# From Calculus to Computers

Amy Shell-Gellasch
Dick Jardine
Series: MAA Notes
Volume: 68
Edition: 1
Pages: 268
https://www.jstor.org/stable/10.4169/j.ctt5hh8dx

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-viii)
(pp. ix-x)
4. Introduction
(pp. xi-xii)

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....

5. ### I Algebra, Number Theory, Calculus, and Dynamical Systems

• 1 Arthur Cayley and the First Paper on Group Theory
(pp. 3-8)
David J. Pengelley

Arthur Cayley’s 1854 paperOn the theory of groups, as depending on the symbolic equation θn= 1 inaugurated the abstract idea of a group [2]. I have used this very understandable paper several times for reading, discussion, and homework in teaching introductory abstract algebra, where students are first introduced to group theory. Many facets of this short paper provide wonderful pedagogical benefits, not least of which is tremendous motivation for the modern theory. More information on teaching with original historical sources is available at our web resource [8], which includes information on the efficacy of teaching with original...

• 2 Putting the Differential Back Into Differential Calculus
(pp. 9-20)
Robert Rogers

The topic of the differential of a function of a single variable is typically relegated to a single section in calculus textbooks and is usually only applied in linear approximation applications. Though the notion of linearization is certainly important, this brief treatment seems to belie the fact that the subject itself is called the differential calculus. While the present treatment of calculus focuses on the derivative as the main topic of study, this was not always the case, as illustrated by the following crude chronology of the ideas encountered in calculus. The reader must bear in mind that it is...

• 3 Using Galois’ Ideas in the Teaching of Abstract Algebra
(pp. 21-24)
Matt D. Lunsford

Joseph Gallian [3, p. 47] in his popular textContemporary Abstract Algebrastates, “The goal of abstract algebra is to discover truths about algebraic systems (that is, sets with one or more binary operations) that are independent of the specific nature of the operations.” While this modern approach, because of its generality, provides a unifying element to the study of mathematics, it conceals from the student many of the great ideas generated by significant problems in the history of the discipline. B. Melvin Kiernan [4, p. 40] asserts that “without a clear historical perspective it is difficult to see or...

• 4 Teaching Elliptic Curves Using Original Sources
(pp. 25-44)
Lawrence D’Antonio

In this paper we give an overview of the subject of elliptic curves and examine some of the original sources that can be used to teach this very important topic in a history of mathematics course. Portions of this material are also appropriate for courses in abstract algebra, number theory, or geometry.

There are two purposes for introducing students to elliptic curves. First, the topic of elliptic curves is a beautiful one combining significant ideas from algebra, geometry, number theory, and analysis; the elements of this topic can be understood by the undergraduate mathematics major. Second, this topic gives students...

• 5 Using the Historical Development of Predator-Prey Models to Teach Mathematical Modeling
(pp. 45-54)
Holly P. Hirst

Many differential equations texts introduce the classic Lotka-Volterra predator-prey model as an application of coupled systems of differential equations. The model is based on a set of very simple premises:

In the absence of predators, the prey population grows exponentially.

Some fraction of predator-prey interactions end in death for prey.

In the absence of prey, the predator population decreases exponentially.

The predator birthrate is dependent on predators interacting with the prey (as the food source).

Using these assumptions as starting points for proportionality arguments, the four terms in the model (prey birth and death, predator birth and death) can be...

6. ### II Geometry

• 6 How to Use History to Clarify Common Confusions in Geometry
(pp. 57-74)
Daina Taimina and David W. Henderson

We have found that students who take our senior/graduate level geometry course usually have very little background in geometry. We have lead many week-long UFE and PREP workshops (funded by the National Science Foundation) for professors on teaching geometry and we found that even mathematicians are often confused about the history of geometry. In addition, many expository descriptions of geometry (especially non-Euclidean geometry) contain confusing and sometimes-incorrect statements — this is true even in expositions written by well-known research mathematicians. Therefore, we found it very important to give some historical perspective of the development of geometry, clearing up many common...

• 7 Euler on Cevians
(pp. 75-92)
Eisso J. Atzema and Homer White

For mathematicians looking to incorporate historical sources into their classroom teaching, Leonhard Euler is of special interest. His mathematical notation closely approximates contemporary use,¹ and his prose style is clear and not forbiddingly concise. Furthermore, he was active just before the era of increasing abstraction in mathematics, so that almost any problem he takes up is of immediate interest to a contemporary undergraduate. The motivation for the problem is usually clear, and it is seldom difficult for the student to come up with lines of further research, either by proving his results using modern techniques, generalizing them, or investigating related...

• 8 Modern Geometry after the End of Mathematics
(pp. 93-98)
Jeff Johannes

InMathematical Thought from Ancient to Modern Times, Morris Kline writes “By the end of the eighteenth century … the mathematicians began to feel blocked.” [5] Several prominent mathematicians expressed concern regarding the future of mathematics. Lagrange fears the end of mathematical evolution, writing that perhaps “it will be necessary sooner or later to abandon it.” Kline continues: “Euler and d’Alembert agreed with Lagrange that mathematics had almost exhausted its ideas, and they saw no new great minds on the horizon.” As final evidence of his assertions, he includes thoughts from both Diderot and Delambre.

I dare say that in...

7. ### III Discrete Mathematics, Computer Science, Numerical Methods, Logic, and Statistics

• 9 Using 20th Century History in a Combinatorics and Graph Theory Class
(pp. 101-108)
Linda E. McGuire

A few years ago, during the first week of the fall semester, I assigned a writing project in an upper-division undergraduate mathematics course. The class was charged with pondering the often-asked question “Was mathematics discovered or invented?” and then crafting a three to four page essay establishing and supporting their personal position. Students were instructed to cite specific and appropriate historical happenings in mathematics to argue their points.

I anticipated that the assignment would serve as an ice-breaker for the class as a whole and that I would gain some early insight into each student’s expository writing ability. While these...

• 10 Public Key Cryptography
(pp. 109-124)
Shai Simonson

When teaching mathematics to computer science students, it is natural to emphasize constructive proofs, algorithms, and experimentation. Most computer science students do not have the experience with abstraction nor the appreciation of it that mathematics students do. They do, on the other hand, think constructively and algorithmically. Moreover, they have the programming tools to experiment with their algorithmic intuitions.

Public-key cryptographic methods are a part of every computer scientist’s education. In public-key cryptography, also called trapdoor or one-way cryptography, the encoding scheme is public, yet the decoding scheme remains secret. This allows the secure transmission of information over the internet,...

• 11 Introducing Logic via Turing Machines
(pp. 125-134)
Jerry M. Lodder

A curious situation has arisen today in the undergraduate curriculum with many computer science majors learning the fundamentals of logic from a memorized list of truth tables and rules of inference, without regard to the original problems whose solutions involved the logic that would become part of the programmable computer. Current discrete mathematics textbooks, which often cover combinatorics, deductive reasoning and predicate logic, present the material as a fast-paced news reel of facts and formulae, with only passing mention of the original work and pioneering solutions that eventually found resolution through the modern concepts of induction, recursion and algorithm. Presented...

• 12 From Hilbert’s Program to Computer Programming
(pp. 135-148)
William Calhoun

The impact of computers on our lives is obvious to everyone, while mathematical logic is an esoteric subject for most people. Yet the histories of mathematical logic and computers are tightly interwoven. I will discuss the connections between logic and computing, and how I use these connections in my teaching. I find that connecting logic and computing helps to motivate both mathematics and computer science majors. The students are surprised to learn that ideas developed by mathematicians studying abstract questions of logic turned out to be fruitful in the design of computers.

The needs, goals and attitudes of mathematics and...

• 13 From the Tree Method in Modern Logic to the Beginning of Automated Theorem Proving
(pp. 149-160)
Francine F. Abeles

In teaching an upper division elective course in mathematical logic for mathematics and computer science students, I have found that the class usually is divided evenly between these two groups of students, both of which suffer from insufficient experience in proving theorems. To remedy this insufficiency, I have chosen as the engine for the first part of the course a proof technique known as the tree method, an intuitively appealing and relatively simple approach for establishing the validity of arguments that works for a large subset of first order logic whose roots go back to the early part of the...

• 14 Numerical Methods History Projects
(pp. 161-164)
Dick Jardine

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...

• 15 Foundations of Statistics in American Textbooks: Probability and Pedagogy in Historical Context
(pp. 165-180)
Patti Wilger Hunter

The last two decades have seen increasing interest within the mathematics community in reforming the undergraduate curriculum. Efforts at reform have embraced a wide range of issues, including teaching methods, content, assessment, and administration. Among the several mathematical disciplines receiving close scrutiny under the lens of reform, statistics—especially at the introductory level—has received much attention, focused particularly on pedagogy, technology, and the content of introductory courses. Recommended changes in content include an increased emphasis on data production and analysis, with less time given to “recipes and derivations” [5]. Some statisticians also suggest treating data ethics and introducing students...

8. ### IV History of Mathematics and Pedagogy

• 16 Incorporating the Mathematical Achievements of Women and Minority Mathematicians into Classrooms
(pp. 183-200)
Sarah J. Greenwald

There are many references for activities that incorporate general multiculturalism into the classroom (e.g., [4, 21, 29]). There are also numerous “women in mathematics” courses that focus on history and equity issues. Yet, except for a few sources such as [19], [23], [24], [26], and [27], sources that discuss women and minorities in mathematics do not include related activities for the classroom that contain significant mathematical content, and those that do are mainly aimed at the middle grades or high school level. This is unfortunate since students benefit from the inclusion of the achievements of women and minorities in mathematics...

• 17 Mathematical Topics in an Undergraduate History of Science Course
(pp. 201-206)
David Lindsay Roberts

An undergraduate survey course in the history of science presents numerous opportunities to discuss the role of mathematics in scientific developments, especially with regard to physics and astronomy. These courses are most commonly taught in a history department, or a history of science department. Although there may be some modest prerequisites in terms of science and mathematics, these are history courses, not mathematics courses, and it is usually inappropriate to treat mathematics topics in rigorous detail, and especially inappropriate to evaluate student solutions of mathematics problems as part of the course grade. But no such course can be considered complete...

• 18 Building a History of Mathematics Course from a Local Perspective
(pp. 207-216)
Amy Shell-Gellasch

One of the challenges of developing a history of mathematics course is deciding what material to cover. The history of mathematics is far too broad a subject to cover in a year-long course, much less a one-semester course. Some options might be to focus on topics such as ancient mathematics, the history of the calculus, great moments in mathematics, great people in mathematics, and so forth. Any course developed must also take into account the audience and intent of the course. Are the students math majors or education majors? Will this be a general education course or a course for...

• 19 Protractors in the Classroom: An Historical Perspective
(pp. 217-228)
Amy Ackerberg-Hastings

The view of mathematics presented in the school supplies aisles of discount stores revolves around concrete aids: flash cards, rulers, protractors, graph paper, and the like. These inexpensive objects, although snapped up by the general public, are often completely foreign to the daily lives of contemporary mathematicians and mathematics educators. Meanwhile, parents, students, and teachers might assume their classroom tools have some sort of eternal existence outside of historical context. However, beneath surface appearances, there are links between mathematical teaching aids and professional mathematics. The history of mathematics can reveal such commonalties. For example, at the turn of the twentieth...

• 20 The Metric System Enters the American Classroom: 1790–1890
(pp. 229-236)
Peggy Aldrich Kidwell

The nineteenth century saw an enormous expansion in American mathematics education. Publicly funded elementary or common schools were established, first in the northern states and then throughout the country. By the second half of the century, public high schools also were becoming usual. The extension of engineering education, much of it modeled after the École Polytechnique in Paris, also encouraged mathematics instruction. At the same time, improvements in printing, cheaper paper, and the national markets created by railroads made it possible to supply students in the new schools with relatively inexpensive, uniform textbooks.

Several teachers and former teachers sought to...

• 21 Some Wrinkles for a History of Mathematics Course
(pp. 237-242)
Peter Ross

In teaching an upper-division history of mathematics course using a traditional text—recently I used Howard Eves’An Introduction to the History of Mathematics, Sixth Edition, 1990—I found it helpful to incorporate several wrinkles in the course. Aside from enlivening the course they permit the inclusion of some recent history, as my examples below illustrate. This inclusion is especially useful in teaching with a traditional text such as Eves, where one is hard-pressed to get to even 18th century mathematics in a single term. I will discuss three such wrinkles, each of which involves some recent history of mathematics:...

• 22 Teaching History of Mathematics Through Problems
(pp. 243-250)
John R. Prather

As teachers, we are always looking for ways to actively engage our students in the learning process. One approach in a history of mathematics course is to have students work on a historically motivated set of problems which are independent of the other requirements of the course. These problems are described, and the effects on the class are discussed.

In devising my history of mathematics course, I had in mind three goals. First, in addition to having a sense of how mathematicswasdeveloped, it is important that students see how mathematics is developed. A history of mathematics class seemed...