Spring 2019: AS.110.211 Honors Multivariable Calculus.
Lecture: MW 1:30 - 2:45pm, in 316 Hodson Hall, and
Section: F 1:30 - 2:20pm, in 309 Maryland Hall.
Fall 2018: AS.110.302 Differential Equations
Spring 2018: AS.110.421 Dynamical Systems.
Fall 2017: AS.110.106 Calculus I (Biology and the Social Sciences).
Spring 2017: AS.110.302 Differential Equations
There is so much to say about how one views the classroom environment that it cannot be fully understood or explained in any single statement or set of principles. However, there are some motivational ideas that help to guide me in my role as an educator. Here are a few in list order, with a more full explanation offered by clicking through:
The monologue as a dialogue: In any standard classroom environment, it is almost always only the instructor that is the one talking. However, information passed between the instructor and the audience is continual and bi-directional. When one is with friends and tells a story or a joke, the impact of the telling of that story is always constantly measured and altered, as it is told, by visual and aural measurement of the audience's reaction to the tale. Confusing parts are elaborated upon, boring or distracting parts are glossed over, entrancing parts are elongated for effect, and overall, the audience of the story becomes part of the story. The information transfer between the orator and the listener is constant, fully bi-directional, and integral (excuse the pun) to the conversation, even if there is only one person talking during the entire episode. The lecture, or monologue, is always a dialog. It just does not always involve two (or more) oral conversants.
Process laid bare: The most important aspect of a mathematics lecture is not really the content of the lecture. Nor is it the accuracy or even the precision of the oration. All of these are important parts of the narrative. However, the process by which a mathematician approaches a problem or concept, boils it down to its essential logical structure, and abuses that structure to better understand or see through the logic is exposed within a lecture only when there is a live audience to see, reflect and react to the argument. And that process cannot effectively be distilled into a step-by-step procedure. Truly, when we teach mathematics at the university level, we are not teaching technique as much as we are teaching the process of thinking. The techniques come along for the ride. But the techniques are only a byproduct of the learning process and not the focus. Here is an interesting article in defense of the lecture. And another.
The injective contract: One effective way to reach a student in the classroom is to engage her individually within the conversation. This is true in any social situation, but certainly in class. However, in a course with upwards of 60-120 students, it is difficult for the lecturer to engage every student actively in the lecture. However, from the student's perspective, the course is more of a one-to-one transaction of knowledge with the instructor. If the lecturer thinks of each student as an individual participant in a one-to-one (in math, a function that is one-to-one is said to be injective) contract with the lecturer, then the lecturer is forced to see, react to and participate with every student in the lecture, if not actively and simultaneously, then at least passively and in turn. This forces the lecturer to actually see the entire audience, rendering the lecture into a true conversation.
Precision versus Comprehension: To be absolutely precise and logically air-tight in one's arguments is a hallmark of a mathematician. Flaws in argument or mathematical thinking can always be traced back to imprecise arguments or glossed-over details. However, when the goal of a mathematics lecture is general understanding or broad comprehension, spending an inordinate amount of time on details can hinder understanding. We have a tendency in mathematics to revisit the same topics and concepts over and over on deeper and deeper levels as we grow mathematically. Upper-level undergraduate analysis cna be viewed as simply calculus on a formal level. (I have a tendency to call analysis "calculus on steroids".) Hence a small dose of imprecision, offered with appropriate care to not deceive the students, can be beneficial to comprehension.
From my reality to yours: Nothing in mathematics is real! Every concept, topic, idea, or thought exists only in the mind of the thinker. There is no tangible number two, or real line, or even a triangle. We draw representations of them as an aid to discussing them, but they really only exists in our heads. Teaching mathematics, conveying mathematics, is simply the process of moving mathematical ideas from one brain to another. One on one discussions, where two participants can focus on each other's reactions and arguments, can be extremely fruitful. And the lecture, as a form of organized presentation of focused mathematical thought to a large number of recipients simultaneously, is just another type of transaction on a broad scale. The power of the lecture to a lecturer, however, is evident when one can actually see statistically the effectiveness of the transaction. One can always reach a few quickly, and many sometimes. But the ability to adapt an argument in real time to increase comprehension, to allow the participation of the audience, and to celebrate, as a group, the occasional true individual epiphany, makes the communal experience of the mathematical transaction much more enriching than any textbook, recorded lecture, or video could.
Some years back, I wrote out a values statement on teaching. It is here.
Lastly, I also train, monitor, and mentor graduate students, newly hired post-doctoral students, and new faculty as they develop themselves as educators. This is mostly to maintain quality in our educational mission. But it is also a way for graduate students and post-docs to self-reflect on (and for me to critique) their role as educators.