Note to Myself - Some Reflections on Teaching
By Dr. Eugene Maier |
A while back I ran across an article entitled "Notes to Myself" that I wrote for the September 1984 issue of The Oregon Mathematics Teacher. At the time I had been teaching for over thirty years. Now its been over fifty years since I taught my first class and I find that these notes have served me well. The article is reprinted here in its entirety.
It's been over thirty years since I taught my first mathematics class. During that time I have evolved a list of notes to myself that serve as guidelines when I undertake a class. Mostly I carry the list in my head. From time to time I attempt to record it. Whenever I do, the list never comes out the same nor does it seem complete. However, there are four items which are in the forefront of my mind right now that have occurred on all my recent lists. These are: 1. Have a story to unfold. 2. Nourish insight. 3. Tell the truth. 4. Be open to change.
1. Have a story to unfold. Robert Davis, director of the Curriculum Laboratory at the University of Illinois, once wrote, "…most first-year algebra courses…—like the Manhattan 'phone directory—contain a great abundance of detail, but no clearly recognizable plot." That is not only true of algebra--school mathematics, in general, tends to be a collection of isolated topics without any apparent continuity or cohesiveness. The result is that, for many folks, the subject becomes boring and pointless, an unending sequence of procedures to be mastered for some purpose that apparently will be made clear in a future course that never seems to arrive.
For me, the antidote is to have a story line, so that each course has a beginning, an ending, and a plot, or is at least a collection of short stories, each with its own integrity. I don't find this easy. Story lines seem absent from most textbooks or, if the authors had one in mind, they aren't willing to divulge it. Recently, I was looking at the seventh grade book of a popular text series and wondered how I could create a story line to fit it. There didn't seem to be any major theme to the book, but some ideas did emerge. I thought about taking all the topics on fractions and combining them into the story of rational numbers—why they were invented, how one operates with them, why the operations are defined as they are, examples of the usefulness of rational numbers, and their limitations, for example, in representing some distances precisely. That could lead into the story of decimals and the real numbers. That's probably a story line I'd use—at least until I had a better idea.
2. Nourish insight. Psychologist Robert Sommer in his book The Mind's Eye maintains that the reason the "new math" failed is that it devalued imagery. Students were not developing images in their mind's eye that they could use in thinking about math. Literally, they had no insight. Without insight, one can learn paper-and-pencil procedures and how to correctly manipulate symbols, but it's difficult to solve problems, apply mathematics, and build conceptual knowledge. Now, if there's any point at all to mathematics education, it's developing these higher order abilities. (We can get machines to do the symbol-pushing for us.) Thus, I want to foster the growth of my students' mathematical insight.
In my view, sensory perception is a critical element for many people in the developing of insight. Hence, creating active learning experiences and using manipulatives, models, sketches and anything else that provides sensory input becomes a critical part of the mathematics classroom. So this statement is a reminder to me to get out the<