Abstracts

Plenary Talks

Elliot A. Tanis, Hope College
Friday, 9:00-9:50 am
102 VanderWerf

Using Maple for Visualization, Manipulation, and Simulation

Students can sometimes manipulate symbols but do not really visualize what the symbols represent. Material "learned" in a mathematics course early in a student's education may be difficult for that student to apply in a later course. Simulation can often lend insight into understanding a theoretical concept.

A variety of examples will show how a computer algebra system, in this case Maple, can be useful. Animation of graphs of functions will be illustrated, problems will be solved symbolically, and simulation will be incorporated in this talk.

Joan Ferrini-Mundy, Michigan State University
Friday Luncheon Address
Haworth Inn

NCTM's Principles and Standards for School Mathematics:
Implications for Mathematics Departments

A brief overview of NCTM's revised standards document, Principles and Standards for School Mathematics. What role did mathematicians have in its development? How are they reacting? Should university and college mathematics departments pay attention to it, and why? Issues related to the preparation of teachers and the transition of students from secondary school to the undergraduate program will be discussed.

Christopher M. Skinner, University of Michigan, Ann Arbor
Friday, 2:00-2:50 pm
102 VanderWerf

Fermat's Legacy

Which integers equal the sum of two squares of positive integers? Can the sum of two cubes of positive integers ever be the cube of a positive integer? Does there exist a right triangle with sides of rational length and area equal to one?

The problems that tickled the mathematical fancy of the 17th century French lawyer Pierre de Fermat have had an influence on mathematics out of proportion to the simplicity of their statements. I will discuss a few of these problems and explain how attempts to solve them led to notions now familiar to all mathematicians and how they are still connected to the frontiers of research.

Edward B. Burger, Williams College
Friday Banquet Address
Piper Restaurant

Personal Thoughts on What to Teach and How Not to Teach It

Here I will share some recent experiments in Pre-Calculus, Calculus, and Math for Liberal Arts courses. Wild departures from conventional wisdom will be confessed and demonstrations of both the high-tech and no-tech varieties will be given.

Edward B. Burger, Williams College
Saturday, 9:00-9:50 am
102 VanderWerf

"How to Always Win at Limbo",
or You can sum some of the series some of the time,
and some of the series none of the time,
but can you sum some of the series ALL of the time?

Have you ever gone out with someone for a while and asked yourself: "How close are we?" This presentation will answer that question by answering: What does it mean for two things to be close to one another? We'll take a strange look infinite series, dare to mention a calculus student's fantasy, and momentarily consider transcendental meditation. In fact, we'll even attempt to build some very exotic series that can be used if you ever have to flee the country in a hurry: we'll either succeed or fail; you'll have to come to the talk to find out. Will you be at the edge of your seats? Perhaps; but if not, then you'll probably fall asleep and either way, after the talk, you'll feel refreshed. No matter what, you'll learn a sneaky way to always win at Limbo.

John Berry, The University of Plymouth (UK)
Saturday Luncheon Address
Maas Auditorium

Developing the Mathematical Feel

One of the aims of my talk is to illustrate some of the mathematics that I enjoy and to suggest that one of the important, and often neglected, aspects of mathematics is its application to real world problems. Not far away though is my belief in the power of technology to develop mathematical understanding and as a tool in problem solving. Thus this talk will feature the twin themes of modeling and the appropriate use of technology and these provide the main aim of my talk, which is to discuss the importance of our students' developing a feel for mathematics. Is it the same feel for mathematics that we have and does it matter?

Contributed Talks

Javad Abdollahi, Lawrence Technological University
Saturday, 10:15-10:35 am
237 VanderWerf

Computational and Statistical Synergy in Data Mining

Modern computational advances (in both hardware and software) have put statistics in a renaissance period. New extensions of statistical methods in computer science have spawned fresh approaches to statistics, especially as it relates to data analysis. Data mining is on the interface of computer science and statistics, utilizing advances in both disciplines in order to make progress in extracting and processing data from very large databases (VLDBs). It is a rapidly emerging field that has attracted much attention in a very short period of time.

We highlight some statistical themes and lessons that are directly relevant to data mining and attempt to identify opportunities where close cooperation between the statisticians and computer scientists will provide synergy for further progress in data mining and analysis.

Reza Akbari, Saginaw Valley State University
Friday, 4:05-4:25 pm
237 VanderWerf

Theta Functions with Half Integer Characteristics

Theta functions with half integer characteristics are introduced. Their elliptic behavior and resemblance with trigonometric functions are investigated. Applications of these quasi-elliptic functions in the study of Riemann theta functions are described.

Chuck Allan, MDE, Tim Husband, Siena Heights University,
and Roger Verhey, University of Michigan-Dearborn
Saturday, 11:05 am-11:50 pm
299 VanZoeren

Standards-based Review of Endorsement Programs
in Secondary School Mathematics

The Michigan Department of Education (MDE) accreditation of higher education teacher preparation programs and standards for the review of their mathematics specialty programs (emphasis on secondary programs).

Stephanie Baar, student, Calvin College
[faculty sponsor: Michael Stob]
Friday, 3:15-3:35 pm
299 VanZoeren

Who is the Winner: BOM versus OBO

In the world of figure skating there exists the problem of choosing one skater to be the winner of a competition. From a mathematician's point of view the problem is not necessarily picking the best skater; rather, it is the dilemma of picking the best method by which to choose the top skaters during the competition. What exactly are the two ways of ranking participants (BOM and OBO)? What criterion should we use to compare the methods? How can we determine which method picks the "right" winner most accurately and most often? Based on my summer research using various areas of mathematics and a bit of computer simulation, a conclusion was quite possible.

Sandy Becker, student & Bette Warren, Eastern Michigan University
[Talk given by Bette Warren]
Saturday, 10:15-10:35 pm
299 VanZoeren

What Do You Mean "Too Close to Call?"

As a nation used to instant information and predictions made confidently before votes are counted, we watched in fascination and disbelief as the Florida vote and the presidency remained "too close to call" for weeks last year. People understood intuitively that close votes lead to uncertainty, but were unclear about why. We present a model that recognizes the inherent uncertainty in casting and counting votes, and answers the underlying question-how likely is it that the "wrong" candidate will "win"- under a variety of assumptions.

Lee Cole, student, Hillsdale College
[faculty sponsor: John Boardman]
Friday, 3:40-4:00 pm
299 VanZoeren

How to Tune a Radio

The sounds we hear on the radio can be modeled using a series of sine or cosine functions. This talk will examine calculus-based exercises which describe the nature of radio frequencies‹their composition, transmission, and reception.

Kathleen Cotrill-Shepherd and Mark Naber, Monroe County Comm. Coll.
Saturday, 11:05-11:25 am
237 VanderWerf

Fractional Differential Forms

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The definitions of closed and exact are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case. Coordinate transformation rules are also computed. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. The metric for the fractional form spaces is given, based on the coordinate transformation rules. All results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is 1.

Amanda Cox and Amy VanderZee, students, Grand Valley State University
[faculty sponsor: Edward Aboufadel]
Friday, 11:30-11:50 am
299 VanZoeren

Bivariate Daubechies Scaling Functions

Wavelets are functions used to approximate other functions or data. They are particularly effective in approximating functions with discontinuities or sharp changes. Wavelets further preserve both the global picture and local detail of a function. Applications of wavelets include compressing images, such as x-rays and the FBI's fingerprint collection, and de-noising data, found in many areas including geology, meteorology, astronomy, acoustics, and economics.

To create a family of wavelets, we begin with a scaling function. In 1988, Daubechies showed how to create a univariate scaling function f(t) with three properties: compact support, regularity, and orthogonality. We will demonstrate how to create bivariate scaling functions, f(x,y), with the same properties. Using a bivariate scaling function we can reproduce planes (linear functions).

Gerry Cox, Lake Michigan College
Friday, 10:15-10:35 am
238 VanderWerf

Relativity of Time (Space-Time)

A simple proof and applications of Space-Time will be demonstrated. This is a very simple proof that can be used in any precalculus or calculus classroom where many students are not familiar with this concept. This proof will be included in a handout with E=mc2 and the shrinking of length at relativistic speeds.

Pamela Cutter, Albion College
Friday, 3:40-4:00 pm
237 VanderWerf

Filling in Some Gaps

By a prime gap of size g, we mean that there are primes p and p+g such that the g-1 numbers between p and p+g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime gap of size 1000 existed. My goal was to be the first to find a prime gap of size 1000, by using a systematic method that would also apply to finding prime gaps of any size. I will describe this method, my results, and related computations with prime triples of the form 6m+1, 12m-1, 12m+1.

Richard J. Fleming, Central Michigan University
Friday, 10:40-11:20 am
299 VanZoeren

Elbert F. Cox, An Early Pioneer

Elbert F. Cox was the first African American to earn a Ph.D. in mathematics. His degree was from Cornell in 1925. We will discuss his life and some of the circumstances surrounding this notable achievement.

Matt Goupell & J.R. Schmidt, students, Hope College
[faculty sponsor: Timothy Pennings]
Saturday, 11:30-11:50 am
238 VanderWerf

Modeling the Flight of a Frisbee

Why is a Frisbee so much fun to throw? How does its size and shape give rise to unique dynamics that allow for strange and interesting flights? Can you make a Frisbee "bounce" off a cushion of air? This talk models the flight of Frisbees and proves the possibility of the air-bounce. Demonstrations will be given for the skeptics. (One of the speakers is a top Frisbee player in the state.)

Jerrold Grossman, Oakland University
Friday, 11:30-11:50 am
237 VanderWerf

Patterns of Collaboration in Mathematics Research

The mathematics research collaboration graph C [resp., C'] has as its vertices all mathematicians who have published research papers. Two vertices are joined by an edge in C [resp., in C'] if the two mathematicians have published a joint paper, with or without other coauthors [resp., without other coauthors]. Using reliable data from Mathematical Reviews, we explore the structure of C and C'. (They are examples of "small world graphs" in the sense of Duncan Watts.) A lot of additional information is available at the Erdös Number Project web site (http://www.oakland.edu/~grossman/erdoshp.html).

Christian Grostic, student, Kalamazoo College
Friday, 3:15-3:35 pm
237 VanderWerf

Connected Sums and Decompositions of Plane Curves - "Puttin' 'em together and takin' 'em apart"

Despite recent advances, the problem of plane curve classification remains unsolved. One possible aid in classification would be to examine compositions and decompositions of curves. We use the connected sum to do just that, examining some of the implications this produces with regard to previous work and introducing some new properties and tools to aid in examination. We then turn to the question of prime decomposition of plane curves, including the proof that a curve has a unique prime decomposition. We conclude with a discussion of some implications and related topics, including avenues for future research.

Lee Kiessel & Brian Yurk, students, Hope College
[faculty sponsor: Darin Stephenson]
Saturday, 10:40-11:00 am
299 VanZoeren

Geometric Characterization of Graded Algebras
Let A = ⊕i≥0 Ai be a connected graded, associative algebra over a field k. The Hilbert series of A is the formal power series HA(t) = ∑i≥0(dimk Ai)ti. If A is generated by elements of degree 1 and the defining relations are quadratic, then the space W of relations is naturally a subspace of A1⊗ k A1. We give some results in low-dimensional cases relating the isomorphism classes and Hilbert series of algebras to the geometry of subspaces of A1⊗A1. In particular, when dimkW = 1, each algebra A naturally determines a point in the projective space P(A1⊗A1). Isomorphisms of algebras determine a group action on P(A1⊗A1), and the Hilbert series and isomorphism classes of algebras can be characterized geometrically in terms of algebraic equations. When dimkW = r > 1, one must work in the Grassmannian Gn,r , where n = dimk (A1 ⊗ A1).

John Kiltinen, Northern Michigan University
Friday, 11:05-11:25 am
237 VanderWerf

A New Hands-on Proof of the Parity Theorem for Permutations

The Parity Theorem says that all expressions in terms of transpositions for a permutation on the set {1, 2, ... , n} must share the same parity. That is, any two expressions for a given permutation in terms of transpositions have the numbers of transpositions used both being odd or even.

All the commonly known proofs are essentially notational and distant from any concrete representation. The speaker will present an apparently new proof, which is more tactile, building on the experience of representing transpositions by swapping numbered markers between numbered boxes. He will use his own software to demonstrate the ideas.

Larry King, University of Michigan, Flint
Saturday, 10:40-11:00 am
238 VanderWerf

A First Proofs Course: Some Problems and Some Success

Most colleges and universities require their math majors and minors to take a first proofs course. There are numerous textbooks from which to choose but none even attempts to teach novice math students how to write proofs. They may devote a section or two to proof-writing and the remainder of the text is devoted to proving some formidable mathematical results. The results for our students are generally not good. I would like to talk about the approach that I have recently developed, which stresses writing, in general, and proof-writing in particular. The results are encouraging.

David Koop, student, Calvin College
[REU]
Friday, 3:15-3:35 pm
238 VanderWerf

Geometric Tomography

Geometric tomography is an area of mathematics that deals with simplified versions of medical x-rays. Specifically, we consider only two-dimensional convex bodies with uniform density and point source x-rays. The problem is to find the minimal number of x-rays necessary to determine a convex body. This presentation introduces geometric tomography and presents progress toward characterizing planar x-rays.

Keneth Kopp, student, Lawrence Technological University
[faculty sponsor: Ruth Favro]
Friday, 4:30-4:50 pm
299 VanZoeren

Blowin' in the Wind

For the 2001 Mathematical Contest in Modeling, we modeled and simulated the evacuation of the coast of South Carolina in the event of a hurricane. [INFORMS prize paper]

Rachel Koskodan, student, Wayne State University
[faculty sponsor: Lawrence Brenton]
Saturday, 11:30-11:50 am
237 VanderWerf

Geometric Applications of a Unit Fraction Equation

The Diophantine unit fraction equation ∑i≤k1/ni - ∏i≤k1/ni arises in the study of groups of symmetry of the regular polyhedra (Platonic solids) and their higher dimensional analogues. Very recently applications have been given in cosmology to models of the universe in which the spatial cross-sections are not simply connected. Our project is to use computer search techniques to determine all solutions to this equation for k ≥ 8.

Machael Kowalczyk, student, Northern Michigan University
[faculty sponsor: John Kiltinen]
Friday, 10:15-10:35 am
299 VanZoeren

Addictive Puzzles and Abstract Algebra

In this talk we will examine puzzles that are most conveniently analyzed with the theory of permutation groups. We will attempt to derive the optimal strategies for solving them and then consider each strategy's meaning and significance in abstract algebra.

Brian J. McCartin, Kettering University
Saturday, 10:40-11:00 am
237 VanderWerf

A Geometric Characterization of Linear Regression

It hardly seems an exaggeration to contend that the fitting of straight lines to experimental data permeates all of science. Would it not seem reasonable to expect a purely geometric characterization of such a straight line? Just such a geometrical perspective was provided by Francis Galton in 1886 if only one of the experimental variables contains error. This was extended by Karl Pearson in 1901 to allow both variables to be subject to measurement error so long as both errors have equal variances. After an intervening century, this presentation will extend the Galton/Pearson geometrical characterization of linear regression in terms of the "concentration ellipse" to the case of unequal variances in the experimental data.

Michael McDaniel, Aquinas College
Friday, 11:30-11:50 am
238 VanderWerf

Old and Valuable

We consider some of the crucial theorems in the undergraduate calculus sequence as museum pieces. The Fundamental Theorem of Integral Calculus certainly ranks as one of the most influential thoughts in the history of recorded intellect, and our students are capable of understanding the proof. Without prescribing a set presentation of this and other theorems, we discuss ways to turn the proofs into memorable events.

For the FTIC, some baroque music sets the tone. A brief summary of life in the late 1600s, the state of mathematics, and a story or two lead into the actual proof. We consider details which, while not mathematically profound, turn the presentation into an icon for the course: the room, the attire, the cookies, etc.

Ronald Mosier, DaimlerChrysler, retired
Friday, 4:05-4:25 pm
238 VanderWerf

Rating Baseball Relief Pitchers

The current method of rating baseball relief pitchers is inadequate. This talk presents a method using Markov Chains. Although it is highly unlikely that organized baseball will adopt the proposed system, it should be of interest to teachers of mathematics and their students because it uses easy theorems from finite Markov Chain Theory and illustrates how mathematics can be both useful and fun. No knowledge of baseball is required to understand this talk.

Melvin A. Nyman, Alma College
Saturday, 10:15-10:35 am
238 VanderWerf

Reading, Writing, Discovering, and Creating Proofs

One of the most difficult transitions for mathematics students is moving from the manipulations of calculus to the proof-oriented abstractions of upper level courses. We will describe our experience teaching a course designed to enhance this transition. The course is prerequisite to most or our upper-level mathematics courses. Student appraisals of the course effectiveness will be discussed.

Timothy J. Pennings, Hope College
Friday, 10:40-11:00 am
237 VanderWerf

Does Devaney's Definition of SDIC Imply a Positive Lyapunov Constant?

Definitions of chaos always involve the notion of Sensitive Dependence on Initial Conditions (SDIC), but SDIC is defined differently by different authors. Some use Devaney's definition involving the eventual separation of orbits to a given distance, while others define it in terms of a positive Lyapunov exponent. Are these two definitions of SDIC equivalent as is sometimes implied? We will show they are not equivalent by providing a function which satisfies Devaney's definition of SDIC, but which has a non-positive Lyapunov exponent. Conversely, is there a function which satisfies the Lyapunov definition, but not Devaney's? We will see. We will also discuss sensitively which definition is "better".

Steven J. Sepanski, Saginaw Valley State University
Friday, 10:40-11:00 am
238 VanderWerf

Stacking the Deck for Learning Statistics

I will describe how I use playing cards for classroom activities that demonstrate nearly all the topics taught in an introductory statistics class. I have used these activities over the past several semesters. They have been well received and some student reactions will be provided. Some sample problems and student solutions will be provided as well.

John R. Stoughton, Hope College
Friday, 4:30-4:50 pm
238 VanderWerf

The Incredible Mr. Euler

Those who know the name Leonhard Euler at all know that he was one of history's most prolific mathematicians. In this talk I intend to present statistics to convince you that this is not a strong enough statement. I expect to see your jaws drop to the ground as I tell you how much mathematics Mr. Euler did. In addition, I will present one of his little-known, but truly elegant, proofs in an attempt to impress you with not only the amount, but also the beauty, of the mathematics he did.

Ted Sundstrom, Grand Valley State University
Saturday, 11:05-11:25 am
238 VanderWerf

Exploration, Proof, and Writing in Mathematics

A "transition course" from the problem solving orientation of calculus courses to the more abstract and theoretical upper level courses is an important course in the mathematics major at many colleges and universities. At GVSU, this course is MTH 210, Communicating in Mathematics. It is also part of the University's Supplemental Writing Skills Program. This talk will focus on our department's approach to this course. This will include a description of the writing requirements, the materials that I have written for the course, and the course home page, which is an integral part of the course.

Akalu Tefera, Grand Valley State University
Friday, 10:15-10:35 am
237 VanderWerf

A Brief Tour of MultInt: a Maple Package for Multiple Integration

Most integral identities in mathematics are usually hard to prove and often require lengthy and tedious verification. The amazing discovery of H. Wilf and D. Zeilberger was that every proper-hyperexponential multi-integral identity with a fixed number of integration signs possess a computer-constructible proof. We will describe a Maple implementation, MultInt, of the continuous version of the WZ method. We will also give various examples of how this package can be used to systematically generate proofs of integral identities (or recurrences) which involve multiple integrals of proper-hyperexponential functions.

John L. VanIwaarden, Hope College
Friday, 3:40-4:00 pm
238 VanderWerf

Proofs Without Words

All mathematics instructors would love to have students gain "geometrical insight" into the operations we do. There is a great collection of proofs of very useful theorems that require only a picture‹essentially no words. This presentation will look at a small collection of these and delight in their simplicity and their applicability. Copies will be provided for the viewers to use in their own ways.

Clark Wells, Grand Valley State University
Friday, 11:05-11:25 am
238 VanderWerf

You Can't Ask Essay Questions in Math, Can You?

A common misconception among students (at all levels of mathematics) is that a question in math must be either "right" or "wrong", and so there is no room for essay type questions in a mathematics class. I will discuss the use of essay questions and writing prompts which deal with deep understanding of mathematical concepts in calculus (over two semesters), linear algebra, and abstract algebra, their purpose, their evaluation, and examples of student responses.

Ping Zhang, Western Michigan University
Friday, 4:30-4:50 pm
237 VanderWerf

Divisor Graphs

For a finite nonempty set S of positive integers, the divisor graph G(S) of S has vertex set S, and two vertices i and j of G(S) are adjacent if i divides j or j divides i. A graph G is a divisor graph if there exists a set S of positive integers such that G is isomorphic to G(S). We present some results in this area.

Matthew Zuckero, student, Central Michigan University
[faculty sponsor: Sivaram K. Narayan]
Friday, 4:05-4:25 pm
299 VanZoeren

Pebbling Numbers of Graphs

The pebbling number of a simple graph is defined as the smallest number of pebbles, such that for any configuration of pebbles on the vertices of the graph, a pebble can be moved by a sequence of "pebbling moves" to any destination vertex. A "pebbling move" involves selecting two pebbles from a vertex, throwing one away, and placing the other pebble on an adjacent vertex. Pebbling numbers of graphs with diameter 2 on n vertices have been shown to be equal to n or n+1. We will present some results in pebbling numbers of graphs with diameter 3.

Building Bridges Workshop

The Building Bridges sessions at the annual meeting of the Michigan Section-MAA and MichMATYC have promoted an ongoing dialogue between high school and college mathematics teachers. The dialogue centers on issues and concerns that impact the transition from high school to college mathematics. This year, the focus will be:

Mathematics for Teaching Standards-based Mathematics Curricula:
The Preparation of Teachers for K-12 Mathematics

Chuck Allan, Tim Husband, Roger Verhey
Saturday, 11:05-11:50 am
237 VanderWerf

Standards-based Review of Endorsement Programs in Secondary School Mathematics

(See abstracts)

Sharon Senk, MSU,
and Roger Verhey, University of Michigan-Dearborn,
with three mathematics professors (teaching first-year mathematics courses)
Saturday, 2:15-3:10 pm
237 VanderWerf

MET and CBMS Recommendations for the Preparation of Teachers of Mathematics

The Mathematical Education of Teachers (MET) and The Conference Board of the Mathematical Sciences (CBMS) recommendations for preparation of K-12 teachers.

Open Discussion

Saturday, 3:20-4:15 pm
237 VanderWerf

We are providing an opportunity for participants to discuss in greater detail the issues, recommendations, and implications for college mathematics put forward during the two earlier Building Bridges sessions.


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