Teaching the Basics
By Dr. Eugene Maier
“City school officials want more basic math,” the headline said. Given the current educational climate, it's a headline that might have occurred in any number of papers. This one happened to be in the Pittsburgh Post-Gazette. I noticed it when in Pittsburgh for a conference.
You can probably guess the gist of the story. Math scores on state tests show little improvement, so the School Board President wants school administrators to drop the elementary math curriculum they have been using—one of the so-called “reform” programs—and look for one "that has shown better results."
“The facts are there” the Board President asserts, “that our children do not have the basics in math,” which, according to her, are “adding, subtracting, multiplying, and dividing.” She doesn't specify how these operations are to be carried out, but I don't think she's advocating calculators. If she's like most folk I know who talk about adding, subtracting, etc., as “the basics in math,” they mean becoming proficient in carrying out paper-and-pencil algorithms.
Why are these “the basics”? I wonder about that. For me, the “basics” of a subject are those things which form its base, its foundation. They are an essential part of the structure of a subject without which the subject would collapse. A criterion not met by any algorithmic procedure.
But suppose one believes that the point of school math isn't learning about the structure of the subject, but about mastering the mathematics that has applicability in one's life. Certainly, in that case, learning how to add, subtract, multiply, and divide is basic. Again I disagree. To make use of mathematics, it's much more important to know when to add, subtract, multiply, or divide than to know paper-and-pencil procedures for doing so. Besides, calculators have rendered the latter redundant. Anything that can be avoided, can scarcely be basic.
The only explanation I have why people believe computational procedures constitute the mathematical “basics” is because that is what they remember being taught in elementary school. We tend to think of elementary school as the place we learned the rudiments of arithmetic. Whatever we learned there, regardless of its relevance, becomes the basics for us. In a kind of role reversal, we let the elementary curriculum of our day define what ought to be taught about a subject, rather than letting the present state of the subject, and its technological tools, define the curriculum.
I admit this is all speculation on my part. I could be wrong about what the School Board President had in mind when she said the basics were adding, subtracting, multiplying, and dividing. Perhaps she was thinking of these arithmetical operations in a much broader context than my narrow interpretation of her statement. Perhaps, she was thinking also
…about the structure of our notational system and how we record the numbers upon which we are operating,
…about the variety of ways in which numbers and their operations can be modeled and the insight one gains from examining and discussing these models,
…of the diverse tools and methods available for performing computations and how one selects those that are best suited for the occasion,
…of the strategies and algorithms students devise for dealing with numbers and how this mathematical creativity is developed and nurtured,
…of meaningful explanations of how the operations on whole numbers extend to the integers, rationals, and other sets of numbers,
…of how to frame situations from everyday life in mathematical terms and deal with them accordingly,
Oh, enough of this. I'm not speculating, I'm fantasizing.