TEACHING STATEMENT
PAUL SIEGEL
Introduction
The purpose of this document is to summarize my experience teaching mathematics. I have taught
24 credit hours of undergraduate mathematics courses as the primary instructor at Pennsylvania State
University and 6 credit hours more this fall at Columbia University. For these courses I was responsible
for preparing and delivering lectures / classroom activities, assigning homework and quizzes, and helping
to write and grade exams. I also have had several additional experiences as a course assistant with lecture
duties; I will discuss these experiences in greater detail below.
Experience as the Primary Instructor
PSU Math 220, Spring 2008. Math 220 is a two credit course on basic linear algebra, and I taught
two sections in the spring 2008 term. It covers linear systems, vector algebra (linear independence, span,
subspaces of Rn ), matrix algebra, determinants, eigenvalues and eigenvectors, and inner products / orthogonality.
PSU Math 231, Summer 2008. Math 231 is a two credit course on differential calculus in several variables. I covered basic vector algebra (dot products and cross products), the geometry of quadratic surfaces,
parametric curves and arclength, differentiation of functions from Rn to Rm , optimization of functions on
R2 , and Lagrange multipliers.
PSU Math 220, Fall 2008. I taught four sections of Math 220 (see above) in the fall 2008 term.
Student Evaluations (out of 7.00):
Section
001
002
004
007
Quality of Course Quality of Instructor Response Rate
4.70
5.70
52.4 %
5.05
6.30
87 %
5.52
6.60
53.2 %
5.39
6.53
76.6 %
PSU Math 230, Fall 2009. Math 230 is a four credt course on multivariable calculus. In addition to the
topics in Math 231 above, it covers spherical and cylindrical coordinate systems, integration of functions of
several variables, and vector analysis (Green’s theorem, Stokes’ theorem, and the divergence theorem). For
this course I periodically used computer aids in lecture to help students visualize geometric constructions in
three dimensions.
Student Evaluations (out of 7.00):
Section
001
Quality of Course Quality of Instructor Response Rate
5.35
5.55
81.6 %
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PSU Math 41, Spring 2010. Math 41 is a three credit course on trigonometry and analytic geometry. I
covered polynomial functions, rational functions, logarithms, exponential functions, and trigonometry, and
applications. At each stage both algebraic manipulations and geometric interpretations were emphasized.
Student Evaluations (out of 7.00):
Section
001
Quality of Course Quality of Instructor Response Rate
4.71
4.59
37 %
PSU Math 18, Fall 2010. Math 18 is a three credit course on elementary linear algebra. I covered
the geometry and algebra of lines in the plane, linear systems, matrix algebra, determinants, the basics of
eigenvalues and eigenvectors, linear programming, and applications. I made frequent use of computer aids in
lecture to illustrate various concepts (such as least squares approximations and basic stochastic processes).
Student Evaluations (out of 7.00):
Section
002
Quality of Course Quality of Instructor Response Rate
5.61
6.11
43.9 %
Columbia Calculus IV, Fall 2012. Calculus IV is a three credit course on integration of functions of
several variables and vector analysis. I covered double and triple integrals, line integrals, Green’s theorem,
Stokes’ theorem, and the divergence theorem. I made considerable use of technological resources during the
course: I prepared slides using Beamer with figures created in Mathematica and I assigned online homework
through WebAssign.
Experience as Course Assistant
UM Math 395 and 396, Fall 2005 and Spring 2006. These courses were four credit honors undergraduate courses taught by Prof. Brian Conrad at the Univesity of Michigan. They covered a range of topics,
including multilinear algebra, basic topology, Riemann integration in arbitrary dimensions, and the basics of
differential geometry (including manifolds, vector bundles, Lie groups, Riemannian metrics, and connections
/ curvature). I served as a course assistant with duties which included grading homework and lecturing once
a week on material supplementary to the main lecture.
PSU Math 527, Fall 2010. This was a first semester graduate topology course taught by my thesis
adviser, Prof. John Roe, at Penn State. The course included the basic theory of metric and topological
spaces, separation axioms, metrizability theorems, differentiation on Banach spaces / the inverse function
theorem, the Tychonoff theorem, topological groups, and the basics of homotopy theory and covering spaces.
My duties included grading homework and lecturing a little more than once every other week, often to go
over graded homework or to deliver the main lecture when Prof. Roe was absent.
PSU REU Program, Summer 2011. Penn State’s summer REU (Research Experience for Undergraduates) program is a six week session during the summer in which students from around the world work on
a variety of research projects. During the summer of 2011 I served as a graduate student assistant for the
program; my primary responsibilities involved meeting with three groups of two students several times each
week in order to provide guidance and offer suggestions. I also gave two supplementary lectures to all of the
students in the program based on topics of interest to me. The three projects that I supervised included a
project involving Poncelet-type theorems via the geometry of elliptic curves, a project aimed at generalizing
the “DNA inequality” which compares the total curvature of a convex curve to the total curvature of an
arbitrary curve inside it, and a project related to 2 and 3 periodic billiards trajectories in non-convex planar
regions.
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PSU MASS Program, Fall 2011. The MASS (Mathematics Advanced Study Semesters) program at
Penn State is a series of intensive semester long undergraduate courses aimed at providing undergraduates
in mathematics with a vigorous mathematical experience not available at many universities. The program
draws talented students from across the country (as well as some international students), many of whom go
on to attend graduate school in mathematics. I participated in the program as an assistant for Prof. Anatole
Katok’s course, “Spaces: From Analysis to Geometry and Back”. My duties included writing detailed lecture
notes for the course, providing a one hour recitation session each week, holding office hours outside of class,
and grading weekly homework. As far as content is concerned, the course served as a basic introduction
to functional analysis: topics included the basics of Banach space theory, Hilbert space theory, convex
geometry, duality, the Stone-Weierstrass theorem, the Arzela-Ascoli theorem, weak / weak* convergence,
the Banach-Alaoglu theorem, and the Krein-Millman theorem.
Statement of Teaching Philosophy
I have benefitted from an unusually wide variety of mathematics teaching experiences as both an undergraduate student at the University of Michigan and as a graduate student at Penn State. I have played
almost every role possible in the American mathematics education system: student, teacher, grader, tutor,
mentor, and assistant. Making these contributions has provided me with a useful and unique perspective
on the learning process, and based on my experiences so far I can summarize most of what I know about
teaching and learning mathematics in two sentences:
• Students learn the most when the teacher responds to their needs.
• Teaching effectively is one of the most useful and rewarding contributions to mathematics (and
society as a whole) that I can make.
My first epiphany about teaching occurred when I was an undergraduate math tutor at the University
of Michigan. I discovered that there are a variety of obstacles to success in math classes and that the
key to helping students overcome them is to be flexible and work within the constraints imposed by their
background. I accumulated a number of useful skills during this period that continue to serve me well,
such as the ability to explain the same concept in a variety of different ways until the student is able to
latch on to one of the explanations. Overall I think I have benefitted from the fact that my qualities as a
communicator of mathematics were forged primarily in one-on-one interactions with students ranging from
atheletes struggling with precalculus to math majors taking linear algebra and probability theory classes.
The lessons I learned as a tutor, grader, and course assistant in my undergraduate years paid off when I
took control of my own classes as a graduate student instructor. Some of my colleagues experience difficulty
communicating with students who did not enjoy thinking about mathematics every day, but this aspect
of teaching came easily to me because I had already learned firsthand which techniques, explanations, and
examples students found most helpful. I have received consistent positive feedback from my students for
my ability to communicate at an appropriate level during class, handle questions well, and help students
outside of class. The key to my success in these areas has been that I approach teaching from the students’
perspective: I organize the material around their strengths and weaknesses, I spend as much time as possible
giving detailed examples, and I diligently warn them about common difficulties and pitfalls.
While my talents as a communicator are an important part of my success in the classroom, I believe my
enthusiasm for teaching mathematics is my greatest asset. Like any other mathematician I have worked
very hard for a very long time to cultivate my understanding of mathematics, and it is extremely rewarding
to share the fruit of that labor with others. My thesis adviser, John Roe, has always emphasized that
a substantial part of my value to the mathematical community lies in my ability to communicate; this is
partially a statement about the papers that I write and lectures that I give about my research, but it is
also a statement about my role in helping to educate future generations of students. I take that role very
seriously, and I hope that it continues to be a part of my career in mathematics.
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