An Eminent Math Professor Says "innumeracy" Rivals Illiteracy as a Cause for Concern in America
Evidencing the growing concern over math education in this country, a book called Innumeracy: Mathematical Illiteracy and Its Consequences (Hill and Wang, $16.95) has been climbing the best-seller list; its author, John Allen Paulos, 43, a professor of mathematics at Temple University in Philadelphia, notes that math phobia may begin in our schools, but its symptoms are pervasive throughout American society. Paulos, who received his Ph.D. from the University of Wisconsin, told associate editor Dan Chu why our cultural attitudes toward mathematics must change if we are to thrive—or even survive—in an increasingly competitive and technologically oriented world.
How do you define "innumeracy"?
It is the mathematical equivalent of illiteracy. Innumeracy is an inability to deal comfortably with numbers, probabilities, logic and other basic notions of math. I emphasize basic. You can be numerate even if you don't understand quantum mechanics.
What kind of people tend to be innumerate?
The term covers a wide range of people, including those who are otherwise highly intelligent and knowledgeable. What's annoying to me is that while illiterates are ashamed of their inability to read, innumerates often take a kind of pride in their mathematical ignorance. People who would never admit to not knowing anything about Shakespeare—even if that were the case—will openly boast that they can't balance their checkbooks.
Where do you see the signs of innumeracy in our society?
Almost everywhere you look. I once heard a misguided TV meteorologist report that there was a 50 percent chance of rain on Saturday and 50 percent chance on Sunday, so, he concluded, "It looks like a 100 percent chance of rain this weekend." And it isn't always clear to some people that a dress whose price has been "slashed" 40 percent and then another 40 percent has been reduced in price by 64 percent, not 80 percent.
Why do people shrink from math?
There are so many popular misconceptions about the nature of mathematics. It is thought to be just computation. Or the feeling that math is "cold" or "irrelevant." Probably the most crippling of all is a notion that there are mathematical minds and nonmathematical minds, and that the latter needn't even bother to try. We need to dispel the notion that math is for scientists, engineers, geniuses, nerds, even the Ruin Man, but not for regular people like you and me.
How might innumeracy cloud the average person's ability to make decisions?
A doctor who was talking about a medical procedure for my wife told us that there was a million-to-one risk associated with the operation. A little while later he said that it was 99 percent safe. He obviously was trying to tell us that the risk was small, but the numbers he used didn't mean anything to him. The difference between 99 percent and one-in-a-million is equivalent to the difference between having $100 in your pocket and having $1 million.
Don't minds boggle at gargantuan numbers?
To many people, a million or a billion or a trillion all mean about the same—that is, "a lot"—when in fact the gap between each of them is vast. I illustrate those differences with the example that it takes only about 11½ days for 1 million seconds to pass but 32 years for 1 billion seconds to pass. And it takes 32,000 years for 1 trillion seconds to pass, a span more than six times that of man's written history.
What may be a consequence of the public's inability to grasp big numbers?
Without some appreciation of large numbers commonly used, you can't respond intelligently to public debates over, say, warheads that carry a megaton (2 billion pounds) of explosive power or discussions of $1 trillion in public debt. Every citizen should be able to make estimates of a rough order of magnitude. When you know there are about 250 million people in the U.S., you ought to be able to estimate that every $1 trillion in public debt translates to $4,000 for every man, woman and child in this country.
What about applying a feel for numbers to personal decisions?
Associated with an appreciation for numbers is a feeling for probabilities. If you are concerned about traveling overseas for fear of becoming a victim of terrorism, for example, you might consider that there is one chance in 5,300 of dying in a car crash on our streets and highways each year. In contrast, even with the terrible headlines lately, the chance of an American being killed by terrorists abroad is approximately one in 1.1 million annually when averaged over the past decade. Innumerates tend to personalize and say, "Yes, but what if I'm that one?" Well, that isn't likely, and you weigh risks on what is likely to happen.
What do you see as some of the other drawbacks of innumeracy?
One is a tendency to overvalue coincidences, which are a lot more common than most people think. If you and I are strangers sitting next to each other in a plane, the chances are better than 99 in 100 that I would know somebody who knows somebody who knows you. We probably won't discover this link between us but it's almost always there.
What is the value of understanding this?
If you don't, you are likely to be more impressed with coincidences than the situation warrants. For example, we've all heard about people who have had dreams that later came true. But with 250 million Americans each dreaming about two hours a night, that's a half-billion hours of dreaming nightly. That some of the dreams are seemingly predictive is not amazing. What would be amazing is if none of them came true. Yet people without a feel for probabilities are likely to attribute such coincidences to some astrological harmony or other parapsychological silliness.
Wouldn't numeracy, then, be likely to breed a certain kind of skepticism?
Yes, and I think that's important because we're inundated by numbers every day, and we have to make some sense of them. When you see a commercial that tell you a new toothpaste reduces cavities by 200 percent, does that mean it is capable of removing all of one's cavities twice over? And you should be able to distinguish between correlation and causation. Suppose that it can be shown that children with bigger feet spell better. Does that correlation mean that big feet have anything to do with spelling? Maybe it is because older children tend to have bigger feet, and they spell better because they are older.
How can the teaching of math be improved in this country?
It's often taught in a rigid, authoritarian way that is divorced from our daily lives. Yet, even in elementary school, you can get across the rudiments of probability by playing board games, or work on decimals by talking about batting averages. Ask whimsical questions: What percentage of pupils in the class have fathers who are bald? I think one way to teach arithmetic is never to write down a number without preceding it with a dollar sign. And that's not entirely a joke because money is not an abstract, esoteric subject but a real one.
Does increasing dependence on computers and calculators hamper the ability of students to manipulate numbers on their own?
No, because computation is only a small part of what mathematics is about. I think computers and calculators should be even more integrated with the study of math, though they are not a panacea. Estimation, comparing, interpreting graphical and statistical data—these kinds of things are much more important than just computation, and too often they aren't being taught.
Is it true that women are especially intimidated by mathematics?
Even today, math is commonly regarded as a masculine subject, and that's really too bad. Women who do everything in their power to avoid a chemistry or an economics course with mathematics or statistics prerequisites may end up in lower-paying careers. An attitude that, in effect, banishes half of society from careers in the sciences right from the outset is plainly absurd.
Are we destined to lag behind other countries in general math education ?
Many countries seem to have a higher expectation of their kids in this regard. It is worrisome but not unalterable. Teachers, by and large, are a dedicated bunch, but they're limited by the misconceptions and cultural attitudes that pervade our society. There isn't much teachers can do if so many of us go around saying "Ugh, I hate math" and think we're being funny.