Charlene Beckmann, Grand Valley State University
Friday Luncheon Address Oakland CenterMathematicians Are from Mars; Mathematics Educators Are from Venus.
What Can We Do to Meet on Planet Earth?The Conference Board of the Mathematical Sciences in The Mathematical Education of Teachers asks mathematicians, mathematics educators, and K-12 teachers who mentor prospective teachers to work together to better prepare teachers of mathematics. When we do come to the table, we often appear to have very different points of view. In particular, sometimes the jargon we use in our respective fields makes it seem as if we have conflicting opinions, when in fact we often seek the same thing.
What can we do to build strong working relationships among mathematicians and mathematics educators who work in the same departments and communities? In this discussion I will share perspectives on our experiences at GVSU in the past five years about how we have built a community where we work together, even amidst some differences, to promote the mathematical education of all students.
Leonhard Euler (1707–1783) was a mathematician of the highest rank, the quantity of whose work is matched only by its extraordinary quality. In this talk, we sample from the Eulerian opus by considering: history’s first venture into analytic number theory (1737); a clever approximation of π (1779); and the derivation of “Euler’s Identity” via integral calculus (1749). Although these represent just a fraction of his total output, they reveal something of the power and insight of this towering figure from the history of mathematics.
An algebraic variety is defined as the common zero set of a collection of polynomials. The simplest algebraic varieties are the complex spaces Cn, and the rational varieties are those that are isomorphic to these simplest varieties “almost everywhere.” Alternatively, one can think of a rational variety as one that admits a parametrization. Despite their importance in both theoretical and applied algebraic geometry, rational varieties are difficult to identity. The talk will begin by reviewing these concepts and end with the most recent discoveries concerning rational hypersurfaces.
“Education is what survives when what has been learned has been forgotten.”—B.F. Skinner. The vast majority of our students soon forget the vast majority of the mathematical details they learn in class (sometimes, in fact, before the final). But what subject is better suited to help students learn to think more effectively than mathematics? Infinity, the fourth dimension, coincidence, topology, and many such topics all can bring students to add the mathematical way of thought to their intellectual strengths.
A fundamental problem in the American secondary school educational system is the dramatic dropout of high school students from college-leading programs. This course captures the interest of high school students previously turned off math by giving them work that they find is both important and interesting.
The authors of MathBC are fully aware of gaps in their students’ precollege preparation that result in trouble in calculus and higher level courses. Many of the topics in MathBC are chosen to fill those gaps before it is too late for the students. Most such topics arise in modern contexts—looking forward to what students will actually do in their futures, they help students see “what this stuff is good for” and they are often fun.
The talk will give detailed demos from the course.
Direct search methods, which optimize without using derivatives, constitute a fascinating chapter, still being written, in the annals of optimization. Their history includes an initial heyday in the 1960s, a fall from grace in the 1970s and 80s, and a resurgence dating from the mid-1990s. Today's research on these methods has highlighted several interesting—in some cases, contentious—issues, ranging from the nature of convergence proofs and the associated assumptions to the proper role for direct search methods in modern optimization practice. We shall consider a selection of these issues, their current status, and their implications, especially for teaching mathematics and scientific computing.
Using elementary computations, we survey the results characterizing the centralizers of a matrix in the full matrix group. The centralizers in GL(k) are also determined. The results are used to do computations of similarity of a matrix over finite fields.
In this paper we discuss the benefits of incorporating technical report writing in undergraduate mathematics courses. The combination of projects and technical writing enhances student learning and sparks a sense of accomplishment and professionalism.
We discuss the existence of nine stationary solutions for a system of linear parabolic equations with coupled boundary conditions. Also, we talk about the stability of those stationary solutions in the linearized system.
Starting from a single function element of an algebraic function one could use analytic continuation to piece together the whole function and in this way study its multiple-valuedness. Here, however, we shall use Riemann’s approach.
Our lecture will explore how a simplified form of a rational function estimates the original function. We will also investigate how the respective derivatives of these two functions resemble one another. Furthermore, we will look at the patterns that emerge at the point when the simple function approximates, to a certain percentage, the original rational function. We may, if time permits, look at the formulas for which our predictions work the best. In conclusion we may explore fractional derivatives.
Let D : Z → Z and f ∈ Z[T] be such that for a fixed positive integer r, f(t) = D(t)r for every t ∈ Z. We show that f must have an rth root in the polynomial ring Z[T].
Let’s modify or simplify the Partial Derivative Chapter in the calculus text. After presenting partial derivatives and their graphs, let’s show that the directional derivative can be derived from the tangent plane. At this point we can present the proof for mixed derivatives and proceed onward to the proof and application of extreme values.
Kristina Lund (an undergraduate student) and I obtained an internally funded grant to spend the summer of 2003 attempting to generalize some recent theorems from Euclidean geometry to spherical geometry. The most successful part of our summer was our work with the Area Principle. The Area Principle from Euclidean geometry is a useful tool in proving many different types of cyclic product relations. Among these are Ceva’s and Menelaus’ Theorems. We figured out how to generalize the Area Principle to the sphere and eventually the hyperbolic plane. Thus, in most cases, the cyclic product type theorems from the Euclidean plane, with appropriate modification, immediately generalize to spherical and hyperbolic geometry.
The dynamics of the polygonal dual billiard map depends on the size and shape of the respective polygon. For a class of polygons, called “large,” all orbits of the dual billiard map escape to infinity (in sharp contrast with the Euclidean case). If the polygon is a regular n-gon with right angles, then all orbits are periodic. We find the periods and give a complete description of the dynamics.
Developmental mathematics courses have taken on a variety of forms over the past 20+ years here at LSSU. This presentation will discuss the background/ history, course content, structure of the implementation, pre-testing and testing, and evaluation.
In search of Gaspard Monge in Paris: from rue Monge to the Ecole Polytechnique (old and new) and points in between.
Presenters designed and implemented data collection and analysis activities involving the TI-73 calculator and Fathom Dynamic Statistics Software in a statistics class for pre-service elementary teachers. A technology pre- and post- survey was administered to determine students’ attitudes towards the technology as well as their intent for its use in their classrooms. Data from the survey will be presented, and sample activities will be made available to those attending the talk.
Last summer the mathematics department at Spring Arbor University (SAU) in cooperation with the Jackson County Intermediate School District (JCISD) and the Jackson Public Schools (JPS) conducted a program for elementary teachers in JPS. The specific goal of the program was to increase the mathematical knowledge of these teachers. The curriculum for the mathematics class these teachers took centered on the Michigan Curriculum Framework mathematics strands and benchmarks. NCTM standards materials were used for this class. The end-of-experience evaluation showed that teachers’ knowledge of mathematics had significantly increased (in a statistical sense). This experience was funded by Title II (Improving Teacher Quality Professional Development Grant Program).
I have been experimenting with a module that recasts my permutation puzzle software, published in 2003 by the MAA, into a competition mode. The competition organizer prepares one set of scrambles and delivers it to all contestants over the Internet. They are timed as they solve the scrambles and when they are done, their scores are automatically uploaded to a web site. The contest can be local or worldwide. I will demonstrate the system and discuss its potential as a math club activity, or even a Michigan Section project to involve students from different campuses in a fun mathematical activity.
In the Department of Mathematics at the University of Michigan several faculty members have become interested in researching and evaluating instructional technology used in mathematics courses. With the assistance of a small amount of University funding a group, with participants from both the Department of Mathematics and the School of Education, has begun to develop instruments to facilitate this work. In the fall semester of 2003, we conducted a preliminary study to measure students’ perceptions and attitudes toward the different technology applications applied in the course. Next, in the current semester, we are conducting a video study in order to be able to characterize and evaluate Computer Algebra assisted teaching practices in lectures. In my talk I will summarize our preliminary results, discuss measurement and analytic instruments, and outline our future research plans.
I will talk about my experience in using technology, such as Blackboard, graphing calculators, and PowerPoint, in my College Algebra and Trigonometry class.
Parrondo’s Paradox originated in nanotechnology and was illustrated by the physicist R. D. Astumian by means of a simple board game. A strategy for playing the game consists of rules used for making a move. If A and B are strategies, let C denote the strategy of randomly deciding to make a given move according to rules A or rules B. Parrondo’s Paradox: it sometimes happens that strategies A and B each lead to a loss with high probability while the combined strategy C leads to a win with high probability. New results on Astumian games will be given.
In A Geometric Characterization of Linear Regression (Statistics, Vol. 37, No. 2, March–April 2003, pp. 101–117), the geometrical interpretation of regression lines by F. Galton and K. Pearson in terms of the concentration ellipse was generalized to permit unequal variances in the experimental errors. Herein, this work is further extended to three dimensions, where the corresponding results have a beautiful geometrical characterization in terms of the concentration ellipsoid.
I'd like to share the enjoyment of one simple problem from Arithmetic. The problem is often called four-number-game, or Ducci cycle problem, and it is about the convergence of the integer mapping,
f(x1, x2, x3, x4) = (|x1 - x2|, |x2 - x3|, |x3 - x4|, |x4 - x1|)
to the fixed point (0, 0, 0, 0) in a finite number of iterations. I'll demonstrate the problem, its proof, and its development to a curious result.
The wide availability of relatively powerful personal computers has caused renewed interest in many computational aspects of number theory. Questions that only a few years ago required the use of a supercomputer can today be explored with a significantly smaller computer budget. In particular, much has been written on generalized pseudoprimes by various authors. This paper explores some of these ideas and gives some limits on numerical questions posed on pseudoprimes relative to specific quadratic polynomials.
Over the past two decades, many faculty have found ways to improve student learning by making better, more active, use of class time. One of the most effective methods—the use of ConcepTests and peer instruction—was developed by Eric Mazur of Harvard. His pioneering work in physics has been successfully replicated in other departments, and its efficacy is clearly supported by data. In this session we will take a look at some sample questions and ways that they can be used in the calculus classroom.
We will consider aspects of randomness in music. An early example of this is Mozart’s Dice Game. Other topics may include the use of Markov chains and/or interacting particle systems to generate music.
Students taking college algebra courses have often seen the topics before. Commonly they find it easier to ignore the underlying structure and focus instead on acquiring a surface knowledge of manipulations. Unfortunately, the fluency they seek often eludes them precisely because they fail to recognize both the underlying algebraic form and the purpose of different forms. This presentation will examine what algebra these students really need to continue on to precalculus and calculus, and how an understanding of symbolic representation can improve learning.
To some, the question of whether one can generalize the Pythagorean Theorem was answered in 1993 when Andrew Wiles proved Fermat’s Last Theorem. In this talk, we will look at another answer, one found deeply buried in book VI of Euclid’s Elements.
Modern technology, Internet commerce, and high-speed communication use increasingly sophisticated data encryption and data protection. Cryptology is now a common (albeit hidden) part of everyday commerce. An understanding of modern cryptology requires both mathematical and technological sophistication, falling in the curriculum somewhere between computer science and mathematics.
At Central Michigan University we have introduced a new undergraduate course, The Mathematics of Cryptology, in order to (partially) meet this need. I will discuss some of the principles and concepts of this course and some of the results achieved by my math and computer science students.
The traditional interdisciplinary mathematics course has focused on connections with science. In this talk, I will describe an interdisciplinary course connecting mathematics with artistic concepts and endeavors for an audience of non-mathematics majors. Both art and mathematics share the basic theme of trying to understand the world around us. Following this theme, we will follow the development of geometry and algebra and their connections to, applications of, and inspirations for artistic creations, both natural and man-made. I will also share some personal reflections and reactions from students and others.
Authors: Eddie Cheng, Ray Kleinberg, Serge Kruk, William Lindsey, and Dan Steffy
In this talk we give a combinatorial approach for solving the exam scheduling problem. Given a set of exams, students and rooms we must schedule the exams into the rooms in time slots while meeting a set of hard constraints, such as no student can attend multiple exams simultaneously, and minimizing an additional set of soft constraints. This problem is NP-hard. We solve this problem in stages using stable sets, weighted bipartite matching, paths in hypergraphs and max flow. In this talk, we will discuss our methods, implementation and results.
Common geometric results with triangles and quadrilaterals can be proven and then generalized to get not-so-common results by utilizing complex numbers. Patterns with ratios of areas within triangles and quadrilaterals will be discussed, and, in spite of the title, they are quite easy to understand with basic knowledge of complex numbers.
What are some examples of conversation and collaboration between mathematicians and mathematics educators? A sampling of different formats will be shared.
Presenters: Charlene Beckmann, Grand Valley State University Will Dickinson, Grand Valley State University Azita Manouchehri, Central Michigan University Mike McDaniel, Aquinas College Pat Shure, University of Michigan–Ann Arbor Ken Smith, Central Michigan University
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