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 blocks, not just for first-graders but for college students as well.

3. Tell the truth. A woman in a methods class I was teaching told me in class one day that her son was learning all about sets in school and she asked me when he would ever use that information. "Truthfully," I replied, "the answer may be 'never'." A guffaw of surprise swept through the class. I suspect the answer was unexpected. What was expected, I think, was some variant of what I call the "Big Lie" of school mathematics: "You have to learn this because you'll need it sometime." I'm familiar with it because I used to say it. Then one day it occurred to me that every time I said it I was probably lying to some member of the class and sometimes, say if the topic was the division of fractions, I might be lying to almost everyone in the class. At any rate, I resolved never again to attempt to motivate the study of mathematics on the basis of future utility.

Life in the classroom has been much better since then. I no longer get involved in the game which begins with a student asking me what use he or she will ever make of the topic of the day, and I responding by listing all the circumstances I know in which the topic in question might be used. The student responds, "I'm never going to do that, so why do I need to know this?"—and 'I'm right back at square one. I find it much more satisfactory to be truthful and say, "Apart from school you may never use this," I believe I'm being honest when I tell students there are certain mathematical skills, like knowing the long division algorithm, that are simply school survival skills. They will need to know them to get out of the fifth grade, get into med school or whatever—and I have no rational explanation for that. I also tell students what I want for them: that, although I can't predict what mathematics may be useful for them to know in the future, I want them to feel confident and competent about learning whatever math they may want to learn at any time of their lives. I suspect much of the math that will prove useful to our students hasn't been invented yet.

4. Be open to change. I think most people get into routines that are comfortable and secure for them. I do. And when something comes along that threatens to disrupt that routine, I resist. I wonder what the resistance is about because I find, when I do open up to change, it's exciting and vitalizing. At least, that was my experience when I finally tried other teaching strategies besides lecturing. Of course, before I could even consider teaching by some other method, I had to be aware of some alternatives. And so this note to myself to be open to change carries with it the charge to be versed in alternatives, to have other options available when what I am doing isn't working or becomes trite or outmoded.

I believe this is especially important as the world of mathematics is drastically changed by the rapid advances in calculator and computer technology. With machines available to perform mechanical mathematics processes, human activity in mathematics becomes conceptual and not computational. This not only affects the content of courses, but also the manner in which they are taught. Conceptual mathematics doesn't lend itself to being taught by drill-and-practice methods. Thus change becomes imperative, and teachers become uncomfortable. Recently, I heard a teacher exclaim that if calculators were allowed in school, the whole fifth-grade curriculum would be destroyed, The teacher knew of no alternatives, he was unable to envision a mathematics curriculum with calculators, and how it might be taught, There are a lot of options, and knowing about them will help overcome the resistance to change.

These four notes to myself are among those which currently set the tone and strategies for my teaching. Perhaps they may be helpful to you as you think about your philosophy of teaching mathematics.