roach to teaching is to illustrate the main points through supporting
examples, and to lead students to these points through an ongoing conversation about our evolving course
goals.
Background
My experience sharing mathematics began at the PROMYS program for high school students interested in
Number Theory, run by Prof. Glenn Stevens at Boston University. Through the six summers I spent there as
a student and counselor, I learned how to explore mathematics independently (embodied in the philosophy
think deeply of simple things), as well as how to understand and explain complex concepts using a few
simple examples and by asking good questions. This ongoing mathematical dialogue forms the basis of my
teaching style, and my interactions with students. More than anything else, I try to help students understand
how to ask the right questions, and hopefully how to begin to answer these questions for themselves.
Lectures
In the classroom I try to maintain an informal atmosphere as much as possible, so that students feel they
are in a safe environment and are comfortable asking even stupid questions (which dont exist!). I am
usually proactive about learning students names and have several times asked students in larger classes if
I can take their pictures to use as ash cards.
In each lecture I focus on conveying one or two main ideas, supported by motivating questions and
simple examples. This illustrates the basic computational skills needed for problem solving, and provides
a few carefully chosen examples to focus on for future discussions. Also, having a few examples/questions
that appear in dierent forms throughout the course provides a valuable reference point and helps students
to feel comfortable as the material becomes more technically challenging.
I also try to keep students engaged by regularly turning to them and asking down-to-earth questions
like Ok, but why do we care? and What is this good for?. These refreshing asides help to break the
all-too-frequent monotony of math classes, and give students a sense of why the things they are learning
are important. This approach is particularly valuable when trying to teach students how to solve problems
whose solution is not immediately obvious.
Whenever we do problems together, I always start with the two
questions: What do we know? and What do we want?. By carefully answering these questions in the
given context, students learn to think more independently and can develop strategies for solving previously
unfamiliar problems using their existing knowledge.
Homework and Tests
Because I hope to teach students both the how to compute examples and how to use the ideas to approach new
examples, my tests are usually fairly balanced with doing-type problems and thinking-type problems.
Though dicult at rst, students nd problems where they are asked to think outside the box very
rewarding in the long run, because it helps them to understand more deeply what it is they know, and how
it can be applied to new situations.
To help students prepare for the exams, I have made an extensive series of review sheets which illustrate
all of the ideas needed to do very well on the tests (though the problems may be dierent). At studentss
request I often also hold late-night review sessions which go over how to do these review problems, and
typically last about 3 hours. These together with homework problems (whose solutions are posted on the
1 course website) provide a rich source of examples for students to test their skills and understandings, and
form the basis of the exams. Finally, when returning graded exams I also distribute hand-written solutions
which work through each question in a motivated step-by-step fashion.
Teaching Experience
In the past 7 years I have been fortunate to have the opportunity not only to teach a variety of undergraduate
courses at three major universities, but also to design several of my own courses and improve existing ones.
These experiences have been very valuable for me personally, and have shown this teaching style to be highly
eective at motivating students to engage course material through their natural curiosity to understand what
is going on. Rather than describe each course in detail, below I will describe some of my most rewarding
and unique experiences.
Cryptography Seminar(Rutgers/Duke)
This course was designed as a more mathematical replacement for an earlier discussion-based math for
poets course at Rutgers discussing cryptography and society. I later proposed and taught this as a freshman
seminar at Duke.
The rst half of the course focused on classical cryptography schemes (letter relabeling and rearrange-
ment) and various statistical attacks for breaking them. Students were responsible for evaluating the security
of these schemes, and for cracking them if they were insecure. These ciphers included the shift cipher, cryp-
togram (without spaces), permutation cipher, Vigen`ere cipher, and the one-time pad. The second half of
the course focused on public-key cryptography and developing the necessary prociency with modular arith-
metic needed to implement these. Mathematically, this included a discussion of primes, units, multiplicative
inverses, the theorems of Fermat and Euler, and primitive roots. With this, we covered RSA, Die-Hellman
secret sharing, and ElGamal. The students were responsible for creation of their own public/private keys,
encryption and decryption, digital signatures, and key management issues. We also discuss related security
issues such as: what is a hard problem, why ciphers are based on hard problems, and the man-in-the-
middle attack.
All computations were essentially done by hand (students were allowed to use a simple calculator, but
no computer programs), which gave a tangible appreciation for the computational complexity of various
cryptographic operations and ensured that students really knew what they were doing. It was impossible
to nd an adequate text for the material, so there were many handouts. To help with this, I created a
course web page with homeworks and many lecture summaries as well as various C programs to compute
the relevant statistics and aid in the creation of the exams. While primarily designed to help students, these
will be a valuable resource for the next instructor and will provide a sense of continuity and a solid starting
point for any future changes.
This more rigorous style in a course for non-math majors comes with the possible pitfalls of discouraging
those with weaker backgrounds and obscuring the issues with mindnumbing mathematics. However students
seemed quite interested in this more hands-on approach, and the course has consistently received high
evaluations, with comments like: Jon allowed me to look at math like I have never seen it before, The
teacher was outstanding, and If I were stranded on a desert island, I would want him with me.
Calculus I and Linear Algebra Course Coordinator (Princeton)
Over two semesters, I was responsible for improving and coordinating the Calculus I and Linear Algebra
courses at Princeton University. In addition to revising the syllabus and homeworks, I made a comprehensive
eort to improve the consistency of the course (across all 10 sections) and to provide more review resources
for students than the usual classroom/oce hours model provides. This involved working with the Director
of Undergraduate Studies to setup weekly lunch meetings for instructors to share ideas about the material
for the upcoming week, and arrange for all of the lectures (from one section) to be videotaped and made
available to students for viewing online. I also created more review opportunities for students in the form
of weekly review sessions, a comprehensive set of review problems for the exams with online answers, and
several days of 2-3 hour review sessions before exams both for course review and problem solving. While
not requiring substantially more man-power to implement, these changes greatly improved the ability of the
course to accomodate students varied study habits, which was particularly important given its predominantly
freshman enrollment. These changes were well-received by students, and will hopefully become a lasting
model for the way these courses are taught in future years.
2 Number Theory Seminar (Duke)
At Duke I had my rst opportunity to teach an undergraduate course in Number Theory for 3 semesters (and
again in Spring 06). This course was modeled on my experiences at the PROMYS program, with the idea of
giving students as much of an idea of exploration and independent discovery as possible. The course was a
hands-on introduction to some important concepts and proofs in classical number theory, focusing on primes
and prime factorization in Z and Z[d], quadratic residues and reciprocity, and representing numbers in
the form x
2
+ dy
2
. Students were also responsible for writing a 10-15 page paper describing/proving some
major result in number theory and giving an hour-long presentation about it to their peers. To help with this,
each student was given an initial project outline with references, and had weekly half-hour meetings with
me to discuss their progress and ask questions. Students found this research-type experience very valuable,
particularly because it allowed them to understand something over time that at rst seemed completely
unintelligible, making comments like: Thanks for a great class. The best Ive ever taken! and I honestly
believe that classes like yours have inspired me to continue to get my Masters and I really do appreciate the
enthusiasm you shared with me during my days at Duke.
Independent Study Projects (Duke)
In addition to the