Practical Problems With Calculators

In an online math discussion group, an Iowa higher school teacher posted this reply to this earlier comment:

Though the district provided calculators this year, a number of Levittown students have owned their own since eighth grade. Over the years, they have learned to plot graphs, calculate exponents, check their computations as well as download games from the Internet.

As a high school teacher, I find that most of my students have calculators, mostly TI-83 and higher numbers. I disagree with all of the above statements with regard to my students.

When a statement is made that "they have learned to plot graphs," I would qualify that to indicate that they (might) know how to punch in the equations on their calculators. However, I do not believe this is equivalent to students' understanding how to plot graphs. In fact, many of mine stare at me when I ask them to name three points on the graph of f(x) = e^(x+2) - 3. Worse, ask them to name three points on the graph of f(x) = ln(x+2) - 3. Ask them to find A and B if f(x) = Aln(x) + B if f(1) = 10 and f(e) = 1, and they stare helplessly at their calculators.

Ask them to compute the log (base 10) of the cube root of .01. Once again, their fingers itch for the calculator, and after consulting it, most would give an answer of -.67, typical of what I call a "calculator math" answer. The correct answer, of course, is -2/3, but most calculator-raised students are wild about rounding off all decimals to two places. To actually figure out the log (base 10) of the cube root of .01, much knowledge is necessary, including knowledge of place value (.01 is 1/100), knowledge of negative exponents (1/100 = 10^-2) and knowledge of equivalent radical and exponential expressions (the cube root of x is equivalent to x^(1/3)) and I am convinced that dependence on the calculator hinders the acquisition of such knowledge.

I assume that "calculate exponents" means something like taking 8 to the 6th power. But I have students in algebra 2 and precalculus who do not understand what 2^-1 means, and who do not know in their heads that 2^4 = 16 or that 4^3 = 64. They didn't bother to learn it, because they know the calculator can do it for them.

Regarding checking of computations, what a joke! My students don't want to check anything even when the calculator is staring at them! They want to ask me if their answers are right. I cannot get them to use the calculator when they ought to! I fear that it breeds lazy minds.

Finally, do not talk to me about games on the calculator. In the last week, I've had to confiscate two calculators and send pointed e-mail messages home to parents of kids who were playing tetris on their calculators during class. Both students are underachieving in my class, and playing calculator games during class does not do anything to remedy this situation.

I want so much to be able to let students use calculators responsibly, for computing compound interest problems, exponential decay problems, etc., and usually end up having to give two-part tests or quizzes, to make sure they do know how to use their calculators, but to also guarantee that they understand the meaning of log(base 10) of a number like 10000.

Most of these pricey calculators are very easy places to store textual information, definitions, and formulas. I have students who have programmed the quadratic formula and numerous other kinds of formulas and theorems into their calculators. It makes it very difficult for me to assess what they really know when they can cheat so easily.

Joye Walker
Iowa City

A posting on an education discussion list:

I think that it is obvious why fuzzies rely on calculators. Do you remember a few days ago when some poster told us how her daughter in third grade arrived at the difference of 15-7? Her fuzzy concatentation of addition and subtraction for this simple problem, was mind-boggling. This used to be a memorized math fact -- no more. Think what the poor student would have to do applying these complex techniques for the probelm 12727897-745986. Of course they won't because of the fuzzy rubric: "if you need to, use a calculator" (see any CMP book). The fuzzies know that their techniques won't work for any problem more complex than simple ones. Their never mentioned supposition is that the students being taught the fuzzy method will never need to do such a problem, or any math problem, for that matter. And their saving fact is that calculators exist: a black box that saves the fuzzies from the charge that their methods won't work well for most problems.

A posted comment:

I don't think it's the toys (or tools if you insist) themselves that are the problem. It's the mistaking the tool for the subject matter. By analogy, carpentry is not about operating a nail gun or a power saw. Carpentry is knowing how to cut the wood to fit and where to put the nails. A skilled carpenter can produce more and better work with power tools, but without the underlying skill, he'll only make more sawdust and noise. With calculators and computers, it's terribly easy for students, and even some teachers, to forget that distinction.

A posted comment:

We have high school students who don't know how to do simple double digit multiplication problems without a calculator! I believe that technology has it's place - but elementary school is not the place. We have to start at the root of the problem - get them to use their brains first - understand the concept of math, then show them the short cuts with a calculator. (To top it off, some kids don't even know how to do long division even with a calculator -- which number do they put in first???)

A posted comment:

"My daughter was utterly baffled by some of her Algebra II homework before I shoved the (required) graphing calculator aside and showed her how to do the work the old-fashioned way. She's since shown some other students in her class who were similarly stumped. The teacher told us at open house that the calculators were great because the students didn't have to 'spend a lot of time' graphing functions, which meant they had time to move on to more advanced concepts. The thing is, graphing the functions is a learning exercise. That's where you gut out the connection between the equation and the graphical interpretation. If you don't do that in high school algebra, when are you going to do it?