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Integrating Teaching and
Research in Mathematics
By Jennifer Fisher Wilson
Panama Geer’s mathematical
research was inspired by an ant colony. Watching
ants traveling one after another from their nest
to a food source, she wondered about their behavior:
How do ants know how to travel in a straight line
when they have minimal vision and no auditory
abilities?
"Ants don’t have
a mathematical notion of a line, and yet they’re
able to organize in this way," Geer says.
"The geometry of that intrigued me."
She set out to learn how the discrete parts (individual
ants) of a complex system (the ant colony) are
able to collectively describe a straight line,
a circle or a curve.
Kristopher Tapp’s inspiration
came when he picked up a textbook on differential
geometry in graduate school. "I loved the
material. It made a lot of sense to me,"
Tapp says.
Geer and Tapp have demonstrated
a talent for both innovative research and creative
teaching, which they are now developing at Bryn
Mawr on three-year Keck Postdoctoral Research/Teaching
Fellowships. The fellowships enable newly minted
Ph.D.s to gain experience in teaching while pursuing
advanced research
Interconnected Systems
Geer received her Ph.D. from
New York’s Rensselaer Polytechnic Institute
in the spring of 2002. A double major in math
and studio arts (sculpture) at Wesleyan University,
she taught high-school math at the Hotchkiss School
in Connecticut for two years before pursuing a
graduate degree. A contributing factor in deciding
to attend graduate school, Geer says, came from
struggling to answer students’ questions
about how one uses math in the real world. She
realized that wanted to learn more about the novel
applications for mathematics.
Along these lines, Geer has
developed a course titled "Emergence"
that focuses on nontraditional ways to better
understand how the natural world works through
mathematics. The interdisciplinary seminar explores
interactions within various complex systems —
such ant colonies, immune systems, earthquakes,
traffic jams, or the inner workings of cities
or economies — by studying them from the
"bottom up," she says. This approach
is based on the assumption that the behavior of
a complex system emerges from the interactions
of its individual parts.
"Emergence is a different
way of approaching things," Geer says. It
differs from the traditional geometric modeling
approach, for example, in which scientists work
from the top down, first designing a mathematical
model of a physical object or system and then
developing an image based on the model, she explains.
This traditional approach assumes the image is
simply a representation that supports the exact
geometric model needed for analysis, Geer points
out.
Geer is particularly interested
in geometric shapes that emerge as a consequence
of interactions in natural systems — for
example, the way lines are generated by the foraging
behavior of ant colonies through the movement
of individual ants. She has used ant behavior
and other natural-systems approaches to simulate
the immune response and formation of memory T-cells
in the human body.
Geer believes that her course
on emergence, to be taught next spring, will appeal
to a wide range of Bryn Mawr students — not
just math and computer science majors, but also
students working in philosophy, architecture,
biology, economics and other fields. "It’s
a great way to study how different ideas interconnect,"
Geer says.
Curved Spaces
Kristopher Tapp says his Keck
fellowship at Bryn Mawr "feels like I’m
coming home." He earned his Ph.D. at the
University of Pennsylvania in 1999 and worked
as a visiting professor at Haverford College for
a year before moving on to the State University
of New York at Stony Brook on a fellowship. Tapp
says he is excited to be back in the Philadelphia
area, particularly at a liberal arts college.
"Liberal arts colleges
are exciting places because the students are strong
and the emphasis is on teaching," Tapp says.
Tapp is known for his creativity
and versatility as a teacher. While at SUNY, he
gave presentations on "discovery learning"
in undergraduate mathematics, developed a course
on mathematical thinking, and helped organize
a creative-teaching seminar. As a graduate student,
Tapp taught a one-year mathematics course for
home-schooled children in Philadelphia, in addition
to various college courses on algebra, calculus
and statistics.
Tapp is enjoying the opportunity
to teach at Bryn Mawr. His advanced-level course
on multivariable calculus, which concentrates
on three-dimensional figures, is one of his favorite
to teach because it connects closely with his
research on visual geometry, Tapp says.
Tapp was hooked on math when
he traveled abroad in his senior year at Grinnell
College to participate in the Semester in Mathematics
Program at the Technical University of Budapest.
He studied with eminent scholar-teachers at Eötvös
University and the Mathematical Institute of the
Hungarian Academy of Sciences.
The experience inspired Tapp
to pursue a doctoral degree at the University
of Pennsylvania, but he struggled initially and
almost quit. "A lot of students had a lot
more background in math than I did, and I wondered
whether I should do something else," he says.
Tapp’s doubts vanished when he was exposed
to differential geometry, which became the focus
of his graduate studies.
Tapp’s research centers
on curved spaces, such as the universe, which
is believed to be a three-dimensional curved space.
He is especially interested in positively curved
spaces that are multidimensional, but Tapp says
that our ability to study higher-dimensional spaces
is rooted in our visual intuitions about the three-dimensional
universe.
Now that Tapp is back in the
Philadelphia area, he’s enjoying the opportunity
to collaborate again with his graduate-school
colleagues at the University of Pennsylvania.
Collaboration in mathematics can be difficult
because the field is so diverse and complex, Tapp
explains. Even within subspecialties of mathematics,
it can be challenging for mathematician to truly
understand another’s work, he says.
"Although some fields
become so super-specialized that cross-communication
becomes difficult, I still believe in finding
every way we can to build bridges," Tapp
says. Tapp and Geer are seeking ways to build
bridges between mathematical specialties by making
the courses they teach more accessible to a wider
range of students, and by involving undergraduates
in their multidisciplinary research projects.
About the Author
Jennifer Fisher Wilson is
a contributing editor for The Scientist.
She writes frequently about science and medicine
for various publications, including Lancet
Neurology, Science and UCLA Magazine.
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