The evil TI-83
Let's work an example: Let's say that you are calculating the frequency of green light, whose wavelength λ is 500 nanometers. The formula is
The problem is, you have no idea how to do scientific notation on a calculator. The speed of light, for instance, is something 3.00 EE 8. On the TI-83, the "power-of-ten" key EE is inconspicuously placed as the second function of another key (the comma I think), which may be part of the problem. But not to worry; you can do what you've always done, which is basically to enter the number as a formula, like so: 3.00 * 10^8. Your calculation becomes
Where did you go astray? Of course, it has to do with the order of operations. You might be thinking of your expression as one number divided by another, each number expressed in scientific notation. But that is not what your calculator is thinking. First it does the exponents; then it multiplies, divides, and multiplies, in the order that those operations appear in the formula. The factor of 10^-7, in effect, winds up in the numerator instead of the denominator.
This mistake happens a lot. Now, of course, it is quite possible to do things right on the TI-83. You can make use of the secret EE key. You can put parentheses in your formula to make sure that the operations happen in the right order. But the point is that the easy thing to do, the obvious thing to do, the thing that a typical college English major who hasn't used the calculator in two or three years will very likely do, is the wrong thing to do.
Do I blame the students? Not really. OK, it would be nicer if our liberal arts students were better at quantitative stuff. But they are actually a fairly smart bunch, as college students go. Ours is a pretty selective college with a good academic reputation that is largely deserved. I would wager that a more representative sample of college students would be even more helpless about this.
Do I blame Texas Instruments? Somewhat. I might have designed the keypad for the TI-83 to put the EE key in a more prominent place. After all, this is a calculator that is designed and advertised for mathematics and science. Scientific notation, calculator style, should be more straightforward. But of course, the problem is that the TI-83 is designed to do so many things. It solves equations! It graphs functions! It does linear regression! And the TI-83 Plus can store specialized libraries of useful routines! But the number of buttons is limited. The machine is already uncomfortably large in the hand. I would never use a TV remote or a cell phone this fat. Anyway, with all these features, some things -- like the EE key -- will have to be less prominent than some folks would like. And TI does make some perfectly fine scientific calculators that are far cheaper than the TI-83. (Of course, it is the TI-83 that they market to the high schools. In round terms, the devices cost $100 apiece. I bet they make a bundle from the program!)
Mostly, I blame the high school teachers and the adminstrators that have decided to put a TI-83 in the hands of each of their college-bound students. What is the attraction of this? I suppose the TI-83 is cool. I mean, it has all these nifty things it can do. It solves equations! It graphs functions! It does linear regressions! (Did I mention that it solves equations?) And so it looks like just the thing you need to do all sorts of math and science stuff. But does it actually help?
As far as I can tell, the actual effect of the TI-83 and its ilk is not to give the students a greater capability to handle the mathematical side. They either never learn how to use it well or they are unable to retain this knowledge. Their quantitative skills otherwise are not noticeably improved. And who in his right mind wants to use a lousy little LCD display to graph a complicated function or a big data set, anyway? If you need to do that for real, you'll probably use Maple or Mathematica or Minitab or SAS or Origin or something, on a real computer.
I suspect that the many, many hours devoted to TI-83 use in high schools are largely wasted. That's the optimistic scenario. On the other hand, they may do actual harm. The use of these graphing calculators does nothing but encourage a dependency on a tool that the students do not understand. The skills gained do not appear to be readily transferable to other tools. That is, students who have grown up with the TI-83 do not seem to have an advantage in learning to use either a simpler calculator or a computer algebra system.
When we do a mathematical problem without a calculator or a computer, we sometimes say that we are doing it "by hand". But in fact, we are doing it "by brain". Calculators are labor-saving devices, but we should never forget exactly what kind of labor we are saving.
With this in mind, here are my own rules for using calculators and computers.
- When solving a mathematical problem, use the minimum convenient technological aid. That is, if you really need a super-powerful computer algebra system, go for it. But if you're doing a few simple numerical calculations, for heaven's sake just use an ordinary calculator. Why use a hundred-function graphing calculator to find the average of seven numbers (a real-life example that my daughter mentioned to me today)?
- Do not teach the use of a high-power feature-laden technological tool before something simpler is mastered.
- Never use a toy to do a real computer's job. There are real hand-held computers, and it might be worth while to teach their use. But the TI-83 is not one of these.
Nevertheless, there were a few advantages to those old-fashioned tools. First, when you used them you probably kept track of the "power-of-ten" part of the calculation separately, which gave you a sense for orders of magnitude. You found out on your own whether the answer was really big or really small. Second, you were much less tempted to write down 1.779342994135776 rather than 1.78. When you worked for every decimal place, you had a better sense of what was a significant figure and what wasn't! Third, you learned to organize your calculation in an efficient way, and to re-use parts of calculations whenever possible. This reduced the number of opportunities for mistakes. And, of course, your batteries never, ever died. The fact is, I was a lot better -- and faster -- at physics problems with my log and trig tables than the other guys in my class were with their calculators.
Note #2: My daughter, who had to buy a TI-83 Plus for her school work, has a sensible defense of it. She says that the good thing about the graphing calculator with the fancy multi-line display is that you can go back and correct typos in the formulas you enter. With a regular scientific calculator, if you make a mistake, you have to do the calculation over from the start. This is an argument nicely constructed to please Dad, who is happiest when life provides a healthy capability for error-correction.
But the reason that error-correction is so important in the graphing calculator is that its design encourages you to put in the whole formula and then evaluate it once, at the end. In other words, you tend to input a lot more keystrokes before you get a recordable output. More keystroke means a greater likelihood that one of them went astray. Also, an error in the result will be harder to recognize, since it is the result of a more involved calculation. It is not at all obvious to me that the net error rate on final answers is actually lower.