After serving in the U.S. Navy, I enrolled at Western Illinois University to get certification to teach high school math. In addition to taking the required education courses, I took a few pure math courses to see if I could still do high level math. The math professors at Western were exceptional math educators, and professors Joseph Stepanowich and James Calhoun were especially influential.
Joe Stepanowich was very friendly, the most down-to-earth person you could meet. On first meeting Joe, you wouldn’t guess he was a legend in math education circles and with former students. Joe taught me some number theory. I can still hear him saying, “9 bundles of x-squared minus 4 bundles of x-squared equals 5 bundles of x-squared. Bundles of x-squared are not the same as bundles of x- cubed.” What a wonderful way to explain to kids how like terms in an expression are combined! I was dumbstruck after Joe showed our class the method of finite differences, which is an algorithm for finding a formula for the nth term of a sequence when the nth term is a polynomial. When I later taught finite differences to my advanced math students in high school, some of them told me they found finite differences to be fun and easy. A fundamental activity of mathematicians and scientists is to find a set of equations that express relationships between two or more variables, the rules of nature. The finite difference algorithm is just one of many pattern or rule finding tools.
James Calhoun taught me the development of the real numbers. Jim was not one of those professors who used proof by intimidation to prove a theorem. If a concept was subtle, he explained the concept from a variety of viewpoints. I remember his discussion of the concepts of equivalence classes and a well-defined operation. Students could easily see that he was deeply committed to helping all his students gain a clear understanding of these important concepts. Jim’s explanation of equivalence classes and well-defined operations served me well in other graduate level math courses. I was so impressed with his teaching style, I tried to adopt it as my own.
Experienced math teachers know that about every ten years a new method of teaching math to kids comes about. The new method is supposed to be the grand elixir. But there is no grand elixir! If there was, we would have discovered it many years ago. Only hard work by students and creative teachers will move math education forward.