What's Basic?

What's Basic?

By Dr. Eugene Maier

What is a basic skill in mathematics seems to be an imponderable that defies description. On the one hand, some people use the phrase as if everyone knows what it means, as in an editorial that appeared recently in the Sunday Oregonian concerning the adoption of mathematics texts in Portland Public Schools. A subhead said the Portland schools were going to put "problem-solving and discovery before basic skills." The head left me wondering, "So what is a basic skill?" The editorial didn't provide an answer other than the parenthetical comment "number manipulation" after the term was used in reference to mathematics.

On the other hand, one can find long lists of "basic mathematical skills." These are invariably open to debate. In an apparent attempt to find consensus among math educators on what ought to be on a list, the National Council of Teachers of Mathematics April issue of Dialogues includes a questionnaire containing 44 mathematical tasks and asks readers to check whether or not each task is a basic skill. To clarify what is meant by "basic", they offer the following definition, "Skills are usually called "basic" when they (1) are deemed necessary for later mathematics or (2) they are deemed so important that everyone should learn and be tested on them."

I found there isn't any item on the list I was willing to check as "basic." The list, for the most part, is a collection of procedures—calculating this, graphing that, solving things in particular ways—and I am hard pressed to think of any mathematical procedure that I would say everyone should learn and be tested on. One could lead a meaningful and productive life with little knowledge of any one item on the list; and one could be quite adept at a variety of mathematical endeavors with hardly any knowledge of the listed items. On the other hand, to pass the required math courses in a typical school math program, one would need a number of the listed skills. Over the years I've noted that many lists of basic mathematical skills could be more aptly described as school survival skills—you need them to get through school but not for much else in life.

An inherent danger I see in most lists of basic skills is they encourage carrying out formulaic procedures at the expense of meaning and insight. The development of conceptual understanding and a problem-solving mentality has far greater applicability and obviates the need of formulas and procedures, other than those a student adopts on their own. For example, one of the proposed basic skills on the NCTM list is "using formulas to find the area and volume of common shapes." I've watched people with a formula knowledge of area go through all kinds of machinations trying to find the area of a common geoboard triangle like that shown in Figure 1, where the area of each of the squares shown is 1.

Finding a height and base of this triangle is challenging for most folk, while anyone who knows what area means and has a bit of ingenuity can readily find that the area is 51/2 without the use of any formulas. (Encase the triangle in a rectangle of 12 squares as shown in Figure 2, and subtract away the areas of corners A, B and C. Corner A is half of a rectangle of 6 squares and hence has area 3, similarly the areas of B and C are 2 and 11/2, respectively. Hence, the area of the triangle is 12 - (3 + 2 + 11/2).

Conceptually, finding the area of a plane figure is straightforward: One decides what the unit of measure will be (one inch, one centimeter, one mile, or any other length of one's choosing) and then determines how many unit squares (i. e., squares whose side is the unit length„square inches, square centimeters, etc.) fit into the figure at hand. For rectangles whose sides are multiples of the unit length (as in Figure 3), one sees the area is the product of its dimensions.

For other rectangles (e.g., one whose dimensions are ¿2 and 1.769), one makes the axiomatic agreement that their area, too, is the product of their dimensions. Given this and a supply of figures, students, working individually and with one another, will devise their own methods for finding areas, including the development of formulas they find useful. They also will have developed a conceptual understanding of area, as well as ownership of a variety of methods for determining it that will serve them far better than a knowledge of area that's no more than a collection of memorized formulas.

Another negative aspect of lists of basic skills is the impression they give that mathematics is a hierarchy of techniques and manipulations, devoid of any plot, rather than a cohesive, developing body of knowledge with a rich history „ a vibrant subject that can be found in all kinds of human endeavors, from the mundane to the exotic. Perhaps, we should abandon our efforts to compile such lists and concentrate instead on unfolding the story of numbers and shapes and how we measure things. Any school version of this story ought to begin with the intuitive knowledge the child brings to the classroom, but, other than that, there're many ways to tell the story. So rather than fragmenting mathematics into a bunch of tasks to be mastered, how about setting a course or, as we say these days, establish a curriculum, through which we can guide our students as they relive this fascinating story for themselves.