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.