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Iowa Section of the Mathematical Association of America |
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| 3:10-5:00 |
| 3:40-4:00 | Proof in Geometry: Euclid and a Class Journal |
| Theron Hitchman, University of Northern Iowa | |
| I'll discuss how I use Euclid as a text, and a class journal as assessment in a Euclidean Geometry course aimed at pre-service teachers. |
| 4:10-4:30 | Using Toilet Paper to Help Students Make Generalizations |
| Martha Ellen Waggoner, Simpson College | |
| When students are given a specific problem to solve, they do not naturally create a general solution method that could be applied in other situations. In this presentation, I will discuss a project that I use to help students learn the value of generalization and give them an introduction to sensitivity testing. The project starts by having students find the number of sheets of paper on a specific sealed roll of toilet paper, but they must take that method and produce a formula that could find the number of sheets of paper on a general roll of perforated paper. They then test the various models created by the class for sensitivity to measurement error to find the "best" method. I have used this assignment in an introductory modeling course with a calculus prerequisite, but the material could be adapted for use in pre-calculus courses. Students find the assignment engaging and surprising. Some of the students struggle with the process of creating a single, simple formula based on measurements. The students always want to know the "real answer" so they can check their work, but instead of telling them the answer I use this opportunity to introduce them to measurement error and sensitivity testing. In this talk, I will give the details of the project, which includes in-class work, use of Excel and a final written report. I will also discuss the pitfalls and surprises the students encounter. Finally, I will give a brief description of how I plan to use the project in Quantitative Reasoning in the fall. |
| 3:40-4:40 | Mathematics Courses for Elementary Education Majors |
| Bridgette Stevens, University of Northern Iowa | |
| At the recent IMSEP Summit for math and science educators in August, it was discussed that mathematics educators should begin a dialogue regarding a set of core competencies (content) for teaching elementary mathematics in the state of Iowa. To in part meet that need, this is a working group session in which participants will discuss a variety of issues around the mathematics courses offered for prospective elementary mathematics teachers at Iowa's small colleges and community colleges. Topics may include curriculum, instruction, best practices, and dilemmas. |
| 7:30-8:30 | MAA's American Mathematics Competitions: Easy Problems, Hard Problems, History and Outcomes |
| Steven Dunbar, University of Nebraska-Lincoln | |
| How do you get bright students hooked on mathematics? How do you keep teachers intellectually engaged and pedagogically innovative? A proven way is to involve them both in mathematics competitions with great problems that span the curriculum. The Mathematical Association of America has continuously sponsored nationwide high-school level math contests since 1952. The sequence of contests now has 5 different contests at increasing levels of mathematical sophistication. Students who succeed at the top level on these contests become the team representing the U.S. at the annual International Mathematical Olympiad. I'll showcase some interesting, easy and hard contest problems, and a little bit of history. Along the way, I'll comment about the intersection of these contests with the school mathematics curriculum. |
| 8:30- |
| 8:00-11:25 |
| 8:30-9:30 | Introduction to Knot Theory |
| Louis Kauffman, University of Illinois at Chicago | |
| The theory of knots is a recent part of mathematics. It originated in the tabulation of tables of knots by the mathematicians Tait, Kirkman and Little in the 19th century. These tables were prepared at the behest of Lord Kelvin (Sir William Thompson) who had developed a theory that atoms were three dimensional knotted vortices in the luminiferous aether. Along with these speculations came the development of geometry and topology in the hands of Gauss, Riemann, Poincare and others. As the knotted vortex theory declined (it has never entirely disappeared!), the mathematics of topology ascended, and the theory of knots came into being as part of the study of low dimensional manifolds, using the fundamental group of Poincare and early versions of homology theory. Max Dehn used the fundamental group to show that a trefoil knot and its mirror image are topologically distinct. J. W. Alexander in the 1920's found a polynomial invariant of knots that bears his name to this day. Kurt Reidemeister, in the 1920's, discovered a set of moves on diagrams for knots that made their classification a (difficult) combinatorial problem. In the 1980's there came a rebirth of these combinatorial schemes in the discovery of the Jones polynomial invariant of knots and links (and its relatives and descendants). Along with the new combinatorial invariants came new relationships with physics and with many fields of mathematics (combinatorics, graph theory, Hopf algebras, Lie algebras, von Neumann algebras, functional integration, category theory) and new kinds of mathematics such as higher categories and categorification. This talk will discuss the history of knot theory and then it will concentrate on describing the Jones polynomial, its relationships with physics, and recent developments related to categorification. |
| 9:30-9:40 |
| 9:45-10:05 | Calculus for the 21st Century |
| Mariah Birgen, Wartburg College | |
There are several deep issues with the way we have been delivering calculus to this generation of students. First is the issue of the audience. With the extreme growth in pre-health science majors, the majority of students in our first semester calculus courses are Biology majors who are taking the course because it is required for the MCAT. They are not particularly excited to be there and are not afraid to share that opinion with the rest of the class. Second, which is tied to the first, is the issue of AP calculus. Many students coming to college who do want to study in the mathematical sciences are not in that first semester calculus course because they have AP credit. Thus, we often do not see OUR students until after that critical first semester in college. Additionally, the AP calculus curriculum is not equivalent to the material we deliver in college, so the students are always lacking, especially in sequences and series. The last issue has to do with engineering. The fundamental reason calculus is the first mathematics course taught to STEM (science, technology, engineering, and mathematics) majors in College is because of the space race. As a nation, we sent a majority of our mathematically talented and gifted students into engineering for decades. However, Wartburg does not have an engineering College and neither do most of our liberal arts college kin. We do have a small number of engineering science majors every year, but they are dwarfed by the health sciences students who are probably not taking calculus based physics until their third or fourth year at Wartburg. Our new applied calculus - foundational calculus sequence is designed to address these three issues. Most students take applied calculus, the SENCER (science education for new civic engagement and responsibility) course first. Often this is the only calculus course taken by the health science students. Now, the students learn the what, how, and why of calculus. These are our civic issues. How do we mathematically manage disease and epidemics? How do we mathematically manage ecological systems? The study of differential equations and mathematical modeling is essential material for the professional students who make up the majority of our first course. Additionally, these are exactly the topics that are not covered in the typical AP calculus course in high school. Finally, students who are coming to study engineering and physics at Wartburg are encouraged to take the foundational calculus course first at the same time as calculus based physics. In fact, because of the pace of this foundational calculus course, students at Wartburg are seeing the techniques of calculus in their calculus course before they see them in the classical physics course. We have only taught the sequence for one year, but we have preliminary evidence that students are persisting in applied calculus a greater rate than in our previous calculus one course. |
| 10:15-10:35 | A Teachers Circle for Middle School Math Teachers |
| Elgin Johnston, Iowa State University | |
| Last year I partnered with Jean Krusi, an Ames Middle School Mathematics teacher, and Gail Johnston, ISU Mathematics Lecturer, to organize and run a Teachers Circle for Middle School Mathematics Teachers. We followed up with a one week Teachers' Circle workshop in June 2009. This talk will describe our experience and supply good references for those interested in trying something like this in their own areas. |
| 10:45-11:05 | Isometries of a Giant Product Space |
| * Corey Gevaert, University of Northern Iowa | |
| I'll be discussing the isometries of the product space Y which is formed by an infinite amount of hyperbolic plane fibers lined up from 0 to 1. I'll be discussing how the hyperbolic isometries are carried over and the Lebesgue transformations that influence these isometries. |
| 9:45-10:05 | A p-adic Euclidean Algorithm |
| Eric Errthum, Winona State University | |
| A brief introduction to the p-adic numbers will be given. Then a p-adic Division Algorithm and a p-adic Euclidean Algorithm will be defined in such a way that they mimic the classical algorithms. Lastly these methods are used to compute a generalized GCD and a p-adic simple continued fraction. |
| 10:15-11:15 | The 1/P Pseudo-Random Number Generator |
| * Donald Peterson, Iowa State University | |
| Seemingly suitable for encryption, the 1/P pseudo-random number generator quickly outputs a long, well-distributed sequence of digits from a small seed. However, without any prior knowledge of the seed, it can be determined and the sequence can be predicted both forwards and backwards by careful examination of 2|P| + 1 digits of the sequence. This article examines how to develop the generator, and more importantly given a small bit of any sequence, how to predict the remaining sequence. |
| 11:25-11:45 | Sequences and their annihilators |
| Christian Roettger, Iowa State University | |
| Annihilating polynomials have been widely used in geometry and to study sequences over fields and over the integers Z. We use the same simple ideas to study sequences over Z modulo n. There are surprising difficulties, surprisingly nice results and an open conjecture. We can demonstrate some applications to recurrence sequences like the Fibonacci and Lucas numbers, or discrete dynamical systems. Joint work with John Gillespie. Prerequisites: ring, ideal, quotient ring, Chinese Remainder theorem - suitable for undergraduates with a first course in algebra. |
| 10:15-11:15 | An overview of Mathematics in the Iowa Core Curriculum |
| Catherine Miller and Megan Balong, University of Northern Iowa | |
| Information about the Iowa Core Curriculum's mathematics component will be shared. Focus will be on the grades 9-12 component as it is to be implemented in Iowa classrooms first. We will also discuss some ways in which the Iowa Core Curriculum may affect college mathematics curriculum and instruction. |
| 11:25-12:10 |
| 12:10-1:30 |
| 1:35-1:55 | A Survey of MAA Study Tours and the Iowa Section |
| Joel Haack, University of Northern Iowa | |
| Highlights of the MAA Study Tours, with special attention to the participation of members of the Iowa Section. |
| 2:05-2:25 | Essay Questions on Calculus Exams? |
| Jonathan White, Coe College | |
| How should a Calculus class be different at a liberal arts college? I present one aspect of my own answer to this question, namely that assessing students' abilities to communicate verbally is particularly important. Included are lessons learned over more than a decade of evaluating students' written expression on exams, selections from my bank of past questions, and student reactions. |
| 2:35-2:55 | Cool combinatorics arising on a cohomology hunt! |
| ** Aba Mbirika, University of Iowa | |
| Can cool combinatorics arise in a hunt for the cohomology ring of a variety? Yes indeed! In 1992, De Mari, Proces, and Shayman introduce Hessenberg varieties. These are a natural generalization of the famed Springer variety. Much is known about the cohomology ring of the Springer variety, but little is known in the case of a general Hessenberg. We provide a step in this direction by inspecting a certain subfamily of Hessenbergs called the Peterson variety. We conjecture that the cohomology ring of a Peterson variety has the presentation of a graded quotient of a polynomial ring modulo a special ideal with very nice combinatorial properties. Along the way, cute combinatorics pops up in the form of Dyck paths, Catalan numbers, etc. We also discuss tantalizing recent work that might help confirm our conjecture. |
| 1:35-1:55 | Hankel Operators and Combinatorial Identities |
| Eugene Herman, Grinnell College | |
| We show that every bounded Hankel operator H on the Hilbert space of square-summable sequences can be factored as H = MM^*, where M maps a space of square-integrable functions to their corresponding moment sequences. By expanding these functions in a Fourier series of orthogonal polynomials, we obtain identities that connect the entries of the Hankel matrices with the orthogonal polynomials. |
| 2:05-2:25 | The classification of finite-dimensional Hopf algebras |
| ** Michael Hilgemann, Iowa State University | |
| Hopf algebras can be considered generalizations of groups, and group algebras are basic examples of such objects. In recent years there have been developments in the classification of finite-dimensional Hopf algebras over an algebraically closed field of characteristic 0, which include many examples which are neither group algebras nor the linear dual of group algebras. In this talk, we will highlight these classification results and some of the useful properties that general finite-dimensional Hopf algebras share with finite group algebras. In particular, we will discuss recent joint work with Richard Ng that completes the classification of Hopf algebras of dimension 2p^2, for p an odd prime. |
| 2:35-2:55 | Classification of Hopf algebras 4p-dimension |
| ** YiLin Cheng, Iowa State University | |
| In recent years, there have been much development on the classification of finite dimensional Hopf algebras over a field of characteristic 0. The 4p-dimensional Hopf algebras when p=3 was classified 10 years ago and there are very few results for classification of nonsemisimple Hopf algebras with the dimension which is a mutiple of 4 during this period. In this talk, I will discuss some progress joint work with Richard Ng about 4p-dimensional nonsemisimple Hopf algebras when the odd prime p is less than or equal to 11. |
| 1:35-2:35 | Mathematics Courses for Prospective Secondary Teachers at Small Colleges |
| Mariah Birgen, Wartburg College | |
| At the recent IMSEP Summit for math and science educators in August, it was discussed that faculty should have more opportunities to share with each other what is going on in their classrooms. To in part meet that need, this is a working group session in which participants will discuss a variety of issues around the mathematics courses offered for prospective secondary mathematics teachers at small colleges . Topics may include curriculum, instruction, technology, best practices, challenges, and dilemmas. |
* denotes an undergraduate speaker and ** indicates a graduate student speaker
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