My Teaching Philosophy

Students can be much more capable than many instructors/mentors give them credit for, but instructors/mentors must be cognisant of the fact that they might care much more about what they teach and work on than students ever will. That’s not a bad thing.

The two most memorable courses I have ever taken with Beginning Russian and Introductory Differential Geometry, both at Cornell. The former was an extremely immersive, extremely fun, sometimes extremely painful experience that met every day of the week. We were forced to wrestle with, for almost all of us (a few students had previous exposure), a new alphabet, a new grammar, and a new way of thought. I have never learned more in such a short amount of time (though, unfortunately, have forgotten just as much since then). Despite that our instructors clearly knew we were struggling, the method of teaching never changed: they knew we could deal with it, and they knew we would be all the better for it.

My differential geometry class was taught by David Henderson, a strong believer in the Moore Method, in the last year before his retirement If you’re not familiar, you can think of the Moore Method as the analogue of a total immersion approach to teaching a language course, the same as I experienced for learning Russian. We were given a textbook our instructor specifically wrote for the class, and over the course of the semester, were forced to come up with definitions, theorems, proofs, and basic examples using our varied backgrounds, some paper, pencils, scissors, and tape. Homework exercises were more conversations with our instructor than graded problems: if you didn’t get a question right, he would ask for clarification/correction, and so long as everything was done in a timely manner, everyone would get full credit. At the end of the semester, my classmates were divided in their thoughts: though all had enjoyed the intuitive theorem building.many wished we had developed more formalism and done more difficult computations as in Manfredo do Carmo’s Differential Geometry of Curves and Surfaces. I was one of the many, but I could not help but realize that, in retrospect, I was never forced to engage and understand the material in any math course before, regardless of homework or quality of texts and lectures. This statement still holds true today.

When I had finally gotten a chance to fully control a calculus course, I had wantonly gone all in: I knew that doing a full Moore Method style course was unfeasible for a first semester calculus course, but I knew I could incorporate concepts into building somewhat of a hybrid. I remembered what I loved and hated when I first took calculus, I knew that most people’s experiences with math in high school tended to be subpar, I came in with an abundance of knowledge common struggles of students at that level from years of tutoring. Past experience in both education and tourism (I had worked as a tour guide in high school) helped me formulate clear, intuitive explanations and engaging, entertaining lectures by appealing as much as possible to general human experience, meticulously paying attention to vocal inflection and segmenting material to give people who fell out a way back in, and little bits of humor to ease tension and make class a more enjoyable experience for all. homeworks I developed gave students the chance to explore new concepts that we either didn’t have time to go over in class (basics of limits were removed in favor of devoting time to introductory ODEs) or were fundamental concepts introduced in future math courses that many wouldn’t take (e.g. curvature). I wanted my students, many of whom would never take another math course, to be exposed to mathematical phrasings of ideas fundamental to human existence that they might never otherwise see, to make their last math course as enjoyable and interesting as possible.

Overall, this course was a success. There were instances where I knew I could have done better in terms of making and phrasing lectures and materials, but I had received a large amount of positive feedback. It wasn’t universal, which was not surprising given how difficult it is to please everyone, especially as they transitioned from high school style math courses to college style ones. By and large, they were up for the challenge, even just coming out of high school! Left to their own devices, when asked questions in a proper guided manner, they could prove basic statements, they could investigate concepts on their own reserved for higher level courses that they wouldn’t otherwise see unless a lecturer or a textbook forced them to investigate it. I had even gotten a number of students that stayed after class (well when it was dark out) to ask me more things about math that they learned from their homework that would have never been touched on otherwise!

However, those that gave me negative feedback finally clicked the switch on something that I had been long exposed to, even to the point of factoring it into my thought process in building the materials, but had never fully clicked until then: I was more interested in everything I was teaching than any of the students in my class. It’s a basic statement, one that most everyone accepts on some face value, but can be surprisingly hard to process; this applies to courses at every level. I am the type of person that decided to devote five years of their life to earning a PhD. I am the type of person to be interested in my intellectual pursuits to the point of spending large swaths of time devoted to them and nothing but them. Other people like to turn off their brains and not think about certain things, and that’s okay.

I have not been an instructor of record since then (though I am currently a TA for an applied deep learning course for which I have significant input), not out of lack of interest, but because my external funding situation put me on a low priority in terms of need to teach compared to other students in more dire need of funding. Despite this, I was still fortunate enough to assist in mentoring a number of undergraduate and graduate students on different research projects, and I’ve applied what knowledge I gained from my instructorship to doing so. My approach has been to prepare enough material suitable to a number of different learning styles to properly motivate and provide necessary background to completing the tasks at hand. Outside of a few nudges (with no pressure) into areas of possible future interest that I think the students will enjoy, I am otherwise laissez-faire. I know that I have never succeeded in doing things well that I wasn’t interested in, the students that I have never succeeded with lots of pressure on things they don’t care about, and I cannot imagine a future where this will ever be reversed. I am always around to make sure the basic objectives are being accomplished, and let my students know to make as judicious as necessary use of my availability/any office hours, but that’s it.

To say that this method has proven effective is an understatement. All of my students have done impressive work; this has led to important empirical observations that have guided my personal research for my PhD, as well as develop software that led to state of the art numerical techniques used in understanding partisan gerrymandering that have been used in analyses for cases in multiple state Supreme Courts as well as the U.S. Supreme Court. They did this on their own, myself and the other mentors helped out as needed, but many discoveries were driven by the curiosity and ideas of the students. I will soon be supervising multiple student project groups in my deep learning course, and I am confident that they will once again impress me. I will nudge them and help them as they wish, but I am sure they can and will do most everything on their own and develop their own appreciation of the material they investigate, at least so long as I don’t force too much upon them.