Editor’s Note: Because the final issue of the fall 2012 semester was devoted to commemorating Mr. Stephenson, the following article could not appear in print. However, because of its time-sensitive content, we are printing it here for interested Gadfly readers.

SCI Minutes: Calculus in Junior MathematicsMichael Fogleman, A’13

This past Thursday, December 6, the SCI convened in the Private Dining Hall to discuss the first semester of junior mathematics. What are the goals of the calculus sequence, with respect to both the mathematics tutorial and the laboratory? How well do we accomplish these goals?

The aims of junior mathematics seem to parallel the language tutorial, where we seek a balance between instruction and inquiry. On the one hand, students need to acquire an understanding of calculus for junior and senior laboratory, so that they can read and understand authors like Young, Maxwell, and Schrödinger. On the other hand, we study Newton’s lemmas and Leibniz’s papers for their own sake. These texts introduce the development of calculus as it actually happened, complement the discussions of Newton and Leibniz in lab and seminar, and raise provocative questions about mathematics and natural philosophy.

It was agreed that neither of these goals are being fully met. While Leibniz’s papers are original and inventive, they lack the elegant, unified, and didactic character that Newton’s lemmas have, so that we are forced to learn calculus by means of notes that are longer than the papers themselves. While these notes introduce skills and exercises for deriving and integrating, it seemed that this practice was inadequate preparation for lab. Additionally, the manual also offers a partial but less than thorough introduction to the practical and theoretical issues present in transforming Leibnizian and Newtonian calculus into a modern, mathematical language.

Because skill in calculus is not expected of students in the same way that algebra is, some feared that the students’ lack of skill with calculus might be preventing them from considering or enjoying the more speculative issues that the Leibniz papers are supposed to raise in the first place. Students who had not studied calculus before St. John’s stated that the Leibniz manual was competent in helping them to read calculus, but they felt uncomfortable with using this new language, so that they often end up relying on their more experienced peers for demonstrations in class.

Some tutors pointed out that the lab’s ambitious curriculum was originally designed with the assumption that students have developed skill in calculus in the math tutorial; without that skill, students cannot fully understand or appreciate the wide range of material. Students confirmed that when most students in a class lack a firm grounding in the technical aspect of an author’s argumentation, discussion can quickly devolve into empty or vague speculation.

When the discussion turned to possible solutions, several options were offered. If the new manual had problems with respect to both the Leibniz papers and the explanatory notes, perhaps it would be better to develop the necessary skills by using the old manual before reading selections from original texts. The one tutorial that did so this year had mixed results not because of the old manual itself, but because they could not study or discuss the material with their classmates or the available math assistants. Others favored keeping the new manual. If some of Leibniz’s papers could be subtracted, the extra class time could be spent reading a supplementary paper relating the Leibnizian calculus to its modern descendant. Alternatively, tutors could provide students with a combination of online resources,

problem sets, and quizzes to help ensure that all students develop skill in calculus. While no one believed that the current state of junior mathematics was entirely satisfactory, there was no unanimous endorsement of a specific solution.