Teaching is a rewarding and enriching experience. My experiences have given me ﬁrst hand knowledge of the
many sides of the learning process, from how students respond to lecture, to how to best help them outside
of the classroom. I value teaching and my ideal position would maintain a balance between teaching and
My goal is to help the learning unfold, watching students struggle and through their own hard work ﬁnd
the Eureka moment when the math clicks and they gain a new insight. I was once an undergraduate and I
remember well what it was like to have an inspiring teacher who helped me see the “cool” in math. Years
later I like serving that same role as mentor.
II. Teaching Philosophy
Although it is always a pleasure to get that impassioned and gifted student, we should not forget that
for many students there are substantial barriers to learning. Students are busy with other classes, their
personal lives, and many have had previous experiences leading them to believe that math is boring or worse.
Teaching is an opportunity to help students overcome their inhibitions and develop their analytic abilities,
ﬁnd applications of math to their own interests and inspire a love for pure math as an end in itself. This is
achieved in the following ways:
The syllabus needs to cover the core course material while being ﬂexible enough to meet the particular
needs of the students. Many students take math classes to fulﬁll requirements for other departments, and
so their interests may be quite diverse. On the ﬁrst day of class, I ﬁnd out the backgrounds of the students
and why they are in the course. From there I try to modify the lectures and examples to match the interests
and skills the students need to develop in their coming careers. For example, one of my Calculus III classes
was ﬁlled with concrete-thinking Bio-Engineering students who greatly appreciated the physical examples I
gave them (see Projects). Of course, I keep in mind that I must serve the needs of all the students taking the
While at Mt. Holyoke I developed a new diﬀerential equations course. Most of the students were taking the
course because of a requirement for another department and are interested in mathematics primarily through
applications. I contacted professors outside of the Math department (for example in Biology, Chemistry and
Physics) to provide diﬀerential equations that come up in their own research thereby making the course more
important and relevant to students from those departments. I also included a computer lab portion of the
course so that they have tools to tackle unknown diﬀerential equations when they come across them in the
future. Some of the faculty who suggested lab projects agreed to give “mini lectures” before the labs to explain
how the equations arise in their research..
There is also a subtle art to making lectures engaging. I try to keep the atmosphere fun and relaxed so
the students feel free to ask questions (or point out mistakes). It is also important for lectures to go beyond
what is presented in the textbook so that the students see real value in coming to class. In my lectures, I
share with students my own interpretation of the material, give diﬀerent pictures, more examples, and try to
connect the material to previous topics. I wanted to brieﬂy share with you an experience I had teaching at
Mount Holyoke College of which I am quite proud.
In my diﬀerential equations course we were studying eigenfunction solutions to diﬀerential equations. I
assigned a homework problem that asked for the shape of a string under tension rotating on its axis (not
like a jump rope - at low speeds the string in this example will be totally ﬂat). When you setup and solve
the diﬀerential equation eigenvalue problem you ﬁnd that there are non-zero solutions for a discrete set of
rotational speeds ωi .
Imagine a string stretched between two points with a variable speed motor attached. When you start the
motor the string is stretched ﬂat (the zero solution). As you slowly speed up the string stays ﬂat at ﬁrst, but
then when it hits ω0 it jumps out to the ﬁrst eigenmode (one arc of a sin function). The really interesting
(and I would argue counterintuitive!) part is that as you speed up and pass ω0 the string should snap back
to ﬂat again until it hits ω1 - since your diﬀerential equation tells you that only the zero solution exists in
between the ωi . When you hit ω2 it pops back out and then back in as you speed up further.
It occurred to me that it should be possible to build a physical demonstration of these eigenfunctions -
to convince the students that they really do exist. I went to speak with the lab technician in the physics
department. As I explained the idea he immediately realized that if it worked as I claimed it would be better
demonstration of eigenfunctions then what they were currently using - exactly because the string going to zero
in between critical rotational speeds is counter intuitive. Twenty minutes later he was in my oﬃce to tell me
that the demonstration worked exactly as predicted.
The following class I brought the students down to see it. They were just as excited as I. Some of the
students commented that his was the ﬁrst time they felt math really told them something (unexpected!) about
the real world. News of the demonstration quickly spread to other physics and math faculty where it’s been
used to demonstrate the ”reality” of eigenfunctions in both departments. With the encouragement of some
of the physics faculty I am writing a small paper for the American Physics Society so that the demonstration
becomes more widely known.
Availability and Feedback.
I want there to be as little barrier to learning as possible. As a teacher, I therefore schedule my time to be
available to students outside of lectures in ways most convenient to them. The aim is to simply eliminate easy
excuses for not mastering the material. I also use my oﬃce interactions with students to gain vital feedback
on how well they understand the material and which topics need reinforcement.
In Calculus III at UPenn, I handed out ﬁve index cards to students on the ﬁrst day of class and throughout
the semester would ask them to anonymously return the cards with a question or comment. Also, my Web
page had a comment box that allows students to anonymously post. In both cases, I want to give my students
an avenue to communicate their diﬃculties without intimidation or fear of retribution. All these techniques are
useful ways for me to gauge student understanding so they are prepared for exams and no one ﬁnds unpleasant
Being available is not only important for the struggling students but for the more talented students as well.
In my Calculus III course at Mount Holyoke there was ﬁrst year student who is both talented and motivated
to go beyond the course material. I identiﬁed her at the beginning of the year and proposed that we work on
some extensions to the material covered in the course (for example discussing diﬀerential forms in general and
working towards the generalized Stoke’s theorem). We met weekly to discuss reading assignments and to take
a look at extra problem sets.
Outside projects are an important tool to help students own the material by using research to actively
engage in the learning process. I see my role as teacher to assist students in ﬁnding that applicability whether
it’s connecting the course to the wider mathematics curriculum or ﬁnding examples relevant to the student’s
interests. For my Calculus III Bio-Engineering students, I supplied project ideas after canvassing recommenda-
tions from graduate students and professors in Biology and Bio-Physics. For the non Bio-Engineering students
I created some Math and Physics problems on my own. I presented the ideas as extra credit problems for
interested students. Several of the students took on these projects and I worked with them outside of class.
When I taught ”Ideas in Mathematics” most of the students were from the nursing school at the University
of Pennsylvania. The course is designed as a terminal course in math and covered elementary topics in game
theory, statistics, and computer science. Many of the students had previous negative experience with Math
and the professor and I put eﬀort to make the course fun, while at the same time giving them mathematics
tools that they could apply to their interests. Rather than a ﬁnal exam the students wrote term papers on
topics of their choosing. We received papers on a wide variety of topics, from how to make the electoral college
more “fair”, to a report on the classic book “On Growth and Form” by D’Arcy Wentworth Thompson. At
the end of the class some of the students commented that this was their ﬁrst interesting math course.
Computers are an important part mathematics research and instruction. For the diﬀerential equations
course at Mt. Holyoke I developed a series of computer labs the students complete in parallel with the
regular homework assignments. The main purpose is to give the students tools to analyze diﬀerential equation
encountered outside of the course. The secondary purpose I did not discover until the course was already
underway. At the start of the course I planned to use Maple, but I found that students did not understand
basic concepts of programming and could not think critically about limitations of computer solutions. Instead
I chose to use Python, a general purpose programming language, so that students had to develop their
own routines to numerically ﬁnd integrals and solve diﬀerential equations. Most of the students had never
programmed before and said that this was a rewarding experience, learning for the ﬁrst time how a tool they
use daily works.
Project assignments allow students to push beyond the structure of the course into topics of individual
interest. In doing so, the breadth of the overall coursework is complemented with a necessary depth, giving
the students a well-rounded learning experience.
Involving students in active research projects is an important part of developing young mathematicians.
As an undergraduate I took part in a number of research projects both at my home institution and at other
institutions via the REU program. These experiences gave me conﬁdence that I could go on to advanced study
and a sense of participation in the larger scientiﬁc community.
In collaboration with a researcher at the NIH I am currently working on a project in applied topology
to identify new treatments for cancer. The basic topological ideas (persistent homology for instance) are
very intuitive and the problem would beneﬁt from more systematic exploration of the parameter space by an
My research in theoretical physics entails writing computer programs to explore possible solutions to
Hermitian Yang Mills equations and construct explicit examples. While the physical concepts can be diﬃcult
the problems can be distilled into simple and concrete mathematical problems suitable for an undergraduate
project such as ﬁnding integer lattice points in a region of space.
I also see my work as being a fusion of mathematics and physics and I have interests in applications
of math to real world problems. I would enjoy sponsoring and overseeing cross disciplinary research that
involved collaborating with faculty in other departments particularly Physics, Chemistry, Computer Science,
Outside Help Resources.
Sometimes we forget that teaching is not about teaching; it is about learning. And for learning to be
eﬀective for all students, a variety of options need to be available, including resources outside the classroom.
I see my role as teacher to help students meet their diverse learning needs. Such resources include but are not
limited to tutoring, math workshops, and math question centers. My experience working in outside support
arenas has given me valuable insight into how they may help students who, for whatever reason, are not
learning well in the classroom environment. As the manager of Penn’s math workshop program, I trained
undergraduate students to run active learning problem solving sessions and provided continued supervision
throughout the semester, helping to troubleshoot mathematical and pedagogical problems.
As professional mathematicians we know that math is an active pursuit, that you never understand the
material until you have engaged and wrestled with it. The math workshops provided a venue for the students
to actively engage in the material as well as develop their abilities to work in groups and communicate
mathematical concepts to others. A typical session would divide the students into groups, assigning a diﬀerent
problem to each group to solve. The students explained the concepts to each other making sure that all
the students in the group understood the end result. After solving the problems, a student from each group
explained their solution on the blackboard to the rest of the class.
Perhaps most importantly, at no time are the students being lectured to; instead, the undergraduate
workshop leaders are trained to guide the groups to the correct solutions without telling them directly how to
ﬁnd it. The process of discussing the problem, solving it in their group, and ﬁnally explaining the problem on
the blackboard gives the student multiple chances to engage with the material, increasing their understanding
each time. I believe this kind of program can be an invaluable way to increase student comprehension.