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 last quarter of the nineteenth century, and the wave of enthusiasm which attended this introduction was unfortunately not sufficiently tempered by emphasis on good teaching and breadth of knowledge, especially as regards applications. In fact, the leading mathematician in America during the early part of this period was glaringly weak along these lines. By means of his bountiful enthusiasm he was able to do a large amount of good for the selected band of gifted students who attended his lectures, but some of these were not so fortunate in securing the type of students who are helped more by the direct enthusiasm of their teacher than by the indirect enthusiasm resulting from good teaching.
The need of good mathematical teaching in our colleges and universities began to become more pronounced at about the time that the wave of research enthusiasm set in, as a result of the growing emphasis on technical education which exhibited itself most emphatically in the development of the schools of engineering. While the student who is specially interested in mathematics may be willing to get along with a teacher whose enthusiasm for the new and general leads him to neglect to emphasize essential details in the presentation, the average engineering student insists on clearness in presentation and usability of the results. As the latter student does not expect to become a mathematical specialist, he is naturally much more interested in good teaching than in the mathematical reputation of his teacher, even if his reputation is not an entirely insignificant factor for him.