Twelfth Annual
VSU Mathematics Technology Conference
ABSTRACTS for
Workshops and Talks
WORKSHOPS:
Prime Numbers and Online Security: From
Spreadsheet to Web Technology
Prime
numbers are mysteriously beautiful objects in mathematics, playing a vital role
in maintaining online security. Yet in
school mathematics they have been approached in a rather traditional way. We present a technology-based instructional
sequence that contributes to the sense-making of prime numbers in middle grades
mathematics and the results of a pilot study. The development is based on the
Theory of Realistic Mathematic Education, featuring realistic starting points
and extensive use of web technology. It includes over 20 interactive activities
appropriate for both individual and group work. The global goal of the sequence
is to provide a conceptually enriched environment, where students can
reconstruct and then apply the RSA cryptosystem in a realistic context. The
topic is suitable for anyone interested in reliving the excitement of
mathematical understanding and the affordances of
technology in mathematics education. Our
present work is a significant upgrade on a spreadsheet sequence presented at
the 2006 VSU Conference and is available at http://garnet.acns.fsu.edu/~lb04f/prime/sec_1_intro.html
Lingguo
Bu, Maria Fernández, Rob Schoen
Visualizing a Dynamical System, Some
Geometry Behind Pascal’s Triangle
We
present what first seemed a novel way to visualize a simple dynamical system
used to model the sharing of money. Use of basic MAPLE programming increased
our interest and understanding of the problem, leading to unexpected properties
and interpretations of the Pascal’s triangle. An interesting family of
sequences is found and investigated.
Jonathan Bryant* and Katie Klimko*
MathZone Prospectus
MathZone is McGraw-Hill’s complete course management solution
for Electronic Homework and Tutorial system for Mathematics. With MathZone, you
can easily:
(1) Measure the strengths and weaknesses of all students in
your class
(2) Create and assign homework that is automatically graded
for you.
(3) Provide additional help and support for your students 24
hours a day.
For this demo, we will be
using MathZone for Calculus.
John Schmidt
McGraw-Hill Higher Education
* Undergraduate Students
TALKS:
Powerlinking Technology:
Getting the Best from
The presenter will show how Georgia Perimeter
College Faculty and IT staff, along with Pearson Education, WebCT(Blackboard),
and the University System of Georgia’s Advanced Learning Technologies have
worked together to provide students with the rich teaching and learning
features of MyMathLab in a single sign-on environment
in Vista. The Powerlink was tested on both the USG
Production and Development Servers in 2006 and will undergo further pilot
testing in 2007 so that this technology can soon become widely available to USG
Faculty who wish to use MyMathLab along with features
of Vista with a single-sign on process
for students.
Sandee D. House, Ed. S.
The Seven Deadly Sins of Modeling
It starts out with the concept of a model of an entity, system or process, the
power of a model, pM, is then defined. The first
deadly sin recalls (yes) real estate. We then go on to characterizations
and examples of the rest of the deadly sins.
Ben Fusaro
Calculating
the number of equilateral triangles in {0,1,...,n}^3
The number of equilateral triangles in {0,1,...,n}^3 has the entry A102698 in "The On-Line
Encyclopedia of Integer Sequences!". Here, we denote it by ET(n). The first 30 values have been computed using a brute force
program written by J. Zucker and extended by H.
Pfoertner up to ET(34).
We
have written a Maple program that calculates these numbers using a different
method based on number theoretic results from "A parametrization
of equilateral triangles having integer coordinates" (arXiv
math.NT/0608068). This program allowed us to compute ET(1),...,
ET(52). However, the results are based on the non-existence of a Diophantine
solution of the equation a^2+b^2+c^2=3d^2 with gcd(a,b,c)=1 and min(gcd(a,d),gcd(b,d),gcd(c,d))> 1 for all values of
d, an odd integer between 1 and n^2. We checked this property as being true for
n=1...63 which implies that our program gives the correct values for ET(1),..., ET(63). We present the main ideas of this program
and some peculiar things about it.
Eugen J. Ionascu
Algorithm
for Medial Axis Determination of Polygons
The medial axis of a convex polygon
is a "stick" figure in the form of a tree, as seen in graph
theory. All points on the edges of that tree, including the vertices, are
centers of maximal circles, circles that are inside the polygonal contour and
are tangent to two or more polygonal edges. We outline a simple algorithm
for determining the skeleton by first generating bisectrices
which are going to be centers of maximal circles for consecutive and
non-consecutive edges and second by determining how much to keep of these bisectrices to create a coherent skeleton. We
implemented the process in a Maple worksheet and found the interaction of
simple geometry, graph theory, and proper use of data structures an interesting
combination for curious undergraduate students to investigate. We then
show the way for an extension of the algorithm to general polygons.
Bruno Guerrieri
The presenter will discuss the development of
the hands-on Informal Geometry course currently being taught at NFCC, where
students develop comprehensive portfolios. Geometers Sketchpad is used as
a tool for discovery, exploration and for development of many of the required
portfolio activities. Although work intensive this has been a highly
successful course where students are immersed into the study of geometry.
Examples of some student portfolios and materials/activities from the course
will be displayed and shared with attendees.
Sharon Erle
How the Brain Learns: Implications for Teaching Math
Our scientific understanding
of how the brain learns has increased significantly since many of us were in
graduate school. This presentation will include a very brief overview of what
we know followed by a discussion of the implications for learning mathematics.
Analysis of effects of tablet PC technology in
mathematical education of future teachers
This
presentation describes the authors’ work using Tablet PCs mobile computer lab
in future teachers’ preparation classes. Faculty from the
Olga Kosheleva, Ana Rusch, Vera Ioudina
Empirical Derivation of
the Volume Formulas for Certain Solids
We show how the familiar formulas for the
volume of solids with certain symmetry such as the sphere, cube, cylinder,
etc., can be obtained by submerging them into water and measuring its vertical
displacement. The volume functions are derived through a simple regression
analysis without any prior knowledge of them. In the process we actually show
why the volume of the cylinder must be equal to area of the base times its
height. Once this is known, we apply it to derive the formula for the other
cases. This work was prepared with the help of Laura Nunley
( CSU math ed student) as an undergraduate project
that provides an alternative method (which does not require the use of
calculus) to high school teachers of mathematics to give their students an
opportunity to experimentally derive the volume formulas they usually know by
memorization.
Using Highlighting
Software in Teaching Mathematics
Hightlighting software that has been
used in language arts is being developed for use in mathematics. The current
focus is at the level of high school algebra, often a roadblock for students in
community colleges and universities. The presentation will consist of a
demonstration of some of the current features of the software and an open
discussion relating to its potential in teaching mathematics.
Steven Blumsack
Scientific Notebook & Geometer’s SketchPad as Deductive Reasoning Aides
How
can computer programs with interactive, direct
mathematics interfaces (e.g., symbolic interfaces to CAS, or interfaces
providing manipulative geometric constructions, transformations, etc.), affect
student’s deductive approaches to their mathematical reasoning? One sophomore
mathematics major working with Scientific Notebook,
and one elementary school education major working with Geometer’s SketchPad were studied during a summer term project.
Concrete positive examples are provided which show
(a) an interactive, pragmatic, inductive approach providing
individualized student-controlled feedback,
(b) the interface’s mathematically proper representations, which
invited students to recognize, and formulate their own abstract, well-defined
deductive premises,
(c) learning explicit mathematical representation and
interpretation required in using a logical machine (e.g., demand for proper
mathematical syntax),
(d) implicit mediation between student, instructor and machine
facilitating maximal mathematical communication and problem solving in minimal
time on task, and
(e) direct pedagogical cogency of “natural mathematics” symbolics and diagrams.
(f) presentation of highly specific, as well as general
mathematical semantics (e.g., interpreting meanings within definitive
mathematical-interface organized formats, or carefully reasoning through one’s
mathematical interpretations via “mathematical word processing”).
Three
negative aspects observed were:
(a) a higher demand to motivate the purpose of a proof when the
machine already (inductively) provided such convincing results.
(b) students confusing a reasoning through of certain
mathematics with merely running through memorized interface procedures.
(c) students get lost in the aesthetics—just playing with
content to an insufficient depth.
Erich Nold
A Visual Syllogism
Validity Tool
We
present a visual, web-based tool developed by the presenters for teaching
syllogism validation. The tool allows the user to activate and deactivate
validity rules and displays the effect on the validity of each syllogism type
in the set of all possible syllogism types. The tool can also search the set of
rules and create subsets that properly partition the set of all possible
syllogisms based on validity.
Chris Eason and
Bennie Coleman*
* Undergraduate Student
An Unexpected Surprise For
Students
The
presenter will demonstrate the benefits of using the internet and math software
(My Math Lab) in a learning support class. Web
site information and material used in a course using computers will be provided
to session attendants.
Jacqueline Spann
Problem BY Problem Solving Techniques
At
Pat Bezona
Synchronous Online Teaching of Calculus
The talk will describe technologies developed
by WebALT and used in synchronous on-line teaching of
calculus at the
Mika Seppälä
A Branch Switching Technique in Numerical Path
Following
Numerical
path following solves parameterized non-linear equations by tracing out points
on the solution curves. Starting from a
point on the solution curve, the path following method finds the next point on
the curve by determining the unique solution to an augmented system that
includes the original function and an arc length constraint. In the
neighborhood of a bifurcation point, where two branches of solution intersect,
this method must solve a bifurcation equation to determine the two directions
so that path following can be continued on these branches. Traditional
procedures require calculation of all second derivatives of the original
function. So the complexity is O(n^3), where n is the
dimension of the system. This complexity can be overwhelming. Based on directional
derivatives, we propose a difference scheme that determines the coefficients of
the bifurcation equation with only nine function evaluations, thus dramatically
reduce the computational complexity of branch switching.
Numerical path following provides an
excellent application to teach calculus, linear algebra and numerical methods.
With modular programs from Lapack or Numerical
Recipes, students majoring in mathematics or computer science can integrate
knowledge from multiple areas together to solve complex non-linear equations.
Running these programs on lab computers can help the students appreciate the
differences of good and bad algorithms. Visualizing the computational results
can stimulate the students’ interests in exploring the world of computing.
Chunlei Liu and Jin Wang
Invited Address:
The Joys and Virtues of Obsolete Technologies
The use of mathematical
technology does not come for free. Each mathematical device (whether stones,
abaci, slide rule, log table, idiot savants, computer, or calculator of today
or the human calculators of world war 2) requires learning certain skill sets
and often these skill sets are useful to other areas in math and science. Often
a technology obsolescent will have both expected and unexpected on these skill
sets. Many of the skills atrophy among the general population, but are still
maintained by specialists. What did we lose by technology changing before we
started blaming it all on the calculator?
Steve
Bellenot
Banquet Talk:
"Lucy in the Sky with
Diamonds...," Pencil Lead, Soot, and Bucky Balls
In
the early 1980's a British chemist became interested in space dust -- carbon
dust, in particular. What forms of carbon could exist in the space between the
stars? There were only two forms of solid carbon known at this time: graphite
and diamonds, and both had been observed in interstellar space. Harry Kroto wondered if chains of graphite could also exist in
the extreme conditions of outer space. His journey led him to discover a brand
new form of solid carbon here on earth in the laboratory -- the first new form
in 2,000 years of carbon chemistry. The discovery has changed the world, and in
fact, is arguably one of the greatest discoveries of the 20th
century. He and his colleagues named the new form of carbon
Buckminsterfullerene. The results of this discovery are changing virtually
every kind of technology that we have. This is a story about how pursuing basic
science can lead to unimagined technology. Harry Kroto's
question about carbon dust in outer space has lead to a revolution in
technology that will continue to change our lives throughout this century and
beyond. In this talk I will also describe a scheme to find Buckminsterfullerene
in space.
Cecilia
Barnbaum
Dept
of Physics, Astronomy & Geosciences
Mathematical Modeling of Simple Mechanical Systems
Simple
mechanical devices can be helpful in enabling students to visualize the concept
of mathematical modeling. Basic levers
and merry-go-'round systems are easy to comprehend and to construct, and they
offer a remarkable opportunity to illustrate applications of the vector product
as well as equations of the form xy = constant. The presentation will involve the use of a
lever and a merry-go-'round model to illustrate uses of the vector product and
equations of the form xy = constant.
Michael D. May