I would like to thank Cindy Chapman for her game of beating the calculator. Unfortunately, there are a number of horror stories from various directions about the excessive dependence on the use of calculators. Here are some examples:
1. From the flagship university of Texas at Austin, a faculty member was chatting with a student about the calculus course. That student said the course was going okay for him but there was an episode he wanted to mention. It seemed that he was in a group project with about 10 other students in an Engineering project. At one point, they needed to take the average of 0.12 and 0.12. The group leader whipped out his graphing calculator and punched in 0.12 + 0.12/2 and got the answer of 0.18. Everyone was ready to put down 0.18 in their report until this student said: "Hey, fellows, the average of 0.12 and 0.12 is 0.12!". In chorus, the group members said: Yeah, we thought so too, but the computer said it was 0.18!. This particular student had to go through the entire process and explain to the group that division was done first on the calculator. At that point, the group members were finally convinced that they should put down the answer 0.12.
2. Another colleague reported that engineering students in his class no longer knew how to divide by 10 without the aid of a pocket calculator. One student in the class was able to divide by 2, but not by 10.
3. Last summer, I was asked to mentor two high school students working on math projects for a math fair. Both are very bright. One worked on a maximum-minimum problem in geometry. To get a rough idea of what was happening, I asked him to examine some numerical examples. He had to look at an inequality where one side was sqrt(3) and the other side was bigger than 2. At this point, he whipped out his pocket calculator. The second student was working on some problems involving continued fractions. Again, to check out some simple examples, he whipped out his calculators. In both cases, the line of thought was broken and we had to start all over again.
Of course, calculators are useful when dealing with long calculations. They should be viewed as a "supplementary tool" that should be used *at the appropriate* place. Above all, the user has to understand the problem of *round off errors*. The main shortcoming of a pocket calculator is the fact that the display window only shows a limited amount of information and the user has very little idea if an error had crept in and there is no easy way to see just what basic fact was not mastered by a student.
On a recent exam. students in a beginning math course were asked to write down the value of sin(15 degrees). A hint was given that 15 = 45- 30 and the addition formula for sine and cosine were prominently displayed on the blackboard. Calculators were permitted.
Many students wrote down the answer 0.2588 and received full credit for the problem. Many others wrote down:
sin(15) = sin(45) - sin(30) = 0.2071
and received no credit. It was not a good test problem. For all one can tell, many of the students who punched into the calculator sin(15) received the correct answer may still believe that sin(15) = sin(45)-sin(30).