of the prominent aims of college mathematics should be to keep in close touch with the other sciences. That is, the idea of rendering direct and efficient services to other departments should animate the mathematical department more deeply than any other department of the university. The tendency toward disintegration to which we referred above has forcefully directed attention to the great need of emphasizing this aspect of our subject, since such disintegration is naturally accompanied by a weakening of mathematical vigor. It may be noted that such a disintegration would mean a reverting to primitive conditions, since some of the older works treated mathematics merely as a chapter of astronomy. This was done, for instance, in some of the ancient treatises of the Hindus.
Mathematics and technical education
The great increase in college students during recent years and the growing emphasis on college activities outside of the work connected with the classroom, especially on those relating to college athletics, would doubtless have left college mathematics in a woefully neglected state if there had not been a rapidly growing interest in technical education, especially in engineering subjects, at the same time. Naval engineering was one of the first scientific subjects to exert a strong influence on popularizing mathematics. In particular, the teaching of mathematics in the Russian schools supported by the government began with the founding of the government school for mathematics and navigation at Moscow in 1701. It is interesting to note that the earlier Russian schools established by the clergy after the adoption of Christianity in that country did not provide for the teaching of any arithmetic whatever, notwithstanding the usefulness of arithmetic for the computing of various dates in the church calendar, for land surveying, and for the ordinary business transactions.[[9]]
The direct aims in the teaching of college mathematics have naturally been somewhat affected by the needs of the engineering students, who constitute in many of our leading institutions a large majority in the mathematical classes. These students are usually expected to receive more drill in actual numerical work than is demanded by those who seek mainly a deeper penetration into the various mathematical theories. The most successful methods of teaching the former students have much in common with those usually employed in the high schools and are known as the recitation and problem-solving methods. They involve the correction and direct supervision of a large number of graded exercises worked out by the students on the blackboard or on paper, and aim to overcome the peculiar difficulties of the individual students.