In this Issue An Evening with the MythBusters

Transcription

20 Mathematics Spring/Summer 2008
The University of Arizona
In this Issue
An Evening with the
MythBusters
page 2
The hosts of the Discovery Channel’s
hit show, MythBusters, Adam Savage,
left, and Jamie Hyneman, center, came
to Centennial Hall on The University
of Arizona campus in October to share
personal experiences and answer
questions from fans. As a local science
advocate and Mathematics Associate
Professor, Bruce Bayly, right, served as
moderator for the event, An Evening
with the MythBusters. The show came
courtesy of NewSpace Entertainment
and the Biosphere 2 B2 Institute. Bayly
said, “It turns out Adam and Jamie are the same offscreen as on: unpretentious, good-natured, and full of
energy and enthusiasm for intellectual challenges. The themes of curiosity and inquiry constantly recurred,
as did the phrase ‘If you do the math . . .’”©
Photo by Kip Perkins
Errata and Clarifications
The following error crept into Mathematics,
Spring/Summer 2008, Volume VII, Issue 1.
In the article, New Solutions to an Old Puzzle,
page 13, the equation should have read:
a4 + b4 + c 4 + d4 = (a+b+c+d)4
View from
the Chair
page 3
Math Department
gets a new chair:
Roy F. Graesser
Endowed Chair
in Mathematics
page 4
New plans for
an old dilemma
page 5
Two examples of
undergraduate
research
assistantships
page 8
Exploring the
geometry of
homogeneous
spaces
page 9
Department of Mathematics
PO Box 210089
Tucson, AZ 85721-0089
NONPROFIT ORG
US POSTAGE PAID
TUCSON ARIZONA
PERMIT NO. 190
Physics Factory
bus makes a run
for the border
page 10
ConcepTests:
Understanding the
path to the right
answer
page 15
Middle school
mathematics
teachers study
hard too
f
n
k
m
x
g
o
Fall/Winter 2008
Volume VIII, Issue 2
2 Mathematics Spring/Summer 2008
A View from the Chair
The Lohse Connection
Nicholas Ercolani
Professor of Mathematics
By Christa King
Administrative Associate
I am very pleased to be announcing
two significant milestones for our
Department. The first of these is the
endowment of a Chair in Mathematics
commemorating Dr. Roy F. Graesser,
one of our former Department Heads and a lifelong advocate
of the central importance of mathematics education.
It will be the first endowed faculty appointment in our
Department’s history. This milestone comes to us through
the leadership and generosity of Linda Lohse, who manages
the trust that was originally established by Dr. Graesser.
The second milestone is the recent announcement by
the University and the Arizona Board of Regents that a
significant portion of our Department will, in the near
future, be moving into a new building, the Earth & Natural
Resources (ENRII) Building currently slated for completion
in 2011 (See related article on page 4). This will be a
transitional occupancy until funds can be raised for the
construction of a building that will accommodate all of the
activities, across campus, of mathematical sciences. In the
meantime, we look forward to the opportunity of forging new
fundamental scientific collaborations between Mathematics
and the climate & environmental science units that will
occupy ENRII with us.
You will also find in this newsletter many updates on the
activities of our faculty, new and old, staff and students
as they push forward at the frontiers of the research,
educational and outreach missions in the mathematical
sciences.
As many of you already know, I have stepped down from my
position as Head of Mathematics. In looking back over the
past seven years, I feel that I am truly privileged to have
been able to represent such an outstanding Department. The
many achievements this Department has made, reported on
in past and current issues of this newsletter, have been the
result of the collegial collaborations of many talented faculty,
staff and students. The Department is currently under the
strong stewardship of Professor Tom Kennedy, pending the
outcome of an internal search for a new Head. I look forward
with great anticipation to the achievements that will be
reported in future issues and which will, no doubt, outstrip
those of the past.©
The Lohse family has a long
history of generosity to the
Department of Mathematics.
In 1999, Ashby Lohse gave a
significant amount of money
to Dr. Stevenson to begin the
Center for Recruitment and
Retention of Mathematics
Teachers. After his death, the
Lohse Family continued to
support this initiative and that
support has enabled the Center
to grow over the past eight
years.
Ashby Lohse was Graesser’s
attorney and when Graesser
died, he named Ashby Lohse as
executor of his estate. Ashby
Lohse created the Graesser
Foundation, which began
supporting Math Department
outreach.©
3
Math Department gets a New Chair
Roy F. Graesser
Endowed Chair in Mathematics
In looking back at the notes from Roy Graesser’s
years as Department Head, some of the following
items stand out:
By Christa King
Administrative Associate
1932: Associate Professors make an annual salary of
$2,790.
Linda Lohse, Secretary of
the Graesser Foundation
and long-time supporter
of departmental outreach
programs, recently
announced donations
toward a Roy F. Graesser
Endowed Chair in
Mathematics. The gift will
honor long-time faculty
member and former
Department Head Roy F.
Graesser.
Prominent Tucson attorney
Ashby Lohse, began supporting
the Mathematics Department
outreach efforts in 1986,
when he gave Professors Fred
Stevenson and Dan Madden
funds to begin a middle school
math camp for outstanding
mathematics students in the
state.
A second math camp was
begun in 1988 when students
from Whiteriver Junior High
in Whiteriver, Ariz., applied en
masse to the camp. Ashby Lohse
increased his financial support
to accommodate two camps. He
continued this support, not only
financially but with his presence
each time the students gave
their final presentations.
Mathematics Spring/Summer 2008
Courtesy of Special
Collections, The University
of Arizona Library,
Graesser Biographical File
Graesser, who received
his bachelor’s and
master’s degrees and Ph.D. from the University
of Illinois, joined the Mathematics Department at
The University of Arizona in 1926 as an Assistant
Professor. He was promoted to Associate Professor in
1932 and to Professor and Head in 1938. He served
as Department Head for 21 years until 1959.
In 1938, when Graesser became Department
Head, there were 17 undergrad math majors and 4
master’s students (the first PhD was not awarded
until 1962). That same year, the Math Department,
which was a part of the College of Liberal Arts,
moved into the renovated Pima Hall from Old Main.
In addition to his duties as Head of the Department,
Graesser published frequently in the journals School
of Science and Mathematics and Mathematics
Magazine, along with frequent articles in Mathematics
Teacher. Graesser also wrote a radio program,
The Invention of Zero, that aired on KVOA.
The majority of Dr. Graesser’s service to the
Department occurred during the years of World War
II and the Korean War. In 1941, annual student
registrations were 1,557 students; and in 1943,
annual registrations were 636. In 1947 they had
increased to 3,461 students and five new faculty
members were added—four of whom were women!
1941: Faculty members complain to Graesser that in
order to have more time for scholarship, they would
like to have their teaching loads reduced from 24
units per year.
1943: An adding machine is requested for the use of
the Statistics students—estimated cost, $100.
1944: Graesser refinishes a used cabinet for the
math models at a cost of $5. That same year, he was
able to purchase a U.S. military slide rule for the
Department—an estimated value of $20—for only 98
cents at war surplus.
1947: “We need more OFFICE SPACE.”
1953: For the first time, all full-time faculty
members have PhDs.
In 1959, Graesser stepped down as Department
Head to return to teaching, which he did on a
part-time basis until his retirement in 1964. Roy F.
Graesser died in 1972 in Tucson. Richard A. Harvill,
University of Arizona President at the time, said
Graesser was “highly exacting in his requirements
and extremely well-liked and admired by students.
He was a very able administrator” and “one of the
great teachers of his generation.”
The Mathematics Department is privileged to be the
recipient of the Roy F. Graesser Endowed Chair in
Mathematics.©
Editor/Writer:
Karen D. Schaffner, Department of Mathematics
Cover photo by Joceline Lega
The Department of Mathematics Newsletter
Fall/Winter 2008, Vol VIII, Issue 2, is published
twice yearly by The University of Arizona,
Department of Mathematics, PO Box 210089,
Tucson, AZ 85721-0089.
All contents ©2008 Arizona Board of Regents. All rights reserved.
The University of Arizona is an EEO/AA - M/W/D/V Employer.
4
such, V is a complete metric space, T maps V
into V, and T is there a contraction with
52
1. It is easily2008
verified5
contraction constant
Mathematics
Spring/Summer
51
that the unique fixed point is x f x 0 u 1 u 1 .
New Plans for an Old Dilemma
Twoexamples
Examples
of Undergraduate
Two
of undergraduate
research
assistantships
Research Assistantships
by
Greenlee,
Emeritus
Professor
By Marty
Marty
Greenlee,
Emeritus
Professor
By Nicholas Ercolani
Mathematics Professor
This past summer the State Legislature passed a $1
billion Stimulus Plan for Economic and Educational
Development (SPEED) to support capital projects
across Arizona’s universities. This plan allows for
construction of the Earth and Natural Resources II
(ENRII) building, which has been at the top of The
University of Arizona’s Capital Improvement Plan
(CIP) for a number of years.
This plan also advances a long-standing University
commitment—dating back to the 1980s—to provide
quality space for Mathematical Sciences, that
further develops the central role it plays across the
University. A significant part of the allocation coming
to the UA has been earmarked for the ENRII project.
The ENRII building will be located on the 65,000
square foot area just west of the Sixth Street Garage
and east of the existing Environment and Natural
Resources Building (ENRB) near the corner of Park
Avenue and Sixth Street on the UA Campus.
At this time, personnel and activities of the
mathematical sciences are spread over five buildings
across campus. The move to ENRII will allow for
the majority of the administrative, personnel,
and interaction spaces (currently occupying two
buildings) to be consolidated within two floors.
Importantly, it will also allow for the demolition
of the structurally obsolete Math Tower and
construction of a new Mathematics Building. This
new building will bring together, all personnel and
activities in the Mathematical Sciences. President
Shelton has recently added this project to the
University’s Capital Improvement Plan. This new
Mathematics building will be the natural home
for the School of Mathematical Sciences, recently
proposed as part of the Provost’s Transformation
process (http://provost.arizona.edu/node/123).
The major goal of the ENRII project is to promote
collaborative, interdisciplinary research focusing
on earth science and environmental programs.
In its initial phase, the new building will house
the Institute for the Study of Planet Earth, the
School of Natural Resources, and the Department
of Geography and Regional Development, along
with a portion of the Mathematics Department.
Since mathematical modeling and simulation of
The Department of Mathematics at The
University of Arizona offers Undergraduate
Research Assistant (URA) appointments to
mathematics majors on a competitive basis.
These are one semester projects proposed by
the department faculty. Selected students
receive a stipend, but not academic credit, and
are obliged to submit midterm and final reports
in PDF format. The following is a description of
two URA projects under my supervision.
The proposed Earth and Natural Resources building
(ENRII) will be located next to the Sixth Street Garage.
The building will house Mathematics administrative
and personnel offices, and interaction space. The
65,000 sq. ft. building will be shared with The Institute
for the Study of Planet Earth, the School of Natural
Resources, and the Department of Geography and
Regional Development.
climate, population growth and their combined
impacts on earth environments is a rapidly growing
field of mathematical research and education
(see: http://www.msri.org/calendar/workshops/
WorkshopInfo/462/show_workshop), this location
will allow for further development in this field here at
The University of Arizona.
Our Department is grateful to many in the
University’s administration who have recognized this
great potential. In particular, acknowledgment for
this should go to College of Science Dean Joaquin
Ruiz, Graduate College Dean Andrew Comrie, former
Provost Gene Sander, Provost Meredith Hay and
President Robert Shelton.
We are in the earliest design stages for the new
space; however, we can already anticipate the need
for resources to secure furnishings appropriate
for a state-of-the-art research and educational
environment in the mathematical sciences. The
“wish list” in another part of this newsletter
describes opportunities for our friends and alumni to
contribute to our future home. Upcoming issues of
this newsletter will keep our community informed as
designs and plans evolve.©
In the fall semester of 2007, Amanda Schaeffer
completed her report entitled "Iterative Methods
for Eigenvalues of Symmetric Matrices as Fixed
Point Theorems." Amanda was then a junior,
and her primary mathematical interest is
abstract algebra. Her starting point is to recast
the classical power method for calculation of the
dominant eigenvalue of a real symmetric matrix
as a contraction map iteration.
To be specific, let A be a real symmetric matrix
with eigenvalues |5 1 | |5 2 | u |5 3 | u u |5 n | u 0,
and corresponding orthonormal eigenvectors
u 1 , u 2 , . . . , u n . 5 1 is called the dominant eigenvalue,
and u 1 , the dominant eigenvector. The power
method consists of iterating the matrix A on an
initial vector x 0 , renormalizing at each iteration.
Provided that the initial guess, x 0 , is not so bad
as to be orthogonal to u 1 , the iteration aligns
with u 1 .
To define the contraction map, T, write x h n as
n
x % ) i x i , and let T 5A1 so that
i1
Tx ) 1 u 1 % ) i
i1
5i
51
u 1 . There is one more step to
defining T, namely specifying its domain. Let
x 0 h n be the initial guess, and let
V £x h n : x u 1 x 0 u 1 ¤, i.e., the
component in the direction of the dominant
eigenvalue is fixed to that of x 0 . V is a
translation of a subspace of h n and is closed. As
such, V is a complete metric space, T maps V
into V, and T is there a contraction with
contraction constant 55 21 1. It is easily verified
that the unique fixed point is x f x 0 u 1 u 1 .
This theory is a bit cleaner than textbook
presentations of the power rule, but is silly as an
algorithm, due to the explicit use of the
dominant eigenvalue 5 1 . This is alleviated by
considering the Schwartz quotients (terminolgy
of Collatz in the 1940s) defined by
A m1 x 0 A m x 0 " m A m x 0 A m x 0 where A m x 0 is of course calculated recursively
with renormalization. Provided that x 0 u 1 p 0,
2m
" m 5 1 O 55 21
. Amanda exhibited two
different ways to incorporate Schwartz quotients
into the algorithm, obtaining the same rate of
convergence as with the power rule. The
approach extends immediately to the shifted
inverse power rule. She also proved that
Schwartz quotients for definite matrices are
monotone.
This fall semester, Johnson Truong is working on
a project entitled, "Convergence Rates of
Quadrature Rules for Non-Smooth Data." The
usual error formulas for quadrature rules, based
on polynomial approximation of the integrand,
require at least two continuous
derivatives–though for practical purposes, two
continuous derivatives can be replaced by a
Lipschitzian first derivative if one invokes the
Lebesgue theory. This is the case for the
midpoint and trapezoidal rules, to which this
discussion will be confined–though the following
extends to higher order formulas.
Johnson, who is a double major in mathematics
and physics, is working on error estimates for
less smooth data. The idea is to view quadrature
rules as Riemann-Stieltjes (RS) integral
approximations to ordinary integrals. Thus one
thinks of approximating the integrator, rather
than the integrand. The relevant integrators are
step functions whose first (distribution)
derivatives are linear combinations of Dirac
delta functions. Due to Johnson’s physics
background, he’s familiar with delta functions so
a crash course in RS theory isn’t needed. The
basic tool in this project is integration by parts
as permissible–grubby but effective.
To fix ideas, consider the trapezoidal rule for a
function f which is continuous for a t x t b. Think
continued on page 13
of f as defined on the line via any continuous
extension, and define the integrator )x as
6
FACULTY NEWS
New Faculty
Youri Bae is an Adjunct
Instructor in the Department.
She has worked as a Research
Associate at the Arizona Center
for Mathematical Science
and as a Visiting Assistant
Professor at the University
of Toledo. Her research
focuses on a combination of
mathematical models and
numerical techniques with nanosciences and optical
sciences. She received her Ph.D. from the University
of California at Los Angeles in 2005; her master’s
degree from Seoul National University, Korea, in
2000; and her bachelor’s degree from Ewha Womans
University, Korea, in 1997.
Crystal Kalinec-Bartels,
Adjunct Instructor, is from
Houston, where she graduated
from the University of Houston
with a bachelor’s degree
in mathematics and was a
Teaching Assistant in linear
algebra. She has taught middle
school mathematics and, in
addition to her part-time work
with our Department, is currently employed with the
Office of Early Academic Outreach and the Tucson
Gear Up Project. She is the lead math specialist for
the Math through Mariachi Program and collects
data on a Math through Mariachi elective course
with approximately 20 students at Cholla High
School. She is interested in topics of social justice in
mathematics education and hopes to continue as a
researcher/professor at a US or German university.
Her hobbies include sewing quilts and dresses, going
to concerts, and remodeling her home in Vail.
Vita Borovyk, Teaching
Postdoc, was born in Kharkov,
Ukraine. She received her
specialist degree from Kharkov
National University in 2001 and
her Ph.D. from the University
of Missouri-Columbia in 2008.
Her undergraduate thesis was
in dynamical systems and her
current research area is analysis, spectral theory,
and operator theory. She loves to read and plays
volleyball in her free time.
Javier Diaz-Vargas is a
Professor at the Universidad
Autonoma de Yucatan, in
Merida, Mexico. Dr. DiazVargas finished his Ph.D. at
The University of Arizona in
1996 under the supervision
of Dr. Dinesh Thakur. His field
of interest is number theory.
He is here at The University of
Arizona for his sabbatical year.
Alyssa Keri is working with
the Arizona Teacher Initiative
as part of the high school
teacher component. Keri is
co-teaching the Statistics
ATI course for middle school
teachers with John Palmer.
She is on leave from Catalina
Foothills High School and
plans to go back next year to
continue teaching high school.
Sangjib Kim, Visiting
Assistant Professor, was born
in Seoul, South Korea. He
received his Ph.D. from Yale
University in 2005 under the
supervision of Roger Howe.
After graduation, he spent
three years at the National
University of Singapore and at
Cornell University. His research
focuses on the representation theory of the classical
groups and related geometric and combinatorial
problems.
Alexander O. Korotkevich,
Visiting Assistant Professor,
was born in Bryansk, USSR,
in 1977. He received his
bachelor of science and master
of science degrees from the
Moscow Institute of Physics
and Technology, Departments
of Physical and Quantum
FACULTY NEWS
Electronics and Problems of Theoretical Physics.
He defended his Ph.D. thesis during the summer
of 2003 at the L.D. Landau Institute for Theoretical
Physics under the supervision of Professor Vladimir
E. Zakharov. During 2006 and 2007, he worked
as a postdoctoral research associate here in the
Department of Mathematics. His scientific interests
include numerical simulation, wave turbulence,
nonlinear waves in different media, and optics of
metamaterials. He enjoys hiking, mountaineering,
and other sports, and the German language.
Derek Moulton, the Hanno
Rund Postdoc, was born and
grew up in Colorado and
received a bachelor’s degree
from the University of Denver
in 2003. He obtained his
Ph.D. from the University
of Delaware in May 2008,
supervised by John Pelesko.
His research involves the
development and analysis of mathematical models
for problems arising in the physical sciences. His
thesis work focused on experimental, analytical, and
numerical analysis of electrostatic fields interacting
with elastic membranes. Recently he has been
working on fluid dynamics problems with ferrofluids,
as well as modeling phase separation in biological
membranes. In his spare time Derek enjoys reading,
hiking, and skiing, and is always on the lookout for a
good soccer game.
Pan Peng, Assistant Professor,
received his Ph.D. in 2005 from
the University of California,
Los Angeles. After graduation,
he worked as a postdoctoral
fellow at Harvard University
from 2005 to 2008. His area of
research is geometry, topology
and mathematical physics. He
is currently focusing on the
duality between Chern-Simons gauge theory and
topological string theory. In addition to his work, Pan
enjoys photography, music and history.
Robert Sims, Assistant Professor, received his Ph.D.
from the University of Alabama at Birmingham,
which is near where he grew up. After graduation,
he spent a year as a postdoc at the University of
7
California at Irvine, two and
a half years as an NSF fellow
at Princeton, three years as
a postdoc at the University
of California at Davis, and
a year as a postdoc at the
University of Vienna. His
research interests lie in the
area of analysis, with specific
emphasis on problems related
to mathematical physics. Bob enjoys traveling,
eating well, and playing poker.
Pham Huu Tiep, Professor,
comes to The University of
Arizona from the University of
Florida. After winning a Silver
Medal at the 21st International
Mathematical Olympiad in
London, he studied at the
Moscow State University in
Moscow, Russia, where he
obtained both his Ph.D. and
Doctor of Science degrees. He was an Alexander von
Humboldt Fellow at the Institute for Experimental
Mathematics (Essen, Germany), and a Zassenhaus
Assistant Professor at the Ohio State University.
In 1998 he joined the faculty of the University of
Florida and became a full professor in 2004. He was
a member of the Mathematical Research Sciences
Institute (MSRI), Berkeley, during the Spring 2008
Semester. His research interests include finite
groups, representation theory, algebraic groups and
Lie algebras, and integral lattices and linear codes.
Andrea Young, VIGRE
Postdoc, grew up in Pittsburgh,
Pa., and received her bachelor
of science degree from Penn
State University. She then
attended the University of
Texas at Austin and earned her
Ph.D. under the supervision
of Karen Uhlenbeck. She
is interested in geometric
analysis, and in particular, in geometric partial
differential equations such as Ricci flow, cross
curvature flow, and the Yang-Mills heat flow.
She is also an actress and a singer and, for the past
six years, has devoted any spare time to being an
improvisational comedian.©
8
Exploring the Geometry
of Homogeneous Spaces
By Philip Foth, Associate Professor
My research involves different flavors of geometry:
algebraic, differential, symplectic, and others. Some
of the main objects that I am dealing with are
the so-called homogeneous spaces. These are the
quotients of Lie groups by closed subgroups. Among
possible examples of such spaces are many familiar
ones such as Lie groups themselves, spheres,
projective spaces, grassmannians, hyperbolic
spaces, coadjoint orbits, and many others.
Additionally, one can consider classification
problems, or moduli problems related to structures
of different kinds on these homogeneous spaces.
The most interesting are those that naturally appear
in different areas of geometry as well as in Lie
theory, representation theory, harmonic analysis,
mathematical physics and other disciplines. One of
the most exciting aspects is to see many significant
benefits in discovering new features arising from
such cross-border interactions.
Let me describe in simplified terms three of the
directions I am currently pursuing. The first has
to do with using Poisson and symplectic geometry
to study integrable systems on homogeneous
spaces and moduli spaces as well as the possible
relationship of these with toric geometry and
representation theory. If G is a Lie group and
O1,. . . , On are its coadjoint orbits, then the quotient
spaces like O1×• ×On //G naturally appear as moduli
spaces of different kinds in physics and geometry.
To give one simple example, if G=U(2), then these
can be thought of as moduli spaces of polygons with
fixed side-lengths in 3-dimensional Eucledian space.
There are many interesting and natural integrable
systems living in these spaces and they are closely
related with representation theory as well, for
example, to the question of finding irreducible
submodules in the tensor products of irreducible
representations of G. Typically, one can treat these
moduli spaces as algebraic varieties, consider their
coordinate rings being generated by certain natural
monomials and study the geometric meaning behind
the combinatorial relations in these rings.
Another direction that I am working in is various
convexity results related to the geometry of the
aforementioned spaces. A classical result by Schur
says that the diagonal entries of a (hermitian or
real) symmetric matrix always lie in a convex
polytope whose vertices are defined by possible
permutations of its eigenvalues. In the early 1970s
Kostant proved a general Lie theoretic result of this
form. Subsequently, in the early 1980s Atiyah and,
independently, Guillemin and Sternberg showed
that all these convexity results follow from a general
theorem in symplectic geometry. Moreover, the
convex polyhedra carry important information about
the geometry of the spaces in question.
Schubert calculus and the projective coordinate
rings of the flag varieties, as well as for his iterative
character formula. A good solid understanding of
these degenerations can bring new perspective
into representation theory and connections with
symplectic and Poisson geometry, as well as mirror
symmetry.©
Physics Factory Makes a Run
for the Border
By Bruce Bayly, Associate Professor
Finally, I will mention an ongoing work with
Sangjib Kim on toric degenerations of BottSamelson varieties, which are important objects
in the representation theory closely related to
flag varieties. They were originally defined as
desingularizations of Schubert varieties, and later
used by Demazure to derive important facts about
9
As word of our program has spread through the
teaching community, The Physics Factory has been
in demand from further afield. In 2007 we took
programs into Prescott, Yuma, and the Phoenix
area, and in early 2008 to San Manuel, Sierra Vista,
Nogales and Douglas. Besides working with schools
and after-school programs, we have developed ties
with the Boys and Girls Clubs organization in Tucson
and other cities. We also began collaborating with
the Biosphere 2, now operated by The University
of Arizona College of Science, in its public outreach
programs. The biggest single enterprise of 2008,
however, was our first international tour.
As with our national tour in 2006, the main
destination of the 2008 tour was the summer
meeting of the American Association of Physics
Teachers (AAPT), this year held from July 19 to 23
in Edmonton, Alberta, Canada. This made it possible
for us to leave Tucson on July 5, travel north through
Nevada, Utah, Idaho, Montana, and Alberta for two
weeks before the conference, then west through
British Columbia and south through Washington,
Oregon, and California for the two weeks afterward.
It was enough time to stop and present plenty of
science events along the way.
In the case when we deal with homogeneous spaces,
these also have an intriguing relationship with
representations of the Lie groups in the spaces of
sections of holomorphic line bundles. Quite naturally,
some generalizations are also valid in the case of
non-compact Lie groups. In particular, together
with Michael Otto we gave a symplectic proof of
a very general van den Ban’s convexity theorem
for semisimple symmetric spaces. There is also a
classical XIX Century problem about finding possible
eigenvalues of the sum of two matrices A + B, if the
spectra of A and B are known.
More recently, I solved a generalization of this
classical problem to admissible elements for
real non-compact Lie algebras. The result can
be expressed in terms of convex polytopes and
polyhedral cones defined by the given spectra
together with certain Lie algebraic data. To give
a simple example, an inequality like this in a
Minkowski time-like space would say that the
third side of a triangle is always bigger than the
sum of the other two. I find these questions quite
fascinating also because they bridge many areas
such as combinatorics, invariant theory, symplectic
and algebraic geometry, representation theory and
others.
Among its many stops along the way on its tour to the
north, The Physics Factory stopped at The Science
Factory, a hands-on science museum in Eugene,
Ore. From left are Christina Pease, Chris DiScenza,
Stephanie Tammen, and Devin Bayly.
What’s a Physics Outreach Program doing in a
mathematics newsletter?
The University of Arizona Department of
Mathematics has a distinguished history of
collaboration with physicists: in our friendly
neighborhood Physics Department next door,
elsewhere on campus, and at other institutions.
My research centers on fluid mechanics (an area of
mathematics with links to physics), and has taken
my collaboration to the K12 science education arena,
working with several local teachers to develop a
mobile demonstration laboratory called The Physics
Factory. It is now well established in the greater
Tucson area.
Specifically we presented our events in Chandler,
Ariz.; Las Vegas; Provo, Salt Lake City, and Logan,
Utah; Bozeman and Great Falls, Mont.; Edmonton,
Alberta; Portland and Talent, Ore.; and Grass Valley,
Santa Barbara, Los Angeles, and Santa Ana, Calif. In
addition, the AAPT conference Demonstration Show
featured The Physics Factory in a specially-created
performance piece entitled “The Physics Fairy,” which
received a standing ovation from several hundred
physics teachers. We reached more than 2000
children during the five week tour.
Besides myself, participants included Devin Bayly, a
junior at Basis Tucson High School; McCabe Bedell,
an engineering freshman at UA; Mike Fenwick, a
physics senior at UA; Stephanie Tammen, a recent
nutrition graduate from UA; Christina Pease, a UA
physics graduate and Pima Community College
physics instructor; Chris DiScenza, a UA math
graduate, now working at The American Physical
Society; Erik Herman, a UA graduate in science
teaching, now at Wildcat School; and Kip Perkins, a
science teacher at Gateways School, Tucson.©
10
ConcepTests
Understanding the path to the right answer
by Professor David Lomen, distinguished
professor of mathematics
Although the concepts of mathematics remain the
same, the pedagogy continually changes. Here,
Dr. David Lomen, whose current work focuses on
mathematics education, explains what he is working
on and how it functions in his classroom.
Reporter: So what have you been so busy doing
the past few years?
Lomen: I am enjoying the challenge of trying to
understand how students learn mathematics.
Reporter: Please explain what you mean.
Lomen: Well, an excellent way to check students
understanding of mathematics is to have them
explain the reasons they use to obtain their answer
to a problem. This easily takes place one-on-one
during office hours, but in the classroom this can
happen to just a very few students because of time
constraints. However, imagine a classroom period
where every student has the opportunity to explain
his or her reasoning several times. The probability
for any student to be bored or disinterested in such
a setting would be lowered considerably. The odds of
a student hearing an explanation that makes sense
to them would greatly increase. This is the situation
when an instructor uses classroom voting.
Reporter: How can you have students vote in a
mathematics class?
Lomen: By using ConcepTests in the following
manner:
1)Using a transparency on an overhead projector,
I project a ConcepTest (usually a multiple choice
question) on the front screen.
2)Students are then given a short time (usually less
than a minute) to think about the question.
3)Students then vote, giving their choice (or
choices) for the answer (answers) they think is
(are) correct.
4)Providing all of the students do not vote for
the correct answer, students are then given a
short period of time to discuss the question with
adjacent students and then are asked to vote
again.
5)I then call on various students to explain the
reasoning they used to obtain their answer. If
necessary, I summarize.
Reporter: Why do you have students explain their
reasoning? Why don’t you just tell them what is
correct?
Lomen: Having students discover and hear a
variety of reasoning methods used in answering
the question is the reason this method is effective.
For one thing, the vocabulary of most students is
usually quite different from that of an instructor. For
another, students often use correct reasoning that
instructors would not have considered. The more
ways to approach a problem to which a student is
exposed, the more likely one way will be presented
that sounds logical to them.
Reporter: You have mentioned ConcepTests a
couple of times. What exactly are ConcepTests?
Lomen: ConcepTests are questions designed to
promote the discussion and learning of a topic. The
questions are usually conceptual, often multiple
choice or true/false, with some free response as
well. I use them as an aid in promoting student
discussion and learning of mathematical concepts
rather than as a means of determining a student’s
grade. It is really for formative assessment—giving
both the instructor and students a means of
assessing how well the students are understanding
a concept or procedure. (Note that some questions
have more than one correct answer, and some
answers need qualification.)
Because of the variety of forms these questions
take, instructors can use them in a manner that fits
comfortably with their teaching style. Three possible
ways are:
1.As an introduction to a topic. This works
especially well if the topic is closely related to
a previous lesson, or is something that most
students have some familiarity.
2.After presentation of a specific topic. Here a
ConcepTest may be used to see if the students
have grasped the concept, or if the topic needs
more discussion or examples.
11
3.As a review of material that has been thoroughly
discussed.
Reporter: How did you become involved with
ConcepTests?
A fourth way of using ConcepTests is to have
students work on them outside of class, either
individually or in groups. Then in class, students
can check their results by first voting, and then
discussing the strategies used in obtaining their
answers with the instructor and each other.
Lomen: A number of years ago a colleague at
Arizona, Deb Hughes Hallett, became aware that one
of Mazur’s students, Dr. Scott Pilzer, was teaching
calculus and writing ConcepTests for his class. She
thought it was a good idea to have a supplement to
our calculus book (D. Hughes Hallet, A. M. Gleeson,
W. G. McCallum et al.) that contained an expansive
collection of such questions, some for each section.
I ended up working with several other authors and
a few teaching postdocs at Arizona to edit Scott’s
questions, and create many, many new ones. This
was done in time for the 3rd Edition of the book.
How ConcepTests Work
Here is a typical question
I would ask using ConcepTests:
1. Consider the rational function
y = (x2 – 4)/(x2 + 2). Adding 2 to the
numerator of this rational function
changes which of the following (more
than one may apply)?
(a)
(b)
(c)
(d)
(e)
The x-intercept
The y-intercept
The horizontal asymptote
The vertical asymptote
None of these
2. After having students explain the
reasons for their answers, I pose the
same question for y = (x2 – 4)/(x2 – 2).
3.Finally I would ask the same question
with y = (3x2 – 4)/(x2 – 2). The same
choices apply for questions 2 and 3.©
Reporter: How did ConcepTests get started?
Lomen: A physics professor at Harvard, Eric
Mazur, noted that his physics students would do
well on exam questions that were similar to those
in the homework, but not when he gave dissimilar
questions which used the same concept. To remedy
this situation, he developed a number of multiple
choice questions (which he labeled ConcepTests)
that stressed concepts. He used them as I described
earlier, and discovered that student scores on
a national physics exam increased dramatically
when taught using this method rather than with a
traditional lecture.
Reporter: Are these ConcepTests the only ones in
existence for mathematics?
Lomen: No, besides the ConcepTest supplement
to our calculus book, there are also questions for
calculus that were developed by Maria Terrell at
Cornell under a National Science Foundation grant
and by Mark Schlatter for vector calculus. Other
NSF grants for such questions have been made to
mathematicians at Carroll College for linear algebra
and differential equations and at Oklahoma for
statistics.
Two years ago, Mariamma Varghese, Erin
McNicholas, Rick Cangelosi (UA instructors at the
time) and I were all teaching College Algebra. We
met once each week to discuss some ConcepTests
I had written and individually, and as a group,
developed some new ones. This was the start of
a collection of ConcepTests for algebra. Now I am
working with Maria Robinson, Erin McNicholas,
and Sacha (Swenson) Forgorson developing more
ConcepTests for an algebra book that Bill McCallum
and several others are writing. We have incorporated
many questions that were developed by Barbara
Armenta at Pima Community College.
Reporter: How do you know that ConcepTests are
effective?
Lomen: We have not conducted any controlled
experiments here, but a paper giving results at
Cornell University using their “Good Questions with
classroom voting” (what they call our teaching
method using ConcepTests) provides evidence
12
ConcepTests – continued from page 11
that this teaching method is more effective than
traditional teaching methods if it is used to motivate
students to participate in small group discussions
about key conceptual issues before a vote is taken.
(see Miller, Santana-Vega & Terrill, 2006)
Other results are by Pilzer (2001), who reported
that his class using this method did far better
on concept questions on a final exam than his
students who were taught with a traditional lecture
method. Pratton & Hales (1998) demonstrated
that techniques which require the students to
actively engage in the material during class produce
substantial improvements in student comprehension
and retention of concepts when compared to
presentation techniques that allow the majority of
students to remain as passive observers who are
simply taking notes.
The Field-tested Learning Assessment Guide
produced by the National Institute for Science
Education (2005) reports how classroom voting
not only results in better class performance, but
higher attendance, lower attrition, and also reduced
performance differences among different population
groups: men, women, and students of varying
ethnic backgrounds. A study including 6,000 physics
students also demonstrated the effectiveness of this
method of teaching (Hake, 1998).
Reporter: Does educational research support your
idea that classroom voting using ConcepTests works?
Lomen: Phrases such as “active learning,”
“discovery learning,” “inquiry-based teaching,” and
“peer instruction” may be used to describe how I use
ConcepTests. It is almost common sense to realize
that students learn best when they are doing the
mathematics for themselves, rather than passively
following the instructor’s work. The more deeply
students are involved in the lesson, the more they
will understand, and the more they will remember.
Chickering and Gamson (1991) list the following
Faculty Inventory for good teaching practices:
1.
2.
3.
4.
5.
Encourage Student–Faculty Contact
Encourage Cooperation Among Students
Promote Active Learning
Give Prompt Feedback
Emphasize Time on Task
6. Communicate High Expectations
7. Respect Diverse Talents and Ways of Learning
The very nature of classroom voting as described
above clearly results in achieving items 1-5 above.
It also happens that the inherent competitive nature
in many students gives the class an unstated, but
real goal to do as well as possible, even though their
answers are not recorded for a grade.
Reporter: Are any disciplines other than physics
and mathematics using ConcepTests?
Lomen: I Googled ConcepTests on Sept. 24 and
obtained 14,700 hits, including ones in astronomy,
biology, chemistry, computer science, engineering,
geology, medicine, pharmacology, and psychology.
Reporter: This has obviously taken up a lot of your
time, but this has nothing to do with the NSF grant
you have held for the past three years.
Lomen: That is correct. On this grant I have been
working with a former graduate student here, Dan
Magee, and two other mathematicians to produce
kits containing manipulatives to help students
visualize vector calculus and analytic geometry
concepts in three dimensions. To go along with
the kits, we are also developing an accompanying
workbook full of projects and laboratory exercises.
After developing a couple of prototypes, by early
next year we will be producing kits for students
to use. (NSF #0442365 Full Development of
Visualization Tools for 3D).©
It Can be Solved:
The Putnam Problem
On the first Saturday of December, twenty
University of Arizona undergraduate math
majors will spend the day matching wits
with 12 math problems so difficult, so
abstruse, sometimes it’s hard to imagine
why they do it. Competing against other
undergraduate mathematics majors from
around the United States and Canada, and
having spent the semester preparing for
the event, these students will wrestle with
the problems of the universe until their six
hours are up.
It’s worth an investment of a Saturday,
however. The top prize in the William
Lowell Putnam Competition is a fellowship
for graduate study at Harvard University.
Begun by Elizabeth Lowell Putnam in
1938 in honor of her husband, who valued
academic intercollegiate competition,
the Putnam asks participants to think
creatively about math while approaching
problems in a new way. It is recommended
that participants be familiar with
differential equations.
Although they have yet to place first in the
competition, last year’s UA team placed
34th out of 516 colleges and universities,
the highest ranking for the UA since 2002.
“Field-tested Learning Assessment Guide” http://www.flaguide.org
Hake, Richard R. (1998) “Interactive-engagement versus Traditional
Methods: A Six-thousand Student Survey of Mechanics Test Data for
Introductory Physics Courses” Am. J. Phys. v.66 p. 64-74
Hughes Hallet, D., A. M. Gleeson, W. G. McCallum et al.(1993) Calculus
(3rd Edition) New York, John Wiley & Sons, Inc.
Miller, Robyn L., Everilis Santana-Vega, & Maria S. Terrell(2006). “Can
Good Questions and Peer Instruction Improve Calculus Instruction?”
PRIMUS v. 16, p. 12-21.
Pilzer, Scott (2001) “Peer Instruction in Physics and Mathematics”
PRIMUS v. 11, p. 185-192
Pratton, J. & L.W. Hales (1986) “The Effects of Active Student
Participation on Student Learning” J. Educational Research v. 79, p.
210-215
Schlatter, Mark (2002) “Writing ConcepTests for a Multivariable
Calculus Course” PRIMUS v.15, p.305-314.
So here’s the question: Are you Putnam
worthy? This is Problem A4 from the 2007
William Lowell Putnam competition.
A repunit is a positive integer whose
digits in base 10 are all ones, such as
1, 11, 111, 1111, and so on. Find all
polynomials f with real coefficients such
that if n is a repunit, then so is f(n).
The answer will be found in the
newsletter’s Spring ’09 edition.
Assistant Professor David Savitt helped
prepare this story and provided the
problem.
13
0, x t a,
)x h
2
, a x b,
h, x u b,
where h b " a. Then
b
; fx d)x a
h ¡fa fb ¢
2
b
is the trapezoidal approximation to ; fx dx. Let
a
*x x " a, so that ) and * vanish at a (a helpful
normalization) and d*x dx. If f is continously
differentiable on ¡a, b¢ integration by parts gives
b
;
a
References:
Chickering, A. W. & Z.F. Gamson(1991) “Seven Principles for Good
Practices in Undergraduate Education’ in New Directions for Teaching
and Learning #47, San Francisco, Jossey-Bass Inc.
delta functions. Due to Johnson’s physics
background, he’s
familiar with delta functions so
a crash course in RS theory isn’t needed. The
basic tool in this project is integration by parts
as permissible–grubby
effective.
Two
Examples of Assistantships but
– continued
from page 5
To fix ideas, consider the trapezoidal rule for a
function f which is continuous for a t x t b. Think
of f as defined on the line via any continuous
extension, and define the integrator )x as
b
b
a
a
fx dx " h ¡fa fb ¢ ; fx d*x "; fx d)x 2
b
; ¡)x " *x ¢f U x dx
a
and again f need not be quite this smooth. This
formula is easily checked without any RS theory,
and gives the error estimate Mh 2 , where M is a
bound for |f U x |. If one can integrate by parts
once more, the standard error estimate for the
trapezoidal rule results. If h b " a is small,
successive integrations of )x " *x approach
zero so that the smoother f is, the better the
approximation, but with derivative terms in the
quadrature rule. For the composite trapezoidal
rule, just add the above over adjacent
subintervals. This is messy, but exact, and uses
nothing about polynomial approximation.
For the composite midpoint and trapezoidal
rules with mesh width h, this produces an order
h estimate with one bounded derivative, and the
usual Oh 2 rate of convergence with two
bounded derivatives. The project includes
extending the above to the composite midpoint
rule for fx x p over 0, 1 with 0 p 1, getting
convergence rates Oh p1 , so that the rates
interpolate, and similarily for the improper
integrals where "1 p 0. The analysis gets
tricky, and I am unaware of such previous
results. For smooth functions, continued
integration by parts is available, and should
result in an elementary derivation of Romberg
integration for acceleration of convergence of
14
Going the Distance: Mathematics
Education Professor takes
Online Learning to the Next Level
By Linda Simonson, Associate Professor
of Mathematics and Suzanne Weinberg,
Education Specialist, CEMELA
What exactly is distance learning?
For most, the term conjures up the dreary image of
a professor lecturing on a video screen to an inert
audience of faceless students—a stopgap measure at
best and not a desirable education tool.
But that would be underestimating its potential.
“Most people have no idea how powerful distance
learning can be to address a variety of academic/
access issues,” said Linda Simonsen, Associate
Professor in the Department of Mathematics.
Today Simonsen is working hard to change the old
perceptions by showing how useful and practical
online learning can be. She spent her early career
teaching mathematics classes the conventional
way—on the campus of Montana State University in
face-to-face classes. In 2000, she began addressing
the acute shortage of mathematics teachers in the
K-12 system.
“Montana is a rural state. We wanted to offer
certification and professional development courses,
but commuting hundreds of miles to the nearest
“...teachers discuss ideas
with the class online, instantly
test the ideas in their own
classroom, then enlighten
the course discussion with
results...the ideal way to meld
theory and practice,”
campus was not an option for teachers working
in tiny towns. But they all had computer access,
so why not offer mathematics courses online?”
Simonsen said.
She plunged in, developing some of the first distance
education courses designed for rural mathematics
teachers. Simonsen set up her courses based on a
growing body of literature about how asynchronous
online discourse—communication mediated by
technology and not dependent on instructors and
students being in the same location at the same
time—can enhance learning. In fact, her current
research focuses on how to evaluate effective online
mathematical discourse.
A recent grant from the Arizona Board of Regents
is allowing Simonsen to share her expertise in
distance learning at the UA. Funding provides for
the development of additional online mathematics
courses for teachers. The development and teaching
of these courses will not be done by Simonsen,
however. The grant allowed Simonsen to conduct a
workshop to train UA faculty to modify their existing
courses for online delivery.
One participant in the workshop was surprised by
the depth of information she got. Lecturer Virginia
Horak, developer of the Geometry for Teachers
course, said, “I expected the workshop to give
me valuable techniques and strategies to use in
an online course. The wonderful surprise was that
it provided me with the opportunity to rethink
how valuable learning experiences from a faceto-face course could be redesigned for an online
environment.”
Simonsen’s vision for online learning does not stop
with mathematics courses for teachers. “I am all
about finding creative solutions to chronic problems,
whether they are reduced budgets, lack of space
or academic access. I hope people will consider
distance learning as a viable component of the
solutions that we are seeking to make UA relevant
and vital in the 21st century,” she said.©
15
There’s also a considerable difference in the way
material is presented. Previously, classrooms were
teacher-centered—the teacher would stand at the
blackboard and work problems while explaining
the theory. Today’s classroom is student-centered,
Adams said, with more student interaction,
engagement, and exploration.
“One of the best parts about it,” Simonsen said, “is
that teachers discuss ideas with the class online,
instantly test the ideas in their own classroom,
and then enlighten the course discussion with
their results. It’s the ideal way to meld theory and
practice.”
When Simonsen arrived at The University of
Arizona in 2007, she found the same problems
she had encountered in Montana—a mathematics
teacher shortage that is especially severe in rural
communities. In the spring of 2008, she taught her
online mathematics course to teachers throughout
southern Arizona.
Enter Arizona Teacher Initiative, the inaugural
program of the Institute for Mathematics and
Education. Under the watchful eye of Professor
Daniel J. Madden, the ATI offers a solution to this
sticky math problem: a Master’s Degree in middle
school mathematics and leadership. The threeyear degree program, designed for the working
middle school math teacher, focuses in-depth on
mathematics, math pedagogy, and professional
development and leadership.
Donna Rishor, left, and Kathy Temple confer on a
problem in the evening class, Math 505 D Data
Analysis and Probability for K-8 Teachers. The pair is
part of the first cohort in the Arizona Teacher Initiative,
a program at The University of Arizona, designed to
extend middle school math teachers’ knowledge of
advanced mathematics and prepare them to take a
leadership role in their schools.
Middle School Mathematics
Teachers Study Hard Too
By Karen Schaffner, Admin. Assistant
If today’s parents are having difficulty helping their
children with middle school math, they’re not alone.
Some of their children’s teachers are having the
same problem.
“A vast majority of middle school mathematics
teachers are elementary school-certified; and many
of them get coerced into teaching math,” said Sue
Adams, Co-Director of the Center for Recruitment
and Retention of Mathematics Teachers at The
University of Arizona. “Unless you are a somewhat
recent graduate you may be ill-prepared for today’s
math.”
According to Adams, even as recently as the midnineties, middle school students (those in the sixth,
seventh, and eighth grades) were taught primarily
arithmetic, fractions, decimals, percents, and word
problems. Today’s students see problems in algebra,
geometry, probability, and data analysis. “For some
teachers, that’s a big leap,” Adams said.
“The object is to have a solid Master’s Degree
that covers the mathematics of middle school and
educational leadership to really improve the quality
of instruction in middle school,” Madden said. “It is
not a typical Master’s Degree.”
To be accepted into the program, potential students
had to undergo a rigorous interview process
that included teaching a math lesson. “We were
excited by the lessons that were student-centered,
engaging, and used manipulatives. We felt those
teachers were most likely to be open to new ideas
and ready to move forward,” Adams said.
Cassie Gribble is typical of the teacher the ATI
is targeting. With 20 years’ experience teaching
first, second and third grades at Pueblo Gardens
Elementary School in Tucson, Gribble took on middle
school mathematics three years ago.
“With that change, I needed to improve my
mathematics for the kids,” she said.
Gribble said the mathematics and pedagogy she is
learning in the program is now being implemented in
her classroom.
“I’m asking better questions, pushing (students)
further. We’re going beyond what they did before,”
she said.©
16
Middle School Mathematics Teachers – continued from page 15
The first cohort of ten students, including Gribble,
began school in August of 2007. They are now
almost halfway to their degree. As an incentive
to encourage teachers to enroll, ATI offered full
scholarships to The University of Arizona along with
a small stipend. At the end of three years, graduates
will have a Master’s Degree in Middle School
Mathematics Leadership. Program funding comes
from a $4.8 million, five-year grant, awarded by
the National Science Foundation Math and Science
Partnership. After that, the program has to find its
own funding.
Because the participants already lead challenging
lives—they work fulltime and often have families
and other responsibilities—ATI classes accommodate
their schedules; most are taught at night. It is also
hoped that in the future classes will be offered
online. “We want to create a curriculum that anyone
can take,” Madden said.
For the first two years, classes are divided into
four- and three-unit classes. Two instructors teach
the four-unit classes, which cover mathematics and
pedagogy. The three-unit classes focus on methods
of research and mathematical investigation. The
third year is dedicated to research in the students’
middle school mathematics classroom, although this
could change as the first cohort has not yet reached
its final year.
The element that runs through all three years,
however, is leadership.
“The idea is that when the student comes out of this
degree, they are able to take a leadership role in
their school,” Associate Professor Rebecca McGraw
said. She’s a Co-Principal Investigator for the ATI
grant. That role can take any number of forms. “It
could be an after-school study group, or a districtwide leader in middle school mathematics. It varies
according to the person.”
Because the goals are so far-reaching, the work can
be a challenge.
“I think for some people it’s been a huge pedagogical
shift; for others it’s more of a confirmation,” Gribble
said. “It does require work on our part but we
don’t have to do a lot that doesn’t have to do with
learning.”
Math Majors Add Experience
Outside the Classroom
Presentations,
Symposia, Conferences
Still, Madden feels program participants are smart,
inquisitive people.
By William Yslas Vélez, Associate Head,
Undergraduate Program and University
Distinguished Professor of Mathematics
For
the composite
midpointmiddle
and trapezoidal
“In Arizona,
people teaching
school math are
rules
with
mesh
width
h,
this
produces
order
trained to teach kindergarten through
the an
eighth
hgrade.
estimate
with
one
bounded
derivative,
and
the
They are good teachers. They weren’t afraid
2
convergence
withoftwo
usual
Oh rate
of a challenge
andofthey
weren’t afraid
math,”
bounded
derivatives.
The
project
includes
Madden said. It was a very short jump
to teaching
extending
themath
above
to the composite
midpoint
middle school
full-time.
“A lot of people
looked
p over 0, 1 with 0 p 1, getting
rule
for
fx x
at that and said, ‘These people are the problem,’ and
p1
convergence
ratesWe
Oh
so thatthe
thesolution.”
rates
throw them away.
think ,they’re
©
interpolate, and similarily for the improper
integrals where "1 p 0. The analysis gets
tricky,
and I am unaware of such previous
Two Examples of Assistantships – continued from page 13
results. For smooth functions, continued
integration by parts is available, and should
result in an elementary derivation of Romberg
integration for acceleration of convergence of
the trapezoidal rule. Though the integrations by
parts are messy, patterns emerge quickly.
Both of these projects use techniques that I
have not seen used on these problems in the
mathematical literature. To that extent, they are
new, at least pedagogically. In olden days when
I was young, they could have been publishable
as original research. When I first made
Amanda’s project available, I was hopeful that it
would lead to new understanding of the
behavior of the Rayleigh quotient algorithm.
That was too far to go in a one semester
project. Her development, however, applies
equally to integral operators with a positive
kernel and a dominant positive eigenfunction,
and to bounded self adjoint operators in Hilbert
spaces. The catch is that the iterations may be
difficult to carry out.
My motivation for Johnson’s project is a desire
to see what happens if one applies quadrature
rules to compute integrals from graphical data.
One can fit curves first, but based on a summer
job as a data reductionist while in college, such
data can be quite bumpy–with corners and
cusps. The idea here is to get an accurate notion
of what happens without the intermediate step
of curve fitting.©
The undergraduate mathematics major at The University of
Arizona is involved in a wide variety of activities, as evidenced
by the two different student experiences that follow:
“What does one do with a mathematics degree?” is a question
that we have often heard. For most of society, mathematics is
equated with teaching. Through the hard work of the Center
for the Recruitment and Retention of Mathematics Teachers,
our graduating classes have a strong contingent of wellprepared mathematics teachers. The mathematics department
is involved in a wide variety of creative—and nationally
funded—activities to improve the teaching of mathematics.
The career path of mathematics majors is now much more
complex than it was just 20 years. Summer internship
opportunities now attract our finest mathematics majors
and these students spend exciting summers applying their
analytical skills to help companies develop their ideas and
promote their businesses. Patrick Valenzuela’s write-up attests
to the excitement that comes from participating in these
activities. A large percentage of the mathematics majors who
had summer internships worked in an applied setting, using
their mathematical skills to investigate phenomena in the
biological sciences or on a NASA project.
Of course, mathematics is a fascinating subject in itself and
many of the mathematics majors spend their summers in
mathematics research projects. Kyle Marshal describes a
wonderful experience that he had in Budapest (every year
several of our students spend a semester there). Kyle decided
to spend a year of intense mathematical study, but that was
not enough mathematics for him. After Budapest, he was
accepted into the summer program at Mt. Holyoke College in
Massachusetts, where he participated in a research project in
mathematics.
Travel and the intense study of mathematics and its
applications are now part of the landscape of undergraduate
mathematics. The Mathematics Department is proud of the
accomplishments of its undergraduates and continues to look
for more resources to support the undergraduate program. If
your firm hires summer interns, please contact me at velez@
math.arizona.edu and I will encourage the undergraduate
mathematics majors to apply to your programs. We hope that
you will join us in our efforts to produce a mathematically
literate society.©
17
David Lomen, University Distinguished
Professor, was on a six-member
committee of the College Board that
organized a meeting of 51 university
mathematics department heads in
Chicago on Oct. 4 and 5. The purpose
was to discuss the current and future
Advanced Placement Calculus program
and how it aligns with college calculus
courses.
Alain Goriely, Professor, will be a
plenary speaker and course lecturer in
November at BIOMAT-2008 in Brazil.
He will be a plenary speaker in January
at Dynamics Days-2009 in San Diego;
Biomechanics of Growth in March in
Bristol, England; and SIAM-Dynamical
Systems in May in Snowbird, Utah, where
he will also give a mini-symposium talk.
Goriely will be a workshop participant
at Computational Morphodynamics in
December at the Biosphere 2, a course
lecturer and school organizer at New
Trends in Biomechanics in July in Les
Houches, France; and will be giving
the Physics Colloquium at Princeton
University in January.
William G. McCallum, University
Distinguished Professor and Director
of the Institute for Mathematics and
Education, is one of the organizers of
the Mathematical Sciences Research
Institute, Dec. 11 and 12, at MSRI on the
campus of the University of California at
Berkeley. He will give two presentations
on Jan. 5 and 6 at the Joint Mathematics
meetings in Washington D.C. He will also
be one of the keynote speakers at the
Chicago Symposium Series on Excellence
in Teaching Mathematics and Science:
Research and Practice, which will take
place March 6 at Loyola University.
Douglas Ulmer, Professor, will be giving
a graduate course at the 2009 Institute
for Advanced Study/Park City Math
Institute in June.©
18
Google Success for this Undergrad
By Patrick Valenzuela
DEPARTMENT OF MATHEMATICS
I’m not quite sure how it happened, but I spent
the last two summers working for today’s coolest
employer (see photo on the next page).
Laptops for instructors (each) $1650
It all started after attending a January 2007 Google
recruiting event in the Math building. I went to
the talk without much hope of working for Google
and left feeling the same way. A few weeks later,
I received a call from a recruiter. I, a sophomore
at the time, was told that it was doubtful I could
get a software engineering internship due to my
lack of experience. Fortunately, a new position was
in the works. Long story short, I became the first
Information Technology Field Technician intern in the
Phoenix office, assisting employees around the globe
with technical matters. It was the most fun and
rewarding summer I could have imagined.
This past summer, I returned to Google Phoenix as
a software engineering intern. My host was Daniel
Norwood, a University of Arizona alum (Math and
Computer Science) that I had worked with in the
previous year. I was put on a team with two other
interns (both Arizona State University students,
but I survived). We created an application from
scratch (with the help of Google’s enormous load of
infrastructure) and came out with a working system.
My role was creating the application’s frontend using
WebWork, a Java-based web application framework,
and Google’s recently open-sourced GXP (Google
XML Pages). There was an immense learning curve
involved with such rapid development, but the
environment was just right.
I don’t enjoy sounding like a recruiting
advertisement, but Googlers are a diverse group of
amazing people. It’s quite a humbling experience
to be surrounded by experts in the industry and
programming geniuses. Realizing that you’re the
dumbest person in the room is not a negative. It is
an opportunity to learn and grow. Conversations at
the office range from new ping pong techniques to
Zeno’s Paradoxes. Most important of all, working at
Google is fun. Perhaps the best way to put this into
perspective is a quote from one of my coworkers,
Trevor, after he had just finished a new feature of
our application’s backend: “I just legitimately used
hyperbolic tangent!”©
take weekend trips into Austria, The Czech Republic,
Romania, or other nearby countries. One of my
favorite experiences was a weeklong trip to Croatia
that I took with a group of other students. On
another trip, I visited Bulgaria and Romania. When I
wasn’t traveling, I spent my weekends taking in all
that Budapest had to offer; from visiting parliament
and the castle to going on picnics in City Park,
exploring the caves, or soaking in one of Budapest’s
many mineral baths.
WISH LIST
Smart Boards and accessories
for classrooms (each)
$5000
Conference Room furnishings
Table (each)
Chairs (each)
Projector (each)
$1200
$150
$1800
Commons Room furnishings
Chairs (each)
$100
Water coolers (each)
$200
Scholarships
• Daniel Bartlett Memorial Fund
• Clay Travel Fund
• Graesser Foundation Math Scholarship
• Lusk Scholarship in Mathematical Sciences
• Rick Peet Memorial Scholarship
Google Success for this Undergrad — It has been two
busy summers for Patrick Valenzuela of Tucson, a
senior mathematics major at The University of Arizona.
As an intern at Google in Phoenix, Valenzuela put
his mathematics knowledge to use first as an IT field
technician and then as a software engineer.
• Richard Pierce Memorial Fund
Discovering Math in Budapest
By Kyle Marshall
What brought me to Budapest was the Budapest
Semesters in Mathematics program (BSM), a math
intensive program designed by a handful of famous
Hungarian mathematicians, including Paul Erdős. The
courses are taught in the Hungarian style, with a
strong focus on problem solving. Inside and outside
the classroom, I collaborated with top mathematics
students from schools throughout the country and
developed close friendships that I know will last for
the rest of our lives (see photo on next page).
I also worked closely with Hungarian mathematicians
and took various interesting courses that are not
Mathematics Spring/Summer 2008 19
When I reflect upon my experiences in Budapest,
I don’t think of the intense mathematics courses I
took, the amazing people I’ve met, or the vibrant
Hungarian culture. What I think about is actually
quite strange, and certainly unexpected. I think
of the street that runs by my apartment, Erszebet
Korut, and I always picture myself walking along
the tram route, as I had done hundreds of times
over the course of my stay. This image is more
powerful to me than any of the other memories I
retain because of what Budapest represents to me.
For an entire year, this was my home. It was my
silent sanctuary in times of study and my launching
pad during nights of dancing and sampling exquisite
beers. Out of this apartment I experienced the best
and worst of Budapest, and came to appreciate it for
all its beauty.©
typical for an American undergraduate curriculum.
One of the most famous classes offered at the BSM
program—and one of my favorite—is Conjecture
and Proof, a course designed with the help of Erdős,
which focuses on exploring and solving difficult
problems in mathematics. The quality of the classes
was rivaled only by the quality of the professors
who taught them. One of my favorite professors
was Czaba Szabo, who would, to make a point, hurl
chalk at the blackboard from across the room. His
enthusiasm was contagious, and learning with him
was never dull.
Though the program was a mathematically intensive
one, the mathematics is only one facet of the
Budapest experience. The program also offers
courses in Hungarian Culture, including a popular
class in Hungarian film. Located in the center of
Europe, Hungary is a perfect place from which to
travel. Access to Western Europe is only a cheap
flight away, and trains are a great and easy way to
Discovering Math in Budapest — Kyle Marshall
of Chandler, a University of Arizona senior
mathematics major, sits by the Statue of
Anonymous, a statue in City Park in Budapest,
Hungary, where renowned and quirky mathematician
Paul Erdõs also used to visit regularly. Marshall
spent a year at Technical University Budapest
studying mathematics but his most vivid memory of
Budapest is of Erszebet Korut, the street where he
walked daily in the Eastern European city.

In this Issue An Evening with the MythBusters

Transcription

Similar documents

Drawing Math Stories: Linking Language, Art, and

NCCTM Central Region Conference - NC Mathematics

Mathematical Modeling as a Means to Demonstrate the

Tracy Curran, Cloneen NS - INTO

The perspectograph - For the Learning of Mathematics

Presentation

Math Tier 2

program overview - Art of Problem Solving Foundation

parents! - First In Math

HKUST`s 4-year UG Programs and Admissions