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 mathematical textbook
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.