Wednesday, December 31, 2008

Some Parametric Curves

If you aren't following my other blog, you just missed two fun posts trying to find some parametric curves. I'll try not to spam this blog too much with links to my other one. But I enjoyed these curves, and thought you might too.

Happy New Year!

Monday, December 29, 2008

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?

Speed Limit Memory

I've got what I consider to be a poor memory. Or, at least, I try not to rely on it too much (perhaps that's why it's no good?). But anyway, I have a hard time remembering what the current speed limit is when driving. On my drive home yesterday from my holiday travels I got an idea for a helper. I was thinking, if you make a felt cover for the top half of your steering wheel (say), and indicated on it mileages from 25-65 in increments of 5, then with another felt ring you slide around to the appropriate place, you'd have a handy way to remember the speed limit. As long as you remember to update the position of the ring when you see a new speed limit sign, of course. I suggest felt, because it seems to me that two layers of felt might hold eachother together mostly, while still allowing some freedom of motion.

I dunno, maybe it's a poor idea. But as of right now, I like it. Plus when I tell my mom about it, it'll give her something fun to do (she sews, rather a lot).

Of course, it's only a temporary solution until the speed limit signs communicate wirelessly with a sensor in our cars that changes a heads-up display on the dashboard which always indicates the current speed limit. Unless, of course, self-driving cars come around before that, and the issue becomes moot.

A Year of Running

Yeah, yeah, everybody is doing end of the year review posts, or posts looking forward to the next year. But I wanted to mess around with the Google Charts API, and figured my running mileage data was as good a data source as I'd find. So here's the chart I made with my monthly miles:
The lightest blue indicates months with less than 100 miles run, and the one darkest (August) was the only month I passed 200 miles.

Perhaps I should have waited through the next two days to post this, in case I decide to run. But even if I go both remaining days in December, I probably won't go more than 8 miles total. That won't affect my totals too much. Currently I've chalked up 1259 miles this year, occupying a little over 168 hours (= 1 full week). That puts my average pace at just slower than 8 minute miles. I thought about making another chart for my average pace each month, but don't feel like going through the computations. Another useless stat: I went for 178 runs, putting my average run length at just over 7 miles. That's an extra 250 miles and 28 runs over last year.

After August I was looking to crack 1500 miles this year, and was on pace for it. In September I ran my first 'ultra', though really the shortest ultra, a 50k. At that point, my running was looking pretty good, and I was thinking about a sub-3 hour marathon in Richmond in November, and doing a 50 mile run in the spring. However, my research was, as my advisor put it, 'stagnating'. Around the same time I met a girl (lucky me :)), and these factors put a 2 month halt in my running. I'm working on getting back into it, though still at a lesser volume. My current goal is the Charlottesville marathon in April, for my third consecutive year. I have no delusions of beating last year's 3:13, but I'm still hoping for sub-4 (had a 3:55 my first year). Guess we'll see.

Friday, December 12, 2008

Camera Box

Last weekend I bought a Nikon S550 digital camera (on sale at Staples for a little while still, if you were thinking about it). I've been having a good time taking pictures (and getting them off my camera from the command line using gphoto2, totally sweet). When I went to clean up my room a bit and recycle the box, I noticed that the box itself was also pretty cool. The box had a cardboard insert separator, which, when unfolded, looked like:
What's nice about that is with the cuts it has, and the folds, it folds up through a series of rigid motions (no folding/rolling except for on the pre-scored lines) to the following separator:
The whole box itself was also folded from a single unit, which lies flat (when unfolded):
And here they are together:
I don't know. Perhaps it's not that impressive. But I really liked it. No tape on the box, just lots of nice folding. As an origami fan, it appealed to me. Plus it gave me more excuses to take pictures, and play with Picasa.

A Few Quotes

The other day on slashdot, the article "Twenty Years of Dijkstra's Cruelty" showed up. I was intruiged, and read the pdf it mentioned - a lovely handwritten paper by Dijkstra. I'm not a CS educator, so I don't have much to comment on about that aspect of the paper (the main point of it). However, I pulled two quotes out, and thought I'd share.
"... by developing a keen ear for unwarranted apologies, one can detect a lot of medieval thinking today."

"And when it comes to mathematics, you must realize that this is the human mind at an extreme limit of its capacity."

Tuesday, December 2, 2008

I'm 1!

Yep, my first post was one year ago today (though technically, I suppose, my first actual post was a week later). Four short months later I got my first comment, and two more months after that I'd had a visitor from every continent except Antarctica. I've gotten 60 comments (though nearly half of them are from me, responding), and had 125 posts. A post every 3 days, and a comment once a week... I'll take it. Especially because this blog tends to ramble and be pointless, even when it pretends to have something to say. All the same, it's been fun.

I thought I'd go back through and pick out my favorite posts so far. But then I couldn't decide what to link back to, and felt bad linking to myself. Most of my favorites were math posts, so you can find them here, because they were tagged. Perhaps my most useful post, if you live in or near Charlottesville anyway, was my writeup about the Rivanna Trail. According to my feedjit widget, this is also the most popular post (at least, from the past n days, I think that's how it works), followed closely by GPS in Ubuntu (which I find kinda surprising, but who am I to judge?). The post I wish more people had seen was this one, about a decent travel mug. I'm still interested to hear about a good travel mug (lid, in particular), especially now that it's almost gift-giving season :).

So, anyway, Happy Birthday blog! I'm a mostly proud parent. Don't get too jealous of your younger sibling (even if it does math better than you). For those following along, feel free to follow me on twitter, or my reader shared items, if you aren't already. I'd love to reciprocate.