A Question About Algebra
A Question About Algebra
By Dr. Eugene Maier
Last month I did an afternoon's workshop on teaching algebraic thinking at a conference of adult educators. Midway through the workshop someone asked if I personally knew anyone who used algebra in their job. I couldn't think of anyone at the moment. Neither could anyone else. Someone thought there were some engineers or physicists who did, but they didn't know when or how. When queried, nobody in the room could recall ever using algebra in any part of their lives outside the classroom.
"If in a group of 30 adults nobody uses algebra and doesn't know anyone who does it can't be very important; so why," the question came, "do we try to teach algebra to everyone?" I confessed I had no ready answer to that question, other than it's required to survive school.
The Oregon Department of Education graduation standards as well as Oregon University System admission standards require a year of high school algebra. The reason as near as I can figure out is that one must take first-year algebra in order to take second-year algebra in order to take trigonometry in order to take calculus, the capstone course. Given the vast number of high school freshmen that are squeezed into this algebra-to-calculus pipeline and the few college students that come out the other end, it hardly seems worth the effort. Especially if all one has to show for it is the standard first-year algebra course in which one learns to manipulate symbols according to prescribed rules that are, at best, dimly understood.
At worst the result is boredom or confusion, and a distaste for all things algebraic. Those for whom this happens are in famous company. "I despised algebra," Eisenhower recalls in At Ease: Stories I tell to Friends. "I could see no profit in substituting complex expressions for routine terms and the job of simplifying long, difficult equations bored me. I by no means distinguished myself." In Dreams, Memories and Reflections, Jung tells of his terror as he sat watching his algebra teacher at work: "He would scribble a few letters on the blackboard. I had no idea where he got them and why he did it—the only reason I could see was that it enabled him to bring the procedure to what he felt was a satisfactory conclusion. I was so intimidated by my incomprehension that I dare not ask any questions. Mathematics classes became sheer terror and torture to me."
At the other end of spectrum are those of us who learned the rules and found no difficulty in manipulating symbols and getting things to turn out right, rewarded by the distinction one gained from getting good grades in math. We weren't doing anything that now-a-days couldn't be done more efficiently and accurately by an electronic symbol manipulator. We were simply slower versions of machines, programmed by our teachers to carry out procedures without worrying about meaning. What I gained from the experience is questionable, other than a false impression of what mathematics was about. (I remember being somewhat chagrined when I discovered—after committing myself to majoring in math and deciding that I wanted to be a mathematics professor—that mathematics was something other than mastering evermore complicated algorithmic procedures. But then I discovered it was a much more creative and absorbing subject than I had ever imagined.)
For the most part, things haven't changed much, other than the size of the textbooks. The contemporary 700-page text in use in a local high school is filled with boxes of definitions, formulas and techniques and pages of worked out examples, telling the student what to write down and how to think, followed by pages and pages of exercises to practice what they've been told—I counted over 265 elementary factoring exercises, over 300 exercise concerning the arithmetic of rational expressions and some 260 exercises manipulating radicals. One can understand why the course breeds boredom and aversion, not to mention confusion, if what one is told to do and how to think makes no sense.
Then there are the so-called applications, all those word problems that are supposed to show how useful algebra is in everyday life. In reality, all of the problems in a first-year algebra book can be solved without using algebra at all; some number sense and a good sketch will do. And the problems are all contrived, witnessed by the fact that a group of thirty adults can't recall a single application of algebra in their lives outside of school.
It's not that studying algebra need be a worthless endeavor, or a difficult or boring one. First, one must understand, that much of what one studies in algebra, as in any other mathematical topic, may never be encountered again and isn't necessary to lead a successful and fulfilling life. One might argue that everyone ought to know how to deal with simple formulas and equations and not panic when they see an x, but that's not the point. The point in studying algebra, or any other branch of mathematics, is developing one's mathematical talents so one might deal confidently and capably with whatever mathematical situations arise in one's life, vocationally or avocationally. That doesn't mean one will study in school all the mathematics they someday may want to know—some of it may not be invented yet. Rather, it means that one develops mathematical maturity to pursue it when it does arise. In one sense, it doesn't matter what mathematics one studies in school, as long as one studies mathematics.
I can't predict what mathematics my students will want to know in the future, but I can help them discover that mathematics is not an arbitrary and capricious world; that learning mathematics is not a matter of mastering rules, but sharpening intuitions and developing understanding. Once that begins to happen, questions about the utility of what's being studied disappear, and the satisfaction and sense of accomplishment that comes with developing one's mathematical potential takes over.
Reaching that point in an algebra class isn't accomplished by wading through 700-page textbooks laden with rules and procedures for manipulating mathematical symbols. In algebra, like any other mathematical topic, I find that insight and understanding is most easily built by examining physical settings from which the mathematical ideas and procedures are naturally drawn. Tile patterns provide such a setting that is relatively easy to deal with in a classroom. (See, for example, Picturing Algebra in the Math and the Mind's Eye section of the MLC catalog.) As students examine tile patterns, and their extensions, they are lead to handling algebraic expressions and solving equations in natural ways that inherently make sense to them, without the need for prescribing procedures and repetitious drill.
Of course, if one has always presented algebra in the standard textbook fashion, it takes some getting used to. The week after the workshop with adult educators, I spent three days with a group of teachers going through some of the Mind's Eye algebra activities. At first, as is generally the case, I sensed the resistance of those who had invested years in teaching paper-and pencil textbook techniques. As teachers allowed themselves to experience algebra in a different way, the mood changed. "I was mentally against the tile," one algebra teacher told me. But then, by the end of the second day, she found working with tile made more sense to her than the textbook routines she was accustomed to using. She was seeing how things worked. "I understood more in two days," she exclaimed with excitement, "than I had in ten years of algebraic manipulations!"
There are better ways of developing algebraic literacy than slogging through 700 pages of rote learning and deadening drill.