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.