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 to encourage
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 [example], which would then be presented in a powerpoint format during class as in this corresponding [example]. 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 email@example.com, 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]).