Mathematical research was practically introduced into the American colleges during the last quarter of the nineteenth century, and the wave of enthusiasm which attended this introduction was unfortunately not sufficiently tempered by emphasis on good teaching and breadth of knowledge, especially as regards applications. In fact, the leading mathematician in America during the early part of this period was glaringly weak along these lines. By means of his bountiful enthusiasm he was able to do a large amount of good for the selected band of gifted students who attended his lectures, but some of these were not so fortunate in securing the type of students who are helped more by the direct enthusiasm of their teacher than by the indirect enthusiasm resulting from good teaching.
The need of good mathematical teaching in our colleges and universities began to become more p.r.o.nounced at about the time that the wave of research enthusiasm set in, as a result of the growing emphasis on technical education which exhibited itself most emphatically in the development of the schools of engineering. While the student who is specially interested in mathematics may be willing to get along with a teacher whose enthusiasm for the new and general leads him to neglect to emphasize essential details in the presentation, the average engineering student insists on clearness in presentation and usability of the results. As the latter student does not expect to become a mathematical specialist, he is naturally much more interested in good teaching than in the mathematical reputation of his teacher, even if his reputation is not an entirely insignificant factor for him.
During the last decade of the nineteenth century and the first decade of the present century the mathematical departments of our colleges and universities faced an unusually serious situation as a result of the conditions just noted. The new wave of research enthusiasm was still in its youthful vigor and in its youthful mood of inconsiderateness as regards some of the most important factors. On the other hand, many of the departments of engineering had become strong and were therefore able to secure the type of teaching suited to their needs. In a number of inst.i.tutions this led to the breaking up of the mathematical department into two or more separate departments aiming to meet special needs.
In view of the fact that the mathematical needs of these various cla.s.ses of students have so much in common, leading mathematicians viewed with much concern this tendency to disrupt many of the stronger departments. Hence the question of good teaching forced itself rapidly to the front. It was commonly recognized that the students of pure mathematics profit by a study of various applications of the theories under consideration, and that the students who expect to work along special technical lines gain by getting broad and comprehensive views of the fundamental mathematical questions involved. Moreover, it was also recognized that the investigational work of the instructors would gain by the broader scholarship secured through greater emphasis on applications and the historic setting of the various problems under consideration.
To these fundamental elements relating to the improvement of college teaching there should perhaps be added one arising from the recognition of the fact that the number of men possessing excellent mathematical research ability was much smaller than the number of positions in the mathematical departments of our colleges and universities. The publication of inferior research results is of questionable value. On the other hand, many who could have done excellent work as teachers by devoting most of their energies to this work became partial failures both as teachers and as investigators through their ambition to excel in the latter direction.
=Range of subjects and preparation of students=
It should be emphasized that the college and university teachers of mathematics have to deal with a wide range of subjects and conditions, especially where graduate work is carried on. Advanced graduate students have needs which differ widely from those of the freshmen who aim to become engineers. This wide range of conditions calls for unusual adaptability on the part of the college and university teacher. This range is much wider than that which confronts the teachers in the high school, and the lack of sufficient adaptability on the part of some of the college teachers is probably responsible for the common impression that some of the poorest mathematical teaching is done in the colleges. It is doubtless equally true that some of the very best mathematical teaching is to be found in these inst.i.tutions.
In some of the colleges there has been a tendency to diminish the individual range of mathematical teaching by explicitly separating the undergraduate work and the more advanced work. For instance, in Johns Hopkins University, L. S. Hulburt was appointed "Professor of Collegiate Mathematics" in 1897, with the understanding that he should devote himself to the interests of the undergraduates. In many of the larger universities the younger members of the department usually teach only undergraduate courses, while some of the older members devote either all or most of their time to the advanced work; but there is no uniformity in this direction, and the present conditions are often unsatisfactory.
The undergraduate courses in mathematics in the American colleges and universities differ considerably. The normal beginning courses now presuppose a year of geometry and a year and a half of algebra in addition to the elementary courses in arithmetic, but much higher requirements are sometimes imposed, especially for engineering courses. In recent years several of the largest universities have reduced the minimum admission requirement in algebra to one year"s work, but students entering with this minimum preparation are sometimes not allowed to proceed with the regular mathematical cla.s.ses in the university.
=Variety of college courses in mathematics=
Freshmen courses in mathematics differ widely, but the most common subjects are advanced algebra, plane trigonometry, and solid geometry.
The most common subjects of a somewhat more advanced type are plane a.n.a.lytic geometry, differential and integral calculus, and spherical trigonometry. Beyond these courses there is much less uniformity, especially in those inst.i.tutions which aim to complete a well-rounded undergraduate mathematical course rather than to prepare for graduate work. Among the most common subjects beyond those already named are differential equations, theory of equations, solid a.n.a.lytic geometry, and mechanics.
A very important element affecting the mathematical courses in recent years is the rapid improvement in the training of our teachers in the secondary schools. This has led to the rapid introduction of courses which aim to lead up to broad views in regard to the fundamental subjects. In particular, courses relating to the historical development of concepts involved therein are receiving more and more attention. Indirect historical sources have become much more plentiful in recent years through the publication of various translations of ancient works and through the publication of extensive historical notes in the _Encyclopedie des Sciences Mathematiques_ and in other less extensive works of reference.
The problem presented by those who are preparing to teach mathematics may at first appear to differ widely from that presented by those who expect to become engineers. The latter are mostly interested in obtaining from their mathematical courses a powerful equipment for doing things, while the former take more interest in those developments which illumine and clarify the elements of their subject.
Hence the prospective teacher and the prospective engineer might appear to have conflicting mathematical interests. As a matter of fact, these interests are not conflicting. The prospective teacher is greatly benefited by the emphasis on the serviceableness of mathematics, and the prospective engineer finds that the generality and clarity of view sought by the prospective teacher is equally helpful to him in dealing with new applications. Hence these two cla.s.ses of students can well afford to pursue many of the early mathematical courses together, while the finishing courses should usually be different.
The rapidly growing interest in statistical methods and in insurance, pensions, and investments has naturally directed special attention to the underlying mathematical theories, especially to the theory of probability. Some inst.i.tutions have organized special mathematical courses relating to these subjects and have thus extended still further the range of undergraduate subjects covered by the mathematical departments. The rapidly growing emphasis on college education specially adapted to the needs of the prospective business man has recently led to a greater emphasis on some of these subjects in several inst.i.tutions.
The range of mathematical subjects suited for graduate students is unlimited, but it is commonly a.s.sumed to be desirable that the graduate student should pursue at least one general course in each one of broader subjects such as the theory of numbers, higher algebra, theory of functions, and projective geometry, before he begins to specialize along a particular line. It is usually taken for granted that the undergraduate courses in mathematics should not presuppose a knowledge of any language besides English, but graduate work in this subject cannot be successfully pursued in many cases without a reading knowledge of the three other great mathematical languages; viz., French, German, and Italian. Hence the study of graduate mathematics necessarily presupposes some linguistic training in addition to an acquaintance with the elements of fundamental mathematical subjects.
Historical studies make especially large linguistic demands in case these studies are not largely restricted to predigested material. This is particularly true as regards the older historical material. In the study of contemporary mathematical history the linguistic prerequisites are about the same as those relating to the study of other modern mathematical subjects. With the rapid spread of mathematical research activity during recent years there has come a growing need of more extensive linguistic attainments on the part of those mathematicians who strive to keep in touch with progress along various lines. For instance, a thriving Spanish national mathematical society was organized in 1911 at Madrid, Spain, and in March, 1916, a new mathematical journal ent.i.tled _Revista de Matematicas_ was started at Buenos Aires, Argentine Republic. Hence a knowledge of Spanish is becoming more useful to the mathematical student. Similar activities have recently been inaugurated in other countries.
=History of college mathematics=
Until about the beginning of the nineteenth century the courses in college mathematics did not usually presuppose a mathematical foundation carefully prepared for a superstructure. According to M.
Gebhardt, the function of teaching elementary mathematics in Germany was a.s.sumed by the gymnasiums during the years from 1810 to 1830.[5]
Before this time the German universities usually gave instruction in the most elementary mathematical subjects. In our own country, Yale University inst.i.tuted a mathematical entrance requirement under the t.i.tle of arithmetic as early as 1745, but at Harvard University no mathematics was required for admission before 1803.
On the other hand, _L"Ecole Polytechnique_ of Paris, which occupies a prominent place in the history of college mathematics, had very high admission requirements in mathematics from the start. According to a law enacted in 1795, the candidates for admission were required to pa.s.s an examination in arithmetic; in algebra, including the solution of equations of the first four degrees and the theory of series; and in geometry, including trigonometry, the applications of algebra to geometry, and conic sections.[6] It should be noted that these requirements are more extensive than the usual present mathematical requirements of our leading universities and technical schools, but _L"Ecole Polytechnique_ laid special emphasis on mathematics and physics and became the world"s prototype of strong technical inst.i.tutions.
The influence of _L"Ecole Polytechnique_ was greatly augmented by the publication of a regular periodical ent.i.tled _Journal de l"Ecole Polytechnique_, which was started in 1795 and is still being published. A number of the courses of lectures delivered at _L"Ecole Polytechnique_ and at _L"Ecole Normale_ appeared in the early volumes of this journal. The fact that some of these courses were given by such eminent mathematicians as J. L. Lagrange, G. Monge, and P. S.
Laplace is sufficient guarantee of their great value and of their good influence on the later textbooks along similar lines. In particular, it may be noted that G. Monge gave the first course in descriptive geometry at _L"Ecole Normale_ in 1795, and he was also for a number of years one of the most influential teachers at _L"Ecole Polytechnique_.
A most fundamental element in the history of college mathematics is the broadening of the scope of the college work. As long as college students were composed almost entirely of prospective preachers, lawyers, and physicians, there was comparatively little interest taken in mathematics. It is true that the mental disciplinary value of mathematics was emphasized by many, but this supposed value did not put any real life into mathematical work. The dead abstract reasonings of Euclid"s _Elements_, or even the number speculations of the ancient Pythagoreans, were enough to satisfy most of those who were looking to mathematics as a subject suitable for mental gymnastics.
On the other hand, when the colleges began to train men for other lines of work, when the applications of steam led to big enterprises, like the building of railroads and large ocean steamers, mathematics became a living subject whose great direct usefulness in practical affairs began to be commonly recognized. Moreover, it became apparent that there was great need of mathematical growth, since mathematics was no longer to be used merely as mental Indian clubs or dumb-bells, where a limited a.s.sortment would answer all practical needs, but as an implement of mental penetration into the infinitude of barriers which have checked progress along various lines and seem to require an infinite variety of methods of penetration.
The American colleges were naturally somewhat slower than some of those of Europe in adapting themselves to the changed conditions, but the rapidity of the changes in our country may be inferred from the fact that in the first half of the nineteenth century Harvard placed in comparatively short succession three mathematical subjects on its list of entrance requirements; viz., arithmetic in 1802, algebra in 1820, and geometry in 1844. Although Harvard had not established any mathematical admission requirements for more than a century and a half after its opening, she initiated three such requirements within half a century. It is interesting to note that for at least ninety years from the opening of Harvard, arithmetic was taught during the senior year as one of the finishing subjects of a college education.[7]
The pa.s.sage of some of the subjects of elementary mathematics from the colleges to the secondary schools raised two very fundamental questions. The first of these concerned mostly the secondary schools, since it involved an adaptation to the needs of younger students of the more or less crystallized textbook material which came to them from the colleges. The second of these questions affected the colleges only, since it involved the selection of proper material to base upon the foundations laid by the secondary schools. It is natural that the influence of the colleges should have been somewhat harmful with respect to the secondary schools, since the interests of the former seemed to be best met by restricting most of the energies of the secondary teachers of mathematics to the thorough drilling of their students in dexterous formal manipulations of algebraic symbols and the demonstration of fundamental abstract theorems of geometry.
=Relation of mathematics in secondary schools and college=
Students who come to college with a solid and broad foundation but without any knowledge of the superstructure can readily be inspired and enthused by the erection of a beautiful superstructure on a foundation laid mostly underground, with little direct evidence of its value or importance. The injustice and shortsightedness of the tendency to restrict the secondary schools to such foundation work would not have been so apparent if the majority of the secondary school students would have entered college. As a matter of fact it tended to bring secondary mathematics into disrepute and thus to threaten college mathematics at its very foundation. It is only in recent years that strong efforts have been made to correct this very serious mathematical situation.
Much progress has been made toward the saner view of letting secondary mathematics build its little structure into the air with some view to harmony and proportion, and of requiring college mathematics to build _on_ as well as _upon_ the work done by the secondary schools. The fruitful and vivifying notions of function, derivative, and group are slowly making their way into secondary mathematics, and the graphic methods have introduced some of the charms of a.n.a.lytic geometry into the same field.
This transformation is naturally affecting college mathematics most profoundly. The tedious work of building foundations in college mathematics is becoming more imperative. The use of the rock drill is forcing itself more and more on the college teacher accustomed to use only hammer and saw. As we are just entering upon this situation, it is too early to prophesy anything in regard to its permanency, but it seems likely that the secondary teachers will no more a.s.sume a yoke which some of the college teachers would so gladly have them bear and which they bore a long time with a view to serving the interests of the latter teachers.
As many of the textbooks used by secondary teachers are written by college men, and as the success of these teachers is often gauged by the success of their students who happen to go to college, it is easily seen that there is a serious temptation on the part of the secondary teacher to look at his work through the eyes of the college teacher. The recent organizations which bring together the college and the secondary teachers have already exerted a very wholesome influence and have tended to exhibit the fact that the success of the college teacher of mathematics is very intimately connected with that of the teachers of secondary mathematics.
While it is difficult to determine the most important single event in the history of college teaching in America, there are few events in this history which seem to deserve such a distinction more than the organization of the Mathematical a.s.sociation of America which was effected in December, 1915. This a.s.sociation aims especially to promote the interests of mathematics in the collegiate field and it publishes a journal ent.i.tled _The American Mathematical Monthly_, containing many expository articles of special interest to teachers.
It also holds regular meetings and has organized various sections so as to enable its members to attend meetings without incurring the expense of long trips. Its first four presidents were E. R. Hedrick, Florian Cajori, E. V. Huntington, and H. E. Slaught.
An event which has perhaps affected the very vitals of mathematical teaching in America still more is the founding of the American Mathematical Society in 1888, called the New York Mathematical Society until 1894. Through its _Bulletin_ and _Transactions_, as well as through its meetings and colloquia lectures, this society has stood for inspiration and deep mathematical interest without which college teaching will degenerate into an art. During the first thirty years of its history it has had as presidents the following: J. H. Van Amringe, Emory McClintock, G. W. Hill, Simon Newcomb, R. S. Woodward, E. H.
Moore, T. S. Fiske, W. F. Osgood, H. S. White, Maxime Bocher, H. B.
Fine, E. B. Van Vleck, E. W. Brown, L. E. d.i.c.kson, and Frank Morley.
=Aims of college mathematics: methods of teaching=
The aims of college mathematics can perhaps be most clearly understood by recalling the fact that mathematics const.i.tutes 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 princ.i.p.al 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 circ.u.mstances. 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 cla.s.sroom, 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 const.i.tute in many of our leading inst.i.tutions a large majority in the mathematical cla.s.ses. 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.
The lecture method, on the other hand, aims to exhibit the main facts in a clear light and to leave to the student the task of supplying further ill.u.s.trative examples and of reconsidering the various steps.
The purely lecture method does not seem to be well adapted to American conditions, and it is frequently combined with what is commonly known as the "quiz." The quiz seems to be an American inst.i.tution, although it has much in common with a species of the French "conference." It is intended to review the content of a set of lectures by means of discussions in which the students and the teacher partic.i.p.ate, and it is most commonly employed in connection with the courses of an advanced undergraduate or of a beginning graduate grade.
A prominent aim in graduate courses is to lead the student as rapidly as possible to the boundary of knowledge along the particular line considered therein. While some of the developments in such courses are apt to be somewhat special or to be too general to have much meaning, their novelty frequently adds a sufficiently strong element of interest to more than compensate losses in other directions. Moreover, the student who aims to do research work will thus be enabled to consider various fields as regards their attractiveness for prolonged investigations of his own.
=Preparation of the college teacher of mathematics.=