Historical Notes for the Calculus Classroom

Readers of Convergence probably agree that, for many reasons, it is a good idea to include some history of mathematics in a calculus course. The questions that many of us have are what history should be used and how to implement it. As a long-time believer in the necessity of history in teaching mathematics, V. Frederick Rickey has successfully integrated history into the classroom in a wide variety of ways. His list of techniques includes [Rickey 1996, p. 254]:

• History of specific topics
• History of notation
• Etymology of terms
• Pictures of mathematicians
• Quotations by famous mathematicians
• Biography
• Anecdotes
• Title pages from famous books
• Problems from old textbooks
• Historical errors

Fred has further noted that history may organically arise from several typical classroom situations, such as introducing new topics or motivating discussion of advanced, modern topics. Regardless of how one uses history in the classroom, he has always affirmed that “it needs to be tied very closely to the material being discussed in class” [Rickey 1995, p. 124]. This tenet has led him over the years to pursue research into historical details that are missing in most history of mathematics books . . . and to use what he has learned both to enrich his teaching and as a lens for reflection on what we should (and should not) teach to our students.

Beginning in the late 1980s, Fred wrote a number of short pieces on specific calculus topics based on his experiences with historical research and teaching. That collection, Historical Notes for the Calculus Classroom, has thus far been distributed primarily to participants in the Institute on the History of Mathematics and its Use in Teaching (IHMT), which Fred co-directed with Victor Katz and Steven Schot from 1995 to 1999. Even today, IHMT fellows remain passionate about these Notes. On one level, they provided IHMT fellows with a set of interesting problems and historical tidbits to use in their own calculus and analysis classes. The article “Perrault and the Tractrix,” for instance, not only examines the history and solution of one of the earliest inverse tangent problems, but also paints an image of Perrault’s death as a result of a camel dissection that “is too vivid to forget.” On another level, Fred’s Notes gave IHMT fellows a model for how to research the history of a topic on one’s own. Examples of various approaches to using primary sources are found, for example, in Fred’s analysis of a letter written by L’Hospital to Bernoulli on 17 March 1694 in the article “L’Hospital’s Rule,” as well as in his journey through Leibniz’s struggle to discover the correct form of the product rule in manuscripts written during 1675–1677 in the article “The Product Rule.” Many of the articles in Fred’s Notes also identified resources for finding out more about the historical development of a given topic, thus providing instructors with a foundation for further exploration and learning in the area. Perhaps most importantly, Fred’s Notes offered IHMT fellows specific strategies for and thoughtful insights into how and why to bring the results of that historical research into the classroom itself.

V. Frederick Rickey discusses the background of Historical Notes for the Calculus Classroom
in an interview conducted 18 April 2023 via Zoom by Amy Ackerberg-Hastings.
Uploaded to YouTube by Convergence Editors, 1 May 2023. Closed captioning available.

As a collection of independent topics, the articles in the collection Historical Notes for the Calculus Classroom can be perused in any order, making them ideally suited for serial publication. In this series, Convergence is pleased to share a selection of articles from the collection with our readers. While the pieces that will appear in this series have been updated to reflect more recent historical research, every effort has been made to retain the original intentions and distinctive voice of their author. Fred described those intentions in an interview for Convergence, which you can view by clicking on the image above.

Please note that the articles in this series are not intended to represent a comprehensive history of calculus, and some topics from a standard year-long calculus sequence are missing from the collection. The individual articles are also not “plug-and-play” lessons; rather, they provide impressionistic and idiosyncratic encounters with past developments that, as stated above, reflect Fred’s priorities in teaching calculus. Using this series as a guide to including history in a calculus course will thus require an investment of time from readers:

• to decide which articles align with topics they plan to cover in their course;
• to read and understand the historical background;
• to work through and gain firm control over the mathematical content;
• to decide on a format for including a historical episode in the classroom (e.g., a lecture, a class discussion, a homework assignment, a short group project via a handout);
• and, of course, to decide how much time to devote to it (both for preparation and in class).

As Fred has noted elsewhere, however, even “an incomplete historical note can captivate students” [Rickey 1995, p. 133].

In some cases (but not all), Fred has also described how he has personally used the history discussed in a particular article when teaching his own calculus classes. It has long been his hope that sharing his experience using history in the classroom “will inspire others to try [these] ideas in the classroom and also to share their own successes in using history” [Rickey 1995, p. 124]. The editors of this journal invite all those who are so inspired to share their own stories and successes in Convergence!

V. Frederick Rickey and Victor J. Katz at the Institute on the History of Mathematics and its Use in Teaching.
 

References

Rickey, V. Frederick. 1995. My Favorite Ways of Using History in Teaching Calculus. In Learn from the Masters!, edited by Frank Swetz, John Fauvel, Otto Bekken, Gengt Johansson, and Victor Katz, 123–134. Washington DC: Mathematical Association of America.

Rickey, V. Frederick. 1996. The Necessity of History in Teaching Mathematics. In Vita Mathematica: Historical Research and Integration with Teaching, edited by Ronald Calinger, 251–256. Washington DC: Mathematical Association of America.

Selected Additional Articles by V. Frederick Rickey in MAA Publications

Huber, Michael, and V. Frederick Rickey. 2008, March. What is 0^0? Convergence.
This look at the history and meaning of the expression 0^0 leads to the conclusion that 0^0 = 1, regardless of what is said in some textbooks.

Rickey, V. Frederick, and Amy Shell-Gellasch. 2008, May. Mathematics Education at West Point: The First Hundred Years. Convergence. A survey of the mathematics education of cadets in the first century after the founding of the U.S. Military Academy.

Rickey, V. Frederick, and Philip M. Tuchinsky. 1980, May. An Application of Geography to Mathematics: History of the Integral of the Secant. Mathematics Magazine 53(3):162–166.
Fred’s first publication in the history of mathematics offers a brief history of a calculus topic that Fred and his co-author describe as an “ideal soapbox for discussing the nature of mathematics, the process of mathematical discovery, and the role that mathematics plays in the world” (p. 166).