The Need to Understand How Functions Fool Graphing Calculators

Graphing calculators are relatively recent and innovative tools in mathematics education. There is quite a debate in today's mathematical community concerning their proper use. Proponents of the use of graphing calculators claim that the technology allows their students to investigate the behavior of more graphs than traditional means allow, to investigate more complicated graphs reflecting more realistic problems, and to develop valuable technological skills. Opponents claim that that use of the graphing calculator really has nothing to do with mathematics and, worse, the graphing calculator creates a cruch that the student will always depend upon.

Let me say a couple of things to prevent a flood of protest from either side:

  1. I refer to and use not only graphing calculators regularly in my classes, but also, powerful computer algebra systems. I, also, encourage the responsible use of these tools by my students.
  2. I believe that serious steps need to be taken to make sure that mathematics is being taught, rather than a course on how to use a calculator, and to prevent students from becoming dependent on the technology.
So, I guess I come down in the middle somewhere. My belief is that technology in education is here to stay and educators need to make sure that our students understand how to use this technology correctly. In particular, students must understand that computers are stupid and make stupid mistakes. Understanding what types of mistakes graphing calculators make helps students use them more intelligently and, also, requires the students to understand the mathematics involved, which is the whole point behind mathematics education in the first place.

Understanding the types of mistakes a graphing calculator will make is, also, beneficial to instructors of mathematics. The knowledgable instructor is able to pose questions on a test, quiz, or homework which will fool a graphing calculator in a subtle way. Continually doing this is a great way to combat dependence on the graphing calculator. Personally, I like to go over such examples in class in the following sequence:

  1. Graph the function using standard analytic techniques (pencil, paper, and brain).
  2. Graph the function blindly with a graphing calculator.
  3. Compare the two (there should be differences).
  4. Attempt to refine the calculator's graph by changing the viewing rectangle and/or settings of the calculator.
I've found that this approach encourages students to experiment with graphing calculators, while imparting an understanding of the mathematics involved.

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