The Christopher Columbus Stuff
By Dr. Eugene Maier
"It's not that they don't know the Christopher Columbus stuff, they lack the common sense and judgment to be successful in the workplace." That, as reported in the local paper, was the response of an administrator in a local engineering firm to a reporter's query. The reporter was asking about the on-the-job performance of adults who had acquired high school educations as part of an effort to move welfare recipients into the workforce.
I was reminded of those programs which stress the "Christopher Columbus stuff" of mathematics: adding, subtracting, multiplying and dividing whole numbers, common and decimal fractions, and signed numbers; calculating percents; evaluating geometric formulas; etc., etc. Procedures that one can learn by rote, test on successfully, and come away with hardly an iota of mathematical common sense or perception. Unfortunately, without the latter, the Columbus stuff isn't of much value—especially these days when there are machines that can do all of it more efficiently than human beings can.
One can argue about the worth of knowing the Columbus stuff and whether mathematical sense and insight is really important. Let's assume that it's all valuable and ask why it is we can get the former without the latter.
The problem, as I see it, is getting things backward—believing number sense and insight will emerge from mastering the Columbus stuff, rather than focusing on developing number sense and insight and getting the Columbus stuff as a byproduct.
The latter is the natural way to do things. Neuropsychologists working in the rapidly expanding field of mathematical cognition maintain that human beings are endowed at birth with an innate sense of numerosity and the capacity and inclination to develop mathematical procedures without formal education. Formal mathematics education, to be effective, should connect with and build upon this natural intuition, abandoning the rote learning of mathematical procedures. (See, e.g., Stanislaw Dehaene's The Number Sense or the recently published What Counts, How Every Brain is Wired for Math by Brian Butterworth.)
Teaching for mastery of the Columbus stuff, rather than enhancing mathematical sense, can have the opposite effect. One can learn, and become quite skillful at carrying out a prescribed mathematical procedure without having any conceptual understanding of what's happening. If that's the case, the learner is at the mercy of, rather than in control of, the procedure. If called upon to adapt what's been learned to a different setting, the learner is at a loss on how to proceed and loses all confidence in their mathematical ability. Or if what's being taught doesn't connect with the learner's innate mathematical knowledge, the learner may decide, like the young Winston Churchill, that mathematics is a "hopeless bog of nonsense." The result, rather than a feeling of competence and confidence in one's mathematical common sense, is math anxiety and avoidance.
One can understand the appeal of teaching the Columbus stuff. One can organize it neatly into little bits and pieces of so-called basic skills, demonstrate a skill and drill the students until "mastery" has been achieved, and then move on to the next bit. It's all very orderly—and quite objective if one ignores that someone has to decide which bits and pieces are to be included and what passes for mastery.
On the other hand, teaching with emphasis on developing students' innate mathematical sense and understanding isn't as clean cut, but it can be done. It means listening to students discuss their methods for approaching a mathematical situation rather than telling them your method. It means providing experiences that build mathematical intuition rather than exercises for drill and practice. It means allowing students to discover and deal with their false starts and misimpressions rather than rescuing them from their difficulties. Depending on what happens, you may have to change lesson plans in the middle of a class, and you are likely not to cover all the material you had in mind. But your students will be making sense out of mathematics.
If your goal is preparing students to be confident and successful users of mathematics, concentrating on the Columbus stuff is likely to land you oceans away from where you intended to be.