Seaway Meeting Home
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Just for Students
Colm Mulcahy, Spelman College
explore a variety of magical feats based squarely on
mathematical principles. Starting with simple card
forces/prediction tricks--diners are invited to bring their own decks
of cards-- we'll move on to an amazing demonstration of
mind reading as "Mathematical Idol 2005" is selected at random from the
finalists Archimedes, Euclid, Euler, Gauss, Hilbert, Newton, Pythagoras
Mulcahy earned a B.Sc and M.Sc. in Mathematical Science from
University College Dublin, in Ireland, in the late 1970s. A few
years after getting his PhD for research in abstract algebra
under the direction of Alex F.T.W. Rosenberg at Cornell
University in 1985, he joined the faculty at Spelman College,
in Atlanta, Georgia.
Saturday Morning Invited Presentations
Peter Turner, Clarkson University
the effects of the Logarithmic Distribution of Numbers:
Gradual and Tapered Overflow and Underflow
The talk begins with a
discussion, and proof, of the surprising observation that 30% of
numbers have leading significant digit 1. The proof uses elementary
ideas and is based on the effect of repeated multiplications.
Turner grew up in England – in the Liverpool area in
the Beatles era. He received his B.Sc. (Honors) in Mathematics in 1970
and his Ph.D in Pure Mathematics in 1973, both from the University of
Sheffield. After a year as a postdoc in Applied Mathematics at
Sheffield, he joined the faculty of the University of Lancaster as a
Lecturer in Numerical Analysis. During this time his research interests
developed to include systems of computer arithmetic. In 1987, Turner
moved to the U.S. Naval Academy where he stayed until taking up his
current position as Chair of Mathematics and Computer Sceince at
Clarkson in 2002. He has published four undergraduate texts in
scientific computing, edited three volumes of conference proceedings,
(co-)authored more than 40 research papers, and served on national
committees for SIAM.
James Tattersall, Providence College
In 1663, Henry Lucas, the
long-time secretary to the Chancellor of the University of
Cambridge, made a bequest, subsequently granted by Charles II, to endow
a chair in mathematics. A number of conditions were attached to the
Chair. Among the more prominent Lucasian professors
were Newton, Babbage, Stokes, Dirac, and Hawking. We focus
attention on the early Lucasians,
many of whom were very diligent in carrying out their Lucasian
responsibilities, but, as history has shown, such was not always
case. In the process, we uncover several untold stories and some
Tattersall received his
undergraduate degree in mathematics from the University of
Virginia, a Master's degree in mathematics from the University
of Massachusetts, and a Ph.D. degree in mathematics from
the University of Oklahoma. On a number of occasions he has
been a visiting scholar at the Department of Pure Mathematics and
Mathematical Statistics at Cambridge University. He spent the
summer of 1991 as a visiting mathematician at the American
Mathematical Society. In 1995-1996, he spent eighteen months as a
visiting professor at the U.S. Military Academy at West Point. He
was given awards for distinguished service (1992) and
distinguished college teaching (1997) from the Northeastern Section of
the MAA. He is former President of Canadian Society for History
and Philosophy of Mathematics, the Archivist/Historian of
Northeastern Section of the MAA, and the Associate Secretary of
the Mathematical Association of America.
David Poole, Trent University
Drawing on my own
experience, I will describe a successful mathematics course for
pre-service elementary school teachers and argue that such
courses need to be more widespread. Even in universities and
colleges where mathematics-for-teachers courses do exist, they
are almost universally perceived as low-level service courses,
while the "real mathematics courses" are on "the other side of
the fence". I claim that not only is this viewpoint flawed, but
there isn't really a "fence" (or
doesn't need to be). With examples from various mathematics
courses I have taught, I hope to demonstrate that serious mathematical
topics can be fruitfully incorporated into a course for
elementary teachers. Conversely, teaching methods appropriate in
a math-for-teachers course can be successfully used in other
Poole received his
B.Sc. in Mathematics from Acadia University and his Ph.D.
from McMaster University. Since 1984 he has been in the
Department of Mathematics at Trent University,
Peterborough, Ontario. He served as Chair of that
Department for six years and
is currently Trent's Associate Dean of Arts &
Science (Teaching & Learning). Professor Poole has been a
frequent participant in education sessions and working groups of
the Mathematical Association of America (MAA), the
Canadian Mathematical Society (CMS), the Canadian Mathematics
Education Study Group (CMESG), and the Mathematics Education
Forum of the Fields Institute. For the CMS, he has also served on
the Education Committee and chaired the Human Rights Committee. In
Ontario, he is a member of the provincial advisory panel for the
high school mathematics curriculum. Professor Poole's research
interests are in ring theory,
discrete mathematics, and mathematics education. He is the author
of the textbook Linear Algebra: A Modern Introduction
(Brooks/Cole), now in its second edition. While at Trent
University, Professor Poole has received three merit awards for
excellence in teaching as well as the university's highest
award for teaching excellence, the Symons Award. In 2002, he was
awarded an Ontario Confederation of University Faculty
Associations Teaching Award and in 2003, he received a 3M
Teaching Fellowship, the highest award for university teaching
Saturday Afternoon Contributed TalksCristina Bacuta, SUNY Cortland
Proof: A process strand for in-service teachers and an activity for pre-service teachers
This presentation will outline some important results of our efforts to assess proof abilities at SUNY Cortland. This work involves collaboration between the mathematics department faculty and a group of in-service teachers, and is supported by our PMET mini-grant "Assessing pre-service teachers' abilities to do proofs".
Howard Bell, Brock University
Some Ruminations on Wedderburn’s Theorem
This year is the hundredth anniversary of the publication of Wedderburn's seminal theorem on the commutativity of finite division rings. We comment on the historical context in which the theorem appeared, the various proofs, and some applications - old and new.
Daniel Birmajer, Nazareth College
The standard identities and the Quantum planes
Since the standard polynomials play an essential role in the Polynomial Identity (PI) theory of the matrix algebras, it is natural to look for similar situations in other contexts. Let F be a field and q an arbitrary nonzero scalar. The quantum plane is the associative ring generated by the variables x and y, whose product satisfies the following relation yx=qxy. In this talk we explore the role of the standard identities in the PI-theory of the quantum planes.
David Brown, Ithaca College
Experimental Mathematics and Writing for First Year Math Students
In this talk, I will describe a course that combines computer experimentation, mathematical exploration, and intensive writing in a second semester course for mathematics majors and minors. Using open-ended problems and even some famous unsolved problems, such as the Collatz Conjecture, simple questions motivate students to investigate topics. Students produce weekly reports, complete with supporting computer and mathematical evidence. Emphasis is placed on a clear style of mathematical writing. In this talk, I will discuss how this approach helped foster a deeper understanding of why we study mathematics and how mathematicians tackle ideas. I will illustrate how student understanding improved during a semester full of writing. I will also indicate how student attitudes toward mathematics changed over the course of the semester. Many students did not initially care for the open-ended approach, but ultimately came to value it.
Allen Emerson and Kris Green, St. John Fisher College
How a writing assignment transformed a writing intensive mathematics course for business students
Five years ago we were approached to develop an alternative mathematics course for business students that introduced spreadsheets and approached mathematical topics in a way that we were later able to identify as being in line with the CRAFTY report from the MAA. A few minutes from the start of the very first class, we discovered that we had no idea what to assign for the students to do. By nature, the incorporation of the spreadsheet led us to need an assignment allowing students to make arguments involving the development and interpretation of data. This required the development of a communicative context and data for analysis Years later, we find that the course is now based almost solely on such argumentation and analysis assignments. These assignments give insight into student approaches to problem solving and have led us to a completely new system of grading that focuses on the standards we set for student learning.
Jeff Johannes, SUNY Geneseo
Mathematical Teaching Soundtrack
For the past two years I have begun every single class session by playing a song that has a connection (at times extremely tenuous) to the content of the day. I will share a ten-minute mix of songs that I have used for this purpose.
John Maceli , Ithaca College
Fairness, The Talmud and Pascal
Topics of fairness make terrific subject matter for a contemporary mathematics course. We will illustrate a few problems related to fairness and equity. In particular, we will discuss a bankruptcy problem arising in The Talmud and show a connection with the solution of this problem and Pascal’s solution of the "Problem of Points".
Neils Hanson, Matthias Youngs, Anthony Macula**, SUNY Geneseo
Pooling Designs for DNA Code Validation
We describe a random group testing method used to design laboratory experiments to validate a DNA code. A DNA code is a collection of synthetic DNA strands closed under complementation. DNA codes serve as universal components for biomolecular computing and biomolecular nanotechnology and also have been used in other areas of molecular biology (e.g., single nucleotide polymorphism genotyping, gene expression profiling and DNA microarray development).
Facilitator: Anthony Macula, SUNY Geneseo
Discussion of Undergraduate Biomathematics
An informal discussion about biomathematics at our institutions as presented by participants. Some topics for discussion could be:
1. What is a biomathematics program?
2. How (not) to organize interdepartmental programs.
3. External funding opportunities.
4. Organization of a local biomathematics group.
5. Organization of a local undergraduate/faculty meeting.
James Marengo, RIT
The Noiseless Coding Theorem
The Noiseless Coding Theorem gives a lower bound in terms of entropy on the expected length of a binary coding of the values of a random variable or vector. The concept of entropy will be discussed and a proof of this theorem will be presented. The author will then examine some implications of this result with the help of some examples.
Donald Muench, St. John Fisher College
A Mathematical Tour of St. Petersburg
In the summer of 2004, my wife and I were in Tallinn, Estonia at a Bridges for Education English Conversation “Camp” for 4 weeks. This led to taking a 5-day tour of nearby St. Petersburg and spending one morning visiting the sites of Euler’s tomb and apartment house. A nearby house was the residence of a number of other famous mathematicians. We will mention a few facts about each of these mathematicians and those “descendents” connected to St. Petersburg. We will show pictures of these sights and the plaques on the walls of these two buildings. This should whet one’s appetite for the MAA Mathematical Tour in 2007 to celebrate Euler’s 300th birthday.
Colm Mulcahy, Spelman College
Wavelets made easy
We will give a Fourier-free introduction to wavelets by presenting a linear algebra approach to the simplest example, Haar wavelets, motivated by a real world application to data/image compression. Anybody who can add, subtract and divide by two will be able to sing along in no time at all.
Moderator: Olympia Nicodemi, SUNY Geneseo
Whither the Textbook? [90 minute roundtable discussion]
A round table discussion wherein we discuss the changing role of the textbook in the college classroom. Questions to be addressed include: What should a good textbook do for a course? Must authors and publishers be at odds with the students and the used book market? How can authors shape the college math curriculum in the current publishing environment? What are or should be the alternatives to textbooks? What about the cost to the students? Representatives from all sides of the issue will attend and contribute to the discussion.
Sam Northshield, SUNY Plattsburgh
Summing across Pascal’s triangle and its continuous analog.
It is well known that parallel row’s of Pascal’s triangle sum to 2^n and that sums along another family of parallel lines sum to the Fibonacci numbers (asymptotic to p^n where p is the Golden ratio). The generalization to families of parallel lines with a fixed rational slope are also known to be asymptotic to c^n for some c dependent on the slope; we shall discuss a proof. The entries of Pascal’s triangle are the binomial coefficients and can be extended, via the gamma function, to have positive real arguments. We shall discuss the analog of the results above involving integration, over families of parallel lines, of the extended binomial coefficients.
Morris Orzech, Queen's University
Matter and mindset in the pre-service classroom
There is much current discussion, but little consensus, about the role of our courses and programs in turning out teachers with mathematical and intellectual attributes that will make them effective. My approach to courses for pre-service mathematics teachers is conditioned by a goal selected from a palette of current prejudices about teacher training, namely that it is desirable to intertwine mathematical, pedagogical and career goals. My exercise of this prejudice is constrained by the context of the courses in our program: the material should be appropriate for an advanced course without duplicating other courses, and should introduce new mathematical ideas while keeping an orientation more for prospective high school teachers than for prospective graduate students. One strategy is to use material that is accessible at high school level (but new) as a vehicle for engagement and surprise, and to follow up with reflection about knowledge, beliefs, and habits that students have developed in previous courses. The classroom interaction that supports the introduction of this material is intended to help develop students’ pedagogical sensitivity and (if I am lucky) their later professional practice.
Joseph Petrillo, Alfred University
Counting Subgroups of a Direct Sum of Cyclic Groups
Recall, a cyclic group Zn of order n is an abelian group which is generated by a single element and has exactly one subgroup for each divisor of n. For example, Z18 has 6 subgroups. By The Fundamental Theorem of Finite Abelian Groups, a finite cyclic group can be decomposed into a direct sum of cyclic groups of relatively prime orders. Thus, Z18 = Z2 Z9, and the number of subgroups of Z18 can be obtained by calculating the product of the number of subgroups of Z2 and Z9. Unfortunately, this method for counting subgroups fails for Z3 Z9. The number of subgroups of Z3 Z9 is actually 10 (not 6). We will derive a formula for calculating the number of subgroups of the direct sum of cyclic groups of orders pm and pn.
Jamar Pickreign and Bob Rogers, SUNY Fredonia
Right Question, Wrong Answer, or Wrong Question, Right Answer? Teacher Evaluation of Constructed Response Test Items.
Much speculation is given to what New York State's new mathematics tests for their new standards will look like. Assuming that the types of questions remain essentially the same, this talk will present samples of actual teacher evaluations of student constructed responses on past Math A exam items. We will discuss the challenges teachers face in evaluating these responses and the implications for mathematics teacher preparation.
Gabriel Prajitura, SUNY Brockport
What we talk about when we talk about Calculus
We will present some examples of differentiable functions having properties which do not look like what we teach in Calculus.
Jennifer Roche, Hobart and William Smith Colleges
Radices and Matrix Rings
Work has been done on characterizing normal subgroups of symplectic groups via nonassociative algebra. The approach is entirely algebraic and hinges on Jordan ideals and specifically the structure called a radix in matrix rings. We have generalized previous results about Jordan ideals and radices in matrix rings over graded algebras in future hope of a result for normal subgroups.
Hossein Shahmohamad, Rochester Institute of Technology
Coefficients of Flow Polynomial of Kn
While the coefficients of the chromatic polynomial of the complete graph Kn is known, the coefficients of the flow polynomial of Kn is not known. We determine some of these coefficients and offer some conjectures on the rest.
Xunyang Shen and Peter Turner, Clarkson University
A Hybrid SLI System with Taylor Approximation
Symmetric level-index (SLI) arithmetic was introduced to overcome the problems of overflow and underflow in the floating-point system. Our current research is to improve the algorithm performance and to promote its implementations. In this talk, I will first give a brief summary of the SLI system and its arithmetic algorithms, and then introduce an approximation scheme using Taylor’s expansion, together with its software implementation in a hybrid system. Finally, two applications will show how the system would facilitate some real life computational problems.
Joseph Smith, SUNY Binghamton
Lattices and Groups
We will begin with a brief discussion of what a lattice is and then look at some small subgroup lattices and the lattice of normal groups within a group. We will conclude by looking at the normalizers in a group and see that in general the set of all normalizers do not form a lattice. Then briefly discuss some cases where they do.
Barbara Stewart, SUNY Geneseo
What's New? The Entire State Math Curriculum!
In March 2005, the New York Board of Regents approved a new Mathematics Core Curriculum for all students in pre-kindergarten to high school. The document outlines math requirements for each grade from pre-K to 8. Courses in Algebra, Geometry and Algebra 2 will replace Math A and B. This session will be an overview of the "who, what, where, when and why" of the new program. This update should be useful to college educators who work with pre-service teachers and may also be of interest to anyone who has children in school in New York State.
Karen Wells, Monroe Community College
Sand, Water, Chocolate Chips: Infusion of mathematics in an early childhood curriculum.
I will discuss how to create an effective early childhood math program through both unplanned and intentionally planned math learning activities across most areas of an early childhood curriculum.
Julia Wilson, SUNY Fredonia
Music as a Mathematical Subject: the Backstory
By the early Medieval period, music was considered a subfield of mathematics, along with number theory, geometry, and astronomy. How does this make sense? We will discuss the origins of this classification among the ancient Greek philosophers, and trace the influence of their theories through the early Christian era and into the Renaissance. This will shed light on how early scholars viewed math and its role in their quest to understand the universe.
Megan Cullen, Tim
Ellis and Laura Jenkins, SUNY Geneseo