stic phases of the youth's mentality were
originally called into their most intense exercise.

T. C. CHAMBERLIN
_The University of Chicago_

VIII

THE TEACHING OF MATHEMATICS

=Recent changes and some of their sources=

In recent years the teaching of mathematics has undergone remarkable
changes in many countries, both as regards method and as regards
content. With respect to college mathematics these changes have been
evidenced by a growing emphasis on applications and on the historic
setting of the various questions. To understand one direct source of
these changes it is only necessary to recall the fact that in about
1880 there began a steady stream of American mathematical students to
Europe, especially to Germany. Most of these students entered the
faculties of our colleges and universities on their return to America
It is therefore of great importance to inquire what mathematical
situation served to inspire these students.

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]

=Influence of researches in mathematics on methods of teaching=

Mathematical research was practically introduced into the American
colleges during the