In math, "elegant" means "cool"!

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On the popular TV game show The Weakest Link, the acerbic host peppers the contestants with questions, and they have only a few seconds to answer if they want to win any money. The questions cover many topics, but every so often one is mathematical. A typical math question would be, "In math, how much is one eighth of one thousand?"

The contestants don’t have calculators. They don’t even have pencil and paper, and even if they did, they don’t have time to do long division. They have to think fast and use whatever shortcuts they can. Isn’t 250 a fourth of 1000? So an eighth would be 125? "125!" "Correct!" and it’s on to the next question about who was the second President of the United States.

Why do people watch The Weakest Link? Although the show’s environment seems very unrealistic at first, the fact is that life does hurl questions at us every day — all kinds of questions, and some of them mathematical. We need a lot of number sense to get along, and the earlier we started developing that number sense, the better.

Our national standards document, Principles and Standards of School Mathematics (National Council of Teachers of Mathematics, 2000), defines problem solving to be "engaging in a task for which the solution method is not known in advance." I have a small problem with this definition: the word the. It implies that for each problem, there is a single solution method, and if we knew what it was, we wouldn’t have a problem.

In fact, if a mathematical problem has a solution (and some do not), then almost certainly it has many methods of solution. In school, the best solution to a problem is often considered to be the one the teacher said to use or the one just taught. But in real life, isn’t the best solution to a problem the one that gets the answer fastest and with the least work?

I started my career as a university-level research mathematician. I can tell you that in mathematical research, the best solution to a problem is always the one that gets the answer fastest and with the least work. Professional mathematicians call it the "elegant" solution. Saying that a calculation or proof is "elegant" is the highest compliment mathematicians pay one another.

At the same time, Ph.D. mathematicians don’t have much regard for a solution that’s obtained by grinding through a standard algorithm. They call this kind of calculation "turning the crank," and turning the crank ranks at zero on the elegance scale.

For example, long division, necessary though it is for us to know, is not an elegant solution to any problem. It’s too much work, and it takes too long. If we want to take one eighth of 1000, we can recognize that taking an eighth of something is the same as halving it three times. "1000, 500, 250, 125, and there’s our answer." The mathematician says "elegant." Kids probably say, "Cool!"

Recently, a Chinese mathematics educator, Liping Ma, published a study of U.S. and Chinese math instruction, Knowing and Teaching Elementary Mathematics (Lawrence Erlbaum Associates, 1999). One of the problems she used in her study was a division of a mixed number, 1³/₄ by a fraction, ¹/₂. What’s the standard approach here? First we convert the mixed number to a fraction, ⁷/₄, and then we "invert and multiply," obtaining the answer (⁷/₄) times; (²/₁) = ⁷/₂.

Kids have a lot of trouble doing this, and some of the U.S. teachers did, too. Liping Ma doesn’t think U.S. teachers are any dumber than Chinese teachers, but she does think we are not flexible enough in our thinking.

We can do a better job with problems like this if we recognize that in mathematics we are teaching problem solving all the time. Can you restate this problem in terms of something more familiar? What does it mean to divide 1³/₄ by ¹/₂?

I happen to love geometry and measurement, so it’s natural for me to recast problems like this in geometric terms. Suppose I have a little half-foot ruler (I used to have such a thing, actually). If I have a board 1³/₄ feet long, how many of these half-foot units are there in the length of this board? Now I can see the problem. There will be two units in a foot, three in 1¹/₂ feet, and another half unit to make up 1³/₄, so the answer must be 3¹/₂, or ⁷/₂. Isn’t this a more elegant approach? Even a cool approach?

I believe teaching mathematics includes teaching the search for cool or elegant solutions to problems. We certainly can’t ignore the algorithms; we have to teach "invert and multiply" as the way to divide fractions. It’s an important piece of pre-algebra. But we ought not to ignore the shortcuts and the nonstandard visualizations, either. We should encourage students to think about problems in as many ways as possible and to find as many paths to the solutions as possible.

The search for elegant solutions leads to what was called "mathematical power" in the earlier stages of the curricular reform movement: the ability of students to engage confidently with a broad range of problems. The more shortcuts and tricks we know, the bigger our problem-solving toolkit becomes.

So if the search for elegant solutions leads to confident students, we want to encourage that process in every way we can. If you’ve ever said in a math class, "That’s not the way we do it," I recommend making a New Year’s resolution never to say such a thing again.

Instead, say things like this:

• "Do you know another way to do that?"
• "Did anyone find a shorter way to do this?"
• "Is there something special about this number we can use?"
• "Can you tell me another reason why you think this is the right answer?"

Quick (you’re on The Weakest Link), what is 11 times 19? Can you think of several ways to do it?

If you thought of 11 times 19 as being 19 more than 190 (or as being 11 less than 220), then you are looking for that elegant solution. (And if this didn’t happen for you, it’s not because you’re "not good at math," it’s because you haven’t had the practice you need to solve problems with agility.)

Most of the mathematical problems that arise in real life require only a good feel for numerical relationships, or what folks sometimes call "a head for numbers," rather than knowledge of formal procedures. Here’s an example. My county has just completed a property revaluation, and the taxable value of my house has been raised about 50 percent. Does this mean I’ll pay 50 percent more in taxes? No, because the County Commission will lower the tax rate. I might have to pay more in taxes, or less, or the same, depending on the new rate. How much must they lower the rate in order for me to pay the same in taxes?

Did someone say 50 percent? No, that’s not right. The county multiplied my taxes by 1.5, or ³/₂, by raising the valuation. If we multiply something by ³/₂, then to put it back where it was we need to multiply by ²/₃. So I hope the new tax rate will be ²/₃ of the old one, or 33 percent less.

All we needed to know here was that ³/₂ times ²/₃ is 1, information we have in our heads. No fancy procedures, no calculators. This is the real mathematics, and everything we do in the classroom should be helping students develop this kind of feel for numbers, the agility some folks are calling "numeracy."

Every semester, students come into my office apologizing. "I was never very good at mathematics," they say. Of course, some people are better at mathematics than others, as is true for every human endeavor, but I see no reason for any student to speak with so little self-assurance. What the students really mean is, "I never built a big enough toolkit to deal confidently with mathematics." To help your students build that confidence, treat all of mathematics as problem solving. Help them search for illuminating shortcuts. Value the ideas they have that work, and suggest as many new ideas as you can.

I promise you, everybody will have more fun in your math classes, they’ll learn more, and they’ll remember more.