Mathematical Illiteracy Knows No Social Or Class Boundaries

Why is innumeracy so widespread even among otherwise educated people? The reasons, to be a little simplistic, are poor education, psychological blocks, and romantic misconceptions about the nature of mathematics. My own case was the exception that proves the rule. The earliest memory I have of wanting to be a mathematician was at age 10, when I calculated that a certain relief pitcher for the then Milwaukee Braves had an earned run average (ERA) of 135. (For baseball fans: He allowed five runs to score and retired only one batter.) Impressed by this extraordinarily bad ERA, I diffidently informed my teacher, who told me to explain the fact to the class. Being quite shy, I did so with a quavering voice and a reddened face. When I finished, he announced that I was all wrong and that I should sit down. ERAs, he asserted authoritatively, could never be higher than 27. At the end of the season, the Milwaukee Journal published the averages of the Major League players, and since this pitcher hadn't played again, his ERA was 135, as I had calculated. I remember thinking of mathematics as a kind of omnipotent protector. You could prove things to people and they would have to believe you whether they liked you or not. So, still smarting from my perceived humiliation, I brought in the paper to show the teacher. He gave me a dirty look and again told me to sit down. His idea of a good education, apparently, was to make sure everyone remain seated. Though not dominated by martinets like my teacher, early mathematics education is generally poor. Elementary schools by and large do manage to teach the basic algorithms for multiplication and division, addition and subtraction, as well as methods for handling fractions, decimals, and percentages. Unfortunately, they don't do as effective a job in teaching when to add and subtract, when to multiply or divide, or how to convert from fractions to decimals or percentages. Seldom are arithmetic problems integrated into other schoolwork--how much, how far, how old, how many. Older students fear word problems in part because they have not been asked to find solutions to such quantitative questions at the elementary level. Although few students get past elementary school without knowing their arithmetic tables, many do pass through without understanding that if one drives at 35 m.p.h. for four hours, one will have driven 140 miles; that if peanuts cost 40 cents an ounce, and a bag of them costs $2.20, then there are 5.5 ounces of peanuts in the bag; that if 1/4 of the world's population is Chinese and 1/5 of the remainder is Indian, then 3/20, or 15 percent, of the world is Indian. This sort of understanding is, of course, not the same as simply knowing that 35 x 4 = 140; that (2.2)/(.4) = 5.5; that 1/5 x (1 - 1/4) = 3/20 = .15 = 15 percent. And since it doesn't come naturally to many elementary students, it must be furthered by doing numerous problems, some practical, some more fanciful. Estimation is generally not taught either, aside from a few lessons on rounding off numbers. The connection is rarely made that rounding off and making reasonable estimates have something to do with real life. Grade-school students aren't invited to estimate the number of bricks in the side of a school wall, or how fast the class speedster runs, or the percentage of students with bald fathers, or the ratio of one's head's circumference to one's height, or how many nickels are necessary to make a tower equal in height to the Empire State Building, or whether all those nickels would fit in their classroom. Almost never is inductive reasoning taught or are mathematical phenomena studied with an eye toward guessing the relevant properties and rules. A discussion of informal logic is as common in elementary mathematics courses as is a discussion of Icelandic sagas. Puzzles, games, and riddles aren't discussed--in many cases, I'm convinced, because it's too easy for bright 10-year-olds to best their teachers. The intimate relationship between mathematics and such games has been explored most engagingly by mathematics writer Martin Gardner, whose many charming books and Scientific American columns would make exciting outside reading for high school or college students (were they but assigned), as might mathematician George Polya's How to Solve It or Mathematics and Plausible Reading. A delightful book with something of the flavor of these others, but at an elementary level, is I Hate Mathematics by Marilyn Burns. It's full of what elementary math textbooks rarely have--heuristic tips on problem solving and whimsy. Instead, too many textbooks still list names and terms, with few if any illustrations. They note, for example, that addition is said to be an associative problem since (a + b) + c = a + (b + c). Seldom is any mention made of an operation that is nonassociative, so the definition seems unnecessary at best. In any case, what can you do with this piece of information? Other terms seem to be introduced with no rationale other than that they look impressive in boldface type inside a little box in the middle of a page. They satisfy many people's conception of knowledge as a kind of general botany in Which there's a place for everything and everything has its place. Mathematics as a useful tool or as a way of thinking or as a source of pleasure is a notion quite foreign to most elementary-education curricula (even to those whose textbooks are adequate). One would think that, at this level, computer software would be available to help communicate the basics of arithmetic and its applications (word problems, estimation, and so forth). Unfortunately, the programs we have at present are too often transcriptions onto television monitors of unimaginative lists of routine exercises taken from the textbooks. I'm not aware of any software that offers an integrated, coherent, and effective approach to arithmetic and its problem-solving applications. Some of the blame for the generally poor instruction in elementary schools must ultimately lie with teachers who aren't sufficiently capable, and who too often have little interest in or appreciation of mathematics. In turn, some of the blame for that lies, I think, with schools of education in colleges and universities that place little or no emphasis on mathematics in their teacher-training courses. It's been my experience that students in secondary math education (as opposed to math majors) are generally among the worst students in my classes. The background in math of prospective elementary schoolteachers is even worse; in many cases, nonexistent. A partial solution might be the hiring of one or two mathematics specialists per elementary school who would move from room to room throughout the day, supplementing (or teaching) the mathematics curriculum. I sometimes think it would be a good idea if math professors and elementary schoolteachers switched places for a few weeks each year. No harm would come to the math majors and graduate students at the hands of the elementary schoolteachers (in fact, the latter might learn something from the former), while the third-, fourth-, and fifth-graders might greatly benefit from exposure to mathematical puzzles and games competently presented. Too many adults--including highly educated professionals--have no grasp of how big is a billion or how small are the chances of meeting a terrorist. They are afflicted, says mathematician and writer John Allen Paulos, with innumeracy--the mathematical counterpart of illiteracy. Innumerate people make bad personal decisions, support insipid government policies, have little appreciation for the risks involved in certain undertakings, and are susceptible to the shenanigans of assorted quacks, charlatans, and practitioners of pseudoscience. In some cases, they may also be slow to comprehend a joke or pun. The reason has to do with the strong correlation between mathematics and humor, one of Paulos's favorite topics of conversation and the subject of his first two books, Mathematics and Humor and I Think, Therefore I Laugh. In the following excerpt from Innumeracy: Mathematical Illiteracy and Its Consequences, his most recent and most successful book (it was on the New York Times best-seller list for five months in 1989), Paulos outlines some of the causes of what he considers a nationwide epidemic--an epidemic that, one may conclude, is contributing to the current difficulty in attracting young people into science careers. In the accompanying interview, he also offers some possible solutions and talks about what it's like to be a professional mathematician and, consequently, a career humorist. A native of Chicago who grew up in Milwaukee, Paulos, 45, earned both his undergraduate degree and his Ph.D. in mathematics from the University of Wisconsin. Since 1973, he has taught mathematics at Philadelphia's Temple University, where he is a full professor. In March 1991, Knopf will publish his latest book, Beyond Numeracy: An Uncommon Dictionary of Mathematics.

Mathematical Illiteracy Knows No Social Or Class Boundaries

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