Today was the start of yet another semester at the University of Virginia. The last few days I've been depressed and frustrated and probably a whole host of other negative things, but I think "the routine" starting back up again is helping me out of it. Nice to know that I still love teaching. It also helps that I have friends that treat me far better than I deserve, but that's a story for another day, perhaps.
I had my first class this morning. In fact, I had an 8am class, so I was one of several instructors who taught students their first college class ever. Kinda a fun thought. I hope I didn't screw it up too bad.
I'm teaching Math 132 again, which is Calculus II for math/science folks. I enjoy the course. Which is nice, since this is my fourth time in a row teaching it.
Since I've taught this course several times recently, I'm pretty comfortable with how the material flows, and how I want to make it flow. That, coupled with whatever nonsense I was going through this last weekend, means I hardly prepared for today at all. I had in my mind that I was going to walk in and ask students to tell me about integrals. Just whatever they knew, completely open ended. About 20 minutes before class started I thought maybe it'd be good to review a couple basic derivative rules too - power, e^x, sin, cos. We also talked about some trig identities, since students brought it up.
It's always fun to ask a group of 30-40 students what an integral is. You're bound to get many answers, and I've always been pleased that many of them were ones I had hoped for. It's also interesting to hear ones you weren't expecting. Today we talked through antiderivatives being derivatives "the other way". When asked what antiderivatives were, there was a lovely pause after I fained ignorance about what "the other way" meant ("write it right to left? bottom to top?"). I could feel the class thinking about another way to say this. We got around to it in time. Then we moved on to definite integrals as areas under the curve (but, really, between the curve and the axis, and really the signed version of that). And how they are related to Riemann sums, and how that was different from just a finite sum of rectangles (and did rectangles have to be below the curve? all the time?).
It was a great discussion. I had a wonderful time, and the students seemed to be reasonably content as well. I was shocked to hear what sounded like an entire classroom full of students tell me the derivatives of several functions, basically in unison. At 8:15 in the morning. I hope they can maintain this energy throughout the semester.
I also hope that the course setup can be like this many days, where it's not so much a lecture as a group discussion, with them leading the way. We'll see how that goes. I'm not expecting to try it all of the time. I'm sure that carefully planned examples will be a necessity before too long. I get kinda frustrated thinking about which examples to work. No matter what I do, the problems will always be easier when the students watch me do them, or we work through it together in class. They have to try problems themselves. They have to get stuck. It's the pain about math.
We meet again tomorrow, though, thankfully, not nearly as early. Tomorrow we start techniques of integration, kicking things off with "undoing the chain rule" (u-substitution), and perhaps hinting at "undoing the product rule" (parts), depending on timing. After that, I'm making them learn the other techniques on their own. The other techniques frustrate me. They're not techniques of integration - they are algebraic manipulations to try that come up in lots of integral problems (the problems that show up in calc textbooks anyway). And I would have no idea how to respond if a student asked why they are useful. They're doable by machine (wolframalpha.com, we've been through this before). My hope with making them learn this by themselves, from the textbook (and I encouraged them to find things online, and share what they found with the class), is that the exercise becomes "learn how to learn math by yourself" (well, starting off by yourself, and finding help where you can) instead of "learn these particular tricks". In the long run, that's a more useful thing anyway. In the short run, it might cause some issues, since the midterm will have a hard time testing if they've learned how to learn math by themselves. After they've turned in their first assignment on these techniques, I'll do some review in class, and then give them some more practice problems before the exam.