So someday I want to be a teacher, and so I find myself thinking about teachery things every once in a while. I've decided to record some of these on this here blogomathing.
I had an interesting philosophical discussion with a student in the tutoring center about the use of calculators in math classes - or, specifically, to what extent calculators should be allowed. On the one hand, calculators are fantastic for helping you grind through tedious calculations that sometimes get in the way of actual learning. On the other hand, I was just tutoring a girl who said, and I quote, "I took calc 2 back in high school, but we always did it with our calculators, so I don't remember how to do it" - so sometimes calculators can become a crutch and prevent the student from learning.
So the stage is set - you find yourself on a battleground with the champions of the calculator trumpeting their technological battle-cry on the one side, and the partisans of pencil-and-paper standing defiant on the other. I think that the best solution is, as is often the case, to be found somewhere in the middle, and my views will be explained through the rest of this post.
First of all, I believe in quantitative literacy, and by this I mean that I think people should have a good idea of numbers and sizes and should be able to do mental "ballpark calculations" to get a rough idea of an answer. (This is a skill that will serve you well throughout your life. If oranges are $1.29 a pound, you should be able to figure out whether or not buying a 4-lb bag for $4.79 is a good deal.) I also believe, especially in a math-class setting, in algebraic literacy - you should know about factoring numbers and canceling top and bottom and when you can split up a fraction. These are basic skills that should be a common denominator, if you'll pardon the pun, for all math classes, and I think that this is where calculators turn into a crutch. If you're taking a math class, you should be able to multiply, subtract, divide, etc. by hand - BUT I don't think you should be compelled to. If a class does not allow calculators on tests (and more on that later), then the teacher should be willing to accept unsimplified answers, or at least be appropriately lenient on simple calculational mistakes (especially when that's not what the class is about). It's a calculus class, after all, not an algebra class, and the student should be graded on how well they do calculus, not algebra.
On the subject of calculators on tests, the student I was talking to yesterday had a really great idea. He said, why doesn't the math department get together and come up with a standard calculator that can be used on all the tests (a TI-83, say), then buy 150 of them, nuke the memory, and keep them in the department office? It'd be really hard to tamper with them, but you'd still be able to do all the basic things. And if you wanted more control, I'm sure you could get your friends in the computer science department to write you some new firmware so you could lock out, say, the graphing capabilities on demand. I really like this idea, and I think it solves the problem of calculators on tests once and for all.
Now, with modern calculators and computer algebra systems and so forth, we find ourselves in an interesting quandary. Things that were computationally intractable ten years ago (indefinite integrals, Taylor expansions, etc.) are now easy to do on your plain vanilla TI-89. This lets lazy students type things into their calculators and get answers without knowing the math behind it. The underlying phenomenon is nothing new, and we even have a word for it. It's called cheating, and this kind of thing is easily detectable (if you bother to look at their paper for 3.7 seconds) and doesn't really worry me. What worries me is that these "middle-ground" students are the ones who would probably learn the subject if pushed to do it, but they are instead letting the calculator do it for them.
Instead of dwelling on why calculators can be bad, let's talk about appropriate uses of technology. I think the most powerful thing about technology is the opportunity it gives students to explore. I've learned a heck of a lot about math by dinking around with a calculator. The beauty here is that again, calculators take care of tedious calculations for you, and allow you to see the Big Picture without a ton of needless effort (you ever try to plot a slope field by hand? Yeah, have fun with that). When you play around with a grapher, you can quickly develop your visual intuition for the rough shape of common functions, how polar coordinates work, and get a good handle on parametric equations. It's also super cool to graph a function along with its first five or so Taylor polynomials - you can immediately see how it converges, and what exactly that radius of convergence means.
For these reasons listed above, I find myself right in the middle of the road when it comes to technology in math classes. I think both of the extrema, whether it be the "CALCULATORS ARE EVIL" school or the "USE CALCULATORS FOR EVERYTHING" school, are rather dogmatic - and rather silly.