The “Aha!” Factor

[Rescued from my old blog.]

For young children, mathematical concepts are part of life’s daily adventure. A toddler’s mind grapples with understanding the threeness of three blocks or three fingers or one raisin plus two more raisins make three.

Most children enter school with a natural feel for mathematical ideas. They can count out forks and knives for the table, matching sets of silverware with the resident set of people. They know how to split up the last bit of birthday cake and make sure they get their fair share, even if they have to cut halves or thirds. They enjoy drawing circles and triangles, and they delight in scooping up volumes in the sandbox or bathtub.

And Then Comes School

After a few sessions of “3 + 1 = 4, 3 + 2 = 5, 3 + 3…,” the little ones begin to whine. Older children recoil from long division. By the time they reach high school, students face torture like: “The product of an integer and the next greater integer is 20 less than the square of the greater integer.” Homework becomes a tedious chore to put off as long as possible or finish with slapdash speed.

Why, as they go through school, do so many children learn to hate math? And why do so many home school moms feel inadequate to teach math?

School ruins mathematics for most people, distorting a discipline that is half art and half sport by turning it into boring lecture and drill. Imagine a piano teacher who insisted her students spend six years on scales and exercises of gradually increasing difficulty before she let them attempt a piece of actual music. Or a football coach who made his team run laps and do sit-ups every day, but only let them play two to three games a year — and scrimmage games at that. How many people would become bored with music or learn to hate football under such instruction?

And then there are the modern “reform math” programs, which avoid lecture and drill, keeping the children busy with hands-on group activities while teaching neither the how nor the why of math. These programs are like a piano teacher who never gave practice assignments at all, but told her students to listen to a CD and fingerpaint their feelings about the music. We don’t want our students to be bored or to hate math, but is this our only alternative?

As every coach knows, skill grows through practice, practice, practice. But practice is meaningless unless the team has a real game to play. And the best practice takes advantage of the benefits of cross-training by emphasizing variety rather than repetitive drill.

Instead of drudgery, mathematics should be a game of discovery. It should give children the same Eureka! thrill that sent Archimedes running through town in his birthday suit. I call this the “Aha!” factor, the delight of solving a challenging puzzle. The “Aha!” factor is what I aim for when I bring home a brainteaser book from the library or a new strategy game from the store. And it is what I find emphatically missing from most math textbooks.

Mathematical Cross-Training

Real mathematics is an exploration of patterns and mysterious connections. School math is page after page of arithmetic. Real math is an M. C. Escher drawing of interlocked fish splashing across a page. School math is long division by hand. Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1+3, 1+3+5, etc.), and the satisfaction of seeing why it must be so. School math is a rousing chorus of, “Ours is not to reason why, just invert and multiply.”

Mathematical cross-training is games, puzzles, story problems, and the challenge of thinking things through. Of course, our students do need to learn how to perform those routine operations, just as piano players do need to practice scales and football players to do push-ups. Just as important, our students need to learn why those operations work. But they should never be led to think that routine operations are the point of mathematics.

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations, he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students, he may give them a taste for, and some means of, independent thinking.

-George Polya
How to Solve It

As a homeschool mom who loves math, I want to help other homeschoolers see the variety and richness of the subject. I encourage parents to look beyond their textbook — a useful tool, but such a limited one — and explore the adventure of learning real mathematics, math as mental play, the essence of creative problem solving. This is what we need to teach our children.

Mathematics is not just rules and rote memory. Math is a game, playing with ideas.

I hope this blog will be a place where we can play around with ideas about learning, teaching, and understanding math. For me, it usually happens in that order. There’s something about teaching: it forces the teacher to work through a topic again and again until she understands it.

And if we can find a way to give our children a good taste of that “Aha!” feeling, the thrill of working through a challenging problem and figuring it out, then we won’t be able to keep them away from math.

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, now available at your favorite online book dealer.