In trying to lead students to a deeper understanding
of both the beauty and utility of mathematics, much of my work
in curriculum development has been focused
on the transition to doing more theoretical mathematics. In addressing the
needs of students wherever they may lie I have also worked on methods
mastery of basic mathematics, both at the college and precollege levels.
Most recently, I've had success with a new lecture structure for my
business calculus class,
which has many students who lack confidence in their mathematical abilities.
Before each class I posted lecture outlines as in this
which would then be presented in a powerpoint format during class
as in this corresponding
The outlines state examples but do not work them out, which I did instead
on the chalkboard or overhead projector.
By giving students the logical
framework for the lecture ahead of time, students are better able to
focus on the critical task of following the examples during class.
Email me at firstname.lastname@example.org, and I will
be happy to share my full set of notes.
This format was motivated by my experience in teaching a similar class
previously, where in one-on-one discussions with students I found that
even those who actively participated during class time would have poorly
structured and sometimes incorrect notes.
I also give inspirational credit to Jeremy
Wolfe, who structured his
introductory cognitive science class at MIT in
a similar way. Having implemented this structure,
I found that students did master both the logical framework
and the basic examples better than in my previous teaching
experience. Students both self-reported a greater
sense of mastery in evaluations
and showed such ability in exams.
I have also encouraged development of basic skills of students outside the college
setting. For three years I served on the executive committee of the
board of directors for the
Mt. Hope Learning Center, a community-based educational
center located in a disadvantaged neighborhood near Brown University.
In my work there, I found that most volunteers felt insecure in their ability
to offer help or programs in mathematics, so with an undergraduate funded by
Brown University's VIGRE grant
we developed some
mathematics enrichment projects for K-12 students.
At the undergraduate level, I have been strongly concerned with the transition
mathematics majors must make from technique-oriented learning to proof-oriented
learning. Based on my four years at the
PROMYS program in mathematics for gifted high-school students, I like to start
with exercises which motivate the theorems to be proven.
I took such an example-first philosphy as
the primary focus
in developing a class on experimentation and proof in
mathematics, based on the
mathematics laboratory class at Mt. Holyoke college.
Elements of this style are also present in classes such as linear
algebra, where I introduce iterated
function systems as a way in which affine-linear transformations can encode
remarkably intricate geometry, which provides strong motivation to analyze and prove
facts about such transformations.
At the graduate level I am currently advising one PhD student, Matthew Miller,
who is working on configuration spaces in three-manifolds, training
in both algebraic and geometric topology.
Both at the University of Oregon and at Brown, I have taught
material related to the study of loop spaces and classifying spaces as a
way to start with students whose knowledge is the level of first-year topology texts,
such as that of
Hatcher, and develop more advanced techniques. At some point
probably far down the road, I hope to write a book which treats these
topics at that level (see here for an [outline]).