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Program Schedule (Gaps in times indicate break times, * indicates undergraduate student speaker, **graduate student)

Friday May 4, 2007	Speaker, Institution, Title
9:00 - 9:10	Daniel Little, Chancellor of UM-Dearborn, Welcome
9:15 - 9:45	David Austin, Grand Valley State University, Circle Packings from Penrose Tilings
9:55 - 10:15	Dale Winter, UM-Ann Arbor, Students' Mental Models and Calculus Optimization Problems Steve Blair , Will Dickinson , Paul Yu, GVSU, The Use of Metaphor in Geometry: an Example Involving Curvature Jack Crowell, Delta College, How and why the mathematics community should help create mathematics library in Kenya.
10:45 - 11:15	Lisa DeMeyer, CMU, The Zero Divisor Graph of a Semigroup Dan Isaksen, WSU, Computations with Quaternions, Octonions, and Beyond
11:20 – 11:40	Daniel Drucker,WSU, A Generalized Approach to Polygons and Morphing Jack Rotman, LCC, Retaining All/More of Your Students: Math Success, Part I Homa Ghaussi-Mujtaba, LCC, Technology and Cooperative Learning: What Works
11:45 - 12:05	Amy Hlavacek, SVSU, The Swapping Number of a Graph Jack Rotman, LCC, Retaining All/More of Your Students: Math Success, Part II Kim Rescorla, EMU, The Effects of Spiraling Homework on Mathematical Concept Development and Skill Retention
12:15 - 2:00 Luncheon address	Richard Hill, Michigan State University, The Experiences of a Mathematician Getting Involved in Mathematics Education Issues
2:10 - 2:30	Jan Hlavacek, SVSU, On the Vitali Covering Lemma Fatih Celiker , WSU, Superconvergence of the numerical traces of discontinuous Galerkin methods for convection-diffusion problems Joseph Matti, SVSU, Friday the 13th
2:35 - 2:55 2:35 - 3:20 (Panel Discussion only)	*Ovidiu Furdui, WMU, On a class of integral operators related to the Fock spaces* Tim Pennings, Hope College, Do Dogs Know Bifurcations? Joseph Kelly, Aquinas College, Inverse points using a non-Euclidean metric* Louise Paquette, Nan Jackson, Homa Ghaussi-Mujtaba, Dan Harned, Kay Barks, Jing Wang, Lansing CC, Panel Discussion: College Algebra or Precalculus – Directing Our Students down the Right Path
3:00 - 3:20	William Dickinson, GVSU, The Right* Right Triangle on the Sphere* Joseph Spencer, Aquinas College, A Look at Deal or No Deal
3:50 - 4:50 Plenary address	Ronald Solomon, The Ohio State University, What have we learned from the Classification of the Finite Simple Groups?
7:15 - 9:45 Banquet Address	Joe Gallian, University of Minnesota Duluth, The Mathematics of Identification Numbers

Title: The Experiences of a Mathematician Getting Involved in Mathematics Education Issues

I have been involved with three different kinds of mathematics education programs/studies:

I will present a survey of various interesting things I have learned (and found surprising) from my experiences in each of these programs and how they are interrelated.

Title: What have we learned from the Classification of the Finite Simple Groups?

The Classification Project was a "hundred years war" with interludes of relative peace. What have it taught us? The names of all the finite simple groups. But, in fact, much more. Our entire way of thinking about finite groups has been transformed by the insights achieved during this project: factorizations, quadratic action, the generalized Fitting subgroup, signalizers, balance, etc. I will survey some of the highlights of this journey of discovery.

Because of the existence of inexpensive, fast, and reliable technology, comsumer products are identified with bar codes and identification numbers that have a built-in ``check'' to partially ensure that the numbers have been correctly entered into a computer or have been correctly scanned by an optical device. In this talk we examine some of the common bar coding and check digits schemes that you encounter everyday. Among them are the UPC bar code, the ZIP bar code, and the check methods used on credit cards, airline tickets, money orders, travelers checks, personal checks, pop cans, books and magazines.

From 1938-1942, the Dutch graphic artist M.C. Escher carried out experiments in printing repeating patterns with small carved wooden squares that contained a simple motif of crossing bands, using a purely algorithmic scheme. He asked and answered combinatorial questions about these patterns, and opened the door to many tantalizing questions that have recently been addressed by mathematicians and computer scientists.

Vibrio cholerae is a pathogen that causes pandemic cholera. In this talk, I will elaborate on the dynamics of V. cholerae infection by describing a mathematical model that governs the colonization process for the bacterial dynamics. The results indicate that both host and bacterial factors contribute to colonization. This model can be used to test therapeutic strategies against V. cholera.

Circle packings begin with a combinatorial object---a triangulation---and impose a geometric structure on it. In this talk, I'll describe some joint work with an undergraduate, Matthew Stamps, in which we studied circle packings that arise from triangulations created by Penrose tilings and compared the geometry of the tilings to the geometry of the circle packings. I will introduce the audience to both circle packings and Penrose tilings.

One challenge to understanding complex analysis is the difficulty one can have in forming an intuition for analytic functions. Frank Farris suggested a new way to visualize complex functions. The method is called domain coloring. In this talk we present two implementations of domain coloring and we contrast it with the usual transformational approach. We also use domain coloring to illustrate some of the deeper theorems in complex variables.

Locally nilpotent derivations (LNDs) generalize algebraically everything we love about partial derivatives of polynomials. LNDs have gained popularity over the last few decades as algebraic geometers find them useful in tackling deep problems such as Hilbert 14 and the Jacobian Conjecture. I will explain the basic algebra behind LNDs and their connection to such problems. I hope that the talk will peak interest in the subject but also inspire those teaching abstract algebra to incorporate LNDs as a special topic for students learning the definition of a ring.

Let S be a commutative semigroup with zero element. We associate a simple graph G(S) to the semigroup S as follows: let the vertices of G(S) be the nonzero zero divisors of S and connect two vertices a and b in case their product is zero. There are many results describing the possible shape of such a graph. The talk will include recent results by undergraduate students as well as several open questions.

The Cayley-Dickson algebras are a sequence of algebras of dimension 2ⁿ. The first three are well-known; they are the real numbers, the complex numbers, and the quaternions. The next algebra, called the octonions, is more obscure but still classical. I will describe the octonions and explain their uses. I will also discuss the Cayley-Dickson algebra of dimension 16.

Contributed Session Abstracts

*Warren Beard and *Louis Leskoviansky, Lawrence Technological University

Turning Around 'Turn Around Time'

The 2007 Mathematical Contest in Modeling, Problem B, asks the team to design an optimal airplane boarding scheme to decrease flight turn-around time and to maximize the overall experience of the customer. The model developed is compared to other models, and its implementation on aircraft of various sizes is contrasted.

*Zachary Beamer and *Erica Manoppo, Michigan State University

Dynamics of the Tent Map: Iteration, Chaos, and Fractals

In this project we investigate the dynamical behavior of the tent function through iterations. Different results occur for various ranges of the parameter , including chaotic behavior and fractal structures. We will characterize these behaviors for different values and provide graphical representations to illustrate these ideas.

Steve Blair, Will Dickinson, and Paul Yu, Grand Valley State University

The Use of Metaphor in Geometry: an Example Involving Curvature

We will discuss an episode from a larger study involving the teaching and learning of non-Euclidean geometry at Grand Valley State University. In particular, we will consider how a teacher and undergraduate students used the metaphor of a bug driving a car along a track to connect intuitive and abstract conceptualizations of planar curvature.

*John Camardese, *Neil Ganshorn, *Richard Geyer, Lawrence Technological University

An Adventure in Boarding Airplanes

Fatih Celiker, Wayne State University

Superconvergence of the numerical traces of discontinuous Galerkin methods for convection-diffusion problems

We study a new superconvergence property of a large class of finite element methods for one-dimensional convection-diffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov-Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are single-valued. In particular, for a local discontinuous Galerkin method, we show that the superconvergence is order when polynomials of degree at most are used.

Jack Crowell, Delta College

How and why the mathematics community should help create mathematics library in Kenya.

Creating a mathematics library in Kenya--- Utilizing the free resources of the Internet and the surplus text books in our offices, we can create a mathematics library that will have a big impact on the teaching and learning of mathematics in Kenya. Creating the library will provide some exciting opportunities for us, the mathematics community, to assist in the development of mathematics education in Africa. Jack Crowell is Professor of Mathematics at Delta College and the Director of the Kenya Rural School Computer and Book Program. In the past 15 years the Program has provided nearly 6,000 computers and thousands of books to over 500 schools throughout Kenya.

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William Dickinson, Grand Valley State University

The Right Right Triangle on the Sphere

Despite the obvious differences between the Euclidean plane and a unit sphere, spherical geometry has many similarities to Euclidean geometry. For example, lines in spherical and Euclidean geometry both divide their respective geometries into two equal pieces and the isosceles triangle theorem is valid in each. Therefore, we might expect right triangles in spherical geometry to behave like Euclidean right triangles. However, if we follow the natural generalization of a right triangle to spherical geometry, the two cousins behave differently. In this talk we explain a different way of generalizing right triangles to the sphere that is much more harmonious.

Daniel Drucker, Wayne State University

A Generalized Approach to Polygons and Morphing

If an -gon is regarded as a vector in rather than as a set of points in the complex plane , then extra generality can be achieved by applying a transformation of to instead of applying a single transformation of to all of its vertices. Definitions of morphs and derivations of formulas for their areas can be simplified by the use of matrix identities and the inner product of .

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Ovidiu Furdui, Western Michigan University

On a class of integral operators related to the Fock spaces

For real parameters and , where is a positive real number we determine exactly when the Bergman type integral operators

and are bounded on , where is the Gaussian probability measure on and is the ordinary Lebesque measure on . These integral operators induced by the kernel function on Fock space were introduced recently by professor Kehe Zhu.

Homa Ghaussi-Mujtaba, Lansing Community College

Technology and Cooperative Learning: What Works

This presentation will benefit participants who want to improve retention in their online college algebra course and want to improve success rate in their calculus courses. Participants learn how cooperative learning is incorporated in online college algebra to create a sense of community in an online environment. Examples of what have worked with data will be shared. Participants also learn which cooperative learning techniques have improved the success rate in honors calculus courses at Lansing Community College.

Amy Hlavacek, Saginaw Valley State University

The Swapping Number of a Graph

We define the swapping number of an arbitrary simple graph, which involves the weakening of the concept of a graph automorphism. We proceed to classify all 1-swappable trees.

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Jan Hlavacek, Saginaw Valley State University

On the Vitali Covering Lemma

The classical Vitali covering lemma states that from any Vitali covering of a set by balls, we can select a pairwise disjoint subset of balls that still covers almost all of the set . In this talk, we will present a survey of some generalizations of this lemma.

*Josua Illian, Saginaw Valley State University

Cyclotomic and Related Polynomials

This paper explores the cycotomic polynomials as minimal polynomials of the roots of unity, . Then, we specifically examine the nature of the minimal polynomials of the less studied element for a given nth root of unity. By calculating these related polynomials, we will determine a formula for their coefficients based on n and the power of the indeterminate. Using those coefficients we will determine and prove a formula for this polynomial, and notice an interesting pattern

Barbara Jur, Macomb Community College

The Relationship between Reading Ability and Performance in Developmental Mathematics

Is there a relationship between the ability to read and the ability to do mathematics? Students at MCC are given the COMPASS placement test to advise them about math courses and also testing reading level. Literature has suggested a relationship to performance. The correlation for students in Fundamentals of Mathematics and Beginning Algebra will be discussed. Implications for teaching strategies will also be offered.

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*Joseph Kelly, Aquinas College

Inverse points using a non-Euclidean metric

The text College Geometry from Key College Press introduces non-Euclidean geometry first by covering a non-Euclidean metric û the taxicab metric û then the Poincaré disc model of hyperbolic geometry. Hyperbolic geometry in the disc relies on a Euclidean construction of inverse points to determine hyperbolic lines. Inverse points may also be constructed given a taxicab metric and a taxicab disc. We will look at two of these constructions.

Joseph Matti, Saginaw Valley State University

Friday the 13th

Paraskevidekatriaphobia or Friggtriskaidekaphobia (fear of Friday the ) are specialized forms of triskaidekaphobia (fear of the number 13). There are 14 distinct calendars in the current (since 1582) Gregorian system - are there any "`lucky"' ones amongst them (i.e. one in which there is no occurrence of Friday the )? NO! Find out which calendars are the least unlucky (1 occurrence) and the most unlucky (3 occurrences) and when they will occur during this century.

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Brian McCartin, Kettering University

A Tale of Two Ellipsoids

In two dimensions, the concentration ellipse of statistics and the inertia ellipse of mechanics coincide. Yet, in three dimensions, the concentration ellipsoid and the inertia ellipsoid are distinct. This talk will review these facts, introduce a new relationship involving the eccentricities of their respective principal cross-sections, and provide an exhaustive treatment of their shared degeneracies.

Tim Pennings, Hope College

Do Dogs Know Bifurcations?

It has been established that dogs - at least Elvis - knows calculus. That is, Elvis can find the optimal - fastest - route to a ball thrown down the beach and in the water. But what happens when Elvis is positioned in the water and retrieves a ball that is also in the water? When should he swim the entire distance to the ball, and when should he swim in to the shore, run along the shore, and then swim back out to the ball? What is the bifurcation point for the change in optimal strategy? Does Elvis bifurcate? Does his fur bicate?

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Kim Rescorla, Eastern Michigan University

The Effects of Spiraling Homework on Mathematical Concept Development and

Skill Retention.

Spiraling homework, which spirals back to previous lessons, may improve retention. Two groups of EMU students in Linear Models and Probability were compared. The spiraling group received spiraling homework, while the nonspiraling group worked conventional exercises. Other variables such as student preparation, exercise difficulty, instructor, and grading were kept uniform. The spiraling group scored significantly higher -value = on the cumulative final exam. Other interesting comparisons are discussed.

David Reyes-Gastelum, Macomb Community College

Reforming Precalculus At Macomb Community College, The Precalculus Reengineering Project. A Case Study

Disappointment in student performance, a realization that the student population had significantly changed, and an awareness of current trends in reforming precalculus mathematics throughout the nation led Macomb Community College to conduct an extensive study of its three-course precalculus curriculum. Reforming precalculus mathematics takes into account not only what is taught but also who the students are and how the teaching-learning process should be organized.

Jack Rotman, Lansing Community College

Retaining All/More of Your Students: Math Success (Part I and Part II)

What can we do to help more students succeed at a high level? What factors contribute to ‘failures’ in mathematics? This session will present specific techniques that you can use to improve student performance! Whether you teach “developmental”, “applied”, or “intensive” mathematics, you will come away with new ideas.

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Steven Schlicker, Grand Valley State University

Numbers Simultaneously Polygonal and Centered Polygonal

Recently, while teaching a course in number theory for practicing secondary mathematics teachers, I introduced the topic of polygonal numbers and centered polygonal numbers. While doing so, a question came to mind. Which numbers are simultaneously - polygonal and -centered polygonal? To my surprise, I could not find any information on this topic in any source, so I investigated this problem and discovered an infinite family of previously unknown integer sequences.

Saturday May 5, 2007	Speaker, Institution, Title
9:00 - 10:00 Plenary Address	Doris Schattschneider, Moravian College, Escher's Combinatorial Patterns and Their Aftermath
10:05 - 10:25	Brian McCartin, Kettering University, A Tale of Two Ellipsoids Barbara Jur, MCC, The Relationship between Reading Ability and Performance in Developmental Mathematics Josua Illian, SVSU, Cyclotomic and Related Polynomials*
10:50 - 11:20	Mike Bolt, Calvin College, Paint by number: a visualization of complex functions Anthony Crachiola, SVSU, Locally nilpotent derivations and their applications
11:25 - 11:45 11:25 - 12:10 (Panel Discussion only)	Steven Schlicker, GVSU, Numbers Simultaneously Polygonal and Centered Polygonal *Zachary Beamer and Erica Manoppo, MSU, Dynamics of the Tent Map: Iteration, Chaos, and Fractals* Warren Beard and Louis Leskoviansky, LTU, Turning Around 'Turn Around Time' Roger Verhey, UM - Dearborn; Joanne Caniglia, EMU; Jerrold Grossman, Oaklunad U; Panel Discussion: Being a STEM Faculty Member in Supporting Secondary Teachers of Mathematics
11:50 - 12:10	Tanweer Shapla, EMU, Confidence Intervals of the Attributable Risk with Intermediate Base-level under Cross-sectional Sampling Scheme John Camardese, Neil Ganshorn, Richard Geyer, LTU, An Adventure in Boarding Airplanes* David Reyes-Gastelum, MCC, Reforming Precalculus At Macomb Community College, The Precalculus Reengineering Project. A Case Study
12:15 - 2:00 Luncheon Address	Anna Spagnuolo, Oakland University, A Model for Vibrio Cholerae* Colonization of the Human Intestine*

The University of Michigan-Dearborn, Dearborn, MI May 4-5, 2007

The University of Michigan-Dearborn, Dearborn, MI
May 4-5, 2007