Physical Sciences and Mathematics Commons^™

Gödel's Theorem In The Continuing Education Of Mathematics Teachers, Ana J. Lemes

Journal of Humanistic Mathematics

The notion of dépaysement épistémologique (epistemological disorientation) aims to capture the sense of disorientation when a learner is led to question their prior assumptions and understandings, generating uncertainty in a context in which they thought they had certain knowledge. This article describes an activity used with a group of practicing mathematics teachers in Uruguay that integrates elements of the history of mathematics related to Gödel’s incompleteness theorem, with the aim of provoking in the participants the experience of dépaysement épistémologique. Results show that several of the teachers participating in the activity felt dépaysement épistémologique, and this feeling triggered …

The Nothing That Really Matters, Szilárd Svitek

Journal of Humanistic Mathematics

Zero has (a) special role(s) in mathematics. In the current century, we take negative numbers and zero for granted, but we should also be aware that their acceptance and their emergence in mathematics, and their ubiquity today, have not come to happen as rapidly as, for example, that of natural numbers. Students can quickly become confused by the question: is zero a natural number? The answer is simple: a matter of definition. The history of zero and that of negative numbers are closely linked. It was in the calculations of debts that the negative numbers first appeared, where the state …

From Historically First "Unary" Numbers, Through Egyptian Fractions, Roman Numerals, Leibniz's Binary Numbers And Kepler's Fractions To Modern Ideas Such As Calkin-Wilf Tree: A Unified Approach To Representing Natural Numbers And Fractions, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

In elementary mathematics classes, students are often overwhelmed by different representations of numbers and corresponding operations: usual fractions, decimal representations, binary numbers, etc. What often helps is when students learn the history of these representations, see the limitations of seemingly reasonable representations like Roman numerals, and how other representations overcame these limitations. Still, history was developed somewhat randomly, so the historical sequence is still somewhat chaotic. We believe that providing a unified approach for all these representations would help describe their sequence in a more logical way and thus, help the students even more.

In our analysis, we explore the …

Archimedes Of Syracuse And Sir Isaac Newton: On The Quadrature Of A Parabola, Wyatte C. Hooper

Journal of Humanistic Mathematics

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that …

“It’S All For The Best”: Optimization In The History Of Science, Judith V. Grabiner

Journal of Humanistic Mathematics

Many problems, from optics to economics, can be solved mathematically by finding the highest, the quickest, the shortest—the best of something. This has been true from antiquity to the present. Why did we start looking for such explanations, and how and why did we conclude that we could productively do so? In this article we explore these questions and tell a story about the history of optimization. Scientific examples we use to illustrate our story include problems from ancient optics, and more modern questions in optics and classical mechanics, drawing on ideas from Newton’s and Leibniz’s calculus and from the …

Mathematical Conquerors, Unguru Polarity, And The Task Of History, Mikhail Katz

Journal of Humanistic Mathematics

I compare several approaches to the history of mathematics recently proposed by Blåsjö, Fraser–Schroter, Fried, and others. I argue that tools from both mathematics and history are essential for a meaningful history of the discipline.

In an extension of the Unguru–Weil controversy over the concept of geometric algebra, Michael Fried presents a case against both Andr ́e Weil the “privileged observer” and Pierre de Fermat the “mathematical conqueror.” Here I analyze Fried’s version of Unguru’s alleged polarity between a historian’s and a mathematician’s history. I identify some axioms of Friedian historiographic ideology, and propose a thought experiment to gauge its …

Design And Evaluation Of An Adventure Videogame Based In The History Of Mathematics, Mariana Rocha, Pierpaolo Dondio

Conference papers

The present paper describes the design and evaluation of an adventure videogame developed to cover the mathematics primary school curriculum. The narrative of the game is based in the history of mathematics and, to win, the player needs to travel through time, starting from the ancient Egypt and finishing at the modern world. To achieve that, the player interacts with real-life characters, such as Pythagoras of Samos, learning about their contributions to the field and using this knowledge to solve puzzles. The aim of the research presented in this paper is to understand the effects of the game on students’ …

Does Teaching The History Of Mathematics In High School Aid In Student Understanding?, Anne Campbell

Undergraduate Honors Thesis Projects

This research will study the effect teaching the history of mathematics in a high school classroom has on student understanding. To accomplish this, lessons both including and excluding historical background on different topics were taught in an Honors Algebra 2 class in the high school setting. This research aims to engage student learning and investigation of topics that normally do not draw a lot of student focus and spark a new or revived interest in mathematics for students by broadening lessons to include material of which students would not otherwise be exposed. The lessons themselves aim to engage other current …

Teaching History Of Mathematics: A Dialogue, Benjamin Braun, Eric Kahn

Journal of Humanistic Mathematics

Many colleges and universities offer a course in the history of mathematics. While the potential benefits for students taking such a course might be apparent, it is often less clear how teaching a history of mathematics course can be a transformational experience for faculty. We present a dialogue between the authors regarding their experiences teaching history of mathematics courses, including their motivation for doing so, the impact these experiences have had on their classroom practices and assessment methods, and the opportunities history of mathematics courses offer for incorporating social justice, equity, and inclusion into the study of mathematics. Our goal …

Ways Of Relating To The Mathematics Of The Past, Michael N. Fried

Journal of Humanistic Mathematics

Historians of mathematics, by definition, look at mathematics of the past. But mathematicians, too, often look at mathematics of the past; mathematicians of the past themselves often looked very closely at mathematics of their own past. Is their relationship to the past the same as that of the historians? Is every view of the past an historical view? Indeed, is every historical view historical in the same way? Or is it possible that there are different kinds of relationships to the mathematics of the past? This paper will suggest that there are in fact a variety of such relationships. It …

How To Calculate Π: Machin's Inverse Tangents, A Mini-Primary Source Project For Calculus Ii Students, Dominic Klyve

Mathematics Faculty Scholarship

Almost every mathematical culture through history seems to have proved, trusted, or suspected that the area of a circle is a fixed constant times the square of its radius. It is maybe not surprising, then, that the last two millennia have seen a seemingly endless array of attempts to calculate this constant (today usually called π" role="presentation">π) with increasing precision.

History Of Computing, George K. Thiruvathukal, David B. Dennis

David B. Dennis

The social and organizational history of humanity is intricately entangled with the history of technology in general and the technology of information in particular. Advances in this area have often been closely involved in social and political transformations. While the contemporary period is often referred to by such names as the Computing and Information Age, this is the culmination of a series of historical transformations that have been centuries in the making. This course will provide a venue for students to learn about history through the evolution of number systems and arithmetic, calculating and computing machines, and advanced communication technology …

History Of Computing, George K. Thiruvathukal, David B. Dennis

George K. Thiruvathukal

The social and organizational history of humanity is intricately entangled with the history of technology in general and the technology of information in particular. Advances in this area have often been closely involved in social and political transformations. While the contemporary period is often referred to by such names as the Computing and Information Age, this is the culmination of a series of historical transformations that have been centuries in the making. This course will provide a venue for students to learn about history through the evolution of number systems and arithmetic, calculating and computing machines, and advanced communication technology …

History Of Computing, George K. Thiruvathukal, David B. Dennis

Computer Science: Faculty Publications and Other Works

The social and organizational history of humanity is intricately entangled with the history of technology in general and the technology of information in particular. Advances in this area have often been closely involved in social and political transformations. While the contemporary period is often referred to by such names as the Computing and Information Age, this is the culmination of a series of historical transformations that have been centuries in the making. This course will provide a venue for students to learn about history through the evolution of number systems and arithmetic, calculating and computing machines, and advanced communication technology …

The Derivatives Of The Sine And Cosine Functions: A Mini_Primary Source Project For Calculus I Students, Dominic Klyve

Mathematics Faculty Scholarship

This curricular modular guides students through a method of calculating the derivative of the sine and cosine functions using differentials. It is based on one primary source: Leonhard Euler's Institutiones calculi differentialis (Foundations of Differential Calculus) [2], published in 1755.

The Search For One As A Prime Number: From Ancient Greece To Modern Times, Angela Reddick, Yeng Xiong

Furman University Electronic Journal of Undergraduate Mathematics

It has often been asked if one is a prime number, or if there was a time when most mathematicians thought one was prime. Whether or not the number one is prime is simply a matter of definition, but definitions are often decided by the use of mathematics. In this paper we will survey the history of the definition of prime as applied to the number one, from the ancient Greeks to the modern times. For the Greeks the numbers (αριθμος) were multiples of the unit, and for this reason one did not fall into the category of …

History Of Mathematics From The Islamic World, Asamah Abdallah

All Student Theses

Learning the history of mathematics is crucial to fully understanding the world of mathematics today. This paper will explore the history of mathematics from the Islamic world. It will focus on the contributions of well-recognized mathematicians including, Al-Khwarizmi, Al-Khayyam, Uqlidisi, Kushyar ibn Labban, and Abu Kamil. It will also concentrate on the contributions that the Islamic world had on algebra, beginning with Al-Khwarizmi and his contribution to the developmental of algebraic equations, and Khayyam and his contribution to the geometrization of algebra. This paper will also discuss the ways in which the Muslims applied the mathematics they learned into their …

The World Before Calculus: Historical Approaches To The Tangent Line Problem, Lindsay Skinner

WWU Honors College Senior Projects

Pierre de Fermat and René Descartes were two brilliant 17th century mathematicians who have had lasting impacts on modern mathematics. Descartes laid the groundwork for the Cartesian coordinate system that is frequently employed in modern mathematics and Fermat’s last theorem vexed the mathematics community until Wiles’ proof was published in 1995. Amidst their many ground-breaking accomplishments these two men produced solutions for another mathematical problem - developing a general method to find the tangent line to a curve.

In spite of their apparent genius, neither man’s method had the lasting impact of their other works. Descartes’ and Fermat’s methods were …

The Bernoulli Family: Their Massive Contributions To Mathematics And Hostility Toward Each Other, Dung (Yom) Bui, Mohamed Allali

e-Research: A Journal of Undergraduate Work

Throughout the history of mathematics, there are several individuals with significant contributions. However, if we look at the contribution of a single family in this field, the Bernoulli probably outshines others in terms of both the number of mathematicians it produced and their influence on the development of mathematics. The most outstanding three Bernoulli mathematicians are Jacob I Bernoulli (1654-1705), Johann I Bernoulli (1667-1748), and Daniel Bernoulli (1700-1782), all three of whom were the most influential math experts in the academic community yet very hostile to each other. Their family structure and jealousy toward each other might have fueled their …

The Discipline Of History And The “Modern Consensus In The Historiography Of Mathematics”, Michael N. Fried

Journal of Humanistic Mathematics

Teachers and students of mathematics often view history of mathematics as just mathematics as they know it, but in another form. This view is based on a misunderstanding of the nature of history of mathematics and the kind of knowledge it attempts to acquire. Unfortunately, it can also lead to a deep sense of disappointment with the history of mathematics itself, and, ultimately, a misunderstanding of the historical nature of mathematics. This kind of misunderstanding and the disappointment following from it--both raised to the level of resentment--run through the paper "A Critique of the Modern Consensus in the Historiography of …

Count Like An Egyptian: A Hands-On Introduction To Ancient Mathematics (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: Reimer, David. Count LIke an Egyptian: A Hand-On Introduction to Ancient Mathematics. Princeton, NJ: Princeton University Press, 2014. 237 pages. ISBN 9780691160122

Al-Khwarizmi: Founder Of Classical Algebra, Calvin Jongsma

Faculty Work Comprehensive List

Adopting a historically defensible definition of “algebra,” we will begin by exploring a few examples of algebra prior to al-Khwarizmi. We will then examine what algebra became through al-Khwarizmi’s work. In conclusion, we will assess the historical importance of al-Khwarizmi’s contributions for developments in European algebra.

Much More Than Symbolics: The Early History Of Algebra And Its Significance For Introductory Algebra Education, Calvin Jongsma

Faculty Work Comprehensive List

No abstract provided.

Routes Of Learning: Highways, Pathways, And Byways In The History Of Mathematics (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: Routes of Learning: Highways, Pathways, and Byways in the History of Mathematics by Ivor Grattan-Guinness. Baltimore, MD: The Johns Hopkins University Press, 2009. xii + 372 pages, with index. ISBN: 9780801892486.

EΠi + 1=0: The History & Development, Dawne Charters-Nelson

Undergraduate Review

I have on occasion run across the equation in books, articles and in conversation with other mathematicians. In each of these encounters the person alluded to a fascination with this equation which links the five most important constants in the whole of analysis:

• 0 = The additive identity
• 1 = The multiplicative identity
• π = The circular constant
• e = The base of the natural logarithms
• i = The imaginary unit

Being a novice mathematician, I wondered how all these fundamental constants could end up in one equation and what it meant. Along with this thought came the realization that …

Neglected Standard: History Of Mathematics In The Service Of Mathematics Education, Calvin Jongsma

Faculty Work Comprehensive List

An integrated approach that more intrinsically connects the historical development of mathematics with its content enables students to learn how an idea or method emerged while simultaneously exposing (i) the dynamic nature of mathematics along with (ii) its connections to other fields, and (iii) its cultural embeddedness. Examples will be given from a textbook currently in development.

Mathematics And The Divine (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: Teun Koetsier and Luc Bergmans, editors. Mathematics and the Divine. Boston: Elsevier, 2005. 701 pages. ISBN 0-444-50328-5

Sylvester: Ushering In The Modern Era Of Research On Odd Perfect Numbers, Steven Gimbel, John Jaroma

Philosophy Faculty Publications

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester's bound stood …

The Beginnings Of Mathematics, Gail Ray

Honors Theses

Our first conceptions of number and form date back to times as far removed as the Old Stone Age. Little progress was made in understanding numerical values and space relations until the transition occurred from the mere gathering of food to its actual production, from hunting and fishing to agriculture. With this fundamental change, a revolution in which the passive attitude of man toward nature turned into an active one, we enter the New Stone Age. The tempo of technical improvement was enormously accelerated.

Selections From "Mathematics: Our Great Heritage" Edited By William L. Schaaf, Mary Beth Mcgee

Honors Theses

This paper reviews and summarizes several essays within the text, Mathematics: Our Great Heritage edited by William L. Schaaf.