rect 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.

=Preparation of the college teacher of mathematics.=

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 th