Mathematical Time Capsules offers teachers historical modules for immediate use in the mathematics classroom. Relevant history-based activities for a wide range of undergraduate and secondary mathematics courses are included. The genesis of this volume was a Contributed Papers Session on Using History of Mathematics in Your Mathematics Courses, organized by the editors at the Joint Mathematics Meetings, San Antonio, Texas, in January of 2006. That session was very well attended, which prompted Andrew Sterrett from MAA publications to suggest that we put together our second volume for the MAA Notes series.

Purpose

For a wide variety of reasons, instructors are looking for ways to include the history of mathematics in their courses. It is not uncommon to see requests for “how to” posted to the History of Mathematics Special Interest Group of the MAA (www.homsigmaa.org) email list, such as this 2008 posting:

In response to such inquiries, we hope to serve the broader mathematical community by offering practical suggestions on how to use the history of mathematics quickly and easily in the mathematics classroom.

A time capsule can be defined as a container preserving articles and records from the past for scholars of the future. Of course our volume does not fit that precise definition, but readers who open this book will find articles and activities from mathematics history that enhance the learning of topics typically associated with undergraduate or secondary mathematics curricula.

Introduction

Advocates of incorporating the history of mathematics in teaching mathematics do so believing that providing a human element may spark student interest in mathematics. Incorporating biographical sketches or historical anecdotes into instruction has the potential to enhance student interest, with the hope that interested students will learn more readily and retain the content longer. The learning value of the historical activities can be enhanced when explored and presented by the student rather than presented to the student by the instructor or textbook. An effective way to have your students deepen their knowledge of mathematics through its history is to have them do the historical research and presentations. A student-centered approach to introducing history in a wide variety of undergraduate mathematics courses is an effective teaching tool, in large part because most students like doing the presentations [1].

Invite your students to share in the joy of discovering the “who” and the “why” of the mathematics they are learning, and to take an active role in making the connections between the mathematics they are learning and its historical origins. Student-researched historical presentations can be done in any course, at any level, and require relatively minimal preparation by the instructor. How much time you allow for student presentations in class is up to you. You may limit students to 5 minutes, or require longer presentations. We provide ideas for different approaches to the historical presentations your students can do based on what has worked for us. One approach is general and easy to implement; the other requires more planning on the part of the instructor.

Introduction

Move over Riemann and make room for Fermat! Most textbooks on the integral calculus focus heavily on the Riemann integral when introducing integration. This method is very effective in transitioning students from the finite (or macro) world of finding area geometrically to the infinite (or micro) world of finding area by integration. Once the notation and abstract idea of an area made up of an infinite number of infinitely thin slices is mastered, most textbooks move directly on to integration techniques. Finding areas using rectangles is usually not mentioned again except in review, to help students visualize a more difficult example, or when transitioning to finding volumes using double integrals.

Riemann used rectangles of uniform width. This is very handy when letting the width, dx, tend to zero. It also corresponds nicely to the definition of the derivative presented in most textbooks, in which the width h = xi+1 – xi in the denominator approaches 0. I still advocate introducing integration in this manner. However, there is no reason to stop there and move directly on to integration techniques.

Prior to Riemann, even prior to Newton and Leibniz, Fermat and others were finding areas using the sum of thin rectangles. However, Fermat's rectangles were not of uniform width. The width of Fermat's rectangles decreased based on a geometric series. Looking at Fermat's method directly after introducing the Riemann integral broadens the student's perspective on the integral calculus. Also, his techniques can be presented in an analysis course to provide depth to the material.

Introduction

The classic Towers of Hanoi puzzle (Figure 1) is known to many of us, either from our school days or as educators. The puzzle works as follows: given some number of discs (washers) of varying diameters stacked in a pyramid on one of three posts, move the stack one disc at a time to one of the other posts in as few moves as possible. The catch is that no disc can rest on top of a disc of smaller diameter. This is a wonderful problem for school age students and many people first encounter it in elementary or middle school. Though the puzzle can be presented as a thought exercise (without the physical model), having the physical puzzle enhances discovery, and as such is an essential aide in any course that discusses dynamical systems. Outside of this venue the puzzle can be used in courses from elementary school through college to teach reasoning, problem solving and the development of mathematical formulas. It can be used to discuss recursion and induction as well as graph theory with students at the secondary and college level.

The Towers of Hanoi puzzle

Given three pegs and any number of discs of increasing radius stacked in a pyramid shape on the first peg, move the stack to the third peg. Only one disc may be moved at a time and no disc of larger radius may be placed on top of a disc of smaller radius.

With the advent of computers and classroom projectors, educators have a world of resources at their fingertips. However, when we let our fingers do the walking in the virtual world, we and our students miss the physical and creative aspects of learning. Numerous studies have shown that doing (as opposed to simply listening or reading) is the best way to learn.

Educators have always employed physical models to aid in instruction. During much of the nineteenth century and the twentieth up to the advent of classroom computers, makers of mathematical devices for the classroom abounded. See Multi-Sensory Aids in Teaching Mathematics (1966) of The National Council of Teachers of Mathematics, and reprinted by the American Mathematical Society for a look at the types and diversity of mathematical models and apparatus used in classrooms during the first half of the twentieth century.

Historically, scientists and mathematicians have constructed models to help them explore the physical world, be it in engineering, chemistry, physics, astronomy, or the realm of pure mathematics. In addition, the daily practice and use of mathematics was tied to the use of devices such as counting boards and the abacus, and in science with the use of astrolabes, quadrant finders, and the like. The physical items used both in the practice of mathematics and its teaching, are an important part of our mathematical heritage. I believe exploring these items adds a level of understanding that is not available with computer simulations.

While teaching at the United States Military Academy (USMA) at West Point, New York from 2000 to 2003, I had the immense good fortune to work with Fred Rickey on the history of the Olivier String Models owned by the USMA Department of Mathematical Sciences. Working with and researching these models became a joy for me. One question led to another, and I became acquainted with all aspects of the models.

To understand the models and present the mathematical concepts they depict to students, I had to learn the mathematics involved in both current and historical terms. I also had to learn about those who developed the mathematics as well as those who developed and constructed the models. One of the aspects of this work that I found the most intriguing was tracking the history of the actual models: who constructed them, who bought them, where have they been since their construction, etc. In many cases, this was the most difficult part of the research. Along the way, I learned a lot of history about the Academy itself and the mathematics department while researching the models through the USMA archives. Fred and I had great fun searching the department for a few missing models that the archives said we should have. We did find one, only a picture of another, and alas, never did find a third. But we also found many other old and forgotten teaching aides, such as a chalk board globe, which we enjoyed bringing back into use in the classroom.

Introduction

Why is the algebraic process of forming a perfect quadratic expression referred to as completing the square? With simple cutouts for an overhead projector, the geometric underpinnings as well as many aspects of quadratic equations can be exhibited quickly and effectively. This can be expanded to the cubic with the use of five wooden blocks. This chapter will show how to use these items in the classroom to give students a geometrically intuitive as well as an historical understanding of quadratic and cubic expressions.

Creating squares

At several points in the undergraduate curriculum, as well as in some secondary courses, we find ourselves “completing the square.” I found myself in just such a position a short while back while teaching a sophomore level calculus course. On the spur of the moment, I asked the class if they knew why it was called “completing the square”. The response ranged from blank stares to shrugged shoulders. Looking at the clock, I decided I could sidetrack for a few minutes.

I started by drawing the Figure 1 on the board to represent the statement x2 + bx. (In general, you will have some expression of the form ax2 + bx + c. But the actual process of completing the square is done on an expression of the form x2 + bx, after division by a and “removal” of c.) I start with the comment that for the ancient Greeks as well as the Babylonians, geometry was physical; that x (in modern notation) represented a length, x2 an area, and so on.

Using the history of mathematics to motivate interest and understanding in the mathematics classroom has become an acknowledged pedagogical option for mathematics instructors. The knowledge base on how to incorporate history into the classroom has grown by leaps and bounds in the last few decades. A look at the number of recent books published by many academic publishers, to include the MAA, is indicative of this effort to improve mathematics teaching and learning.

However, due to the nature of mathematical advances in the last few centuries, the history of topics such as logic and Galois theory are much less accessible. The level of mathematical sophistication needed to grasp these topics, even at an introductory level, is much higher than that needed for most older subjects. Given that the development of a subject is usually harder to follow than the final theory, the history of more recent topics may be even harder to comprehend. For exactly these reasons, it is important to present the historical evolution of and motivation for these topics. Many higher-level concepts seem remote and arbitrary to students. Presenting the historical development of these more recent and advanced topics gives the students the connections needed to gain a deeper understanding of higher mathematics.

This volume is intended to be a resource for undergraduate mathematics teachers, providing ideas and materials for immediate adoption and proven examples to motivate innovation by the reader.