Books: In Review
Innumeracy: Mathematical Illiteracy and Its Consequences
By John Allen Paulos
Hill & Wang, 19 Union Square West, New York, N.Y. 10003; 135 pp., $16.95 cloth.
At the conclusion of Innumeracy, John Allen Paulos says that his motivation for writing the book was "the desire to arouse a sense of numerical proportion and an appreciation for the irreducibly probabilistic nature of life."
"Innumeracy," as Mr. Paulos defines it, is "an inability to deal comfortably with the fundamental notions of number and chance." The themes of number and probability are played out in dozens of amusing, whimsical, and sometimes serious contexts throughout the book, which is intended for an audience of educated but mathematically illiterate readers.
Along the way, Mr. Paulos manages to suggest some mathematical concepts in such a way that even the most "innumerate" readers should be able to understand them.
But as the book progresses, the author also draws sharper and deeper lines of distinction between the innumerate and the numerate. And as he attributes more and more of society's problems to innumeracy, we become convinced that it is not acceptable to say smugly that we "were never good at math."
Combined with their "tendency to personalize," a misunderstanding of mathematical concepts often causes the innumerate to be misled by their own experiences or by the media's focus on dramatic individual cases, according to Mr. Paulos.
For example, he observes, if we only knew more about probability, we would not be so impressed by the "it's a small world" phenomenon of meeting a stranger on an airplane whose aerobics instructor knows our spouse's best friend. In fact, the chance of such a "coincidence" is actually quite high, if we assume that each of us knows about 1,500 people.
A faulty understanding of coincidences can also have more serious consequences, as the probabilities involved in tossing a coin illustrate. If we flip a coin a large number of times, a surprising number of consecutive heads or tails come up. And at almost any point, significantly more heads or tails will have appeared.
Similarly, Mr. Paulos observes, some people become known as "winners" and others as "losers" in life "though there is no real difference between them other than luck." People are much more sensitive to streaks and absolute differences than to long-term patterns, he explains. Winners or losers, he says, may simply be those who happen to be stuck for a while on the right or wrong side of even.
Mr. Paulos blames innumeracy for nearly every type of illogical reasoning, nonscientific thinking, superstition, and plain stupidity. The foibles he attributes to the innumerate include belief in pseudosciences--such as astrology, numerology, biorhythmic analysis, and parapsychology--and high susceptibility to fraudulent medical cures and treatments.
He cites mathematical examples to show how a numerate person would be able to apply logical, scientific, or mathematical reasoning--and thereby debunk pseudosciences or provide alternative explanations for unusual phenomena.
For instance, using numerical reasoning about the size and age of the universe, the numerate individual would be skeptical about8the possibility of visitors to Earth from outer space. Such thinking, he writes, would lead to the conclusion that any extraterrestrial neighbors are probably more than 2,000 light years away--and may not care about visiting us anyway.
If only people were more numerate, Mr. Paulos implies, they would not be so easily taken in by charlatans, faith healers, and fortune tellers, or by casinos and supermarket tabloids; in his unspoken view, only the ignorant masses fall prey to gambling, irrational advertisements, and television evangelists.
But do the mathematically literate among us always reason clearly in conducting their lives? If so, why do so many mathematicians, scientists, engineers, and doctors smoke cigarettes, live near earthquake fault lines, or buy gas-ng cars?
Maybe, "on the average," these highly numerate folks do lead more rational lives than everyone else, as Mr. Paulos would have us believe. But he offers no firm evidence or data to support this conclusion.
Perhaps the most disappointing facet of the book is its emphasis on the superficial consequences of innumeracy. Of course, it would be nice if educated but innumerate persons were better able to understand aids testing. And it would be great if so many people weren't so susceptible to fast talkers and pseudosciences.
But the truly damaging consequence of innumeracy--which Mr. Paulos barely mentions--is the extent to which millions of young people, especially women and members of minorities, are denied entry to science-related educations and careers. By contrast, Everybody Counts, the recent report issued by the National Academy of Sciences, treats this topic thoroughly.
Mr. Paulos does discuss the causes of innumeracy--and finds more than enough blame to go around, from elementary-school teachers to textbooks to misconceptions about the nature of mathematics. He even identifies mathematics professors as part of the problem.
But his proposed solutions fall short of the mark. Venturing into unfamiliar territory, he makes suggestions that sound like conversations from the hallways of university mathematics departments.
In faulting teachers and textbooks, he admits that they do a good job of teaching routine exercises. And he correctly points out that many teachers do not teach children mathematics as a way of thinking or as a source of pleasure. But when did anyone last see a standardized mathematics test based on such approaches?
Mr. Paulos overlooks the fact that schools and teachers are required to teach what the local4school board wants children to learn. For the past 10 years, state and local legislators have insisted on basics with such a vengeance that mathematical thinking has been wrung out of the curriculum.
Mr. Paulos thinks that "it would be a good idea if math professors and elementary-school teachers switched places for a few weeks each year" so that 3rd, 4th, and 5th graders might benefit from exposure to competently presented mathematical material. Meanwhile, no harm would come to the universities' math majors and graduate students, he says. In fact, he contends, the elementary teachers might learn something from them. This proposal is exciting--but for reasons opposed to those Mr. Paulos cites.
Mathematics professors would learn a great deal about the realities of teaching in elementary school. Perhaps this exposure would help them design and teach the kinds of mathematics courses for elementary teachers that would help those teachers enjoy the subject and encourage them to study more math.
Similarly, math majors and graduate students--many of whom eventually become mathematics professors--might learn something about teaching from the elementary-school teachers. They might also become spoiled enough by the experience to expect good teaching from their professors.
Reflecting the current national concern for mathematics education, Mr. Paulos's book will strike a chord with many people. If the nation is to regain its scientific pre-eminence, mathematics and science must lose their mystery to the untrained. Books like this one will help accomplish that goal.
Gerald Kulm is project director for the American Association for the Advancement of Science's office of science and technology education. He is editing a book on higher-order thinking skills in mathematics.