Recent standards and
research, published by mathematics education professional organizations, place
a great emphasis on “connections” in all grade levels. Through this emphasis on
interrelatedness, students begin to see the subject not as a collection of
separate strands, but rather as an integrated field of study. When linkages
between diverse domains of knowledge are formed (by comparing, contrasting,
analyzing, and applying), we have increased the likelihood that we develop
deeper understandings within both domains.
Thursday
Jonathan A. Zderad
"Are the Real Numbers Real?"
Is mathematics real or
true is some objective sense? Or is mathematics simply a human or cultural
creation? Leopold Kronecker(1823-1891) tried to answer
these questions with his well-known Creationist claim that “God created the
integers, all else is the work of man.” In this essay, I will assume a weaker
form of Kronecker’s hypothesis, namely, that “God created the natural numbers.”
I will attempt to demonstrate that such an assumption implies the existence of
many other mathematical systems, including the real numbers. As a conclusion, I
will be able to support a much more inclusive Creationist philosophy of
mathematics.
Thursday
Scott Taylor
"Mathematics and the Love of God: An introduction to the thought of Simone
Weil"
Simone Weil
(1909-1943) was a French philosopher, mystic, and activist. Her brother, André
Weil, was a founding member of Bourbaki and one of the twentieth century’s
greatest mathematicians. Simone Weil’s search for truth caused her to wrestle
with the spiritual significance of mathematics. She praises mathematics’
ability to develop concentration and criticizes the modern scientific establishment
for abandoning a religious motivation for mathematical work found in the work
of the ancient Greeks. Her writings have much to say regarding the relevance of
faith to mathematics and mathematics to faith.
One of the principle
adjustments in the mathematics major curriculum in the last thirty years has
been the introduction of a transition to advanced mathematics course to address
the discontinuity between the emphases on objects and processes in the
lower-level courses to the emphases on abstractions and proofs in the
upper-level courses. Scripture points out the importance of giving reasons for
our claims while maintaining integrity.
Defending the
reasonableness of mathematical proof, in turn, honors God. This study examined
the conceptions of proof that undergraduate students have upon entry to a
transition course on mathematical proof, how they develop skill in planning and
reporting proofs, obstacles they encounter, and the effects of instruction on
their performance in solidifying schema for proof-planning and proof-reporting.
Thursday
Brad Whitaker
"Playing the Chaos Game With Any Regular Polygon:
A General Formula for the Simple Rules"
The Chaos Game,
resulting in the Sierpinski triangle, first received broad interest as a result
of James Gleick’s 1987 book, “Chaos: Making a new Science” and a January 1989
episode of Nova titles, “The Strange New Science of Chaos”. Over the past
fifteen years, many have “rolled the dice” and generated a fractal triangle,
with some taking the game further to create the Sierpinski Carpet (rectangle)
as well as the “Sierpinski” pentagon and hexagon. But why stop there? Using
basic algebra and trigonometry, mathematics educators lead high school and
college students in an exploration of the mathematics behind these fractal
images and derive a general formula for the simple rules for playing the Chaos
Game with any regular polygon. Particularly appealing to mathematics educators,
the chaos game provides an avenue through which fundamental concepts of chaos
and fractal theory may be presented to mathematics students from high school through
college.
Thursday
Mary Wagner-Krankel
"Integrating Laptops into a Mathematics Curriculum"
St. Mary's University
has recently adopted a laptop program campus-wide. This session will look at
the impact of this program on hardware and software support, faculty training,
and course offerings in the mathematics department. Research results on student
usage and satisfaction with the program will also be discussed.
One of the major
limitations on processor performance is hard-to-predict branches. Branches that
are incorrectly predicted can incur a significant penalty on execution time.
Predication is a compiler optimization that has been used with general purpose
processors to effectively remove branches, exploiting available parallel
execution slots to execute both paths of the branch. However, modern
applications often include complicated branch structures. Applying predication
to these structures can result in extensive code expansion, creating a
bottleneck in the pipeline and increasing the power consumption of the system.
Digital Signal Processors are a class of specialized processors executing
programs with simple branching constructs that require precise guarantees on
latencies within the system. This presentation argues that in the DSP
environment, predication can be used to its fullest extent to increase
performance while maintaining the high standards of speed, timing and power
consumption required for real-time analysis.
Thursday
David Stucki
"Computer Science, Metaphorically Speaking"
I would like to
explore one way in which we can think about the relationship between our
Christian faith and belief structure and the attitudes and beliefs we hold
towards computers and computer science, namely metaphor. I will lay out an idea
for a stance that may help us in evaluating our responses to developments in our
discipline and then invite discussion.
Thursday
Jeff McKinstry
"Synchonous Activity Binds Visual Stimulus Properties Viewed By A
Brain-Based Device"
Animals can
effortlessly bind the attributes of stimuli to form a coherent scene. A
theoretical proposal to solve this binding problem relies upon reentrant
signaling that synchronizes activity between cortical areas. We tested this
notion with a mobile device that was guided by a simulated visual system based
on the primate brain. In the simulation, area V1 contained neuronal units
selective for color and orientation having small receptive fields that
projected to units in area V4 that had larger receptive fields. V4, in turn,
projected non-topographically to a simulated area IT. Reciprocal connections
were present within V4 and between V4 and IT. Neural activity was simulated
with a mean firing rate model augmented by a phase variable representing the
timing of activity. The device explored an enclosure containing objects of various
shapes and colors. Analysis of the simulated IT activity showed that reentrant
connections were necessary for the device to correctly classify multiple
objects that were viewed simultaneously. In a scene containing multiple
objects, the phase of neuronal group activity in V4 and IT responding to a
given object tended to be similar,yet was distinct
from the phase of other objects. Object selective neuronal units in IT emerged,
similar to those found in the primate,with responses
that were invariant to shift and scale over the range of views encountered.
This study supports the proposal that reentrant connections and synchronization
provide plausible mechanisms for solving the binding problem in the presence of
self-motion and real-world input.
Mathematical models
can serve as powerful and convenient means to represent reality. There is a
danger, however, to view such models as not just presentations of reality, but
as reality itself. This is most evident in physics and cosmology. One must be
careful to distinguish between abstract universals and concrete particulars.
Any set of data can be represented by a variety of models. These models are in turn
subject to various physical and philosophical interpretations. The choice of
model and interpretation is often made on metaphysical grounds, depending on
one's worldview.
Thursday
Andrew Simoson
"A Greater Tantalizer"
After a brief overview
of the five Platonic solids and their lore, we introduce and analyze a
combinatorial game involving a set of six octahedra, each face of which is
colored with one of six colors. The object of the game is to stack the blocks
so that on each of the column faces, all six colors appear. A little bit of
graph theory manages to reduce the search space for the solution down from over
300 million different arrangements to the unique solution. Each member of the
audience will receive a game set to keep, being 6 plastic octahedra,
hand-painted in
Thursday
Eric Gossett
"The Search for the Real Josephus Problem"
The Josephus Problem
is the familiar puzzle where n men are seated in a circle and every mth man is
killed until only one is left alive. You are to determine which is the ideal position in the initial circle. The puzzle was
inspired by an event from the life of Flavius Josephus. However, the actual
event did not involve men in a circle and the killing of every mth man (typically
described as m=3). The talk will present some partial findings regarding the
search for the origins of the problem in its present form. I will also discuss
several (very old) variants on the problem, as well as present a mathematical
solution for the case m = 2.
In educating future
leaders and citizens of our fast-changing society as well as “ambassadors” and
“representatives” of the Kingdom of God in this world, it is very important for
us to teach mathematics with emphases in critical thinking (logic, fallacies,
problem-solving, etc.) and quantitative reasoning (data analysis, statistical
understanding). I have been teaching courses in critical thinking at Christian
institutions and have made efforts to incorporate some contents and methods
originating from a Christian perspective. In my talk I will present some of
these examples as well as what I consider to be goals and objectives in such a
course. More specifically, I will discuss my use of an online discussion board,
logical/propositional display of Bible passages, and other practical examples
interesting from a Christian standpoint.
Friday
Johan de Klerk
"A Christian perspective On Mathematics: History Of
Mathematics And Study Guides"
At previous
conferences of the ACMS I discussed the matter of a Christian perspective in a
mathematics class and a method of giving atttention to such matters in practice
via a contextual approach. In this presentation attention will be given to the
following two topics:
Friday
Richard E. Sherman
"Non-Random ELS Extensions in the Book of Ezekiel"
A set of equi-distant
letter sequences (ELSs) of a pre-defined group of country names was located in
the book of Ezekiel and a control text (a Hebrew translation of Tolstoy's War
& Peace). Each of these initial ELSs were reviewed
by two Hebrew experts for possible extensions. Neither expert had any knowledge
of the source text of any of the letter strings they reviewed. Statistically
significant differences in the number of extensions discovered in Ezekiel were
observed. An extension discovery rate from the control text was used to
estimate the probability of chance occurrence of a large cluster of extended
ELSs located in Ezekiel 37. That probability was too small to accurately
estimate by standard methods.
An emphasis is offered
for the inference portion of an elementary Statistics course: the equivalence
between confidence intervals and tests of hypotheses. This equivalence is
rarely mentioned in basic texts but seems helpful to students. Student
reference sheets which employ this equivalence are available on-line.
Friday
David E. Boliver
"Preserving Significant Past Events and Studies in Mathematics
Education"
Each new generation in
mathematics education is tempted to think that it is the first to be truly
scholarly. Yet, there are significant past events and studies which remain
relevant, but are often forgotten. This presentation will include a selection
of those events and studies spread over the last hundred years and outline a
proposed network of web pages to spread this knowledge. Attendees are
encouraged to prepare by bringing to mind those events and studies they would
like to see in such a network.
Friday
Wayne Roberts
"State Mathematics League: An
In states or broad
geographic areas not having a high school mathematics league, there is a great
opportunity for providing a service in the area of our expertise. Such a league
serves able and interested students, is appreciated by high school
teacher/coaches, and identifies your college as a place supportive of
mathematics in the region.
Wolfram exposes some
ideas about informatics that relate to Christian scholarship: Does Wolfram's
definition of free will permit God to have free will? In what way is computer
software that is moved to new hardware like human souls who are resurrected to
a new body as described by Paul and Aquinas? Jesus'
incarnation as in-form-ation. An overview of
informatics from Aristotle to the present.
Friday
Wayne Iba
"Artificial Service vs. Artificial Servants"
Numerous research
projects in Artificial Intelligence have addressed problems within the context
of providing personalized assistance. Many of the other efforts in AI could be
viewed as developing technologies that are directly applicable to helping
individuals solve problems. However, the Microsoft paper-clip is still the most
prominent representative of assistant technologies and is particularly
unhelpful. Instead of artificial service in the false or vacuous sense, we want
artificial servants in the synthetic or computational sense.
Friday
Kim Kihlstrom
"Men Are From The Server Side, Women Are From The
Client Side: A Biblical Perspective on Men, Women, and Computer Science"
The percentage of
women in computer science is small and has decreased over the last twenty
years. Why is this the case, when computer science is
a wonderful and growing field with many opportunities? I believe that the
situation has its roots in the basic differences between men and women,
differences that were present from the beginning of creation and are part of
the way that God made male and female uniquely. In order to ensure that both
talented men and women are attracted to computer science, we need to understand
the differences between men and women, and how those differences affect the way
we approach computer science.
Far from exhibiting
Monod's dualism of "chance and necessity," our current mathematical
descriptions of the world exhibit subtle forms of order via the laws of
probability, chaos, complexity theory, and ambiguity (fuzzy sets).
Friday
Kevin Vander Meulen
"Mathematics, Stranger Than It Used to Be: Mathematical Contributions to
Postmodernity"
In this presentation I
plan to explore the relationship between mathematics and culture. While there
is a very strong link from mathematics to culture in the development of modern
thought, the link is not quite as clear in the rise of postmodern thought. My
focus will be to examine the role that developments in mathematics played in
constructing a postmodern mindset in culture today.
Friday
Jeremy Case
"What is a Random Event? A Project for Finite Math or Statistics"
Randomization is an
important idea in Finite Mathematics and Statistics. One main idea in these
courses is that events that appear to be performed in a random fashion are
often not random. Here we present a simple project involving
"randomly" opening the Bible. This activity leads to deeper
philosophical questions such as how to study the Bible and whether an event can
be considered random if God intervenes.
Connecting mathematics
in a meaningful way to a Christian perspective is a difficult task. I propose
that of the two ways to consider the connection, use of a Christian perspective
to inform mathematical thinking is an example of an "inverse
problem," the difficult inversion of a direct problem. In this
presentation, I will review the concept of an inverse problem, describe several
mathematical images of Christianity–potential solutions to the direct problem,
and demonstrate a web-resource which compiles these images based on connection
to specific course content.
Saturday
Dave Neuhouser
"Mathematics, Science, Christianity, and George MacDonald"
George MacDonald, nineteenth-century
poet, preacher, novelist, and writer of fairy tales, was a science major in
college and at times taught science and mathematics. This paper will explore
his understanding of the beauty of, and the role of imagination in, mathematics
and science and the relation of those two fields to the Christian faith and
life.
Return to ACMSOnline