One Hundred and Fifty Years of Teaching Calculus

George M. Rosenstein
Professor of Mathematics
Franklin & Marshall College
Lancaster, Pennsylvania

Suppose you wanted to know what a calculus course was like 50 or 100 or even 150 years ago. Where would you turn? What would you study? There are student descriptions of what their classes and teachers were like; we even have some notebooks. But by far the most readily accessible source of information on what was covered and how it was covered comes from textbooks.

Textbooks mold our courses. We may not follow them exactly; we are likely to add and subtract, omitting a section here, adding interesting asides there. By and large, however, our courses are organized around our books. So looking at texts seems like a reasonable way to discover how calculus has changed, at least for the beginner.

I'd like to be able to tell you that I began studying calculus texts some 20 years ago because this was the question that fascinated me. But that's not the case. I began because I happened to have a copy of Elias Loomis's book of 1860, Analytical Geometry and Calculus. What fascinated me was that Loomis appeared to know nothing of Cauchy's work on the foundations of calculus, written some 40 years earlier. While I was clever enough to know that it took time for ideas to filter down, particularly across an ocean, 40 years seemed a long time. What I was to learn is that forty years is not so long and that more of Cauchy than I realized was in Loomis.

What I'd like to do is talk about several topics that are handled quite differently, here at the beginning of the twenty-first century, than they were in the nineteenth, hoping you'll find something novel to take back to your classroom. As an historian, I want to be fair to the materials I present by making it clear that these are not silly men who did not know any better, but conscientious teachers trying to do right by their not always enthusiastic students.

Note: George Rosenstein, Professor of Mathematics at Franklin & Marshall College and a former Chief Reader for AP Calculus, gave this talk on Professional Night at the 2002 AP Calculus Reading at Colorado State University. The entire text is available in the PDF file One Hundred and Fifty Years of Teaching Calculus that can be downloaded from "More," below.

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