by Dr. Mark H. Shapiro
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.".... ...Albert Einstein.
Commentary of the Day - June 7, 2006: Calculator Dependence. Guest commentary by William Kohl.
Although I am not a teacher, I am currently involved with our educational system in a mentoring project, where adults are paired one-on-one with an "at risk" student. (The school prefers "on the verge of success" to "at risk" presumably so as not to injure the student's fragile self esteem.) The mentor is an adult who cares and wants to help the student to be successful both at school and in life.
The student I have been paired with is deficient in math. Mentors are not tutors but they can give tutorial help if it is called for. After learning how deficient my student was in math, I decided to spend some time tutoring him. He is in an Algebra I class that had been studying second order equations. I obtained a copy of a test he had missed. One of the first problems was:y = x2 + 5, if the constant 5 is changed to 1, the curve
a. does not move
b. shifts 1 unit up
c. shifts 1 unit down
d. shifts 4 units up
e. shifts 4 units down
I said, "What would you do to find the answer?" He said, "I have to get my calculator." I said, "Why?" He said, "I need it to work the problem."
I said, "Couldn't we just think about the problem first? Even though it may seem hard, (as it probably did to him), perhaps we can start by finding a simpler problem inside this difficult problem."
I was thinking of analytic strategies I had learned in math, one of the basics of which is, look for a simpler problem inside a complex one. He argued with me, and said it was necessary to have his calculator. It seemed as if he had relegated part of his thinking process to this calculator and his math brain would be incomplete with out it. He got his calculator and had it draw the graphs of y = x2 + 5 and y = x2 + 1. Since the graphs appeared one at a time on the calculator display, we still had to plot them on paper so he could compare them side by side and note what had changed.
After we had done this I showed him what I had in mind. How we could have started with the simpler problem, y = x2. From there we could create a table of y vs. x (where y is the number we get when we square x) and then plot the points. We could then repeat the same process for y = x2 + 1. It would be obvious from the new table of y vs. x that the constant term increased every y value by one. So now we could see that y = x2 + 5 was a curve five units above y = x2 and four units above y = x2 + 1. So changing the 5 to a 1 moved the curve down 4 units. My student argued that it was much easier to use the calculator because you didn't have to multiply or add. In other words, you didn't have to do any math!
I had a similar experience when I was asked by the son of a friend for some help with his college calculus course. He was having some difficulty with a problem that involved plotting an equation with three independent variables. He wanted to find a computer program that would produce the plot. However, I was able to show him that by holding one of the variables constant, he could reduce the problem from that of plotting a surface to plotting a simple curve. The surface could then be built up from a series of these lines. This student recognized the value of this approach; and, hopefully, was weaned a bit from calculator (or in this case computer) dependence.
What I am seeing seems to be that dependence on the calculator has short circuited the learning of math and the development of analytical skills. Most students who take high school algebra are not going to be scientists, mathematicians or engineers. These skills are the most important things they should take from their math courses. The computational and analytical skills learned in math often can be applied to a host of everyday problems in business, personal finance, etc.
Another effect of calculator dependence is that many younger people are not comfortable with numbers. In my generation we learned to do simple arithmetic (addition and multiplication) problems in our heads, and more complex ones with pencil and paper. We can do a quick calculation to check a price in the supermarket or to figure the tip on a restaurant bill without having to reach for a calculator.
Today, many elementary school educators believe that the ready availability of calculators has made learning elementary arithmetic skills like addition and multiplication unnecessary. Working problems without a calculator, in my view, helps students to develop those important analytical skills. Calculators certainly have their place, and they are essential for some problems. However, students who have developed good basic arithmetic and analytical skills can master just about any calculator in a few hours. Perhaps if we delayed the introduction of calculators, our students would learn math better.
© 2006 William Kohl.
William Kohl holds an MS degree in Ocean Engineering from the University of Miami. He works for a major corporation in the Houston, TX area.
The IP comments: The IP could not agree with Bill Kohl more. For the past several years the IP has taught introductory college physics courses for scientists and engineers. Far too many students in these classes are calculator dependent. They have not really learned the properties of trigonometric functions such as sine and cosine or how exponentials and logarithms work. The result is that when they must solve problems that require understanding the properties of these elementary functions rather than simply computing their values, they become totally lost.