College Teaching / Studies in Methods of Teaching in the College

If the stated course in earth-science is limited to the junior and senior year by the existing requirements of the curriculum of the institution or by the rulings of its officers—as is not uncommonly the case at present—it is relatively immaterial whether the sections of the course are marshaled under the single name "geology" or whether they are given separate titles as sub-sciences, provided the special subjects are arranged in logical sequence and in consecutive order. If, on the other hand, the teacher's choice of time and relations is freer, the more accessible phases of earth study, now well organized under the name of "physiography," form an excellent course for either freshmen or sophomores. It opens their minds to a world of interesting activities about them which have probably been largely overlooked in previous years. It gives them substance of thought that will be of much service in the pursuit of other sciences. It has been found that it is not without rather notable service to young students as the basis of efforts in the art of literary presentation, a felicity to which teachers of this important art frequently give emphatic testimony. The secret seems to lie in the fact that physiography gives varied and vivid material susceptible of literary presentation, while the fixed qualities of the subject matter control the choice of terms and the mode of expression.

If geography and physiography are given in the earlier years, the course in historical geology, as well as the study of the more difficult phases of geological processes, of the principles of dynamic geology, together with mineralogy, petrology, and paleontology, may best fall into the later years, even if some interval separates them from the geography and physiography.

One hundred and twenty classroom hours, or their equivalent in laboratory and field work, are perhaps to be regarded as the irreducible minimum in a well-balanced undergraduate course, while twice that time or more is required to give a notably strong college course in earth-science.

A consideration of the sequences among the geological sub-subjects, as also among the subjects that are held to be preliminary to the earth-sciences, is important, but it would lead us too far into details which depend more or less on local conditions. In the experience of American teachers it appears to have been found advisable to put geological processes and typical phenomena to the front and to take up geological history afterwards. The earlier method of taking up the history first, beginning with recent stages and working backward down the ages,—once in vogue abroad,—has been abandoned in this country. It was the order in which the science was developed and it had the advantage of starting with the living present and with the most accessible formations, but this latter advantage is secured by studying the living processes, as such, first, and turning to the history later. This permits the study of the history in its natural order, which seems better to call forth the relations of cause and effect and to give emphasis to the influence of inherited conditions.

Respecting antecedents to the study, the more knowledge of physics, chemistry, zoölogy, and botany, the better, but it is easy to over-stress the necessity for such preparation, however logical it may seem, for in reality all the natural sciences are so interwoven that, in strict logic, a complete knowledge of all the others should be had before any one is begun, a reductio ad absurdum. The sciences have been developed more or less contemporaneously and progressively, each helping on the others. They may be pursued much in the same way, or by alternations in which each prior study favors the sequent one. They may even be taken in a seemingly illogical order without serious disadvantage, for the alternative advantages and other considerations may outweigh the force of the logical order, which is at best only partially logical. It is of prime importance to stimulate in students a habit of observing natural phenomena at an early age. It may be wise for a student to take up physiography, or its equivalent, early in the college course, irrespective of an ideal preparation in the related sciences. It is unfortunate to defer such study to a stage when the student's natural aptitude for observation and inference has become dulled by neglect or by confinement to subjects devoid of naturalistic stimulus. To permit students to take up earth-science in the freshman and sophomore years, even without the ideal preparation, is therefore probably wiser than to defer the study beyond the age of responsiveness to the touch of the natural environment. The geographic and geologic environment conditioned the mental evolution of the race. It left an inherited impress on the perceptive and emotional nature, only to be awakened most felicitously, it would seem, at about the age at which the naturalistic phases of the youth's mentality were originally called into their most intense exercise.

T. C. Chamberlin
The University of Chicago

VIII

The German mathematical developments of the greater part of the nineteenth century exhibited a growing tendency to disregard applications. It was not until about 1890 that a strong movement was inaugurated to lay more stress on applied mathematics in Germany.[3] Our early American students therefore brought with them from Germany a decided tendency toward investigations in mathematical fields remote from direct contact with applications to other scientific subjects, such as physics and astronomy, which had so largely dominated mathematical investigations in earlier years.

This picture would, however, be very incomplete without exhibiting another factor of a similar type working in our own midst. J. J. Sylvester was selected as the first professor of mathematics at Johns Hopkins University, which opened its doors in 1876 and began at once to wield a powerful influence in starting young men in higher research. Sylvester's own investigations related mainly to the formal and abstract side of mathematics. Moreover, "he was a poor teacher with an imperfect knowledge of mathematical literature. He possessed, however, an extraordinary personality; and had in remarkable degree the gift of imparting enthusiasm, a quality of no small value in pioneer days such as these were with us."[4]

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 pronounced 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 institutions 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 classes 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.

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 institutions.

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 classes in the university.

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 analytic geometry, differential and integral calculus, and spherical trigonometry. Beyond these courses there is much less uniformity, especially in those institutions 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 analytic 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 Encyclopédie des Sciences Mathématiques 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 classes 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 institutions 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 institutions.

The range of mathematical subjects suited for graduate students is unlimited, but it is commonly assumed 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 entitled 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.

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 assumed 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 instituted a mathematical entrance requirement under the title 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 pass 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 institutions.

The influence of L'Ecole Polytechnique was greatly augmented by the publication of a regular periodical entitled 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 assortment 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 passage 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.

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 analytic 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 assume 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 Association of America which was effected in December, 1915. This association aims especially to promote the interests of mathematics in the collegiate field and it publishes a journal entitled 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 Bôcher, H. B. Fine, E. B. Van Vleck, E. W. Brown, L. E. Dickson, and Frank Morley.

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.

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.

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.

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 illustrative 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 institution, 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 participate, 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.

The fact that the college teacher has need of much more mathematical knowledge than he can possibly secure during the period of his preparation, especially if he expects to take an active part in research and in directing graduate work, has usually led to the assumption that the future teacher of college mathematics should devote all his energies to securing a deep mathematical insight and a wide range of mathematical knowledge.[10] On the other hand, students prepared in accord with this assumption have frequently found it very difficult to adapt themselves to the needs of large freshman classes of engineering students entering upon the duties for which they were supposed to have been prepared.

The breadth of view and the sweep of abstraction needed for effective graduate work have little in common with accuracy in numerical work and emphasis on details which are so essential to the young engineering students. The difficulty of the situation is increased by the fact that the young instructor is often led to believe that his advancement and the appreciation of his services are directly proportional to his achievements in investigations of a high order. This belief naturally leads many to begrudge the time and thought which their teaching duties should normally receive.

The young college teacher of mathematics is thus confronted with a much more complex situation than that which confronts the mathematics teachers in secondary school work. Here the success in the classroom is the one great goal, and the mathematical knowledge required is comparatively very modest. Possibly the situation of the college teacher could be materially improved if it were understood that his first promotion would be mainly dependent upon his success as a teacher, but that later promotions involved the element of productive scholarship in an increasing ratio.

The schools of education which have in recent years been established in most of our leading universities have thus far had only a slight influence on the preparation of the college teachers, but it seems likely that this influence will increase as the needs of professional training become better known. It is probably true that the ratio of courses on methods to courses on knowledge of the subject will always be largest for the elementary teacher, in view of the great difference between the mental maturity of the student and the teacher, somewhat less for the secondary teacher and least for the college teacher; but this least should not be zero, as is so frequently the case at present, since there usually is even here a considerable difference between the mathematical maturity of the student and that of the teacher.

It may be argued that the future college teacher will probably profit more by noting the methods employed by his instructors than he would by the theoretic discussions relating to methods. This is doubtless true, but it does not prove that the latter discussions are without value. On the other hand, these discussions will often serve to fix more attention on the former methods and will lead the student to note more accurately their import and probable adaptability to the needs of the younger students.

Among the useful features for the training of the future mathematics teachers are the mathematical clubs which are connected with most of the active mathematical departments. In many cases, at least, two such clubs are maintained, the one being devoted largely to the presentation of research work while the other aims to provide opportunities for the presentation of papers of special interest to the students. The latter papers are often presented by graduate students or by advanced undergraduates, and they offer a splendid opportunity for such students to acquire effective and clear methods of presentation. The same desirable end is often promoted by reports given by students in seminars or in advanced courses.

Prominent factors in the training of the future college teachers are the teaching scholarships or fellowships and the assistantships. Many of the larger universities provide a number of positions of this type. It sometimes happens that the teaching duties connected with these positions are so heavy as to leave too little energy for vigorous graduate work. On the other hand, these positions have made it possible for many to continue their graduate studies longer than they could otherwise have done and at the same time to acquire sound habits of teaching while in close contact with men of proved ability along this line.

It should be emphasized that the ideal college teacher of mathematics is not the one who acquires a respectable fund of mathematical knowledge which he passes along to his students, but the one imbued with an abiding interest in learning more and more about his subject as long as life lasts. This interest naturally soon forces him to conduct researches where progress usually is slow and uncertain. Research work should be animated by the desire for more knowledge and not by the desire for publication. In fact, only those new results should be published which are likely to be helpful to others in starting at a more favorable point in their efforts to secure intellectual mastery over certain important problems.

Half a century ago it was commonly assumed that graduation from a good college implied enough training to enter upon the duties of a college teacher, but this view has been practically abandoned, at least as regards the college teacher of mathematics. The normal preparation is now commonly placed three years later, and the Ph.D. degree is usually regarded to be evidence of this normal preparation. This degree is supposed by many to imply that its possessor has reached a stage where he can do independent research work and direct students who seek similar degrees. In view of the fact that in America as well as in Germany the student often receives much direct assistance while working on his Ph.D. thesis, this supposition is frequently not in accord with the facts.[11]

The emphasis on the Ph.D. degree for college teachers has in many cases led to an improvement in ideals, but in some other cases it has had the opposite effect. Too many possessors of this degree have been able to count on it as accepted evidence of scientific attainments, while they allowed themselves to become absorbed in non-scientific matters, especially in administrative details. Professors of mathematics in our colleges have been called on to shoulder an unusual amount of the administrative work, and many men of fine ability and scholarship have thus been hindered from entering actively into research work. Conditions have, however, improved rapidly in recent years, and it is becoming better known that the productive college teacher needs all his energies for scientific work; and in no field is this more emphatically true than in mathematics. Some departmental administrative duties will doubtless always devolve upon the mathematics teachers. By a careful division of these duties they need not interfere seriously with the main work of the various teachers.

The American teachers of mathematics follow the textbook more closely than is customary in Germany, for instance. Among college teachers there is a wide difference of view in regard to the suitable use of the textbook. While some use it simply for the purpose of providing illustrative examples and do not expect the student to begin any subject by a study of the presentation found in the textbook, there are others who expect the normal student to secure all the needed assistance from the textbook and who employ the class periods mainly for the purpose of teaching the students how to use the textbook most effectively. The practice of most teachers falls between these two extremes, and, as a rule, the textbook is followed less and less closely as the student advances in his work. In fact, in many advanced courses no particular textbook is followed. In such courses the principal results and the exercises are often dictated by the teacher or furnished by means of mimeographed notes.

The close adherence to the textbook is apt to cultivate the habit on the part of the student of trying to understand what the author meant instead of confining his attention to trying to understand the subject. In view of the fact that the American secondary mathematics teachers usually follow textbooks so slavishly, the college teacher of mathematics who believes in emphasizing the subject rather than the textbook often meets with considerable difficulty with the beginning classes. On the other hand, it is clear that as the student advances he should be encouraged to seek information from all available sources instead of from one particular book only. The rapid improvement in our library facilities makes this attitude especially desirable.

An advantage of the textbook is that it is limited in all directions, while the subject itself is of indefinite extent. In the textbook the subject has been pressed into a linear sequence, while its natural form usually exhibits various dimensions. The textbook presents those phases about which there is usually no doubt, while the subject itself exhibits limitations of knowledge in many directions. From these few characteristics it is evident that the study of textbooks is apt to cultivate a different attitude and a different point of view from those cultivated by the unhampered study of subjects. The latter are, however, the ones which correspond to the actual world and which therefore should receive more and more emphasis as the mental vision of the student can be enlarged.

The number of different available college mathematical textbooks on the subjects usually studied by the large classes of engineering students has increased rapidly in recent years. On the other hand, the number of suitable textbooks for the more advanced classes is often very limited. In fact, it is often found desirable to use textbooks written in some foreign language, especially in French, German, or Italian, for such courses. This procedure has the advantage that it helps to cultivate a better reading knowledge of these languages, which is in itself a very worthy end for the advanced student of mathematics. This procedure has, however, become less necessary in recent years in view of the publication of various excellent advanced works in the English language.

The greatest mathematical treasure is constituted by the periodic literatures, and the larger colleges and universities aim to have complete sets of the leading mathematical periodicals available for their students. This literature has been made more accessible by the publication of various catalogues, such as the Subject Index, Volume I, published by the Royal Society of London in 1908, and the volumes "A" of the annual publications entitled International Catalogue of Scientific Literature. All students who have access to large libraries should learn how to utilize this great store of mathematical lore whenever mathematical questions present themselves to them in their scientific work. This is especially true as regards those who specialize along mathematical lines.

In some of the colleges and universities general informational courses along mathematical lines have been organized under different names, such as history of mathematics, synoptic course, fundamental concepts, cultural course, etc. Several books have recently been prepared with a view to meeting the needs of textbooks for such courses. College teachers of mathematics usually find it difficult to interest their students sufficiently in the current periodic literature, and one of the greatest problems of the college teacher is to instill such a broad interest in mathematics that the student will seek mathematical knowledge in all available sources instead of confining himself to the study of a few textbooks or the work of a particular school.

G. A. Miller
University of Illinois

References

For articles on the teaching of mathematics which appeared during the nineteenth century, consult 0050 Pedagogy in the Royal Society Index, Vol. I, Pure Mathematics, 1908. For literature appearing during the first twelve years of the present century the reader may consult the Bibliography of the Teaching of Mathematics, 1900-1912, by D. E. Smith and Charles Goldziher, published by the United States Bureau of Education, Bulletin, 1912, No. 29. More recent literature may be found by consulting annual indexes, such as the International Catalogue of Scientific Literature, A, Mathematics, under 0050, and Revue Semestrielle des Publications Mathématiques, under V 1. The volumes of the international review entitled L'Enseignement Mathématique, founded in 1899, contain a large number of articles relating to college teaching. This subject will be treated in the closing volumes of the large French and German mathematical encyclopedias in course of publication.

Footnotes:

[3] P. Zühlke. Zeitschrift für Mathematischen und Naturwissenschuftlichen Unterricht, Vol. 45 (1915), page 483.

[4] Committee No. XII, American Report of the International Commission on the Teaching of Mathematics, 1912, page 9.

[5] Internationale Mathematische Unterrichtskomission, Vol. 3, No. 6 (1912), page 2.

[6] Journal de l'Ecole Polytechnique, Vol. 1 (1896), part 4, page lx.

[7] F. Cajori, Teaching and History of Mathematics in the United States, 1890, page 22.

[8] A. E. H. Love, Proceedings of the London Mathematical Society, Vol. 14 (1915), page 183.

[9] V. V. Bobynin, L'Enseignement Mathématique, Vol. 1 (1899), page 78.

[10] The Training of Teachers of Mathematics, 1917, by R. C. Archibald. Bulletin No. 27, 1917, United States Bureau of Education.

[11] Cf. M. Bôcher, Science, Vol. 38 (1913), page 546.

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