Teaching Mathematics to Mathematicians

Up until this year, I haven’t really had much experience with teaching high level math to mathematicians. Last semester, I gave a series of expository seminar talks, and this semester I’ve been lecturing once a week on algebraic geometry codes. So now that I have some experience under my belt, I want to use this space to organize my thoughts on the matter.

The first thing I want to look at is my goal when teaching in this context. My initial idea is that I want to overcome possible barrier to understanding and appreciating the material. The way I think of it is that, given a sufficient amount of time and desire, a mathematician can work her way through a given paper and have a mechanical understanding of what’s going on. She can verify that the proofs are logically correct, knows the mechanics of the definitions, and can compute some things. Said mathematician may however not know what intuition a particular definition is trying to capture, or have a conceptual understanding of what’s going on in the more unwieldy proofs. Further, unless she had independent reason for believing that the maths happening in the paper is beautiful, this mathematician will probably fail to appreciate the aesthetics and elegance of what’s going on. My goal then is to give folks the ability to go beyond this surface level understanding of the material.

More specifically, what does that actually mean? It means that my job is to somehow take some exceedingly abstract idea, and help people develop a working intuition for this idea. The idea is that, with a working intuition, it will be easier to see what proofs and definitions are trying to accomplish, as well as the beauty inherent in whatever I happen to be talking about. So I know what I’m not trying to do at least. I absolutely do not want to just regurgitate an existing paper word for word. That doesn’t actually add anything.

Now, how do I go about accomplishing this lofty goal? Right now, I’m lecturing over algebraic geometry codes with a paper as the primary resource. Now I assume that the organization of the paper itself shouldn’t be tampered with, as there’s a natural progression of ideas and it’s written by experts in the field. Further, I’m about as far away as one can get from a coding theorist or an algebraic geometer. On the other hand, I just said that I don’t want to regurgitate the paper. What I’ve been trying to do is provide a narration and framework for what’s going on.

The big overarching goal is to explain algebraic geometry codes and how decoding them works. Let’s think of that as the overarching plot for a series of books. In order to get to the ending I have in mind, I have several smaller stories to tell, which are my intermediate goals and the theory that I need to develop in order to do some heavy lifting. So we can think each lecture as a “chapter” in a particular book. Now in each chapter I’m trying to make progress on the plot of the “book” that I’m in, while reminding the audience where we’re going and what we’re doing to get there. I’m also concerned with the big structure, but that’s more to give an idea to the audience of how what we’re doing contributes to the overall goal.

What this process does is organize my own lectures, and provide a framework for the audience in order to help them see the forest from the trees. Additionally, as I’m going along and talking about the material, I tend to explain the intuition that I’ve cultivated for the material and how I see it. Then when I give an example, do a computation, or prove something, I can demonstrate how that intuition is informing the process. That’s the idea at least.

My hope is that people come out of my lectures with some understanding of the “plot” to continue that metaphor. They definitely won’t get everything, but maybe they’ll be equipped to pursue the material further if they choose to.

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