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A history of mathematics course as a senior seminar

A history of mathematics course as a senior seminarA HISTORY OF MATHEMATICS COURSE AS A SENIOR SEMINAR*

ABSTRACT: The senior seminar in mathematics at Hood College is a course in the history of mathematics. A requirement for all mathematics majors, it serves as a capstone course, bringing together students who have been pursuing different tracks in the major, and giving them a glimpse of the Big Picture at last. It is a student-led seminar, with students gaining practice in oral presentation, preparing visual aids, and planning a class. Students also complete projects, both individually and in groups, and gain library research experience. We use films, the Internet, and journal articles, as well as books, to learn about the history of mathematics. At the end of the course, we celebrate Mathematics Awareness Month by sharing what we have learned with the rest of the campus.

KEYWORDS: Senior seminar, history of mathematics, presentation skills, research skills.

INTRODUCTION

Hood College is a small, liberal arts college for women. Some of our graduates go directly to graduate school, but most of them enter the workforce, in teaching or government or industry. For that reason, a capstone course in which students learn to prove new theorems or explore increasingly esoteric areas of mathematics did not seem appropriate for our program. But we did want to offer a seminar experience for our students in their senior year, one which would tie up some of the loose ends of their mathematical education and give them a common experience. We hit upon the idea of offering a course in the history of mathematics as a required senior seminar for the mathematics major. We have taught the course three times in its present form.

The seminar is a three-credit course, offered every spring semester. The prerequisite is junior standing and at least one course in mathematics beyond calculus, but in practice most students have had many courses in mathematics by the time they take the seminar.

We will discuss the course in terms of its learning objectives and indicate why we think they are important, how we address them and, when appropriate, how we measure students' achievement of those objectives.

Students will be able to identify and discuss important people, events and topics in the history of mathematics. This is the heart of the subject matter, what we do in class every week. We have a course outline which is organized chronologically: we start with ancient mathematics and work our way up to the 20th century. Students read books and articles and discuss them in class. They write papers and book reports. We play "Mathematics History Jeopardy," in which they must exhibit their knowledge of those important people, events, and topics.

Students will understand the "big picture" of the development of mathematics - to see major trends, not just isolated discoveries.

Students always get little snippets of the history of mathematics in other courses - the feud between Newton and Leibniz in calculus, Cauchy's rigorization of calculus in real analysis, an algorithm named for some guy named Gauss in linear algebra, etc. But this is the first chance they get to see mathematics as a truly human endeavor, and as part of a larger cultural history. We look at the groundwork done by Fermat and Wallis - and Archimedes - in the development of the calculus, and learn about those giants upon whose shoulders Newton stood. We look at the history of computing an approximation to 7r and see how developments in other areas of mathematics made advances possible in that field of study. We look at the topic of women in mathematics, and talk about barriers to women's participation in mathematics, as well as about a few famous women mathematicians.

Students will gain an appreciation for the development of mathematics in different cultures, at different times.

One of the books we use in this course is George Joseph's The Crest of the Peacock: Non-European Roots of Mathematics [111. We read a chapter or two every week and discuss his alternate theories of the creation and transmission of knowledge. Once students have been sensitized to this subject, they are brutal in their critiques of other authors' Eurocentrism and bias! We spend a lot of time discussing the development of mathematics in India and China and the Arab world, and contrast the mathematical richness of other cultures with Europe's "Dark Ages."

Students will become familiar with standard textbooks about the history of mathematics.

There are so many good standard texts on this subject, and we have such a difficult time choosing just one of them. So we don't. We place several on reserve in the library, and each student checks out a different book every week. If we are studying Greek mathematics, one student may be reading Howard Eves [41, and another David Burton [3], and another Morris Kline [6]. During the course of the semester, students become familiar with the organization and style of different texts, and work problems from all of them. Class discussion shows that they recognize the differences in style, and students definitely have their favorite texts.

For a class session on ancient mathematics, the instructor may assign problems from Eves' text, writing about systems of numeration from other cultures, or solving problems found on Babylonian tablets. When the class discusses Greek mathematics, students work problems from Katz's text on triangular numbers and Pythagorean triples and congruent triangles. And when we study the development of probability theory, they work problems in probability from Burton. Having all the students work the same problems helps to ensure that, no matter which text they have read that week, there is some commonality of experience. And the students are led to review different areas of mathematics (geometry, calculus, probability) in order to work the problems. Students are encouraged to work together on the problems if they like, and to ask for help from the instructor if necessary.

Students will become familiar with nontraditional, alternative texts in the history of mathematics.

As our students complete their college years and prepare to go out into the world, we want them to know that it is possible to walk into a bookshop and buy interesting, fun paperback books about mathematics. Since the students do not have to purchase a standard text, we have them buy several paperbacks for the course instead. It is also possible to cover more nontraditional topics with such texts. The list changes slightly from year to year; some books we have used are: The Crest of the Peacock, mentioned above, Bill Dunham's Journey Through Genius [9] or The Mathematical Universe [101, Teri Perl's Math Equals [14] or Women and Numbers [15], and MAA publications like Learn from the Masters [16] or Vita Mathematica [7]. When we use Journey through Genius, we read a different theorem every week, and a student presents its proof in class. When we read Math Equals, we perform the mathematical activities associated with each mathematician. Students who plan to teach mathematics especially appreciate the suggestions in the MAA books about using history to teach mathematics.

Students will be able to identify and find well-known resources in the history of mathematics: books, journal articles, Internet sites, and films.

The first assignment in this course is to go to the online catalog and find out what books our college library holds in the history of mathematics. That (long) list becomes the students' first list of references for the course. Students actually print it out and put it in their class notebook, which also holds notes on their reading, handouts from class, completed homework assignments, and the notes they take on other students' presentations. We can choose a mathematics history text at random, go to the end of any chapter, and find a list of references which illustrate the wide range of resources available on this topic, from books to professional journals to popular magazines. In another early class, we all walk over to the library together and go into the stacks and look through the shelves of mathematics history books, to experience tangibly the wealth of resources there, both primary and secondary. (Later, each student chooses one book to read and write a book report on.) Then we go to the reference section of the library, to look through scientific encyclopedias and dictionaries which they may not think to look for [21 - 26]. Finally, we go to the periodicals section of the library and find journals with articles about the history of mathematics.

We were also pleasantly surprised to discover how many films our library holds which are appropriate for this course, including the Cal Tech Project Mathematics! films The Theorem of Pythagoras and The Story of Pi [27], MSRI and NOVA films on Fermat's Last Theorem [31,32], European Mathematicians' Migration to America from the AMS [28], and, of course, Donald in Mathmagic Land [30].