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 the young engineering students. The difficulty of the situation is increased by the fact that the young instructor is often led to believe that his advancement and the appreciation of his services are directly proportional to his achievements in investigations of a high order. This belief naturally leads many to begrudge the time and thought which their teaching duties should normally receive.
The young college teacher of mathematics is thus confronted with a much more complex situation than that which confronts the mathematics teachers in secondary school work. Here the success in the classroom is the one great goal, and the mathematical knowledge required is comparatively very modest. Possibly the situation of the college teacher could be materially improved if it were understood that his first promotion would be mainly dependent upon his success as a teacher, but that later promotions involved the element of productive scholarship in an increasing ratio.
The schools of education which have in recent years been established in most of our leading universities have thus far had only a slight influence on the preparation of the college teachers, but it seems likely that this influence will increase as the needs of professional training become better known. It is probably true that the ratio of courses on methods to courses on knowledge of the subject will always be largest for the elementary teacher, in view of the great difference between the mental maturity of the student and the teacher, somewhat less for the secondary teacher and least for the college teacher; but this least should not be zero, as is so frequently the case at present, since there usually is even here a considerable difference between the mathematical maturity of the student and that of the teacher.
It may be argued that the future college teacher will probably profit more by noting the methods employed by his instructors than he would by the theoretic discussions relating to methods. This is doubtless true, but it does not prove that the latter discussions are without value. On the other hand, these discussions will often serve to fix more attention on the former methods and will lead the student to note more accurately their import and probable adaptability to the needs of the younger students.
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.