The PTA Does Fractions

By Dr. Eugene Maier

It was math night. We parents were listening to the fifth-grade teacher describe her approach to teaching fractions. Before showing us the algorithm she taught for dividing fractions, she asked us to divide two fractions using whatever method we were taught. Mass confusion ensued. As I looked about the room, I saw moms and dads conversing quizzically with one another. I heard fragments of hushed conversations: "What's the rule?"; "There's something about inverting."; "Is this right?" No one I could see had any confidence in what they were doing.

"How much time and energy," I wondered, "was spent teaching these folks how to divide fractions? Here's a roomful of well-educated adults from a middle-class neighborhood in a university town. I suspect they have all been 'successful' at school. What went wrong?"

I was reminded of this incident by the morning paper, which brought me face-to-face with what went wrong, and is still going wrong, with the way fractions—and most other mathematics—is taught.

An article on charter schools included a two-column, three-inch photograph of Kevin, a sixth-grader, standing in front of a white board in his math class at ATOP (Alternative Thought Orientation Process), a school in Phoenix, Arizona, emphasizing the "basics." The photo shows Kevin, face buried in his hand, struggling, as the caption says, "for the answer in math class." Behind Kevin, one sees ten exercises in multiplying fractions under the heading "Cross Cancel." Kevin, with the help of his classmates we are told, has done nine of them. A replica of some of his work is shown below.

cross cancelling

Cross canceling, it appears, has something to do with reducing fractions before multiplying them. I don't know how adept Kevin is at carrying out the process, but I'm willing to wager he hasn't any idea of why it works—other than the teacher said so. I suspect, as far as Kevin is concerned, it's just another magic trick among all those he's been taught in his math class. For Kevin—and countless others—acquiring basic mathematical skills becomes mimicking magic tricks.

There's not just the magic of "cross-canceling," but there's also the magic of "inverting and multiplying," "moving the decimal," "cross-multiplying," "adding exponents," and on and on. Kevin and his classmates will learn these magic tricks a few at a time and gain enough proficiency with them to pass the test of the moment, but by the time they have children and are going to math night at the PTA, the magic tricks will have gotten all jumbled up so one no longer remembers quite how things are done or which trick makes aces drop out of sleeves or which pops rabbits out of hats. And so, magic night, oops, I mean math night, at the PTA becomes a mystical mess. 

What's gone wrong? It's a misunderstanding of what's basic knowledge of mathematics. Somehow or other, the misconception has become ingrained in a large segment of the public—certainly in the parents and educators that established Kevin's school—that the basics in mathematics consists of rules for carrying out procedures; rules like cross-canceling that need to be memorized and practiced. To do this, students resort to rote learning which lacks meaning and context and provides no recourse once a single misstep is made or the slightest confusion occurs.

Basic long-term competency in mathematics isn't developed through memorization and drill on proscribed procedures. Its developed through the nurturing and enhancing of the number sense innate in all human beings (see Num·ber Sense/Numb·er Sense). By providing students with a wide range of concrete experiences involving the variety of numbers and operations found in elementary mathematics, their number sense will develop and expand and they will create mental images for how things work—images that can be called upon whenever needed. They will have acquired the most basic competency of all: the ability to devise their own methods for dealing with a mathematical situation.

So when it's math night at the PTA, or any other time a mathematical problem arises, mom and dad will not be faced with extracting the right rule from that mass of rules that have become blurred and entangled over time. Rather, they will evoke their number sense and collection of mathematical models and images to construct their own process for resolving the situation at hand.

And young Kevin will not stand in front of the class, face buried in his hand, trying to remember and replicate by rote memory some prescribed procedure for multiplying fractions. Instead, with eyes wide open, he will describe to the class how he, using his well-developed number sense and his images of fractions and the arithmetical operations, figured out on his own how to multiply fractions.