The direct aims in the teaching of college mathematics have naturally been somewhat affected by the needs of the engineering students, who constitute in many of our leading institutions a large majority in the mathematical classes. These students are usually expected to receive more drill in actual numerical work than is demanded by those who seek mainly a deeper penetration into the various mathematical theories. The most successful methods of teaching the former students have much in common with those usually employed in the high schools and are known as the recitation and problem-solving methods. They involve the correction and direct 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.