Introduction (A Funny Thing Happened on the Way to Calculus)

During the fall semester of 2005, I was slated to teach University Calculus I to a class of mostly incoming freshmen. It had been a while since I taught both the class and freshmen, so on the first day I decided to do some review and pick my students' brains. I wrote y = f(x2) on the board and asked if that was a function. The unanimous answer was yes. Without exploring that too much, I drew Figure 23.1 on the board and asked if that was a function.

The response was overwhelmingly yes and when I asked why, the response was that it passed the vertical line test. I then wrote the following on the board.

When asked if that was a function, about half of the class said it was and about half said it wasn't. From the looks on people's faces, the majority of students in the class were not sure of their answers either. After having students discuss their thoughts with their neighbors, we had a class discussion about whether it was, in fact, a function or not. The gist of people who said no was that it wasn't a function because there was no formula or graph with which to determine specific values. The people who said yes said that it was still a rule that assigned a unique value to each x.

There was still no definite consensus when I drew Figure 23.2 on the board.

Introduction

How often do you have to tell your students to brush up on their notation? When they have dropped limit notation, forgotten critical modulus signs, mixed up their integrals, muddled up their derivatives, how do you convey to them the importance of recording it right?

What of your exasperation as they fail to appreciate the precision that mathematical notation affords them—notation which has been developed and refined over centuries, and notation that will continue to be improved for centuries more. Indeed, mathematics is the one subject in which they can really express exactly what they mean. How can we help them appreciate the symbolism they use?

This paper provides a light-hearted look at various notational conventions concerning the derivative. It will briefly cover the various proposals for its symbolic representation over history and the reasons behind the prevalence of the various manifestations over others. The examination of the history of the development of calculus notations is not only fascinating, but suggests a paradigm for the development of future notations. An examination of the reasons behind the failures and successes of past notation can equip one with a certain foresight regarding the future of newly introduced symbols, as particular areas of mathematics expand.

This capsule is intended for undergraduate calculus classes and should ideally be offered directly before or after covering the derivative and its various applications, or indeed right in the middle when the students need a bit of a break from theory and practice! It may also be suitable for higher level secondary school mathematics.

Introduction

In 1726, John Colson (1680–1759), a British mathematician and member of the Royal Society of London, devised an ingenious way to represent positive integers using what he called negativo-affirmative figures.[2] With his scheme positive and negative digits are intermingled and the basic arithmetic operations of addition, subtraction, and multiplication are as straightforward as in decimal arithmetic. The figures can be used to encrypt integers and have been rediscovered on several occasions. One version makes unnecessary the use of the digits 6, 7, 8,and 9, another rotates the digits 180°. Colson referred to his method as a “promiscuous scheme” to simplify the basic operations of arithmetic. In the process, he discovered a more compact and efficient way to multiply two numbers. This article is appropriate for an advanced elementary or secondary school mathematics class and represents a block of mathematical-historical material.

Historical Background

There are several ways to represent positive integers other than using the standard decimal system. For example, the internal operations of computers are executed using the binary system which is translated into the hexadecimal system making it easier for humans to understand it. Colson's negativo-affirmative figures offer students an introduction to ciphering and a different perspective on the basic arithmetic operations. For example, consider the negativo-affirmative expression 3 5 7 8 4 which represents the positive integer 2 5 6 2 4. To understand why this is true, replace every digit in 3 5 7 8 4 with a bar over it with a zero to obtain 3 0 7 0 4.

Introduction

English mathematician and scientist Thomas Harriot (1560–1621) gave the usual formula for Pythagorean triples using his new algebraic notation but he also started to list them in a systematic way. If he could have continued his list indefinitely, would he have listed all of the Pythagorean triples? In exploring this question, students can recognize and describe patterns, write and use algebraic formulas, and construct proofs, including proofs by mathematical induction. Students also have the opportunity to study an historical approach in which a mathematician seemed to believe that tabular presentation of a result was just as valuable, effective, and interesting as symbolic presentation of that result.

This material could be used in an undergraduate number theory course, in a “proofs” or “transition” course, or as enrichment for bright algebra or general education students. At least part of it could be used in college algebra or other general education courses. In lower level courses, the material should be presented in class, either during an interactive lecture or as an in-class exploration. In more advanced courses, it could be presented in class, as homework, or as a project for a small group (or small groups) of students.

Mathematical Background

Pythagorean triples are ordered triples (x, y, z) of positive integers with the property that x2 + y2 = z2. Such a triple can be viewed geometrically as giving the lengths of the sides of a right triangle with hypotenuse of length z and legs of lengths x and y. The Pythagorean triple (3, 4, 5) has been known since antiquity, with the triple (5, 12, 13) making its appearance fairly early on in history as well.

Introduction

Long before the calculus arrived a medieval philosopher, Nicole Oresme, developed what he called the latitude of forms, a graphical representation that sheds light on the fundamental connection between area and what we now call the integral. In a calculus course, the latitude of forms can be used to introduce the idea of the integral as area, while simultaneously introducing the idea that the distance traveled is the integral of velocity. Of course the two ideas can be addressed separately, if you prefer. In that case, the latitude of forms might be used to connect the two. In any event, you will be reviewing some simple geometry that students have often forgotten.

At the risk of being untrue to the original, I have modernized my presentation. The Commentary section will attempt to partially correct this distortion.

Historical Background

Scholastic philosophers, following Aristotle, were greatly interested in explaining the workings of the natural world. In this sense they appear to our eyes as scientists. They also were interested in precise definitions, careful distinctions between cases, and rigorous logical deduction. To us they appear to be mathematicians and analytic philosophers. Yet when we read their works, we can see they were also trying to explain why things work, and seeing how well their explanations fit their theology. Thus to us they appear to be trying to tackle everything at once.

This article focuses on Nicole Oresme (c. 1323–1382), who was born in Normandy and studied at the University of Paris. Immediately upon receiving his doctorate he became grand master of the University of Navarre.

Introduction

Calculus, like most other well-established branches of mathematics, did not originally appear in the same form as it occurs in modern textbooks. Many mathematicians contributed to the development of calculus over many centuries, using widely varying notation and languages. A proper history of the subject can easily consume a book [1].

Although a thorough study of the history of calculus is completely unnecessary for an introductory calculus student, it is nevertheless of some interest for such students to see an overview of this subject's fascinating and colorful history. Today's calculus students will no doubt consider original papers somewhat cryptic at the very least, and maddeningly cumbersome and obscure in places. Still, there are passages from these writings which will appear comforting in their familiarity. This paper seeks to point out some of these passages and their connections with the modern elementary calculus curriculum. We concentrate on the two mathematicians generally considered to be the fathers of calculus, Sir Isaac Newton (1642–1727) and the German Gottfried Leibniz (1646–1716).

Historical Background

It would be impossible to say authoritatively when the first ideas of calculus appeared. Arguably, many early mathematicians used a form of integral calculus in approximating the area or volume of irregular objects using finitely many or infinitely many recognizable shapes such as rectangles. In particular, the Greek mathematician Archimedes is famous for estimating the area of circle using the so-called method of exhaustion, and effectively computing the value of π.

One can say with confidence, however, that the English mathematician Sir Isaac Newton (1642–1727) and the German Gottfried Leibniz (1646–1716) are most famous for their discoveries of calculus.

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

One of the more important mathematical concepts students encounter is that of the greatest common divisor (gcd), the greatest positive integer that divides two integers. It can be used to solve indeterminate equations, compare ratios, construct continued fraction expansions, and in Sturm's method to determine the number of real roots of a polynomial. For a development of these applications, see [1]. Most of the significant applications of the gcd require that it be expressed as a linear combination of the two given integers. The gcd and its associated linear equation provide an efficient way to find inverses of elements in cyclic groups, to compute continued fractions, to solve linear Diophantine equations, and to decrypt and encrypt exponential ciphers. In order to calculate the gcd and determine the required linear combination, most textbooks present the ancient but effective Euclidean approach putting an algebraic strain on many students. A more innovative technique, Saunderson's algorithm, offers a much more efficient approach. The algorithm can be introduced in number theory, modern algebra, computer science, cryptology, and other courses that require a method to find the greatest common divisor of two integers and its associated linear combination.

Historical Background

Euclid's Elements, written around 300 B.C., consists of a deductive chain of 465 propositions in thirteen “books” or what we would refer to as chapters. The Elements serves as a synthesis of Greek mathematical knowledge and has made Euclid the most successful textbook author of all time. It is one of the most important texts in intellectual history. He was a great synthesizer for it is believed that relatively few of the geometric theorems in the book were his invention.

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

Who first determined the size of the Earth? How did they do it? These fundamental questions arise in studying early Greek, Indian and Islamic mathematical astronomy. In this article we look at the attempts of Eratosthenes, Posidonius, and al-Bīrūnī to determine the circumference of the Earth and ways to use this topic in the classroom. These calculations use only basic knowledge of geometry and trigonometry, so that instructors in many different courses can include this topic in their syllabus. It would be appropriate to discuss the problem in a high school or college geometry class, in a precalculus class, a history of mathematics class, or in a freshman mathematics survey class.

There are three primary methods for determining the circumference of the Earth: using the lengths of shadows, the elevation of stars, or the altitude of a mountain. Explaining these methods can be done in roughly two hours of class time. If an instructor wants to assign students a project to carry out one of these calculations then one or two more hours may be needed to complete the topic (assuming that the students do measurements during class time).

There are certain geographical and astronomical terms that are frequently used in this topic and should be defined for students. The position of a point on the Earth's surface is given by two coordinates, its latitude and longitude. The latitude of a point ismeasured by how far it is north or south of the equator, so that points of equal latitude form a circle parallel to the equator. Latitude is measured in degrees from the equator (0°) to the poles (90°).

Introduction

To many students, differential calculus seems like a set of rules to be applied for solving problems such as optimization problems, tangent problems, etc. This really should not be surprising as differential calculus literally is a set of rules for calculating differences. These rules first appeared in Leibniz's 1684 paper Nova methodus pro maximus et minimus, itemque tangentibus, quae nec fractus nec irrationals, quantitates moratur, et singulare pro illi calculi genus (A New Method for Maxima and Minima as Well as Tangents, Which is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This). A translation of this appears in [5, p. 272–80]. As the title suggests, our students' perceptions are not far off. Indeed, Leibniz's differential calculus is very recognizable to modern students and illustrates the fact that this is really a collection of rules and techniques to compute and utilize (infinitesimal) differences. The fact that Leibniz's notation is so modern in appearance, or rather our notation is that of Leibniz, allows these rules to be presented in a typical calculus class. The author has typically done this while covering the differentials section of the course, as the rules are rules for differentials, not derivatives. Doing this reinforces the rules for computing derivatives and introduces the student to the manipulation of differentials that will be necessary in integration.

A bolder approach, which the author has employed, is to replace the typical “limit of difference quotient” derivations with these heuristic arguments and adopt the point of view that is a ratio of infinitesimals.

Introduction

In our college, we teach a quick, one-credit Calculus Workshop course for students who have received credit for taking first-semester calculus elsewhere (in high school or at another college) but who need a brief introduction to some specific topics they may not have seen before. And so we spend one class on using a computer algebra system, one class on Euler's Method, one class on writing about mathematics … a whirlwind tour of a variety of topics our regular Calculus I students see in more depth. The class meets for one hour and fifteen minutes twice a week during the first half of the semester.

Even in such a condensed course, we want our students to learn something about the history of calculus, and especially about the most famous names associated with its beginnings. They have encountered Isaac Newton's name in solving a murder mystery (Newton's Law of Cooling) and in studying air resistance for a falling body (Newton's Laws of Motion), but few of them know much about his role in the discovery of calculus — and most of them have never even heard of Gottfried Leibniz. Students in our regular calculus sequence read, discuss and write about several articles on the development of the calculus from [2] and [4], but we do not have that luxury in the workshop course. And so on the last day of class, we take a quick dive into the history of the subject. The activity described in this article introduces students to the lives and work of Newton and Leibniz in a way that includes active student participation, collaboration, and the use of technology — in one class period.

Introduction

In most trigonometry courses, the instructor begins by defining the sine, cosine, and tangent of an angle as ratios of certain sides in an appropriate right triangle. She then proceeds to calculate, using elementary geometry, the sine, cosine and tangent of angles of 30°, 45°, and 60°. But once students need to calculate the sine of 27°, they are told to punch some buttons on their calculators. What do students think happens when they do that? Do they imagine that somewhere inside the calculator, someone draws a miniature right triangle with one base angle 27°, then measures the sides and divides? Where do these numbers come from that so miraculously appear on the calculator screen in half a second?

Fifty years ago, no one had calculators. Then, the trigonometry texts simply told the students to consult the table at the back of the book to find the sine of 27°. That took a bit longer, but still, there was little in the text to show students where those numbers came from. They just “were”. Whether one uses tables or uses calculators, it still seems that there is a mystery in these numbers that should not exist. Most teachers certainly want their students to be fluent in calculator use – and these are generally easier to use than tables. But still, we do not want students thinking that calculators are magic.

Introduction

In today's world of electronic clocks and universal calendars, it's easy to forget how important mathematics used to be just for the fundamental task of figuring out what time it was. The standard rigorous approach to the problem involved applying trigonometry to observed positions of the sun or the stars, as described below (“In the Classroom”). But several simpler methods were also developed for use when observations were unavailable or calculation was unappealing. One such practical device was the sinking-bowl water-clock, used for many centuries in India. Students (and teachers) will be impressed by how easy such a clock is to construct and adjust, and how much mathematical labor it can save.

This activity and discussion can be used as part of a module on trigonometry. A more advanced class in calculus may be interested in the theoretical modeling of water-clock construction, and especially in comparing the real mathematics of water-clock design with the artificial assumptions made in typical “related rates” problems about filling and draining water tanks. The construction and testing of the sinking-bowl model can take as little as ten or fifteen minutes (depending on the length of its period): exploring the trigonometry of time-telling may involve fifteen or twenty minutes more.

Historical Background

Any water-clock (or “clepsydra”, Greek for “stealing water”) works on more or less the same principle as an hourglass: it measures a fixed period or interval of time by means of a substance flowing through a hole in a container, and at the end of that interval it must be reset manually to measure another period of the same length.

Introduction

Does anyone care about trigonometry? Certainly many of our students don't, aside from the exigency of getting through their exams. As mathematics teachers, we have passion for our subject for its own sake — but we often justify ourselves to our students in terms of what the mathematics can accomplish elsewhere. For trigonometry as for many other topics, this takes the form of the widespread “word problems”: how high is that pine tree across the street? How far did that motorboat travel when it went across the lake? And here we reach a crucial pedagogical problem: few of us really care precisely how tall the tree is, or how far the boat went. We find ourselves forced into producing “baby” problems like these with little real relevance, assuring our students (with fingers crossed behind our backs) that the genuine applications — too complex for their immature minds — hopefully work kind of like these ones do.

Meaningful contexts are surprisingly hard to find. Some pedagogical efforts are searching for realistic classroom friendly projects, and are having some success. However, one source that might easily be overlooked is the history of the subject. Two thousand or more years of human experience is a powerful resource on which to draw. Mathematical subjects arise for good reasons, and bringing these reasons to light can motivate more honestly what otherwise might appear dull, even deceptive in its fake “applications”.