athematical 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