# Old Texts, New Ways of Teaching Math

#### June 15, 2016

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Nick Scoville: Chair and Assistant Professor of Mathematics and Computer Science

B.S., M.S., Western Michigan University, M.A., Ph.D., Dartmouth College

Ursinus College is one of seven colleges sharing a $1.5 million collaborative National Science Foundation grant called TRIUMPHS, integrating primary sources—original historical writings by mathematicians—to help students see what they are learning in context of how it came to be.

**Q.** Can you explain your part in the grant, using history to inform topology?

Our main reason for writing this grant is to develop materials based on primary historical sources to be used in the classroom in order to teach all undergraduate areas of math. My particular contribution will be in developing projects based on primary sources in point-set topology, my scholarly area of expertise, and on developing in-class materials.

**Q.** Explain the difference between the way math is taught now and the way it could be taught using primary sources.

Mathematics is typically presented in a systematic fashion. We begin with a precise, crisp, and clean definition, follow it up with an example to illustrate, and then prove properties and technical facts about our new definition. While this presentation certainly has its merits, it hides the struggle to come up with “the right” definition or the proper way in which to frame one’s mathematical ideas. Contrast this with philosophy where one reads the original writings of, say, Plato and, then moves on to Aristotle and Aquinas to see what they took and rejected from Plato. Our work with primary sources is somewhere in the middle of the current systematic approach and the purely original source approach of philosophy. While the projects we write include many quotes and excerpts from original sources, we also include commentary, problems, and exercises to help students work through

the original writings.

**Q.** Can you define connectedness and what it has to do with this teaching? I attended a class where you showed how the teachings of German mathematician Georg Cantor were connected to the problem the class was solving. What was the benefit to the class?

Connectedness is a property of an object which intuitively says that the object is in one piece. But how do we make this notion precise? Cantor gave an initial definition of connectedness in 1883. This definition was very geometrical. Over the years, the definition evolved and was finally given the definition that we use today in 1911 by N.J. Lennes. By following the evolution of the definition of connectedness, students can see changes, tweaks, and the culmination in a modern definition. It provides a better motivation.

**Q.** What in your background made you interested in this?

My Ph.D. is in topology and even as a student I realized that the concepts we were learning in topology were so far removed from anything that we would conceivably come up with on our own. I remember my undergraduate topology professor writing down the definition of “compactness” on the board and saying “No one in their right mind would ever write this definition down!” What his comment made me realize was that these definitions do not come to mathematicians out of the blue, but that they arise much more organically in the context of attempting to frame a very particular problem or phenomena. After this realization, I wanted to try and get back to the source of these definitions.

**Q.** How do you write coursework based on primary sources?

I work backwards by starting with a standard concept in topology and seeing if there are any history of math papers written about the concept. This gives me a trail of original sources to begin investigating and so far, I have found these original sources rich with content to draw from. The history gives the math context. So far, I have written three mini-projects based on original sources and plan to write three more mini-projects along with two longer projects under the grant.

**Q.** What is your goal?

My goal is the same as all math professors; that is, to offer the best math education for students and to give them the tools and techniques to do math in the best possible way. Part of math is to ask the right questions. Math textbooks give you the assumptions but not the process of getting there. We believe that primary sources are one way to help realize this goal.