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

Euler's method is a technique for finding approximate solutions to differential equations addressed in a number of undergraduate mathematics courses. Various current texts include Euler's method for calculus [4], differential equations [1], mathematical modeling [9], and numerical methods [2] students. Each of those courses are opportunities to give students an opportunity to read Euler's own brief description of the algorithm, and in the process come to understand the technique and its limitations from Euler himself. This capsule includes historical information about Euler and his development of the approximation method. Additionally, I describe Student Assignments (Appendix A) I use to connect that history to the mathematics the students are learning. The activities are designed to deepen student understanding of Euler's method specifically and reinforce learning of calculus skills in general. I also include a translation of Euler's writing on the topic (Appendix B).

Historical preliminaries

Leonhard Euler (1707–1783) was one of the most gifted of all mathematicians. Excellent biographies of Euler, some identifying the voluminous quantity of his mathematical writing, are available [6], which interested readers are encouraged to explore. One of Euler's many gifts was his ability to write mathematics clearly and understandably.

Introduction

Solving quadratic equations is a topic relevant to modern mathematics instruction, as it has been for thousands of years. As we start the 21st century, more often than not students will use calculators and computer algebra systems to solve quadratics. Today, we associate solving quadratics with curves (parabolas) rather than rectangles and squares (even though the word quadratic is from the Latin quadratum, a four-sided figure). A centuries old method which hopefully will survive in classrooms in this millennium is the method of completing the square. Understanding the process of completing the square is important for our students, for a wide range of reasons including that it provides arguably the best approach to deriving the quadratic formula. In the examples below, we outline the use of completing the square as it was done in four previous millennia.

Over the years, the method has had various representations. Understanding the historical, geometric representation may help students internalize the method when algorithmic or algebraic representations alone may not. Multipleways of learning and knowing are offered by including the historical perspective. The examples given in this capsule are actual problems solved in the past, and your students are invited to solve them today using the methods of antiquity as well as current techniques. In my courses, I present the information as an interactive lecture that extensively involves students, as described below.

Historical preliminaries

About 4000 years ago, Mesopotamian scribes pressed the method of completing the square into clay tablets, the technology used to record information in that time.

Introduction

In a differential equations course, students learn to use integrating factors to solve first order linear differential equations, and in the process reinforce learning of key concepts from their calculus courses. This capsule offers some differential equations solved by the originators of the technique of using an integrating factor, though they did not use that expression. Solving differential equations via integrating factors is difficult for some students, particularly those who try to memorize a formula. We advocate that students learn to derive the method and solve differential equations using the product rule and the fundamental theorem of calculus, as advocated in a number of modern texts [2, 3]. Memorizing the formula would not be in the spirit of the originators of the method, Johann Bernoulli (1667–1748) and Leonhard Euler (1707–1783), nor does formula memorization lead to deep learning of fundamental mathematical processes. Understanding why integrating factors work, as offered in this historical perspective, can deepen student understanding of calculus topics such as the product rule, the fundamental theorem of calculus, and basic integration techniques. This capsule provides some historical information about the work of Bernoulli and Euler, and we offer student activities that will connect that history to enable more thorough learning of the method of integrating factors.

Historical preliminaries

Johann Bernoulli was a colleague of Gottfried Leibniz (1646–1716) and is acknowledged as one of the foundational figures in the development of the calculus. In the early 1690's he prepared lectures in the nascent calculus for Guillaume de l'Hôpital (1661–1704), who is credited with writing the first text on the calculus.

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

Hermite, Runge, Birkhoff, and Shoenberg are just four of the many mathematicians of the past 200 years who contributed to the field of numerical analysis. In the numerical methods course described in this article, students met Charles Hermite (1822–1901) while studying the interpolating polynomials bearing his name. Students became acquainted with Carl Runge (1856–1927) as they did the historical research to learn what applications motivated the development of his popular method for approximating the solution to differential equations. Isaac Shoenberg (1903–1990) and Garrett Birkhoff (1911–1996) had different purposes in developing and applying splines, just one of the mathematical methods created to support the American wartime effort of the 1940s. Students study splines in this and other numerical analysis and mathematical modeling courses ([1] and [5] are typical texts), but in this one they learn the context for the historical development of the mathematics.

Projects and other history-based course activities provide an opportunity to bring to life the topics of a numerical analysis course. A brief overview of history-based projects included in a numerical methods course is given here. All the projects are described in the paper; one of the course projects is contained in an appendix. The historical components of the projects are the focus, however other historical activities included in the course are also presented, hopefully providing the interested reader with resources and ideas for implementation in other courses.

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.

In the summer of 2001, the editors of this volume organized a contributed papers session at the MAA MathFest in Madison, Wisconsin. The topic was ways to use the history of mathematics in teaching. We received many wonderful abstracts. Due to time limitations, we had to make some cuts. Two or three of the abstracts dealt with more recent topics. The majority of the topics however, were more “traditional” in that they addressed ways to use the history of Greek mathematics or calculus for example. We decided to use talks which dealt with topics up to the development of the calculus. The abstracts that addressed more recent mathematics were very intriguing, and made us realize that even though using the history of mathematics in the classroom is becoming very popular, the topic of recent history may be overlooked.

We decided to organize a follow-up session the next year in Burlington. This session was entitled The Use of History in the Teaching of Mathematics, with a focus on roughly the last two hundred years. Again, we had many excellent abstracts. The talks covered a range of topics from Galois theory to using the history of women and minorities in teaching.

The interest at MathFest in the topic of recent mathematical history, and a noticeable lack of information for educators on how to incorporate that history into the classroom, was the motivation for this volume.