(Or, Dear College Board, would you please stop requiring a graphing calculator for the AP Calc test.)
*Warning! Rant ahead!*
It’s hardly a new or insightful observation that calculators are overused in math classes. Still, I feel compelled to rant about this issue today. I have no problem with responsible calculator use. While I keep my running tally in my checkbook by hand and figure tips in my head, I usually use a calculator to total deposit slips, balance my checkbook, and estimate budgets, and I don’t mind at all if my students do the same. In fact, I don’t even care if they use their calculators to figure out 7 times 8 (for the older ones at least, who are long past learning that in school), though I think it’s silly and slow. What bothers me is that so many of my students act as if mathematics cannot be done without a calculator in one’s hand. They seem to see the calculator as the primary weapon in the battle against mathematics (and yes, to them it’s a battle). That concerns me.
If math is a battle, your primary weapon is your mind, followed closely by a pen(cil), some paper, your friends, and Leonhard Euler. Calculators are helpful, but they require responsible use. Your calculator is dumb. It can compute answers faster than you can, but only you can ask it the right questions, and only you can interpret the results it gives you.
So why don’t my students see that? There are probably lots of reasons, starting with having no idea what mathematics is about, but here’s a big one: curriculum planners, textbook writers, test makers, and the teachers who have to use their stuff, seem to have gotten so concerned with preparing students to use the tools of modern life that using a calculator has become a goal unto itself. (Actually, I don’t know if anyone has done this on purpose. It may just be that calculator algorithms are easy to teach and easy to test because they don’t require any understanding. Or it may be that Texas Instruments has done a really, really good job pushing their wares.)
Every high school and college textbook I’ve tutored from in the last six months includes problems with a “calculator symbol.” Once in a while, one of these problems includes real-world data. The symbol is there to warn you that the data is messy and the arithmetic will be much easier on a calculator (as if you needed a symbol to tell you this.) More often though, the problem gives you some numbers, which you are instructed to plug into your calculator in a particular way and then record the result. At no point are you expected to connect the calculator algorithm to the topic of the homework, or to understand the calculation. Generally, I flat out refuse to do these problems with students (except for the ones who are really not getting it). I tell them that they can do the problems on their own, or better yet, tell their teacher or prof that these problems are an offensive waste of their time and that they won’t do them.
Even in the non-calculator problems, I’ve noticed that textbooks have a tendency to throw in difficult numbers that make it nearly impossible to solve the problem without a calculator. For example, in the exercises for a chapter on using the quadratic formula, you might see something like this:
1) Solve for x: 3x^2 + 4x +6
37) Solve for x: 4.47x^2 + 5.98x + 6.23
The trouble with this is that it obscures that actual point of the exercise, namely solving the quadratic (though why you need a whole section to practice using the quadratic formula, I’m still not sure). Arguably, for certain kinds of problems, one could make a case that including, say, fractions in problems is a good way to reinforce skills, but that hardly applies where decimals come in. No one would try to solve that without a calculator, so what skill is being reinforced? (This doesn’t apply if you are actually trying to solve a real world problem, in which case messy numbers are the norm and calculators are surely fair game, as are any other tools at your disposal. But do you actually need to practice that with artificially constructed messy numbers in the middle of learning how to do something else?)
But wasting time and obscuring the point are minor problems. The bigger problem is that the ubiquitous presence of calculators creates an environment in which students are encouraged (if not required) to use a calculator before they’re ready. Your calculator is great for speeding up the tedious process of multiplication, but it’s not going to teach you how to do multiplication or what multiplication means. We seem to understand that pretty well: we don’t send third graders off to school with calculators in their backpacks to do their multiplication problems. At least, we didn’t when I was a kid 20 years ago. For all I know, that’s changed, but I dearly hope it hasn’t.
On the other hand, the same principle applies for older students and more advanced math, and here we seem to have lost sight of the obvious. A graphic calculator (or better yet, Maple or Matlab, since graphing calculators often give really misleading results) can be a great help in quickly visualizing a complicated function, if you know how to read the results. But no graphing calculator is going to teach you the meaning of those bumps and curves, rises and falls, starts and stops you see on a graph. For that, you need to spend hours and hours drawing hundreds and hundreds of graphs by hand. You need to notice that you have a problem when you try to graph f(x) = 1/(x-2) when x=2, and then see what happens to numbers nearby. (By the way, one great use of a non-graphing calculator in this process is to use it to quickly calculate f(x) for numbers really close to 2). You need to get so fed up with the tedium of drawing practically the same graph over and over and over that you start looking for shortcuts. (You’d be amazed at how much mathematical insight is the result of lazy mathematicians looking for labor saving shortcuts.) That’s how you learn to read a graph, and no amount of graphing calculator practice is going to replace that.