Teaching Calc 2

Last semester I had the opportunity to teach Calc 2 here at the University of Virginia. And I get to this coming semester as well. While my course evaluations from last semester went well, there's still plenty of things I want to change. I thought I'd post some of my thoughts here, in hopes of maybe organizing them and, more importantly, getting some feedback.

First off, some parameters. I'm teaching out of Stewart's Calculus book, chapters 7 (Techniques of Integration), 8 (Applications - Arc Length and Surface Area), 10 (Parametric and Polar), 15 (Multiple Integrals), and then 11 (Series (Hurray!)). Yes, we skip the chapter on differential equations (in addition to some scattered sections), and chapter 15 might typically be considered Calc 3 material (that is, after all, where it is in the book). But the course I'm teaching is 'coordinated', meaning that there are several sections of it, and we all have common exams. This also means I'm to cover the above chapters in the order I list them in (or, I suppose, I could switch 7 and 8, or 10 and 15). I get to teach 3 times a week, for an hour each time. Finally, I have a TA who will be running an additional 'discussion section' (alternatively called a '4th hour'). Oh, and I expect to have approximately 40 students.

One of my concerns is finding how to best use my TA. The first (and only) time I had a TA, she got the usual duty (for calc classes here, anyway) of answering whatever questions the students had ("do this problem please") and giving a shortly weekly quiz. I didn't really like the plan, but had no idea what else to do. I still don't like the plan, and don't yet have much of a better plan on what to do. Besides perhaps not having the quiz during the discussion section itself, to save time (which I heard was a complaint among other classes last semester). When I have run my own discussion sections in the past, I haven't given a quiz at all.

So, some thoughts:

• Chapter 7, Techniques of Integration, is concerned with learning four techniques (and when they are mostly likely usefully applied). One thought I had was to break the class into four groups, and have each group teach one section (because you learn best when you teach). Of course, I'd meet with the groups and make suggestions and corrections and oversee to make sure things went mostly ok. Another idea I had, and might be leaning towards, is a bit more complicated. I thought I might write up a bunch of worked examples (some from the text, some solved problems, some examples from other texts) to distribute to the class along with a collection of unworked problems. I would not permit them to look at their book, at all, during this part of the class. Instead, the idea would be that they should look at the worked examples, try to find worked examples that look similar to the unworked problems, and learn their own way through the technique. I rather like this idea of 'pattern finding', which I like to think is part of what mathematics 'is' - as opposed to 'learn these mechanical procedures and repeat'. I may also structure the class so that each student presents a problem to the class. This might get them to start early so they can have more freedom in which problem they do (I'm thinking just ask for volunteers during each class, they get to work any problem that hasn't yet been done). It might also occupy some of the time in the first few discussion sections, which would be convenient.
• In Chapter 10, on parametric and polar equations, I thought I might give them a project: find parametric equations (likely piecewise, discontinuous) that will draw their initials. I'd probably allow piecewise linear functions, but would certainly encourage (5 points of the project for non-linearity?) something a bit smoother. This makes me wonder if it would be a bonus, extra credit project, or count as a weekly homework, or if I should also find some sort of similar project from each chapter (or for each of the exams?) so the course has somewhat more regularity (1 random project seems odd).
• I'm almost certain, though I distrust my memory, that in my calculus class as an undergraduate, we were given an assignment on a section before we ever talked about it in class. We were expected to go home, read the section, go to office hours if necessary (or work in groups), and complete problems from the section before the next class. I loved it. I have friends here in grad school that say they would have hated it. But I was thinking I might structure one chapter this way, perhaps chapter 8 (arc length and surface area). I recall last semester noticing that chapter 15 wasn't the most friendly read, but I wonder if chapter 8 might be better for that. I think if I do this I might pick the first couple of problems from each section and have them due at the beginning of class. These first few problems are typically pretty easy.

So, basically, I have some idea about having them teach themselves techniques of integration by identifying patterns, teach themselves some applications by reading the book, and then I have a random project idea in the section on parametric curves. In some regards, I like the various modes the class will go through, because it makes students experience the mathematics differently. Reading math, versus learning math via lecture, versus finding patterns for themselves. On the other hand, I wonder if it will give the class a very discontinuous feel (which is, clearly, a sin (not a sine) for a calc class (sorry, I had to)). The one random project halfway through the semester doesn't seem to help the continuity. So, do I stick with more the more lecture-based approach of last semester throughout? Do I find more projects to do from more of the chapters?

I'd also like to incorporate technology more. The classroom I'm teaching in has a computer and projector. I'm pretty sure (though I'll have to check) that the computers on campus (in labs and so forth) have Maple and/or Mathematica. I feel like this will help immensely in visualizing parametric and polar curves, and multiple integrals. Plus it gives me more fun things to play with :)

Oh, and I was also debating about making videos of me working through problems, and post them on youtube. I'm not exactly sure why (besides giving me an excuse to play with such things more). My students last semester did ask that I complete more problems during class, but I don't really feel like there's enough time. Plus, I'd rather have them finish up the problems. I like to only work the new parts of a problem, and let them finish up the algebra (or integrals using techniques from older sections). I certainly could write up the solutions (I do love LaTeX), but I wonder if the video approach might be more appreciated by students? Less intimidating, perhaps? Maybe I should have them make the videos? Or allow more freedom - let them make presentations posted online somewhere? Or not have them do any such thing, to avoid copyright worries and things?