Pi Mu Epsilon - Mathematical Association of America

Thursday MAA Session #2 August 2, 2012
9:10–9:25
The History of the Primality of One
Angela Reddick
University of Tennessee at Martin
It has often been asked if one is prime number, or if there was a time when most mathematicians
thought one was prime. Whether or not one is a prime is simply a matter of definition, but definitions
are often decided by the use of mathematics. In this paper we will explore 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 (αριθµoς) were multiples of the unit, and for this reason one did not
fall into the category of primes (a subdivision of the numbers). This view held with few exceptions
until Stevin (c. 1685) argued successfully one was a number, at which point it finally makes sense
to ask if one is prime. This is followed by a period of confusion which begins to dissipate with
Gauss’ Disquisitiones Arithmeticae. Thus, for most of history, one was not considered a prime,
and that there was no point in time where the majority of mathematicians viewed one as prime.
Finally, we discuss who was the last mathematician to consider one a prime.
9:30–9:45
Multiple Choice Versus Open Response Assessment in Calculus
Monica Lee Chalke
Bridgewater State University
The use of multiple choice questions is becoming increasingly common in recent years as an assessment
tool in Calculus. The ease of grading of multiple choice questions has lead to their
widespread use to assess student understanding on homework assignments via online homework
systems as well as on tests. Motivated by the question of whether or not it is appropriate to use
multiple choice questions on quizzes and exams in Calculus, we conducted a study to investigate
the impact of multiple choice questions versus open response questions on student understanding.
The goal of our study is to identify if there is a performance gap between Calculus students
who were tested via multiple choice questions on quizzes and exams versus students whose exams
consisted only of open response questions. We also investigate the performance of students on
particular Calculus topics when assessed using these two methods.
9:50–10:05
Symmetry of the Numerical Range
Shelby Burnett and Ashley Chandler
Cal Poly San Luis Obispo
The numerical range of an n × n matrix A is a convex subset of the complex plane defined by
W (A) = {⟨Av, v⟩ : v ∈ C n , ∥v∥ = 1} . The numerical range is said to have n-fold symmetry
about the origin if z ∈ W (A) implies e 2πi
n z ∈ W (A). If A is a 2 × 2 matrix then A is a possibly
degenerate elliptic disk with foci at the eigenvalues of A. Naturally, symmetry results for this case
have been fully explored. We also know that if the entries of A are real (or A is unitarily equivalent
to a real matrix), then W (A) is symmetric about the real axis. In our talk, we will discuss natural
generalizations of these results to n × n matrices where n ≥ 4. Specifically, we will focus on
n-fold symmetry about the origin and symmetry about particular axes.
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