ONE POINT OF VIEW: Calculators and Common SenseAuthor(s): Jon L. HigginsSource: The Arithmetic Teacher, Vol. 37, No. 7 (MARCH 1990), pp. 4-5Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193850 .

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ONE POINT OF VIEW

Calculators and Common Sense

Jon L. Higgins

NCTM re- leased its Cur-

riculum and Evalu- ation Standards for School Mathematics in March 1989, I looked for the reac-

tion of the press to this very important document. In almost every newspaper the headline or the thrust of the article related to the Standards was that NCTM now recommended that calcu- lators be used in mathematics class- rooms. How discouraging! NCTM members know that in March 1989 that recommendation was very old news. Even worse was the fact that the more substantial recommenda- tions of the Standards were generally ignored.

Because they are tangible, calcula- tors will probably always be more newsworthy than abstract ideas, such as a mathematics core curriculum. Rather than wish that the calculator issue would go away, it is time to try to bring some common sense to the topic. Despite the publicity given to arguments about using calculators in the mathematics classroom, I believe that we are devoting time and energy to a nonissue. The real issue is not whether calculators should be used in mathematics classrooms; it is how cal- culators should be used in class-

Jon Higgins works with preservice teachers and teaches graduate courses in mathematics edu- cation at Ohio State University, Columbus, OH 43210. He is interested in relating learning the- ories to the teaching of mathematics.

rooms. We ignore this important dis- tinction when, as a profession, we proclaim that the use of calculators should be required. Requiring the use of calculators by mathematics stu- dents is a dramatic attention-grabber, but I believe that it is a professional blunder. We need to focus on how the teacher guides students in the use of the calculator, and in some situations we should require that unimaginative mathematics teachers not use calcula- tors.

In many instances, those who are against the use of calculators in the classroom and those who are for the use of calculators seem to be talking past each other. These camps seem not to realize that they are talking about very different ways to use cal- culators. As a result, both sides can point to appropriate ь 'horror stories" to bolster their positions. I recently visited a general mathematics class- room in which some of my undergrad- uate students had been observing. One of them decried the fact that some of the general mathematics stu- dents could not find 10 percent of a number without using a calculator. What a wonderful example for those who believe that calculators "rot the mind"! If calculators are used as a substitute for thinking, of course they rot the mind! It is only human nature to try to minimize difficult thinking, and in today's stressful workplace such strategies may be vital survival skills. Nevertheless, the avoidance of thinking in classrooms is not appropri- ate for learning. The use of calculators to avoid thinking is inappropriate.

Simply keying a problem from a math- ematics textbook into a calculator and pressing the equals key may very well be a way to avoid thinking. Teachers who cannot imagine any other ways to use calculators in classrooms should be required to stop using calculators in mathematics classrooms at once!

But of course, calculators can be used in other ways. One of my stu- dents told me that she had never suc- cessfully taught the concept of pi until she started using calculators. Oh yes, she had had students measure lots of circumferences and lots of diameters, but in the process of doing the long division with paper and pencil every- one got bogged down and lost the idea of ratio. If calculators are used, the time usually required for division is freed for other things. One exciting in- troductory activity is to show stu- dents cross sections of various circu- lar objects - a pill bottle, a soft-drink can, and a wastebasket might be ex- amples. Then ask them to predict the relative sizes of the quotients if one were to measure each circumference and divide by the corresponding diam- eter. Most children, and many adults, will agree that the wastebasket should give a division answer much bigger than the pill bottle, for example. By measuring with string and using calcu- lators to divide, students can very quickly focus on the fact that the dif- ferences are not evident until the first or second decimal place. Using the calculator has shifted the focus from the calculation to the idea behind the calculation. But it has not served as a substitute for thinking.

Consider the calculation of 10 per- cent. The matter of "taking a per- cent" seems very much like an oper- ation, and if one uses a calculator that has a percent key, the process does seem like a new operation. But is "taking a percent" related to other operations? If 50 percent is half of something and 33 1/3 percent is one- third of something, is 20 percent, or 10 percent, also related to dividing? Does one really need a calculator to divide or multiply by 10? If a calcula- tor is used to help answer a series of questions like these, then I would maintain that the calculator is not be-

4 ARITHMETIC TEACHER

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ing used as a substitute for thinking. It is used once again to shift the focus from the calculation to the idea behind the calculation.

How, in general, is this shifting ac- complished? The two foregoing exam- ples, as well as the hundreds more that readers can generate, have one thing in common. They use the calcu- lator to explore. The key to explora- tion in the classroom is finding the necessary time. And time is exactly what can be saved by using calcula- tors - time for searching, conjectur- ing, and testing - in short, time for thinking. If calculators are used to ex- plore and to encourage thought, then and only then should calculators be required in mathematics classrooms. But if calculators are used to avoid thinking about mathematics prob- lems, then of course they should be banned. Perhaps NCTM should issue licenses that would permit mathemat- ics teachers to use calculators in classrooms; teachers who could not give examples of mathematical explo- rations with calculators should be for- bidden to use them in teaching.

Finally, I have suspected for some time now that one of the things that sets good teachers apart from ordi- nary teachers is that the good ones don't take anything for granted! As a profession, we tend to take for granted that students can judge when calculator use is appropriate and when it is inappropriate. This assump- tion is almost certainly false. We need explicitly to teach students when mental arithmetic and paper-and- pencil calculations have an advantage over calculators. Several years ago I had the good fortune to participate in a contest between ten fourth graders with paper and pencil and myself with a calculator on twenty simple subtrac- tion problems. Because the subtrac- tion problems could all be done by mental arithmetic (e.g., 18 - 7 = ?) all ten of the fourth graders beat the pro- fessor, who laboriously had to key every number and operation into his calculator. Every student learned something that day about appropriate and inappropriate uses of calculators ! We need to build a lot of these lessons into our curriculum and have explicit discussions with students about the

best use of calculators. Knowing how to use a calculator should also include knowing when to use a calculator.

''Should calculators be required in every mathematics classroom?" I be- lieve that that question misses the point and is inappropriate. "How should calculators be used in mathe- matics classrooms?" is the appropri- ate question. When one looks at the appropriate question, common sense suggests that some uses of calculators

are appropriate and that other uses of calculators are inappropriate. Instead of globally mandating the use of cal- culators or globally banishing them from the classroom, it's time to use common sense.

Reference

National Council of Teachers of Mathematics, Commission on Standards for School Mathe- matics. Curriculum and Evaluation Stan- dards for School Mathematics. Reston, Va.: The Council, 1989. W

MARCH 1990 s

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