Teaching

Every week I have two recitations sections for my calculus students. The second one always goes better than the first one because by then I have a better idea of what the students are struggling with most and how to best explain the concepts to them. It could certainly be worse. For example, I can't imagine how things go for the poor students who end up with a non-native speaker as a TA. Math graduate students can be awkward enough, but putting in a language barrier can only make situations more uncomfortable. I'm generally doing a good job. There's certainly room for improvement, but seeing that it's only my first semester in front of an entire class, I feel a bit ahead of the curve.

So this week, I was trying to explain a concept in calculus. Basically, if you have the graph of a function and you want to find the area underneath it, you perform an operation called integration. There's an in-depth geometric explanation of what's going on that I can overly simplify here so that either you'll follow and understand what's going on, or you won't and you'll at least get an idea of how difficult the concept is for the students.


We can approximate the area under a curve by drawing in rectangles as seen in the figure. This is not going to give us the exact area because the curvature of the graph makes some rectangles contain more area than is under the graph in that position and some rectangles contain less. But if we were to look at another approximation, this time with twice as many rectangles having half the width, we would have less of an error from our estimation and we would have a better approximation. And the more and more rectangles we use with smaller and smaller widths, the better our approximation is. And what happens is we can take the limit as the number of rectangles goes to infinity and the width of a rectangle goes to zero, and the limit of the approximate areas we get will reach the actual area. In the course, things are much more complicated because there's a nasty equation to deal with and things are much more formal. But at the same time, I can draw pictures to explain things, so that certainly helps.

Anyway, you're probably thinking that this is really complicated. You may have even skipped the second half of that last paragraph. That's because this is very difficult material. And I was struggling so much to try and get the students to understand what's going on here (even though it's the professor's job, he can't explain things sufficiently for the students to understand things). So after explaining some parts of the process three or four times, I muttered to the silent students, "Please ask questions if you don't understand any parts of this. I know it's difficult material, and I'm trying my best to explain things." One of the students with whom I'm a bit friendly said, "No, I think I get it. Anyway, you're doing a much better job than the professor." Figuring she was just being nice, I asked her "Really?" I turned around and saw a sea of heads, all bobbing up and down in agreement.

It's really disheartening that the professor who's supposed to teach them this material is so bad at explaining things that they feel I'm doing a much better job, even when I feel I'm not up to my own standards. It feels like they're being cheated out of a good education, which is a terrible shame. All of these feelings help me know that I definitely want to teach, and hopefully with a few more years of experience, I'll be able to be the great teacher that I wish these students could have.

-Induced Homomorphism