A recent post of mine (Revenge of the calculators) seemed to ring true with quite a few readers. I think that most math teachers would agree wholeheartedly that an overreliance on calculators has done an awful lot of damage to students' computational skills. Most would also agree that calculators simply get in the way in most applications.
Why then did I just spend a few thousand dollars to license Maple and Geometer's Sketchpad for a new math/computer science lab in my school? Because computers, software, and even calculators do have an application in math education since they so obviously have a place in business and industry. I made a very handsome living as a statistical programmer before I started teaching because computers are just so darned good at doing a lot of things quickly. Could I calculate correlation coefficients and p-values by hand? Sure, if I really wanted to and was feeling especially masochistic, but why? If I understood statistics enough to interpret the output (and program effectively) from my application of choice, then there was no need to go through the pain of hand calculations (or cost my employer days of work).
This actually gets at what many of the folks who commented on my last post were saying. Calculators are spiffy tools, but if students don't understand the underlying math, then they are very expensive toys that have no place in a classroom. If, on the other hand, students already understand the math and would like to begin exploring more sophisticated problems, calculators and software can help them do that quickly and easily. For example, I just finished a course on linear models in my master's program. We spent quite a while looking at population models and the linear algebra behind them. Then we broke out Maple and began crunching away at 10X10 matrices that described real populations, performing extensive calculations that would have been suicide-inducing without the software to help. Because of the speed with which we could model population changes, all of us in the class were able to get a much better understanding of the population dynamics/model rather than focusing our efforts on hand computation.
Similarly, one of the other teachers in the program described how she used Sketchpad to model honeycombs for her students. The goal of the class (AP Calculus, in this case) was to show how bees minimized the use of wax, while also minimizing wasted space, through the hexagonal structure of a honeycomb. Instead of getting bogged down in class tessellating different structures on a board, she was able to engage students in modeling a variety of structures on the computer before they turned back to pencil and paper for the calculus minimization problem.
How about periodic functions? Remember the inordinate amounts of time you spent plotting intercepts to learn the effects of various parameters on phase and amplitude? Quick visualization in Maple can show the same lesson more quickly and give students time to explore the math while working in an environment (i.e., sitting in front of a PC) they like and understand.
I sincerely hope that computers never replace paper and pencil for learning mathematics. I get bent out of shape when my students use pens, let alone when they break out their calculators unnecessarily. However, I'd like to believe that my bad math experience at Chuck E. Cheese last weekend can be balanced by an appreciation and understanding of applied math cultivated in a computer lab, used intelligently by like-minded math teachers.