Patricia Lamm, Michigan State University
Friday, 9:00-9:50
T429 AuditoriumMathematical Inverse ProblemsInverse problems occur widely in many applications, including problems of biomedical imaging (CT scans and X-rays), image reconstruction (from satellites or other sources), and geophysical exploration. We discuss some of the theoretical and computational challenges associated with solving inverse problems and give an overview of some recent developments in "local regularization" solution methods.
Glenda Lappan, Michigan State University
Friday Luncheon Address
UTLC GalleryImproving the Mathematical Preparation of Teachers: CBMS RecommendationsIn the fall of 2001 the Conference Board of the Mathematical Sciences issued a publication entitled "The Mathematical Preparation of Teachers." This book was the focus of a National Summit on the Mathematical Preparation of Teachers held in Washington in early November. The luncheon talk will report some major recommendations of CBMS as well as give some examples of the kind of work reported at the summit.
Brian Conrad, University of Michigan, Ann Arbor
Friday, 2:00-2:50 pm
T429 AuditoriumPrime Values of PolynomialsIf is a polynomial with integer coefficients, how often does one expect its values
f(n)at positive integersnto actually be prime numbers? For example, iff(T)factors non-trivially then this won't happen too often (e.g.,n2-1is only prime forn = 2). But iffis irreducible, there is no evident obstruction tof(n)being prime for infinitely manyn. For example, there ought to be infinitely many primes of the formn2 + 1. Can one make such infinitude statements more precise by giving an asymptotic which measures how oftenftakes on prime values? The only case in which results are definitively proven is for linearf, but there is a remarkable conjecture due to Bateman-Horn for what is to be expected in higher degree cases. The numerical evidence is overwhelming, but a proof in general is far out of reach. Despite our complete ignorance of how to prove anything in the non-linear cases, one can ask for more: what about polynomials in several variables? Or systems of such polynomials? For example, how often isx2 + y4prime? How often arex2 + y2zandy5 - x - zsimultaneously prime? We will explain some of the historical background behind such questions (going back to Hardy and Littlewood), present the basic heuristic which leads to a conjectured asymptotic in the general multivariable case, and show lots of convincing numerical evidence in low degree. As a mild surprise, it turns out that the convergence of a certain infinite product in the conjecture rests on Deligne's work on the Weil conjectures, so we will try to briefly indicate the connection if time permits.
Keith Devlin, Stanford University
Friday Banquet Address
Café Lawrence, Buell BuildingOur Public ImageWhat is the popular conception of a mathematician? Do people have any idea what mathematicians do? Has the recent flurry of novels, movies, and stage plays about mathematicians changed that image? If so, was the change for the better?
Keith Devlin, Stanford University
Saturday, 9:00-9:50
T429 AuditoriumHow Did Mathematical Ability Evolve?Mathematics, as we generally understand it, is at most 5,000 years old. (Numbers are at most 10,000 years old.) That's too short a period for any major changes in the human brain. So, when we do mathematics, we must be using mental capacities that evolved long before mathematics came along. What are those abilities and what survival advantages led to their finding their way into the human gene pool? And if everyone has these abilities - as an evolutionary account will imply - why do so many people find math impossibly hard?
Based on Devlin's book The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. (Basic Books, 2000)
Bernard Madison, University of Arkansas and Formerly MAA Resident Mathematician
Saturday Luncheon
UTLC GalleryQuantitative Literacy: Everybody's OrphanThe heavy and increasing use of quantitative ideas and language in civic discourse means that quantitative literacy, or numeracy, is needed by all. However, numeracy education is claimed by no discipline and suffers accordingly in the discipline-dominated school and college curricula. What responsibilities do other disciplines have for quantitative literacy? And what responsibilities do we have as mathematicians?
Bernard Madison, University of Arkansas and Formerly MAA Resident Mathematician
Janet Andersen, Hope College and Chair of CTUM (Committee on the Teaching of Undergraduate Mathematics)
Gretchen Mooningham, Saginaw Valley State University
2:00-3:30
T429 AuditoriumAssessing Learning in Undergraduate MathematicsMAA is making a concerted effort to be of assistance to those faculty members faced with developing or enhancing assessment plans. This session will consists of three parts: a general introduction using a PowerPoint presentation, a mini-workshop on resources for getting started, and an open forum on what's happening in assessment in Michigan.
Hamza Ahmad, Saginaw Valley State University
Friday, 3:40-4:00
T428On Function Fields of Certain Quadratic FormsWe consider the question of whether the two neighbors of the same Pfister form and of the same dimension turn out to be birationally equivalent, and record some affirmative answers.
Reza Akbari, Saginaw Valley State University
Friday, 3:15-3:35 T428Harmonic Functions on Multiply Connected Planar RegionsHarmonic functions on simply connected regions can be investigated through their corresponding analytic functions. The generalization of this approach for the case when the domains are multiply connected planar regions is introduced. Applications of this generalization are discussed.
Steven C. Althoen, Matthew F. Wyneken, and Scott Russell
University of Michigan-Flint
Friday, 4:05-4:25
T428Writing Across the Curriculum for Elementary Education MajorsWriting Across the Curriculum has been successfully implemented in our mathematics content course for elementary education majors through direct, classroom involvement with our Writing Center tutors. We will explain how we go about this project and bring before/after samples of student writings that illustrate the effectiveness of our approach.
Char Beckmann, Pam Wells, John Gabrosek, Ed Aboufadel,
David Austin, Alverna Champion, Phyllis Curtiss, and Will Dickinson
Grand Valley State University
Saturday, 10:15-11:50
T329Enhancing the Mathematical CoreEnhancing the Mathematical Core is a multi-part, multi-year project. The Departments of Mathematics and Statistics at Grand Valley State University are engaged in a study of five core mathematics courses, one core statistics course, and a mathematics course for which one of the core courses is a prerequisite. The core courses are Communicating in Mathematics, Euclidean Geometry, Probability and Statistics, Linear Algebra, Modern Algebra, Discrete Mathematics. Non-Euclidean Geometry, for which Euclidean Geometry is a prerequisite, will also be studied. Through the course studies, the core concepts of each course that align with concepts addressed in K-12 mathematics curricula are determined. The NSF-supported K-12 curricula projects and other NCTM standards-based materials are then examined to find materials to incorporate into GVSU classes to launch the discussions of these core concepts. Pilot courses followed by research studies of each course ensue with papers. This session will describe the work completed in three of the courses, Communicating in Mathematics, Euclidean Geometry, and Probability and Statistics, as well as provide information about next steps and timelines.
Tony Berard, student, Lawrence Technological University
Saturday, 11:05-11:25
T328Lines and Conic SectionsThis is my senior project at LTU titled, "Lines and Conic Sections," and it is an extended look into the conic sections of precalculus. In most situations for the project, we seek to find the equation of the line passing through some specified point in the conic and intersecting it again some specified distance away although this is not always the case. With each new situation, the necessary formula will be derived using precalculus techniques. However, the conceptualization and resulting algebra will be quite intense.
Matthew Boelkins, Grand Valley State University
Friday, 10:15-10:35
T329The Near Orthogonality of PIPCIRsPIPCIRs are Polynomials whose Inflection Points Coincide with their Interior Roots. As the name suggests, a degree n PIPCIR is a polynomial with n distinct real zeros such that every inflection point is also a root. Discovered in 1997, this family of functions has a wide variety of interesting and beautiful properties.
In this talk, we take a further look at PIPCIRs, inspired by a paper that linked these polynomials to solutions of Jacobi's differential equation. We'll show how PIPCIRs satisfy a three-term recurrence relation, are uniform limits of Jacobi polynomials, and form a "nearly" orthogonal class of polynomials.
Gerry Cox, Lake Michigan College
Friday, 10:15-10:35
T428Constants from Kepler's Laws of Planetary Motion: mechanical energy and angular momentumThis presentation will show that Kepler's equations of motion imply that mechanical energy and angular momentum are constant. These are good additions to show students when presenting Kepler's Laws in a calculus class.
Ada C. Dong, Lawrence Technology University
Friday, 10:40-11:00
T428Structural Induction in Discrete MathematicsWhen taking introductory theory of computation or principle of programming languages, most of our computer science undergraduates encounter their first shock in mathematical induction on recursively-defined structures.
I believe that the transition, from natural number based induction to structural induction, can be improved by gently introducing the concepts and techniques in discrete mathematics through discrete structures like trees, graphs, or other nonlinear structures. Unfortunately, most textbooks on discrete mathematics either totally ignore the concept (Rosen, Goodaire), or briefly mention it without any examples (Truss).
In this talk, I prompt a stronger and smoother mathematical preparation for computer science related majors, using my experiences and working examples on structural induction.
Charles Du, student, Delta College
Sponsor: Jim Roznowski
Friday, 4:05-4:25
T328Binomial Series-An Alternate Way of Writing CoefficientsThe presentation will include an introduction to binomial series and the usual way of writing coefficients. A new factorial approach will then be presented. The difficulty of extending the pattern to other binomial functions will be discussed.
Don Faust, Northern Michigan University
Friday, 3:40-4:00
T428The Concept of Negation:
(1) Aristotle's Privatives and Contemporary Elementary Logic
and
(2) Logics of 'Conflict without Contradiction'We use Evidence Logic (EL), containing Classical Logic plus machinery for the representation and processing of confirmatory and refutatory evidence. In ELn (n>1), Pcx1...xk:e asserts confirmation of Px1...xk at evidence level e while Prx1...xk:e asserts refutation at level e (for e in the Evidence Space {i/(n-1): i=1,...,n-1}).
Concerning (1) and using unary predication for simplicity, where Px is a unary predication, non-Px is a privative unary predication. Aristotle argued (e.g. ON INTERPRETATION, X and PRIOR ANALYTICS, Book I, XLVI) for distinction of non-Px and NOT Px, maintaining the former implies the latter but not conversely. However, in contemporary elementary logic the term complement non-Px is widely used in such a way that non-Px is logically equivalent to NOT Px. After some observations about this peculiar situation, we suggest a faithful implementation of Aristotle's view, an axiomatizable extension T of EL2 wherein Px is interpreted with Pcx and non-Px with Prx (and T includes the axiom Prx IMPLIES NOT Pcx ).
Concerning (2), which is related to aspects of negation particularly relevant to paraconsistency, we construct a family of extensions of ELn which axiomatically allow conflict of the confirmatory and refutatory while forcing one or the other to be the case. These theories omit axioms reflecting Aristotle's view while including axioms reflecting the converse of his view. As such, these theories, with respect to confirmatory and refutatory predications, allow violation of 'laws of noncontradiction' while forcing 'laws of excluded middle'.
"The concept of evidence", IJIS 15(2000),477-493.
(abstract) "Evidence Logic", BSL 4(1998),86-87.
"The concept of negation",Logic and Logical Phil. 5(1997),35-48.
(abstract) "The concept of evidence", JSL 59(1994),347-348.
Kalpana Godbole, Citi Commerce Solutions, a division of Citicorp
Friday, 3:40-4:00
T328Transition from Academic to Corporate World -- Lessons Learned and Math UsedOrganizations are constantly on the alert to gain a competitive edge, using the many tools that have long been touted as a way to beat the competition. This talk will present some of the tools utilized at a Citicorp site in Gray, TN and their connections to classroom mathematics.
Sidney Graham, Central Michigan University
Friday, 11:30-11:50
T329The Middle Third-A First Course Involving ProofsMany mathematics departments have a first course in proofs; i.e., a course to help students make the transition from calculus to abstract algebra or real analysis. I have taught such courses numerous times, both at Michigan Tech and at Central Michigan University, and I am currently developing a textbook for the course.
I will talk about common misconceptions that students have in these courses, and I will talk about strategies that I have developed to deal with these misconceptions--some that have failed spectacularly and some that have succeeded modestly.
Garnet S. Hauger, Spring Arbor University
Friday, 4:30-4:50
T428Using Microcase® in Teaching StatisticsRecommendations concerning the teaching of statistics from MAA and NCTM suggest the use of appropriate computer software and real-world data. Microcase® is both a statistics software package and a data bank of several important national and international data files. It is being used at many colleges and universities to teach statistics and provides a rich venue for statistics instruction. This report is about how it is being used at Spring Arbor University to teach statistics for social science students. This computer software is used for weekly lab assignments and statistics projects. In-class exams include printouts generated by this software and for which students are asked to write interpretive paragraphs.
Pamela Lowry, Lawrence Technological University
Friday, 11:30-11:50
T428The Impact of a Laptop Initiative on Student AttitudesThis study investigated the relationship between the impact of the laptop computer initiative on student's attitudes toward mathematics and mathematics achievement at Lawrence Technological University. The dependent variables were student attitudes toward mathematics and mathematics achievement. The independent variables were the use of a laptop computer in the classroom and outside the classroom. Antecedent variables were students' past mathematics courses taken, fulfillment of course prerequisites, and past computer experience. Data was collected at the conclusion of the spring semester 2001 from mathematics courses below Calculus 1. A factor analysis was conducted on the research. There were significant relationships between the four mathematics factors namely, intimidating, useful and necessary, and narrow segment and achievement. There also was a significant relationship between the computer application, course management system and achievement.
Azita Manouchehri, Central Michigan University
Saturday, 11:05-11:25
T428Utilizing Inquiry Based Instruction in Undergraduate Mathematics Education:
An Example from a College Geometry CourseThe session will focus on describing the content, structure and outcomes of a semester long college geometry course designed for undergraduate mathematics majors. In this class, inquiry-based instruction and computer-based explorations were used as the primary teaching methods. Following a description of course activities, I will share data from the students relative to their mathematical work and progress and their reactions to inquiry based mathematics instruction. Since a majority of the students enrolled in class were also pursuing a high school teaching career, I will elaborate on the value they attached to the instructional techniques used in class for their own teaching.
Brian McCartin, Kettering University
Saturday, 10:15-10:35
T428Eisenstein Primes and Equilateral EigenvaluesAnalytical expressions for the eigenvalues of the Laplacian on an equilateral triangle are available for both Dirichlet and Neumann boundary conditions. The multiplicity of these eigenvalues is determined by the number theoretic properties of the binary quadratic form m2+mn+n2. This presentation will provide a complete treatment of such multiplicity using Eisenstein primes, thereby providing a direct link between number theory and the core engineering topics of vibrations, heat transfer, acoustics, and electromagnetics.
Mark Naber, Monroe County Community College
Friday, 10:40-11:00
T329Matrix Order DifferintegrationThe Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to assume matrix values. It is found that the computation of derivatives and integrals to matrix order is well defined for any square matrix over the complex numbers. Some properties are worked out for special classes of matrices. It is hoped that this new formalism will be of use in the study of systems of fractional differential equations and sequential fractional differential equations.
Patrick Pan, Saginaw Valley State University
Saturday, 10:40-11:00
T428Reflexivity and Bounded Reflexivity of Subspaces of OperatorsBounded reflexivity is a new concept in operator theory. In this talk, the definitions and some basic properties of reflexivity and bounded reflexivity will be given. Similarities and differences of the above two concepts will also be discussed.
William A. Sargeant, Central Michigan University
Friday, 11:05-11:25
T428A Survey of Rationales for the Use of Technology in College Mathematics InstructionThe session will focus on describing ways in which collegiate mathematics instructors at one university make use of technology in their classrooms, and the reasons they offered for their choices. Data obtained over one semester from classroom observations and outside interviews of various instructors teaching a wide range of mathematics classes will be described. The question of whether technology was used as an add-on to traditional instructional strategies or as a tool for improving students' understanding will also be addressed.
Jon Stevenson, student, University of Michigan-Flint
Saturday, 11:30-11:50
T328Ramsey Numbers or How to Plan the Perfect Party!We present a puzzle that involves inviting random guests to a party. The solution to this puzzle turns out to be an application of the first non-trivial result of Ramsey theory. First, some necessary terms from graph theory are given, and then we define Ramsey numbers.
A proof establishing Ramsey numbers' existence and giving an upper bound follows. Using colored graphs to illustrate the lower bounds, the exact values of several small Ramsey numbers are calculated. We finish by sketching an outline of what could be done to calculate exact values for larger Ramsey numbers.
Radu Teodorescu, Western Michigan University
Friday, 3:15-3:35
T328Solving Linear Recurrence RelationsIn this note we give a recursive method of finding the general term of a sequence satisfying a linear recurrence relation with constant coefficients. In the first section we are studying completely the situation when the recurrence relation has three consecutive terms and in the next section the situation when the recurrence relation has k +1 consecutive terms. Several numerical applications are also included.
Tara Terry, student, Oakland University
Advising faculty: Serge Kruk, Oakland University
Saturday, 10:40-11:00
T328Derivatives of Matrix FunctionsApplied mathematics, especially optimization with the development of semidefinite programming, has seen a renewed interest in functions of matrices. The applications under current scrutiny are varied and interesting and the solution techniques rely on tools as classical and well known to undergraduates as Newton's method. Yet, the standard definition of derivative is woefully inadequate when one needs, for example, the derivative of the determinant of a symmetric matrix.
We revisit some definitions of the derivative to choose one more appropriate to our needs and we show how it reduces to the standard definition in simple cases but is applicable to a much wider range of cases.
Jack Tower, student, Lansing Community College
Sponsor, JingLing Wang
Saturday, 10:15-10:35
T328Finding Roots of Cubic PolynomialsWe all know that, for a quadratic function, there is a quadratic formula for finding the roots. However, what about a cubic polynomial? How can we complete a cube? Is there a general formula for the roots of a cubic polynomial? If there is, what does it look like? In this presentation, I will answer these questions and demonstrate the construction of a general formula (Cardan's formulas) for the roots of cubic polynomials.
Gerard Venema, Calvin College
Friday, 10:40-11:00
T328The deductive approach to geometry can be enjoyable and worthwhileIn recent years I have been experimenting with a redesigned undergraduate "Foundations of Geometry" course that can better meet the needs of undergraduate mathematics majors, especially those who are in the secondary education program. I have found that it is possible to rework the topics in the course in such a way that the course becomes much more relevant and enjoyable to the students and at the same time to achieve some important goals that go beyond the subject matter of the course itself.
The course works well as a bridge course in which students learn the art of writing proofs, using Euclid himself as a model. The course pays conscious attention to the way in which high school Euclidean Geometry textbooks are organized; this helps students to achieve a much higher level of confidence in their own abilities to understand the high school course. Students are brought to the level of geometric thinking in which they can understand and compare different geometries. Attention to the historical development of geometry leads naturally to a deeper understanding of the revolutions that have taken placed in our understanding of the foundations of mathematics and to an exploration of the connections between mathematics and the real world.
In this talk I will describe the course and the strategies used to achieve some of the goals.
Roger Verhey, Moderator, University of Michigan-Dearborn
Presenter: Sharon Senk, Michigan State University
Panelists:
Al Taylor, University of Michigan-Ann Arbor
Chris Hirsch, Western Michigan University
Mike Lehman, Holt High School
Friday, 3:15-4:50
T329The Mathematical Education of High School Teachers: CBMS RecommendationsThis session is a panel discussion of the CBMS high school recommendations for the Mathematical Education of Teachers from the perspective of a high school teacher, a university mathematics educator, and a university mathematician.
Clark Wells, Grand Valley State University
Saturday, 11:30-11:50
T428One Bad Turn (or Rotation) Deserves Another?Visualizing molecules is extremely useful for chemistry students, and can be especially effective on the Web. MDL's Chime Plugin for Netscape makes this possible, but has limitations. Chime treats composition of rotations as if it were commutative, and applies rotations in an arbitrary order. How do I animate the rotation I want? Using ideas from Linear Algebra, I tell the software to rotate one way so that it "messes up" in the way that I want -- and I have a nice application of Linear Algebra to show students, as well!
P. K. Wong, Michigan State University
Friday, 11:05-11:25
T329An Online AP Calculus - Preliminary ReportMichigan State University recently introduced an online AP Calculus-BC course. The course is available as a CD-ROM containing 12 chapters written as Mathematica notebooks. Students purchase a graphing calculator (TI-89) and use Mathematica or the MathReader (free) to view the notebooks. Numerous examples incorporating the use of the TI-89 and Mathematica are provided. Students can modify the examples and observe the resulting changes to gain better understanding of the concepts presented. Students download their weekly reading and individualized homework assignments and log onto the course website to submit individualized homework. Interaction with the instructional staff is by email or web-based synchronous and asynchronous communications.
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