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]