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.
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9 comments:
I disagree. I love me some good tech as much as the next girl, but I think the point of a calculus course should be to learn calculus. The theorems, why they are true, and how to use them. Solving interesting novel problems sounds more appropriate to an engineering course.
NY makes me let kids use graphing calculators for everything, and I hate it. I do my best to use them well as a learning tool. But I think they also seriously handicap learning in many ways.
I mean, do I have to invoke the Morlock and the Eloi here? Did you see Idiocracy? People relinquish power when they don't know how crap works and trust a machine to do it for them.
Where I went to grad school (SUNY Binghamton) the math department banned calculators on tests as a policy. It didn't bother me much, because I went to high school just as graphing calculators were sort of getting started, and had some tough old school teachers. But those undergrads were beyond freaked. They also couldn't tell you cos(pi) without a calculator. Because they didn't really know what that meant.
So your math department only did trig calculations on angles convienently located at 0, pi/6, pi/4, pi/3, pi/2, etc? Isn't that a bit restrictive?
@Jason we used the standard trig angles almost all of the time (though not always), and I suppose it is mostly a shame. But it seems like a minor issue to me. I'm honestly not exactly sure what else to say about it. If we're just having students push symbols around on paper, the difference between moving something like sin(pi/4) and sqrt(2)/2 isn't particularly interesting. As long as students realize the difference between saying sqrt(2)sin(pi/4)=1 and saying sum_(n=1..infinity) 2^(-n) = 1.
@Kate I'd love to think I was teaching Calculus. When I look at the steps in most of the problems though, there is maybe one line where you are "using Calculus", and the rest are just algebra. I don't know, perhaps I haven't thought about the distinction enough.
As far as solving "interesting novel problems", I may have implied (and been thinking about) engineering/real world problems in my post, but I don't want to imply I wouldn't look for theory problems over real world problems. I guess I should be doing that more anyway?
I did see Idiocracy, and thought it was a fun movie. I also have very little idea how my car works (much less a plane, or ...), and pretty much just trust it to get me from A to B. Should I take back control and only ride my bike around (environmental issues aside)?
We ban calculators on our exams, so it's pretty easy to convince kids not to bother with them on homeworks. Most of mine can, at least without tooo much of a pause, tell me cos(pi) (in <= 2 tries). If they're thinking about it (instead of just going with the first thing that pops in their heads), they'll get it right. And I've alway been happy to ban calculators, it's how I was brought up (mathematically speaking). I should mention, though, that we do a majority of our homework using the online homework system, Webwork, and that this already eliminates a decent portion of mechanical algebraic simplifications. Students can stop a few lines earlier in the solution of a problem, type in somewhat long expressions in webwork, and it will carry out some calculations to see if the answer is correct.
I guess I kinda lost track of where I was, and were I was going with these comments, and I apologize. I feel like perhaps I should be trying to come up with examples of what questions I'd rather be asking my students, compared to questions I'm currently asking. I've got just under a month before I start teaching another calculus class, so hopefully I'll get something together before then...
Ok, I should have read my news first, and then replied. Robert Talbert, at Casting Out Nines, posted yesterday, something that I think captures my thoughts about changing calculus better than I have. I framed things too much from the computers standpoint.
Jason - Yep.
I enjoyed Robert's post as well. He has some great insight.
Anyway, I didn't mean to jump all over your shit. I think I misunderstood at least part of your gist. Maybe you can introduce something like a "lab" or "problem set" component for part of their grade? Where the students have to tackle an unfamiliar, challenging, but do-able problem with what they know, and they are allowed to use Maple or somesuch to crank out the gruntwork? Just a thought.
@Kate no worries. That's what sharing thoughts and opinions online is for, isn't it :) In whatever degree there was any misunderstanding, I'm sure my inability to put thoughts together was a large contributing factor.
I think a lab/problem set component is a good idea. I can do sort of standard homework assignments as I see fit, but then also start getting towards whatever sorts of "interesting" questions it is I think are out there. Now, to find/write some...
This discussion seems to have moved to teachingcollegemath. So head on over there.
I've got too many suggestions for you to write out here. It might take a phone call or skype! (scary, this idea of voice communication)
@Maria tempting... but then you and I are the only ones having the conversation, that surely lots of people could benefit from :) I'll follow along (and maybe contribute sometime) at TCM
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