Reform Now, before It's Too Late!

carefully weeded of difficult generalities; and at a later stage a philosophical theory. To begin with the latter would be to my mind a mistake; the appeal would be to too limited a class. Professor E.H. Moore, the vice-president of the Society, was also concerned with “fundamental ideas” in calculus, delivering an address titled “Certain fundamental ideas which should be emphasized throughout the undergraduate course.” He advocated an intuitive development of the fundamental ideas that would culminate in a more rigorous presentation: Intuition is by no means to be excluded from mathematical work. On the contrary, in every way let intuition be kept alive and active, but when a new idea is introduced as an intuition let it be sharply defined with the understanding that in so far as subsequent work depends on that intuition a new branch of mathematics is being originated. It is interesting to consider, then as now, how the pronouncements of “reformers” actually translate into classroom practice. In his day, Moore was a much sought-after speaker on the issues of reform. For many of Moore’s students, however, the implementation often fell short of the lofty goals. Moore’s classroom is given a vivid evocation in [18]. The setting is a graduate course, and although one might imagine Moore treading more lightly on his undergraduates, the atmosphere is apparent enough: . . . Moore had perhaps the most unconventional teaching style, one which tended to appeal to only the most determined and talented students. Like Sylvester before him, Moore often came to his graduate classroom fresh with his latest ideas on the topic under discussion. While this did not generally make for tightly organized and polished lectures, it did instill in his students a “sense of the hunt.” With Moore at the blackboard, he and his class would work together to resolve some stubborn detail or to prove some theorem. Sometimes they succeeded, and the class came to a satisfying conclusion; sometimes they failed and dispersed scratching their heads in confusion. Yet regardless of their success or failure, Moore demanded the quick attention of his students, and if they fell behind as his thoughts raced, they soon felt the sting of what Mary Winston termed his “peppery temper.” Intending nothing personal by these outbursts, Moore simply forgot about accepted social conventions when overwhelmed by the heat of mathematical thought. Students in his classes may have needed thicker skins than those in his colleagues’ courses, but, as one student put it, “it was a proud moment when one who was ambitious and interested found himself in the relatively small group of those who could stand the pace.” The “fundamental ideas which should be emphasized throughout the undergraduate course” that Moore had lectured about did not, apparently, include “social conventions.” Accordingly, one wonders if the “proud moment” described above, and described by Moore himself in his addresses, might have felt more like a stay of execution to many of his students. We know that the aforementioned Mary Winston, who would survive Moore’s class and then leave the department, wrote home to her mother about the psychological trials of Moore’s “reforms.” [18] (Ms. Winston would go on to become the first American woman to earn a doctorate in mathematics at a German university, in Göttingen, under the direction of the much more encouraging Felix Klein.) THE NEW EDUCATION. In the late 1800s, teachers who were interested in improving the quality and relevance of instruction in their classrooms—and there were