Illinois Section of the MAA

Abstracts of Talks at the 2003 ISMAA Annual Meeting

Assessment Forum

Tom Rishel, Cornell University, Ithaca, NY and Herb Kasube, Bradley University, Peoria
12:45p.m. Fridy, March 28, 2003, McGaw Fine Arts Center-Sibert Auditorium

The sessions include a brief overview of assessment as viewed in the Supporting Assessment in Undergraduate Mathematics (SAUM) project; a discussion of some assessment case studies from SAUM and MAA Notes #49; and a discussion of what's happening in assessment in ISMAA. The second hour of this Forum will be held on Saturday from 1:15-2:15 p.m.

The Preparation of Mathematics Educators in a Standards-Based System

Nancy Patton, Division Administrator for Professional Preparation and Recruitment, Illinois State Board of Education, Springfield
2:00p.m. Friday March 28, 2003, Parker Science Building, Room 106

Effective July 1, 2003, all mathematics education programs in Illinois will be guided by standards that include indicators of knowledge and performance. The presentation will provide a brief history of how the standards were derived and will speak to the implications for higher education, prospective mathematics education candidates, and No Child Left Behind.

Rational Points and Rational Lines

Tom Bengtson, Augustana College, Rock Island
2:00p.m. Friday March 28, 2003, Parker Science Building, Room 107

This talk is for undergraduate mathematics students. We define rational points and rational lines in the plane, and then show that there are lots of them. However, almost all of the points in the plane are not on a rational line, and we show this, too.

Solving the Precalculus Equation: Engagement = Real Math + Nontraditional Teaching

Paul J. Karafiol, Walter Payton College Prep, Chicago
2:00p.m. Friday March 28, 2003, Baxter 106.

The structure and pacing of the traditional precalculus curriculum often feel like they "force" teachers into traditional lecture-problem set teaching. I'll talk about how I use discussion, two- or three- period inquiry projects, and combine discovery with justification to enrich a precalculus curriculum without sacrificing content.

Heyting Algebras: An Introduction with Some Results and Some Questions

Samuel Johnson, Eureka College, Eureka
2:00 p.m. Friday March 28, 2003, Baxter 100.

Heyting Algebras are a generalization of Boolean Algebras. Just as Boolean Algebras model the truth values for standard logic, Heyting Algebras model the truth values for non-standard logic. Also, just as the most common source of Boolean Algebras is the set of all subsets of a set, a standard source of Heyting Algebras is the set of all opens of a topological space. Heyting Algebras, however, exhibit significant algebraic variation, which Boolean Algebras do not. Among the open questions is the extent to which topological Heyting Algebras represent the full variety of all Heyting Algebras. It is certainly the case that not all Heyting Algebras are topological.

Analytic Structure of Ternary Rings of Operators

Manmohan Kaur, Benedictine University, Lisle
2:35p.m. Friday March 28, 2003, Baxter 100

We have used operator space theory to show that, as far as the local structure is concerned, Ternary Rings of Operators (TROs) have many C*-algebra properties. We will discuss the Mobius mapping between two TROs and show that its derivative is not only linear, but also completely bounded.

How the Classification of Linking Pairings Arises Topologically

John Bryden, Southern Illinois University, Edwardsville
3:15p.m. Friday March 28, 2003, Baxter 106

The work of Wall in the early 60's and that of Kawauchi and Kojima in the 80's classifies linking pairings up to isomorphism and proves that any linking pairing is isomorphic to the linking form of a closed orientable 3-manifold. Joint research with Florian Deloup, University of Paul Sabatier, Toulouse and the Einstein Institute of Mathematics, Hebrew University of Jerusalem, on the construction of the abelian topological quantum field theories requires the general construction of the linking pairing for any closed orientable 3-manifold. It follows from this joint work that the linking form of any closed orientable 3-manifold is isomorphic to the liking form of a Seifert manifold whose orbit surface is the 2-sphere. This is surprising because the block sum diagonalization of the linking form of an arbitrary closed orientable 3-manifold possesses hyperbolic blocks along the diagonal.

Axial View of Trigonometric Functions

M. Vali Siadat, Richard J. Daley College, Chicago
3:15p.m. Friday March 28, 2003, Baxter 100

In terms of modern pedagogy, having visual interpretations of trigonometric functions is useful and quite helpful. This axial view should help students develop an intuitive sense of the magnitudes and signs of trigonometric functions. Moreover it presents, pictorially, an easy approach to prove all the single angle trigonometric identities.

An Illinois Community College Board Update

Barbara Risse, Director of Transfer Programs, Illinois Community College Board, Springfield
3:45p.m. Friday March 28, 2003, Baxter 100

An update will be provided on several state level initiatives and their relevance to the higher education mathematics community.

A Calculus Project That Really Makes Cents

Daniel Green, Olivet Nazarene University, Kankakee
3:15p.m. Friday March 28, 2003, Parker Science Building, Room 107

Many math majors understand abstract algebra, but have no practical knowledge about how much money they could save by paying a little extra on their mortgage, or what monthly payment they would need to invest in order to secure a comfortable retirement. We will discuss a calculus project that exposes students to the concept of retirement annuities in both the saving and withdrawal phases, via revenue streams represented by integrals. Students use modeling skills to solve several related problems as the assumptions of the original problem are changed. The project requires them to use a computer algebra system to carry out computations that they could not otherwise perform. Students' reactions, methods, and results are also examined.

The Henstock-Kurzweil Integral

Paul M. Musial, Chicago State University, Chicago
3:50p.m. Friday March 28, 2003, Parker Science Building, Room 107

The Henstock-Kurzweil (HK) integral extends both the Riemann and Lebesgue integrals, yet the construction of the HK-integral is similar to that of the Riemann integral, using partitions of compact intervals. It has the advantage that all derivatives are integrable in this process. Also, many functions which have so-called "improper" Riemann integrals are HK-integrable. I will give the definition and several properties of the HK-integral. Anyone who has taught Calculus may wish to attend.

Discussion on "The Preparation of Mathematics Educators in a Standards-Based System"

Todd D. Oberg, Illinois College, Jacksonville
3:15p.m. Friday March 28, 2003, Parker Science Building, Room 141

This session follows the presentation from 2-3 p.m. by Nancy Patton and is an opportunity for discussion of issues related to the college mathematics preparation of future elementary and secondary teachers.

Soap-bubble Singularities and Deltahedra

John Sullivan, University of Illinois, Urbana-Champaign
8:00p.m. Friday March 28, 2003, Cummings Dining Hall

Over 120 years ago, Plateau observed the geometric structure of soap froths: at any corner where bubbles meet, there are exactly four bubbles, in a tetrahedral pattern. Plateau's rule was not proved rigorously until the 1970s; the proof relies on ruling out seven other possibilities. For instance, when we dip a wire-frame cube into soapy water, the resulting soap film has four Plateau corners instead of one of a new type. We will examine how these eight candidates arise from the eight polyhedra with equilateral-triangle faces (including Platonic solids as well as less familiar ones).

Keystone Project: Past Practice and Results

M. Vali Siadat, Richard J. Daley College, Chicago; 2002 ISMAA Award for Distinguished Teaching of Mathematics recipient, and Paul M. Musial, Chicago State University, Chicago
8:30a.m., Saturday march 29, 2003, Kirby Lecture Hall

The keystone project is an innovative approach in teaching mathematics which improves students' success and retention in lower level mathematics classes as well as improving their study and work skills. Using dynamic assessment of student learning, based on responsive/adaptive teaching techniques, the keystone model has improved students' outcomes in mathematics and in reading comprehension tests. The latter benefit which is attributed to improved concentration skills, has a carry over effect in other subjects. In this workshop, we will present the theory and the most recent statistical results of the project.

Undergraduate Programs and Courses in the Mathematical Sciences: A CUPM Curriculum Guide

Herb Kasube, Bradley University, Peoria; Patrick Dale McCray, Systems Project Leader, Pfizer Inc., Evanston; Mercedes McGowen, William Rainey Harper College, Palatine
9:40a.m. Saturday, March 29, 2003, Parker Science Building, Room 106

MAA's Committee on the Undergraduate Program in Mathematics (CUPM) has produced a set of recommendations for mathematics departments approximately once every ten years since its inception in 1953. The latest draft of the Curriculum Guide is in its final stages. This panel will outline the recommendations and ask for comments.

Ways to Incorporate Technology Into Your Classroom

Jennifer Koehler and Chis Christensen, Addison-Wesley
9:40a.m. Saturday, March 29, 2003, Baxter Room 106

I will be demonstrating our math course management system called MyMathLab. This system allows professors to quiz and test students, assign homework, post their syllabus, and use a gradebook to monitor student's progress. Best of all MyMathLab comes free bundled with any new Addison Wesley text and it is easy to use.

Visualizing the Construction of Hyperbolic Surfaces and Three-Manifolds Using a CAS

Timothy D. Comar, Benedictine University, Lisle
9:40a.m. Saturday, March 29, 2003, Baxter 100

We present a laboratory activity that utilizes basic analytic geometry, trigonometry, and the graphing capabilities of a computer algebra system to construct hyperbolic surfaces and related hyperbolic three-manifolds. This activity helps develop an understanding of the behavior of Möbius transformations in two and three dimensions and relationships between two and three-dimensional hyperbolic geometry.

A Mathematics Course for Majors with a Writing Emphasis

Karen Mortensen, University of Illinois, Urbana-Champaign
10:15a.m. Saturday, March 29, 2003, Baxter 100

This presentation will describe a new mathematics course at the University of Illinois Urbana-Champaign. The course provides mathematics majors a transition between calculus and upper division proof-based math courses and has a special emphasis on helping students learn the complex task of writing mathematics well. The course satisfies a General Education requirement in Advanced Composition. The presentation will describe the writing activities in this course and discuss their effectiveness.

The Trinity Triathlon:Community Service or Self Service?

Sharon K. Robbert and David B. Klanderman, Trinity Christian College, Palos Heights
9:40a.m. Saturday, March 29, 2003, Kirby 108

Trinity Christian College will host their tenth annual mathematics competition, the Trinity Triathlon, for middle school students in April of this year. Students from constituent schools compete in three main events and receive prizes for participation and excellence. The public goal of the event is to promote interest in mathematics and serve community schools; the private goal is to promote interest in Trinity for future study of mathematics. Anecdotal evidence shows that both goals are being met for Trinity. Logistical details for planning a similar competition to benefit your own community and institution will be presented. Sample materials and the web location of competition archives will be distributed.

The Boomerang: A Look at the History, the Beauty, and the Mathematics

Howard Dwyer, Monmouth College, Monmouth
10:15a.m. Saturday, March 29, 2003, Kirby 108

The flight of the boomerang has fascinated people for centuries. However, it was not until recently that scientific principles have been applied to the problem of understanding the flight of the returning boomerang. In this presentation , we will briefly describe the history of this wonderful object, and then present a mathematical model of a simple boomerang and discuss results based on the model.

A Top Ten List of Fresh Ideas for Traditional Calculus Courses

Jim Trefzger, Parkland College, Champaign
10:50a.m. Saturday March 29, 2003, Baxter, Room 106

Concepts and theorems, technology and applications. A fast-paced tour of topics where classroom-ready handouts will fill in many of the details.

Issues Concerning Articulation

Iraj Kalantari and Larry Morley, Western Illinois University, Macomb. Joseph Rosenblatt and Anthony Peressini, University of Illinois, Urbana-Champaign
10:50a.m. Saturday March 29, 2003, Parker Science Building, Room 106

The goal of this session is to develop a position paper formed around axioms and principles consistent with the group's collective view of the current status of mathematics education in Illinois.

Mathematics Course Placement Process

Allen Davis, Eastern Illinois University, Charleston
10:50a.m. Saturday, March 29, 2003, Baxter 100

The process EIU uses to place students into their first university level mathematics course will be presented. Participates should be prepared to present their college or university's placement process, also.

Best Practices

11:25a.m. Saturday, March 29, 2003, Baxter 100

This session was suggested in the 2002 ISMAA questionnaire. Participants should come prepared to share (no longer than five minutes) their best practice in an undergraduate course they teach. What works well for you? Favorite topic, student engagement, technology use, etc. Come prepared to share!

Bounds for the Sum of Principal Divisors of a Positive Integer

Roger B. Eggleton, Illinois State University, Normal
10:50a.m. Saturday March 29, 2003, Kirby, Room 108

The principal divisors of a positive integer N are the maximal prime-powers that are divisors of N; for example, 3 and 8 are the principal divisors of 24, while 2, 3 and 5 are the principal divisors of 30. The product of principal divisors of N is equal to N itself, but how large can the sum of the principal divisors be? If N has n principal divisors, must their sum be no more than N/n? Some elementary arguments will be presented to establish several upper bounds for the sum of principal divisors of N.

Forbidden Symmetry: Relaxing the Crystallographic Restriction

Frank Farris, Santa Clara Universiy, Santa Clara, CA; editor of the Mathematics Magazine
12:00 Noon Saturday, March 29, 2003, McGaw Fine Arts Center-Sibert Auditorium

If you look at enough swatches of wallpaper, you will see centers of 2-fold, 4-fold, and 6- fold rotation. Why not 5-fold centers? They cannot occur, according to the Crystallographic Restriction, a fundamental result about wallpaper patterns, which are defined to be invariant under two linearly independent translations. Even so, we offer convincing pictures that appear to show wallpaper patterns with 5-fold symmetry. What is going on? The talk is intended to be accessible to students.

March 25, 2003