Last Tuesday I started teaching my summer Calc II course. We meet for 2 hours every day (M-F), and then the students also have a 45 minute discussion section with our TA. I decided to set up the course a little differently from how I've done Calc II the last few times, and thought I might try to describe some of it here.
The, say, "standard" way this course would go is to start with techniques of integration, do improper integrals, arc length and surface area of revolution, parametric curves, polar curves, iterated integrals, and then series. Last semester, we had one exam on techniques of integration and arc length and surface area, one on parametric and polar curves and iterated integrals, and one on series.
When I was setting up my course for the summer, I looked at how many days to spend on each topic, and how to try to schedule things best to naturally have exams and things. I decided to shoot for having all of the topics besides series as a single exam, around halfway through the course, and then have series be their own exam. We're just about to the first exam (it'll be Friday), and it looks like we can meet that schedule (with in class review time before the exam even!).
I was able to condense the material down in a few ways. First, the arc length and surface area calculations we do with usual $y=f(x)$ curves, we later redo with parametric curves (in the "standard" class). I decided to start with doing them as parametric and polar curves, and then make a few comments about how to convert a usual $y=f(x)$ curve to a parametric curve. [While I'm talking about surface area... I know that when you rotate around whichever axis, you have (sometimes) two choices for how to set up the integral. You could do it dy or dx. Probably I just haven't dug into enough examples, but is there one when the integral is "doable by hand" one way, but not "doable by hand" the other way, even though both can be set up?]
The other way I crunched things together was to leave more of the responsibility to the students to learn the techniques of integration on their own, outside of class, for homework. I made a few comments here or there, but largely it's been up to them. In my mind, the only real techniques of integration are u-substitution and integration by parts anyway, both of which I did talk about in class. The others, in the text, are trig integrals ("apply lots of trig identities"), trig substitution ("memorize a chart of what substitution to make when"), and partial fractions (I've got no beef with partial fractions). It wouldn't matter if I spent all day for a few class periods working integrals, the students would leave feeling like they weren't too bad, and would get stuck on them at home. So I cut out more of the "they aren't that bad" feeling, and mostly threw the students in to get stuck. My hope was that I'd see lots of kids in office hours, which has not (yet) been the case.
My other thoughts, along the lines of techniques of integration, are concerned with how useful they actually are. This general question is something me and every other math teacher with a blog have brought up before, especially recently with Wolfram Alpha doing integrals (and showing steps!). What is it we should actually be teaching these kids? I'm not convinced I've ever worked a trig-substitution integral outside of calculus class, but would love to hear about it if anybody out there had. Ok, sure, working them gives you lots of good practice with algebraic rules, and so I can now manipulate symbols like it's my job. But I've been trying to set up my class to sort of tend away from this. I really want to spend more time on, and get my students to answer more questions about, how to set up integrals to calculate things you might want to calculate, or what a given integral calculates (in a picture). I also try, when we work an integral, to talk them through any sort of consistency checking I would do for the answer (is it positive or negative? should it be? is it too big, or too small?). I like to think that I am giving them practice taking a problem and converting it to something a computer can easily do, and then thinking about if the answer is reasonable.
Part of my other goal in having them learn techniques of integration with somewhat less guidance, was to give them experience learning from the textbook directly. Probably I want to do this because it is how the calc class I took as an undergrad was set up.
I'm not convinced, based on homework grades, that things are going terribly well so far. I've asked them to give me anonymous feedback about any changes they want to have made, and tried to stress that I want them to ask questions and come to office hours. I've not gotten too much back in the way of feedback yet. The one, really, so far was that I should work harder problems in class. I think I get this every semester, so probably it is something I should seriously look into. But most of me thinks: they'll always look easier in class, you have to get stuck on them yourself. Math is not easy. I don't know, perhaps that's my laziness shining through. But at the same time, I can't really start with hard examples in class, because they won't make sense right away, and there's only so much time during class.
At some point I was preparing lectures and getting frustrated by the examples I had, and with trying to find new ones. (Is there a place online where teachers keep their favorite examples of all sorts of problems? Some sort of examples repository or something?) I came up with an idea I'm still mulling over for how to teach my classes. My thought was maybe I'll start each day with a little discussion about whatever new concepts there are for the day, at sort of the theoretical level. And then I'll just grab the first couple (or just one) odd problems from the text, and work through them. And then the rest of class time will be students working through as many more of the problems as we have time for. They can work in groups if they want, but don't have to. This gives them the chance to get stuck on problems, but quickly start asking questions to get over initial hurdles. They'll have their textbooks with them, to dig through examples, and will also have other students (and me) to ask when they're really stuck. Anybody have any thoughts on this setup?