Charles Allen, Michigan State University, President of MCTM
Rebecca Walker, Grand Valley State University
Leo Paveglio, Lapeer Public Schools
Friday, 12:00 – 1:45, Luncheon Panel Discussion, Van DusenSome Consequences of No Child Left BehindThe new Michigan Mathematics K-8 Grade Level Content Expectations: What they are; How they align; and, How they will be assessed. Possible implications for 9-12 expectations will also be presented.
Undergraduate Programs and Courses in the Mathematical Sciences: A CUPM Curriculum Guide 2004 was developed after four years of intensive work and interaction with the mathematical community. It is accompanied by an online publication, Illustrative Resources for CUPM Guide 2004, which was developed to serve as an existence proof that the recommendations of the Guide are indeed feasible. This talk will discuss the contents of both publications and invite comments and responses from the audience.
In this presentation we will present a look at the history of the project including how it was started and how it has developed over the last eight years or so. We will also relate some of the amusing and not so amusing incidents along the way. Finally we will take a look at some of our future aspirations including some possible research topics for helping us understand our intellectual history.
It is no surprise when two lines go through a point, but when three lines have a point in common, one might call that a miracle. Such miracles occur frequently, however, when we deal with triangles. For example, the three medians of a triangle go through a common point as do the three angle bisectors and the three altitudes. But there are many less well known (and perhaps more surprising) examples, and there are general techniques for proving that such miracles occur.
The talk will discuss Liouville's 1850 theorem to the effect that there is a general paucity of conformal mappings in Euclidean spaces of dimensions three and higher – say in comparison with the situation in the plane, where the Riemann mapping theorem ensures a wealth of conformal mappings.
In this talk we will develop some mathematical insights into BMX bicycle and car drag racing. We will use these to explore - and break - some of the cherished myths in drag racing folklore.
In 1976, Robert M. May published his famous article “Simple mathematicalmodels with very complicated dynamics” in Nature . In this article, May investigated the dynamics of the logistic map f(x) = ax(1-x) where a is a constant. After recent advances in the field of nonautonomous difference equations, the nonautonomous logistic map F(a,x) = ax(1-x) surfaced back as a simple model with complicated dynamics. In this talk, we discuss the dynamics of the nonautonomous logistic map in a periodically fluctuating environment, its cycles and their stability, and its attractors.
Certainly, within your own department there has been considerable debate about whether it is possible to teach a high-quality online math class. We would like to host this round-table discussion with one assumption about on-line classes: It IS possible to do it effectively. If we operate from this assumption, what would our effective, high-quality online class look like? Bring your ideas, research, and success stories to share with others as we try to brainstorm what a good online class would look like. If there’s an online math expert in your department, bring them with! (or at least pick their brain for ideas before the conference) Hopefully, we can all walk away from this session with some ideas for developing our own online math classes.
If your college teaches Beginning Algebra and Intermediate Algebra, chances are you have a lot of sections! Many sections sometimes equates to many headaches. Why reinvent the wheel? We all face the necessity for assessment and the use of learning-based outcomes as a result of accreditation. Let’s share our ideas with each other! In our recent survey of neighboring community colleges, we discovered some great ideas for managing assessment and consistency in these classes. If you have an assessment technique, strategy for fairly assigning courses to adjuncts, or a strategy for measuring learning-based outcomes that your department is particularly proud of, please share!
An essential student activity for building understanding is the act of summarizing class notes. Students often resist doing this, perhaps because they think "well, it's all in the book anyway". But what if students didn't have a book?
In this talk I will share experiences from teaching a course in Euclidean geometry without a text. The course relies heavily on students' work in in-class discovery activities and computer laboratory exercises, most of which include in-depth proofs. We'll discuss how "writing a textbook" can make an outstanding project for a group of students working both independently and cooperatively, examine how to implement such a project, and hear students' own words that describe the impact the project has had on their learning and perspective.
This talk is not about how to write a computer program to calculate the first billion decimal digits of π. Yet such papers abound in the literature. Why? Of what possible use could it be to know that the million and first digit in the decimal approximation to π is 4 and not 5?
It this talk I will give examples of the sort of thing that we can discover by accurately measuring the ratio of the circumference to the diameter of actual circles in the real world. The size and mass of the earth and the shape and destiny of the universe can be inferred from the difference between the experimental value of π and its theoretical Euclidean value.
The Principle of Mathematical Induction is a central topic in any mathematics course emphasizing proof techniques.
The traditional approach to induction has been through examples drawn from elementary number theory (working out a few initial cases, formulating a conjecture for the general case, which is then proved by induction). Other examples are proving formulas for arithmetic or geometric sums.
We propose another approach that has been successful in teaching Discrete Mathematics. This consists of real life examples drawn from computer science that are of immediate appeal to students who have had some exposure to computer science concepts through introductory programming.
Let f ∈ R[T], where R is a UFD. We know we can write f as a product ∏ Firi, where Fi is the product of all irreducible factors of f with multiplicity ri. We will refer to the sequence of multiplicities of f, the ris, as the factorization pattern of f. When R is an infinite UFD with finitely many units, we show that the factorization pattern of f is determined by the set of values of the function f.
In this presentation, I will discuss some ancient Japanese problems in Euclidean geometry, called San Gaku Problems. These were problems that appeared during the Edo period (1603-1867) on colored tablets that were hung in shrines or temples as a form of devotion. They provide a rich source of Euclidean problems that have had very limited exposure in the western world. Therefore, they provide a great launching point for undergraduates to try and generalize them to spherical and hyperbolic geometry. I will present student work from the REU at GVSU over the summer of 2004 and indicate some of the ideas that might be explored in the future.
Call a number a - b√2 with a and b both positive integers tiny if it is closer to zero than any number c - d√2 such that c and d are positive integers with c < d and d < b. It turns out that tiny numbers can be sequenced by a nice general recursive function with closed form solutions. The original tiny number problem was proposed by Dr. Judith Grabiner.
Next to a power series, the simplest type of series expansion involving polynomials is a Legendre series expansion. In this talk, we demonstrate how the Taylor series of a function analytic in a neighborhood of the origin can be expressed as a limit of Legendre expansions, which are valid in varying ellipses that become the disk of convergence of the Taylor series in the limit. The main result itself is easy to state, and the proof utilizes elementary ideas as well as two interesting, albeit admittedly somewhat obscure, facts concerning complex-valued Legendre polynomials.
I will give an overview of the mathematics exams in the College Board's College-Level Examination Program (CLEP), with special emphasis on its soon-to-be released Precalculus Exam. The purpose of the program is to give students the opportunity to demonstrate college-level achievement for credit and/or placement.
For three years now I have offered a pilot version of a two semester, 8-credit Calculus-with-Review course. It covers all of our Calculus I course, with topics from precalculus integrated into the development. The main audience is students who did not quite place directly in to Calculus I via our placement system, and students who are required to take Calculus I and are in some way afraid of doing so. I will describe the approach I have taken, the textbooks available, the results of the experiment as I see them, and relevant research results from the literature.
Topics from cryptology can be used as applications and as theory and proof motivators in the Introductory Discrete Mathematics course. Examples will include: logic - binary numbers, XOR circuits; mod 2 additive ciphers - Feistel, DES; mod cryptosystems - Caesar addition, decimation multiplication, RSA exponentiation; algorithms and their efficiencies - Euclidean Algorithm, Fast Exponentiation, primality testing, factoring algorithms; use of famous theorems - Lame's Theorem, Fermat's Little Theorem, its contrapositive and its "most of the time true" converse; recursion - Fibonacci, linear feedback shift register; software - use of Maple to implement and time algorithms.
A complete list of 9-vertex irreducible graphs that are irreducible for the torus was obtained. First, all the possible toroidal embeddings of certain common graphs are found. An algorithm is then developed to prove nonembeddability of a graph G on the torus; a subgraph of G that is isomorphic to one of the common graphs is displayed, and extensions of each embedding of the common graph to our graph G are systematically eliminated.
To prove irreducibility of a graph G, a list of triangulations on the torus is first given. It is then shown that deleting any edge of G results in a subgraph of one of these triangulations.
Finally, we assert that our list is complete.
We study the asymptotic behavior of the norm of the function in the title for a real-valued function h(t). Under certain conditions on h(t) we can use the method of stationary phase and certain equidistribution theorems to obtain a formula that generalizes a 1973 result by D. M. Girard.
The classic problem is to find a single counterfeit coin from a set of 12 coins and determine whether it is heavy or light in three "weighings" on a balance scale. My paper generalizes this, under various assumptions, to find the maximum size set from which a counterfeit coin can be found in n "weighings".
It is known that there are only Moore Graphs of degree 2, 3, 7, or (possibly) 57. The purpose of this paper is to determine the state of that 'possible' 57. First, we examine the history of the problem. Secondly, we set up a method of generating possible graphs that might fulfill the necessary conditions. Lastly we set up a method by which the graph is tested to verify that it fulfills the necessary conditions.
Baseball is a game replete with its own set of statistics. Players, managers, broadcasters, and fans use baseball statistics to discuss team and player performance. In this talk we discuss some statistical models for baseball. In particular we use a Markov Chain to model a half inning of baseball. There are 25 different out-base runner situations in baseball. We estimate a transition matrix that gives the probabilities of going from one state to another using actual play-by-play data. We discuss the relative importance of pitching and hitting to team performance.
Modeling and Simulation (M&S) uses the analytic, or mathematical, model of the vehicle. An analytic model is a system of coupled differential equations with constraints that describe the behavior of the vehicle being modeled. Simulating the vehicle is equivalent to solving the system of DE’s with initial conditions. Other math that gets used includes: linear algebra, matrix techniques, numerical methods, graph theory, Lie groups, etc. We want to discuss the math used in M&S of ground vehicles. Our goal is to show math students a career where math can be put to good use and inspire thinking about other ways to use math.
Laplace's dictum: "Read Euler, read Euler. He is the master of us all," may be rightfully transcribed as, "Study e, study e. It is the master of all." Evidence for this statement will be provided in the form of a panoramic view of some exhilarating appearances of e throughout pure and applied mathematics.
The moments of a finite discrete probability distribution can be computed using a linear recursion. We shall prove this and also prove that if certain conditions are met, a linear recursion generates the moments of a finite and discrete probability distribution. We will derive the distribution that generates the Fibonacci sequence and one that generates all moments equal to 1. The talk will be elementary and accessible to undergraduates.
In this talk we discuss combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method.
In this talk I will review the standard sets of metric axioms for the geometry that is studied in an undergraduate course on axiomatic geometry. (Axiom systems due to Birkhoff, MacLane, SMSG, UCSMP, etc.) Each of these axiom systems includes an axiom about distance measure (a Ruler Postulate), an axiom about angle measure (a Protractor Postulate), and an axiom that relates the two kinds of measurement. This last axiom is usually Side-Angle-Side, but in a transformational approach SAS is replaced by a Reflection Postulate. While it is well known that the Reflection Postulate can replace SAS, it seems to be less well known that the Reflection Postulate can actually replace both SAS and the Protractor Postulate. I will outline a proof of that result.
Last summer, we wrote and co-taught a Concepts of Calculus course for elementary education math majors at the University of Michigan-Flint. We will discuss (1) our motivation and rationale for such a course, (2) an overview of what we actually did, (3) student response to this course, and (4) what we would do differently.
Let X1,X2,...,Xn be a sequence of i.i.d. random variables with random variables with the common distribution function F. Set Sn = ∑i=1nXi. Assuming that F belongs to the domain of attraction of a non-degenerate stable distribution G with characteristic exponent α ∈ (0,1], we will show that for any p ∈ (0,1), limε↓0(-1/log ε)∑n≥1(1/n) P{|Sn|≥εn1/p} = αp/(α - p). This is joint work with Prof. Yimin Xiao.
By systematically dividing a triangle into smaller and smaller sub-triangles, points of convergence are determined within the triangle. The resulting points naturally lead to area and other geometric relationships to the original, which can be determined with the use of barycentric coordinates.
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