Thoughts on Teaching

This is my second semester teaching precal recitations (what some schools call "labs"). The kids in my course attend a large lecture with their professor three days of the week, and the other two days, they have recitations, which are 50 minutes long and in small classrooms (about 35 students).

It is an interesting experience. Every Thursday I give them a quiz or an exam given to me by the professor, and on Tuesdays, and Thursdays before the quiz, I go over material with them, answering their homework questions and reviewing stuff for the upcoming quiz or exam.

Teaching (the actual teaching part) is really fun, but also weird and frustrating. When I gaze out at my class, many of them look either terminally confused or completely bored, and I can't always tell which. Many of my students totally ace all of the quizzes and exams - they probably don't need me for that. Some of my students struggle around the middle, and hopefully I help them. And some of my students are just consistently confused and their quiz answers are weird and wrong.

One thing that happens is that I don't know how to pitch my class. I know some of my students really aren't getting certain things - do I just hammer on things endlessly, trying to bring everyone along, while boring the others out of their skulls? Or do I sort of blithely move on to the newer, more exciting material, risking leaving some in the dust? There isn't one answer to this question - it is more of a feel thing, and I don't think I have the feel of it yet.

Another thing I find myself doing, and what I really want to talk about, is focusing on mistakes. There are certain persistent mistakes made by my students that sort of drive me nuts. For example, a lot of my students think (at least while taking a quiz) that the square root of (x^2 + 9) is x + 3. Or (the same error) that (x+y)^2 = x^2 + y^2. Or they add or multiply fractions completely wrong.

These are really basic mistakes that they should know better than to make, and it's easy for me to sort of become obsessed with these errors and how I can drive them out. And I'm realizing that focus is completely wrong.

I've had professors who seem more focused on mistakes than on the math they are teaching, and it's made certain classes very negative for me. It starts to feel like the point of the class is to avoid errors, to not fall into certain common traps, and so on, rather than to learn exciting new math. What's the fun of that?

I've also seen professors focus too much (I think) on their fears about students not being prepared for the next course. I don't know if this is universal, but I had a lot of middle school teachers who would say things like, "I can't let you get away with xyz, because your high school teachers will never allow that," and high school teachers who said the same thing about college professors. It never turned out to be true. And I've seen profs at my school wield calculus in the same way - "If you can't do [this particular skill], you will fail at calculus."

It's not that it's false. Your lack of ability to add fractions will increasingly be a handicap as you proceed through the calculus sequence. But I'm not sure threatening people with the upcoming courses is really the way to go. Some level of mistake-making is normal, natural, and not indicative of future failure. People get better all the time, even when they are making mistakes.

Some of them really will go on to fail calculus. Some will fail precal and not even get to calculus quite yet. But making calculus sound dire and horrible to everyone won't necessarily solve that. Calculus is one of the most beautiful inventions of the human mind. Being able to study it is a great luxury. They should look forward to it!

So despite my controlling tendencies I am going to make an effort to keep a positive focus in my teaching and not become obsessed with typical student errors and mistakes. I really don't want to be that kind of teacher.

Socialization

I have experienced an interesting attitude change since starting grad school.

Ed (my ex-boyfriend, with whom I still share an apartment) and I are in grad school together, as most readers of this blog know. Before coming here, we'd had one class together at my undergrad institution, and that was when I learned that we have similar classroom styles (as students), except that his style is more extreme. We both tend to ask more questions (and answer more questions posed by the professor) than other students, and are prone to...well, sort of acting as though we are the only student in the room and can just freely interact with the prof without regard to what s/he is trying to accomplish with the class.

In the math pedagogy class that I took, our professor once gave us a list of problem student types, one of which he called "Mr. Non-Sequitur," giving the example of that guy who always asks you how such-and-such relates to fractals. At the time, this reminded me of Ed, who will often ask tangentially related questions.

Last year, I found Ed's classroom behavior pretty obnoxious, and worried that mine was obnoxious as well. But then this year, I observed that, in talks, many of the professors in our department behave exactly the same way. Whether the speaker is internal or external, they will interrupt with questions, make nitpicky corrections, and ask about strange tangents. And, though there is no way to put this on my public blog without risk, I will say that the professors who act this way are some of the ones I respect the most (independent of their behavior in talks). This is also the talk-watching style of the genius among the grad students of our department.

Maybe this is normal, socially appropriate behavior for my discipline.

I was curious what would happen when Ed took another class with our pedagogy professor, who is a bit strict in his classroom management style. Would he quash Ed's interrupting tendencies? The answer turned out to be a pretty big no. If anything, I think he appreciates being interrupted, nitpicked, and asked weird questions. He told me once that we were his best class in many years because we are so engaged and challenging, and I think he also gave private positive feedback to Ed once.

So I have basically totally revised my opinion of this style of behavior, and now think it must, indeed, be socially appropriate in our field. This led to an interesting conundrum recently, however, when one of my cohort gave a talk.

The talk was very interesting. At one point, though, Ed stopped the speaker to ask, basically, "So what?" He didn't use those words but wanted to know about the motivation for something she was talking about. She didn't have an answer right away, and he said, "I just think I would get more out of this if I knew why we were talking about it."

I felt like it was a bit over the top, given that she is our friend, is a bit early in her career (like we are), and may not have been completely confident in giving the talk. I wouldn't have pushed her in that way myself.

Now, I think Ed just asked the question because it was on his mind. But I wonder...maybe it is our job to socialize each other by asking these kinds of tough questions, even if it makes the speaker uncomfortable. You could argue that we should refrain so that our friends can be more comfortable, or that we should intentionally not refrain so that they can toughen up and not be stymied in (e.g.) a job talk later, when someone in the audience is of this more obnoxious cast.

Fortunately, our department has a good mix of people who like to speak up and people who don't, so I guess it will all just average out. But these are just some thoughts I've been having lately.

Teacher Follies

An amusing/embarrassing thing happened to me today.

Because I write so much, I am very careful to keep paper that I've only used one side of to use as scratch paper. I have a big stack of this on my desk, and when students come to see me, I usually write with/for them on some of this paper, and then if they want, I let them take the paper with them. I figure the graduate math on the other side won't hurt them any (though it hurts me plenty, I can tell you).

Anyway, sometimes when I am working on problems, I write notes to myself in the margins. Often the notes say something like, "I suck at this type of problem :(" or "I will never get this!!!!" I try not to give papers like that to students, but I don't try very hard.

Yesterday, a guy came to my office hours and I sent him away with three sheets of my scratch paper. Today before class, he was showing it to another student, and I overheard this conversation:

Other student: Huh. It says "I am a bad ass" on this side. Did you [to the first student] write this?
First student: No, I think she wrote that. This is her work for some other stuff.

So I wandered over and sure enough, I had written "I am a bad ass" in the margin of my work. We had a good laugh over it.

It's kind of embarrassing, but I'm glad it was positive self-talk for once, and not the negative kind. One of my friends in the program pointed out that it was good that I had at least not written that I was a BAMF, which is also something I might write (either as an abbreviation or written out).

So anyway, there's that.

Measure-Theoretic Probability

This is definitely my hardest semester of grad school so far, for the simple reason that I am taking two core courses (which are the kind here that require the most work) plus a reading course in measure-theoretic probability, which is taking as much time and effort as a core course (and more than either of mine, actually). But I'm increasingly wanting to go in a probability direction with my studies, so I'm thrilled to be doing it, and I'm really enjoying it so far.

The core course I took last year was in measure theory. I've written about it a little before, but today I'm going to (briefly) explain its relationship to probability. Then I'll end the post with my real reason for posting, which is a quote that excited me this morning.

Measure is just (more or less) a generalization of length. Let's say you're on the real line and you want to know how big a set is. If the set is just an interval, like (0,2), it's easy enough to say its length (or measure) is 2. But what if your set is much weirder? Like, what if you want to be able to say what "size" any subset of the reals is, in a sensible way that accords with our notion of length? That is, you'd like the interval from 0 to 2 to still have length 2, but you still want to be able to measure anything you want?

Well, unfortunately that is impossible, but what we can do is come up with a measure that works for pretty much any set that anyone cares about. In fact, coming up with a set that you can't measure requires using the axiom of choice to construct some bizarro thing that just doesn't arise in normal life.

So, let me tie this to probability. Let's say you want to pick a random number between 0 and 1, with all numbers equally likely. I'm just talking about a uniform distribution on the interval [0,1]. It's pretty obvious (I think) that the odds of getting a number between 0 and 1/2 is 50%. This corresponds to the length of the interval that you're talking about.

But what about the odds of getting a rational number? Or the odds of getting a number without a 2 in it anywhere? Or the odds of getting a number whose first three digits (after the decimal point) are repeated 9 times?

These are questions which can be answered with measure theory. In particular, there is only one measure (it's called Lebesgue measure) which both assigns to each interval its length and gives a measure for every Borel set (the ones I'm characterizing as "all the sets anyone cares about").

You might ask, "Who cares about the odds of getting a number whose first three digits are repeated 9 times?" But let's unpack that a little bit.

Let's say we roll a 10-sided die an infinite number of times. If the sides of the die are labeled 0 through 9, then if you wrote out the results, after a decimal point, you'd get a number in the interval [0,1]. For instance, it might look like .98362819501... and so on.

If we assume all combinations of die rolls are equally likely (which is true if the die is fair), then the odds of the die roll meeting whatever criteria we give are exactly the measure of the set in [0,1] that corresponds to the numbers that the roll represents.

For instance, let's look at my question about rational numbers. I should note that we're not going to let anything terminate, so for instance, we wouldn't write 0.4 (because that wouldn't correspond in any obvious way to an infinite series of dice rolls) - we would write 0.399999999..., which is the same number.

So what is a rational number? It's one that repeats forever when you write it out like that. For instance, 1/3 is 0.33333333.... You could also have a rational number like .123123123123... where more than one number repeats.

So, if we randomly choose a number between 0 and 1, the odds of it being rational are identical to the odds that, if we roll a die infinitely many times, eventually we hit some number or group of numbers that then repeat forever (for instance, we start rolling 1,2,3 over and over again for all eternity). Common notions of probability suggest that the probability of that happening is basically nil, which turns out to be right - the measure of the rational numbers is 0. (This is proved in an entirely different way in a measure theory course.)

Or, to use my other example, the odds of having your first three rolled numbers happen again like that 8 more times in a row is the same as the measure of the set I mentioned earlier - numbers with their first 3 digits repeated nine times. (So in other words, you can use this to measure probabilities that happen in a finite number of dice rolls too.)

This leads me to my quote, which is from Section 4 of my textbook, Billingsley:

Complex probability ideas can be made clear by the systematic use of measure theory, and probabilistic ideas of extramathematical origin, such as independence, can illuminate problems of purely mathematical interest. It is to this reciprocal exchange that measure-theoretic probability owes much of its interest.

So there you have it. Nifty, no?