2010
Saturday morning – Chair: David L. Roberts
8:15 – 8:45 Registration on site
8:45 – 9:15
Bob Stein, California State University, San Bernardino
History in a Math Course for Teachers: How and Why
Mathematics courses for teachers are so very full of material that it may seem strange to even consider adding some history of mathematics. However, doing so can be very rewarding in some contexts. We will examine some mathematical topics and present possible uses of history to enhance teaching those topics. This talk will focus primarily
on examples from K-12 mathematics, but the ideas presented may be applied at any curricular level.
9:20 – 9:50
Uffe Thomas Jankvist, Southern University of Denmark
utj@imada.sdu.dk
Designing teaching modules on the history and philosophy of mathematics – a report from an in-progress study in Danish high schools
For the mathematics program in Danish high schools a recent reform has put emphasis on the development of students’ mathematical competencies as well as what is referred to as three types of ‘overview and judgment’. ‘Overview and judgment’ are not themselves mathematical competencies (which is regarded as a form of readiness to act in mathematical situations), but a kind ‘active insight’ into the character of mathematics as both a pure and applied science as well as its relations to nature, society, and culture. The Danish KOM-report on ‘competencies and mathematics learning’ lists the three types of overview and judgement: (1) the actual application of mathematics in other subject and practice areas; (2) the historical evolution of mathematics, internally as well as in a societal context; (3) the nature of mathematics as a subject. The definition of the latter (3) includes various aspects of the philosophy of mathematics, e.g. the characteristics of mathematical problem formulation, thought, and methods; the types of results produced by mathematics; the science-philosophical status of its concepts and results; its construction; its connections of other disciplines and the way it distinguishes itself from these.
By explaining the design of teaching modules using original historical sources, both of the emergence of a mathematical discipline and a later application of this to an extra-mathematical problem, I shall discuss the possibilities of having the students engage actively in discussions of and reflections about the above mentioned types of ‘overview and judgement’ (especially those related to 3). In particular I shall discuss a teaching module which is presently being designed and which is to be implemented and monitored in a high school class later in 2010.
9:55 – 10:25
Chris Rorres, Drexel University
The History of the Archimedes Screw and its Contemporary Applications
The Archimedes screw is one of the oldest machines still in use today. It is presently enjoying a renewed interest because of its proven trouble-free design, its ability to lift wastewater and debris-laden water effectively, and its gentle treatment of aquatic life. In this presentation I will give a history of this device from Archimedes’ time to the 21st century and also discuss its popularity as a teaching tool for engineering, science, and mathematics students.
10:30 – 10:40 COFFEE BREAK
10:45 – 11:15
Victor Katz, University of the District of Columbia
Quadratic Equations: Can We Really Apply Them?
The solution of quadratic equations was at the center of the first algebra textbook, written by Mohammad al-Khwarizmi around 825 CE. Later Islamic authors extended al-Khwarizmi’s work and even attempted, unsuccessfully, to find algebraic solutions to cubic equations. It was this Islamic work that was transmitted to Europe in the late Middle Ages and became the core of Europe’s own work in algebra. But certainly the mathematicians of the Middle Ages and the Renaissance were interested not only in theoretical mathematics but also in applying algebra to solve genuine real-world problems. We will consider the work not only of Islamic mathematicians but also of European mathematicians of the Renaissance and later who wrote on algebra and see how they attempted to find real problems which required quadratic equations.
11:20 – 11:50
John Snygg, East Orange, NJ
Is Copernicus the victim of a bum rap?
After the discovery of a manuscript by al-Shatir in 1957, it was determined that Islamic astronomers originated a mathematical device used by Copernicus. This device enabled Islamic astronomers and Copernicus to eliminate an internal contradiction of Ptolemy’s theory, known as the “equant problem.” Tycho Brahe who never accepted a completely heliocentric theory expressed the opinion that this resolution of the equant problem was Copernicus’ greatest achievement.
Since the device appears complicated and even idiosyncratic, it has become conventional wisdom that Copernicus adopted this device from some unknown Islamic source without attribution. Copernicus claimed that he arrived at the device only after he approached the problem from a heliocentric point of view. In this paper, I will show that a modest attempt by Copernicus to solve the equant problem in a heliocentric context would have led inevitably to a rediscovery of the Islamic solution.
11:55 – 12:25
Frederick Rickey, United States Military Academy
Machin’s Formula for computing Pi.
Around 1706, John Machin found an interesting arctangent formula:
π/4 = 4arctan(1/5) – arctan(1/239)
that he expanded using Gregory’s arctangent series and then computed π “True to above a 100 Places.”
We shall prove Machin’s formula, discuss how it has been used to compute π and discuss its interesting history and ramification. We shall refrain from the “indoor sport” of deriving variants of Machin’s formula but will discuss a few of them.
12:30 – 1:40 LUNCH AND BUSINESS MEETING
Saturday afternoon – Chair: Bob Stein
1:45 – 2:15
Eugene Boman, Penn State, Harrisburg campus
Ghosts of Departed Errors: A look at Bishop Berkeley’s, The Analyst and the scientific community’s initial response to it.
In 1734 Bishop Berkeley criticized the logical foundations of the Calculus in The Analyst and set off a small ‘pamphlet war’. James Jurin and John Walton replied immediately and angrily. Berkeley then responded to each of them. Shortly after this exchange Jurin and Benjamin Robins engaged in a lengthy and eventually acrimonious public debate on the same topic. Somewhat later Benjamin Robins, Collin Maclaurin, Thomas Simpson and others wrote treatises on The Method of Fluxions which were at least in part intended as responses to The Analyst. These exchanges offer a window onto the scientific community’s view of Calculus in its earliest stages. I will attempt to peer through that window.
2:20 – 2:50
Amy Ackerberg-Hastings, University of Maryland University College
John Farrar and Curricular Transitions in Mathematics Education
Historians of American mathematics education from Florian Cajori to Helena Pycior have recognized John Farrar (1779-1853) for introducing the French analytical style into college teaching through his Cambridge Series of Mathematics and Natural Philosophy. These eleven textbook translations asked much more of Harvard students than had the instruction provided by previous professor and college president, Samuel Webber. Farrar’s efforts also reflected the shift to Unitarian leadership and the intellectual and physical expansion of the campus of the John T. Kirkland administration. Yet, Farrar faced opposition throughout his career both from those who appreciated traditional approaches and those who argued his reforms were not substantive enough. By the 1830s, he was pushed aside by Josiah Quincy, Nathaniel Bowditch, and Benjamin Peirce, who advocated financial streamlining, disciplined behavior, and elective courses targeted to talented students in the Harvard curriculum. While his direct influence over the Harvard mathematics curriculum was relatively short-lived, thanks in large part to the emergence of Peirce as the prototype for an American professional mathematician, this paper will argue that Farrar nonetheless filled an essential transitional role in the development of nineteenth-century American mathematics education. Supporting evidence will be drawn from Farrar’s surviving correspondence, Harvard administrative records, and his textbook series.
2:55 – 3:25
Andy Fiss, Indiana University, Bloomington
afiss@indiana.edu
The Effects of the Civil War on College-Level Math Education
In the decade before the Civil War, mathematics classes at American colleges emphasized applications to the point of absurdity. In the first course of the required two-year series (algebra), students had to recite long passages about the relevance of mathematics for a variety of careers, from trading and engineering to history and government. The last course of the series (trigonometry) ended with lengthy units about the uses of mathematics in two practical professions of the day, navigation and surveying. However, in the decades after the war, pure mathematics came to dominate college-level education, to the point in 1893 when seven of the nation’s top colleges and universities offered no applied classes at all. Although it may at first seem paradoxical, this paper discusses ways in which the Civil War can account for this disciplinary change in mathematics. The four-year conflict led to the devaluing of one applied mathematical pursuit (surveying), the increasing autonomy of another (navigation), and the establishment of novel institutions of higher education where pure mathematics could find a home.
3:30 – 3:40 COFFEE BREAK
3:45 – 4:15
J.J. Tattersall, Providence College
The Mathematical Department of the Yates County Chronicle
From February 29, 1872 to August 26, 1880, the Yates County Chronicle, a weekly newspaper published in Penn Yan, New York, featured a mathematics department edited by a local doctor, Samuel Hart Wright. Originally, the column was limited to easy and popular examples supported by local contributors. It advanced gradually in interest to become a medium of higher mathematics supported by mathematicians of distinguished abilities from around the country. Wright’s goal was to make the department acceptable to all and especially interesting to its contributors. We discuss Wright’s interesting life, several mathematical contributions, and other items that appeared in the Chronicle.
4:20 – 4:50
Walter Meyer, Adelphi University
Cajori Two and the History of Undergraduate Mathematics in America in the 20th Century
In the last century or so, the extent of applications in the undergraduate curriculum in America has varied. According to evidence provided by Florian Cajori, the extent was substantial at the end of the 19th century. We also have good evidence, from an MAA survey, that by the middle of the 20th century applications were largely gone from undergraduate curricula. For the intervening period, over half a century, we have little easily available information (as far as I can tell) covering a representative sample of institutions.
It is not that the course information does not exist. It can be found in college catalogs in the archives of the institutions issuing those catalogs. But these hardly ever circulate and are rarely online except for recent years. And hardly anyone has the funds and the yen to travel to far corners of the country to visit those libraries. Cajori Two is a project I am undertaking with 3 colleagues to gather photocopies of the mathematics sections of the catalogs of 20 representative institutions (with the help of archivists at the libraries), to summarize the course data found there in Excel workbooks, and to develop software to provide numerical counts and sums in a flexible way required by the user. This database should give insight into the role of applications, of discrete mathematics, of service courses and many other questions about curriculum. This project is far from done, but a progress report will be given.
4:55 – 5:25
George Rosenstein, Franklin & Marshall College
The Fundamental Theorem of Calculus through the eyes of Granville, Smith, and Longley
What is the Fundamental Theorem of Calculus and why is it fundamental? In this talk, I will examine these questions and their changing roles in the texts of Granville, Smith, and Longley widely used between 1905 and the 1950s.
Sunday morning – Chair: Amy Ackerberg-Hastings
8:45 – 9:15
Ilhan M. Izmirli,. George Mason University
iizmirl2@gmu.edu
Egyptian Unit Fractions – An Existence Theorem
One of the most fascinating aspects of Egyptian mathematics was the way they expressed fractions as sums of unit fractions (inverses of natural numbers and 2/3). It is even more fascinating that the first existence theorem was actually given some millennia later by Fibonacci.
Since this representation (which, from now on I will refer to as unit representations) is not unique (for example, 5/6 = 2/3 + 1/6 = 1/2 + 1/3), later efforts concentrated on finding “better representations” (ones with fewer terms and/or smaller denominators). To this end, several algorithms were developed.
In this presentation, after talking briefly on the mathematics of Egypt, we will give an existence proof of unit representations and discuss some of the algorithms designed to find them. We will also talk about the significance of unit fractions and the possible reasons as to why this was the preferred mode of representation.
9:20 – 9:50
James Propp University of Massachusetts Lowell
Dedekind’s Forgotten Axiom and Why We Should Teach It
Around 1870, Richard Dedekind formulated an axiom characterizing the continuity (nowadays we would say “completeness”) of the real number system. You probably think you know what this axiom was, but if you’re like most mathematicians (including myself a year ago), you’re mistaken. I will discuss Dedekind’s axiom and argue that it makes more pedagogical sense than the usual approaches to axiomatizing the real numbers. I will raise (but not answer) the historical question of why Dedekind’s axiom has been “lost”. Finally, I’ll show how one can deduce the axiom of induction for the natural numbers as a consequence of the axiom of completeness of the reals, and explain why this is not as perverse an approach to induction as one might at first think.
9:55 – 10:25
Robert L Brabenec, Wheaton College, Wheaton, IL
Thinking Philosophically about Mathematics
In my talk, I will present a broad overview of the important periods throughout history, from the ancient Greek culture until the present day, when mathematicians were encouraged to think philosophically about their discipline. While the three major philosophies of mathematics–logicism, intuitionism, and formalism–were proposed during the period roughly from 1880 to 1930, there are several other distinct times when philosophical concerns influenced mathematics. It is instructive to be aware of the mathematical developments during the nineteenth century that led to this emergence of formal philosophies after 1880. These include such items as the discovery of non-Euclidean geometry, the bringing of rigor into calculus, the development of abstract algebraic structures, the definition and construction of real numbers, and Cantor’s theory of infinite sets. To understand the overall development of these major concepts in their historical context can help both faculty and students to better understand and enjoy the discipline of mathematics.
10:30 – 10:40 COFFEE BREAK
10:45 – 11:15
Roman Sznajder, Bowie State University
90th anniversary of emergence of the Polish School of Mathematics; Polish mathematics between the world wars
The aim of this talk is to outline the history and genesis of the Polish School of Mathematics, which was a unique event in European intellectual life after WWI. We will emphasize the founders of this School and the research trends they represented.
The history of the Polish School of Mathematics (PSM) is hardly known to a broader mathematical audience. The 90th anniversary of the creation of the PSM is a good occasion to highlight its tremendous achievements, which positioned Polish mathematics in the mainstream of world mathematics.
11:20 – 11:50
Jerry Lodder, New Mexico State University
Deduction Through the Ages: Teaching Elementary Logic via Primary Historical Sources
A beginning undergraduate course in discrete mathematics is often taught as a fast-paced news reel of facts and formulas with little regard for the vast historical developments preceding the textbook presentation of modern, polished mathematics. In this talk we examine the development of propositional logic, and in particular the truth of an “implication” from ancient Greek sources to the twentieth century. Beginning with Philo of Megara’s (4th century B.C.E.) statement that a valid hypothetical proposition is “that which does not begin with a truth and end in a falsehood,” we examine various other major premises from another ancient Greek philosopher Chrysippus (280–206 B.C.E.) and discuss possible logical equivalences among them. This is followed by George Boole’s (1815–1864) introduction of arithmetic symbols to codify verbal statements. Exhibiting a complete break between arithmetic and logic, the concept script of Gottlob Frege (1848–1925) is introduced, with particular emphasis on the “condition stroke.” Philo’s statement above is contrasted with Frege’s declaration that the condition stroke is a function of two truth values that is false when the zeta (first) argument is true and the xi (second) argument is false. The streamlined notation of the modern writers Bertrand Russell (1872–1970) and Alfred Whitehead (1861–1947) is introduced, and their definition of “p implies q” as “it cannot be the case that p is true and q is false” is contrasted with Frege’s condition stroke and Philo’s statement. Finally the truth tables of Emil Post (1897–1954) are discussed and his table for p implies q concludes the talk. By contrast, many modern textbooks on discrete mathematics begin the chapter on propositional logic with this very table.
11:55 – 12:25
David L. Roberts, Prince George’s Community College
WFF ‘N PROOF and the climate for Mathematical Recreations in the 1950s and 60s
In the early 1960s Layman E. Allen of the Yale Law School began to market “WFF ‘N PROOF, The Game of Modern Logic,” with the stated belief that “learning ought to be fun.” Competing players used dice-like cubes to explore the construction of what logicians refer to as well-formed formulas (WFFs). By working through a sequence of progressively more complicated games, players were expected to learn the rudiments of propositional calculus, in the parentheses-free notation of Polish logician Jan Łukasiewicz. This talk will examine how these games fit into the context of their time, especially the mathematical recreations being promoted by Martin Gardner, and the “new math” educational reforms. I hope to illuminate tensions in this era between promoting mathematics as a pleasurable human endeavor and pushing it for its instrumental value.
12:30 – 1:00
Roger Rosenkrantz, independent scholar
Physics through its history
I will describe an historical approach to high school physics that emphasizes scientific method and mathematical modeling. The opening unit traces the development of applied geometry, cosmology, astronomy and the origins of the experimental method in ancient Greece and outlines the synthesis of Aristotelian cosmology and geocentric astronomy and its ancient critics. This sets the stage for the scientific revolution of the 17th century, the core of the course. Some of the flavor will be conveyed by a brief sketch of the nearly two thousand year quest to find the distance to the Sun.