Icosian Reflections

…a tendency to systematize and a keen sense

that we live in a broken world.

[OGPS] What I Learned from Jacob Lurie

This post was written mostly after week 3 at OGPS. The weeks 4, 5, and 6 posts may or may not be forthcoming.


Learning Education

Earlier this fall, I lotteried for  USW35, "Dilemmas of Equity and Excellence in American K-12 Education". But, like 70% of those who tried to get a seat, I was rejected. So, instead, I'm taking a math course in Functional Analysis (Math 114). It satisfies my Analysis requirement for my joint CS/Math concentration. So there's that.

There's also the fact that professor Lurie has taught me more about how to teach at OGPS than I imagine "Equity and Excellence" ever could. (Aside: In no way do I mean this as a slight against Prof. Merseth. I'm sure that her class is fantastic. But the impression I got from shopping week was that it's a very academic treatment of the education problem in our country, and not "Here's how to teach a kid to program a computer for the first time.")

Now, Jacob Lurie is a frighteningly intelligent man. His undergraduate thesis, for example, contains some words that I know.

My critical reading of "On Simply Laced Lie Algebras and their Miniscule Representations" by J. Lurie, 2000

It also won a first place in the American Mathematical Society's Morgan Prize "for superior mathematical research by an undergraduate student". And he earned a full professorship from Harvard at age 31. So there's that.

He's got a fascinating lecture style. At the beginning of class, he picks a point on the ceiling, and proceeds to deliver the entire hour-long lecture at it. (Quoth Nick Watters: "You should probably convince that ceiling to join your study group.") Now that the class has moved out of a classroom and into a proper lecture hall, though, you can almost convince yourself that he's simply looking towards the back row. Or maybe he's gradually approaching the point where he'll actually be able to make eye contact with his students. It's hard to tell.

But that's just superficial. (Though, 6-second silent video clips strongly predict end-of-semester student evaluations...) As a sometimes math teacher, though, I'm fascinated by the content of his lectures.


Functional Analysis

There was an awkward silence in 114 today. Prof. Lurie had just asked us -- the class -- "Okay, we've produced an upper bound, and now we want to show that the sum converges. What should we do next?"

It's not that it was awkward because we were unaccustomed to speaking up; it was probably the third time today that he'd stopped the lecture until a student suggested a way forward. (And today was no exception; since week three, all proofs in class have been of the form "We'd like ___. So we'll begin with ___. And then what should we do?") It was awkward simply because no one had an answer.

So we waited for two minutes.

student: "Place the epsilons in a geometric sequence?"

JL: "Well, we could, but note that N varies with epsilon."

student: "Oh. Never mind, then."

another student: "But wait. If you arrange the n to ensure a geometric sequence in epsilon..."


JL: "...then you could pass to a convergent subsequence, yes. That might work."

And he proceed to show us, in rigorous detail, how it might work, covering two and a half blackboards, until: "And what should we do now?"

And so it went. (And so it's gone for several weeks now.) Lecture concluded on "Well, we haven't gotten as far as we set out to, but you've done the hard part. It might or might not be on Friday's midterm."


Yes, that might work...

student 1: "How are we supposed to do it?"

Mr. McDonough: "Well, didn't you put together the game board models?"

student 1: "Yeah, but we had instructions!"

me: "Forget instructions. You make the rules. Now, think: What is one of the very-important parts of a robot?"

student 2: "Brain!"

me: "Yes. Here, have a brain." (hands over a NXT controller)

student 3: "Motors?"

me: "Sure. How many?"

student 3: "Two, so you can turn one off and turn one on and turn a doughnut..."

me: "Okay, here." (gives two motors to each group)

"Now, take the rest of the pieces..." (points to the four trays of Lego materials sitting on the floor) "so do what you have to do. I want to see robots that move!"

The rest of the lesson consisted of many, many interactions of this form:

me: "Okay, what are you trying to do now?"

student: "Make it go fast!"

me: "How about we get something that works, first, and make it faster later?"

student: "Okay. But I don't know how."

me: "Well, what's an important part of the robot?"

student: "Wheels."

me: "So, get some wheels."

student: (does)

me: "Do they need to be attached to something?"

student: "Yeah, a motor of course."

me: "What could we use to attach them?"

student: (points to a connector piece)

me: "Okay, let's try it. Figure it out; I'll be back in a sec, but I need to help some other group right now..."

(repeat, for 50 minutes)

By some miracle, we ended the class period with two mostly-constructed robots. (I may or may not have spent another thirty minutes afterward examining the students' designs so I can better nudge their design decisions forward next week...) Which, in retrospect, is a fabulous accomplishment. But in the lesson plan, we expected to finish construction AND do a review committee to compare and critique competing designs. Apparently, the day has not yet come when Diane and I can accurately predict the appropriate scope of a 90-minute lesson.


(Some of) what I'm learning by teaching

(A) Things that are immediately obvious to me (by now) are often only apparent to others after repeated false starts, hints, and strokes of luck. Which makes sense. The contents of my head did not arise from a vacuum, and there's no good reason to expect that just because I try to frame things in the ways that have come to make sense to me now, other people will suddenly be able to make those logical leaps with ease. (How long did it take the Chocobots to realize that we needed low-speed gearings that mounted the wheels directly under the motor housings? Two years, as I remember.)

(B) It's infuriating to teach slowly. I didn't have this problem at HSYLC (where I set the pace of lectures and more or less ignored the kids who were falling behind...), but when faced with the FLL "kids do all the work" model, and resolved not to connect any two Lego pieces myself, I was forced to realize how hard it is for my students to put together pieces without a set of instructions for reference. I really don't know how Professor Lurie handles it in 114.

(C) It's not always apparent what work you should do for your students, and what they should do themselves. This, too, is a much larger topic, and I suppose I'll return to it later, as this post has gotten long enough.