Summer Calc II, Week 1

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?

Walpha Wiki

Just in case you missed it, I thought I'd share the link for the Walpha Wiki. The same evening that I posted asking if anybody had a wiki going, Derek Bruff started this one. I've not contributed as much as I want to have yet, but still intend to. Won't you help?

The blog also includes discussion about the impact of W|alpha on mathematics education, a topic that's been on my mind recently.

A New Kind of Wiki

Well, not really... I'll just explain.

So, when Wolfram|Alpha (referred to as w|a below because I'm lazy) came out, I, like many of you, was pretty excited to play with it. I was primarily interested in its use as a free, online, computer algebra system (CAS). So when I tested it, I gave it the sorts of questions that I give my calculus students (in fact, I essentially tested it with exams I've given students). In many areas it was obvious what to do, in some areas I could mess around and get a reasonable answer, and in a remaining few areas, w|a seemed to come up lacking.

I thought it would be great to have a resource telling how to input questions you might typically ask a CAS, since apparently entering straight-up Mathematica code doesn't always work (I guess Wolfram still wants to sell copies of Mathematica). One of my early thoughts was that I should make one. And then I thought, surely somebody else has already done so. In fact, the folks at w|a probably already have some nice documentation online. I made a note to look into it, and thought it funny that I was hoping to find documentation for such an online system.

Not long after that, and before I did any more playing with things, Maria Anderson, @busynessgirl on Twitter, posted a tweet: "I am toying with the idea of taking a standard algebra TOC and putting up a webpage that shows which topics W|A can do." A fantastic idea (which she quickly refined: webpage -> wiki). Extend it to calculus, and I'm there. And show not just what it can do, but what it can't do, what it does wrong (or oddly), and ways to make it do what it can do.

I think such a thing should come into being. Perhaps it already has, and I missed it? Or perhaps there is some nice documentation for w|a that I've not yet found? If either of these is the case, could somebody point me to it?

If there is no such thing yet, I say it's time to make one. I'm getting antsy. In the comments below, if you want such a wiki to exist, would you please leave some helpful feedback? I'm particularly interested in: (1) What (free, hosted) wiki software would you suggest or suggest avoiding? I think right now I'm leaning toward wikispaces, though I've not looked into things a whole lot. (2) What should it be called? (3) Any other comments or suggestions you have.

To get things rolling, I'll say that this coming Saturday (May 30), if no links are provided to an existing webpage, I'll start a wiki somewhere that seems to fit the consensus of the comments (I hope there are comments, and they have a consensus). I'll then let you know where it is.

Update 20090526: Derek Bruff left a comment that he was starting one, and posted the link http://walphawiki.wikidot.com/calculus-i via twitter. Looks promising!

Octagons

I've been at the University of Virginia for 5 years now. I've walked on campus, then, many hundreds of times. It's a pretty campus. Since my most recent move, I've adjusted my walking path to the office. Now I typically walk along some of the "garden" paths, just off of "The Lawn".


In case you're curious about those wavy walls, that was Mr. Jefferson's idea. And we couldn't be more proud.

But in all of my walks, up until just recently, I never noticed anything interesting about those little posts lining the driveway in the picture above. Then, from above:


Octagons!

The Education and Music Industries

Recently, a friend of mine pointed me at the article, "Psst! Need the Answer to No. 7? Click Here.", from the New York Times. I found it an interesting article, and wondered how many of my students were using online services to look at former students' notes, or for solutions. Every semester, I put a question on my course evaluations asking what outside resources students used. This is mostly to help point me at new and helpful sources. However, I've yet to get a direct reference (besides "Google"). It's always frustrating.

But anyway, after reading the article, and observing, as my friend pointed out, that all of the solutions for the calculus text that we use (Stewart) are on cramster.com, I was struck by a sort of comparison with the music industry. A decade ago (and still) the music industry was put into turmoil because it became easy to put music online, accessibly and freely. Whole albums and individual songs were there for the taking. The parallel to seeing whole solution manuals, and solutions to individual solutions, struck me.

I like to think there is an important difference. Downloading songs online, without paying for them, didn't hurt the consumer (directly - now we've gotta deal with all sorts of crap, but that's not my point). Besides, I suppose some might argue, a sort of moral degradation or something. With solutions being freely accessible online, it seems like only the consumer (student) is being hurt (to me, at first glance, anyway).

Of course, they're only being hurt if they aren't appropriately using the solutions. If they take the easy route on homeworks, and copy a few solutions, they'll likely have trouble at test time (if they don't, more power to them - or not). I don't think there's any argument that these resources can be used to improve the learning experience, instead of cheat it.

Should I spend some of my class time teaching students how to effectively use solutions manuals? Does anybody point their students at solutions manuals (or similar things online)? Do you incorporate them into your teaching? Or does it seem a non-issue, because students don't use them? Are students more likely to come to office hours to get help?

It occurs to me that while cramster might have a certain convenience, the solution manual for our textbook is available for short-term (something like 3 hours I think) checkout at the library.

It's!

Leonhard Euler's Flying Circus!

That will be the home of the little Python/Sage learning group I mentioned recently.

Now, let's see if this thing gets off the ground...

Changing Calculus

Calculus, at least derivatives, are the (a?) study of rates of change. What I've been wondering recently is how instructors are thinking about change - in their curricula.

I know we've had calculators for quite some time that can do lots of the work we assign our kids. There has always been a price barrier for students using them though. I'm thinking Wolfram Alpha is about to change that (when it goes live later this month).

There has always (well, for quite some time, anyway) been integrals.wolfram.com, which will compute integrals (a big part of a calc 2 course). However, no indication is given there about how to obtain the solution. According to the ReadWriteWeb account of Wolfram Alpha, you can ask it to do an integral, and also ask to see the steps in the computation.

I think this is just one sign, of many, that calculus class will be changing. Sure, technology has been around (behind a price barrier) that will give students answers. Teachers could typically rely on "Show All Work" to hopefully get their students to not bother with the calculators. But now, perhaps, "Show All Work" is also done by the machines, and now it's free. How should I be changing the setup of my calculus class to accommodate this shift?

It seems to me that my classes should start spending less time going through the algebra and "doing integrals" (though not completely removing this from the syllabus), and spend more time finding ways to use them to solve problems. Perhaps try to work some more theory into things, besides just "Oh, look, with functions that look like blah, a substitution blah makes them easier to integrate". I need to figure out how to shift my classes from "do the algebra to work out this computation" to "set up a computation that will determine the answer to this 'interesting' question".

Wolfram Alpha, which has brought this issue up most pressingly (in my mind), might also be a useful tool in shifting how my calculus courses are set up. By the looks of things, Wolfram Alpha has access to lots and lots of data, and can do lots and lots of interesting computation with it. So perhaps it will be a great way to find and create new problems, and give students interesting opportunities to find solutions. Of course, it's too soon to say, because the service isn't up yet. But it will be soon.

So, have people already started making these changes, and I'm just behind in my teaching (as it the rest of my school)? If so, how do I get to where you are? What should I be doing? What are the "interesting" problems I should have my students thinking about, instead of the interesting (in terms of symbol pushing) problems they currently do? Perhaps the tools that I'm just starting to see available for free in Wolfram Alpha are already around (anybody have some links for us)? Or is this all a non-issue, because doing 10 steps of algebra in each of 10 problems, each with a different algebra trick, is what we want our students to be able to do after they're through a calculus class (because in the "real world" (which I'm assuming is out there) they'll have to do everything by hand, no computers)?

I know the technology in math classes debate is not a new one. But I think it is getting more pressing. Maybe I've just been reading too much online/tech news.

I also know this is not the only question that should go into changing courses (if a change is going to happen). What is the goal of a calculus course? How does it fit into the entire mathematics curriculum? And what are the answers to these questions in terms of students going into mathematics, versus science, versus the arts? What actual calculus (and other math) should they be getting out of my class? What other things should they be getting out of my class (how to read a math text? how to present a mathematical solution? how to write one?)? What other questions am I supposed to be asking?

Apparently giving a final exam today is making me philosophical.