My Philosophy of Teaching, part 2

Harold Reiter

When I was in high school, as a typical college bound student, I took all the standard math and science courses available. After a good semester of trigonometry and three-dimensional geometry, I was offered the opportunity to take a half-year calculus course.

The course consisted mostly of computing limits of rational functions, f(x)/g(x), where both functions were factorable, and the limit was requested at a point where g(x) was zero. Later in the course, I learned about the power rule for differentiating polynomials, and a few other rules for taking derivatives.

It was not until the next year as a freshman at LSU that I realized that I had not really learned calculus. There I saw that I had not the same understanding of ideas or reasoning skills of my peers in that (very strong) class. What I had 'learned' in the high school class were formulas and procedures, which I had memorized without much real understanding. Not only that, I did not realize that calculus is a collection of concepts that one could learn to reason about and even figure out for oneself. What a shock it was for me to realize that I did not even understand the idea of 'function', the main object of study in calculus.

Of course there is some memory work in mathematics, but every memorized fact must be reinforced by an understanding that itself is not memorizable. That understanding becomes a part of the learner in a way that no memorized fact ever can. It cannot be forgotten. Thus, there is great danger in memorizing without understanding. Students sometimes fail to remember the context of the memorized formula. I will give two examples.

  1. Last semester (fall, 2000), my college algebra students learned the Zero Product Principle (ZPP), which states that a product of numbers can (and must) be zero when at least one of them is zero. We used this principle to solve problems like "find all solutions to (x-3)(x-2)=0". Later, when confronted with "Solve (2x-3)(3x-2)=0," some students wrote "x=3 and x=2" simply taking the negative of the constant term of each linear factor instead of creating the two linear equations 2x-3=0 and 3x-2=0, and solving them.
  2. When asked to find the sum of the roots of (x-2)(x+3)+(x+3)(x-6)=0, some students show that they have misunderstood the ZPP by setting each of the components x-2, x+3, x+3, and x-6 equal to zero to get four roots with sum 2. Of course this would be right if the problem was (x-2)(x+3)(x+3)(x-6)=0. The fact is that (x-2)(x+3)+(x+3)(x-6) is not expressed as a product as it is written, and it becomes a product only after the common factor x+3 is factored from both terms.

As your instructor this semester, I think the way I can be most helpful to you in learning the course content is to work with you to develop your understanding of the ideas of the course.