### The Last Time You Used Algebra Was...

#### By Donald G. McNeil Jr.

(New York Times, December 12, 2004)

In the 1986 movie, "Peggy Sue Got Married," Kathleen Turner, an unhappily married wife and mother, magically returns to relive her senior year as the most popular girl at Buchanan High.

She leaves a math test blank, and when her teacher (described in the screenplay as "an officious little creep") demands an explanation, answers: "Mr. Snelgrove, I happen to know that in the future, I will never have the slightest use for algebra. And I speak from experience."

Audiences and critics loved the line, presumably because they too rejoiced in knowing that they had never, ever used the quadratic formula again. (Disclosure: I squeaked by in calculus while never really grasping it, and can no longer help my ninth-grade daughter solve equations with two variables. The toughest math I tackle now is calculating a tip in a moving taxi.)

Last week, the United States proved, yet again, that its mathematical literacy is abysmal. In a survey by the Organization for Economic Cooperation and Development, it ranked 28th out of 40 countries in mathematics, far below Finland and South Korea, and about on a par with Portugal.

The survey tested simple, "everyday" skills like estimating the size of Antarctica or footsteps in the sand. Nonetheless, as in past comparisons, American 15-year-olds did rather better than students in Mexico, Indonesia and South Africa, and substantially worse than those in rich countries, especially Asian ones.

These annual humiliations produce two consistent reactions.

One set of experts grouses that the surveys are unfair: average American students are compared to distant elites; Americans play sports and hold jobs; foreign countries impose national standards while America believes in local school boards.

Another set gloomily predicts that math malaise will ultimately gut the economy, frequently citing an estimate that American businesses waste $30 billion a year on remedial training. (In 1990, the elder President Bush announced an expensive plan to have American students lead the world in math by the year 2000.)

But there is also the Peggy Sue school of thought, which asks: So what?

In all but the most arcane specialties (like teaching math), the need for math has atrophied. Electronic scales can price 4.15 pounds of chicken at $3.79 a pound faster than any butcher. Artillerymen in Iraq don't use slide rules as their counterparts on Iwo Jima did. Cars announce how many miles each gallon gets. Some restaurant bills calculate suggested tips of 15, 18 or 20 percent. Architects and accountants now have spreadsheets for everything from wind stress to foreign tax shelters. The new math is plug-and-play.

True, those calculators and spreadsheets and credit card machines need to be programmed. But, in between bouts of visa restrictions, American universities successfully import thousands of math whizzes each year because jobs await them, and the tiny percentage of American-born students who do Ph.D. work equal the world's best.

In math, as in chess, countries that produce the most grandmasters per capita - like Hungary and Iceland - not only don't rule the world, they don't even rule chess. Sheer power counts, as it did in chess for the Soviets. America may lose math literacy surveys, but it dominates number-crunching in every sphere from corporate profits to supercomputers to Nobel prizes.

So is it necessary that the average high-schooler spend years nailed to the axes of x and y?

Maybe not, said Robert L. Park, former director of the American Physical Society, an independent group of physicists, who teaches at the University of Maryland.

"As a teacher, I'd like to think it's going to have a huge payoff," he said. "But I'd like to know the answer."

He once calculated that a third of the Americans who won Nobel prizes were born abroad, and said that an open-door policy benefited both sides: American universities get well-trained, driven students, and they in turn flourish in the more creative atmosphere here.

Bob Moses, who developed the Algebra Project in Cambridge, Mass., focuses on the other end of the spectrum: poor blacks and Hispanics who are the first in their families to aspire to college. "No one is going to pay you because you can do division," he said, but added that without a grasp of the concepts his students would be "serfs in the new information age," stuck in dead-end jobs as surely as illiterate Europeans were forced to the bottom of the job heap by the Industrial Revolution.

Most experts point out that careers in science or computers require mathematics, even when it is not a real job skill but a filter for the lazy or stupid, as passing freshman physics is for pre-med students. (Disclosure: me, for example.) Physics requires calculus, calculus requires algebra and trigonometry, and so on. One must start early.

In the age of Googling and spell-checking, noted Diane Ravitch, the education historian, the "so what?" question could be asked about learning virtually any subject.

"But a democratic society demands an educated populace," she said. "Why spend hundreds of billions on public education if we're going to sling it over our shoulder?"

But the best defense - the first to get beyond the utilitarian argument - came from a certain Miss Collins. She is my daughter's math teacher at a school where there are no boys to distract or intimidate calculating young women.

"If you ask the girls," she said, "they'll say it's another hoop they have to jump through to get into a good college."

She feels otherwise.

"What we do isn't exactly what mathematicians do," she explained. "And I know more alums here become artists than become mathematicians. But kids don't study poetry just because they're going to grow up to be poets. It's about a habit of mind. Your mind doesn't think abstractly unless it's asked to - and it needs to be asked to from a relatively young age. The rigor and logic that goes into math is a good way for your brain to be trained."

## The Last Time You Thought Clearly Was ...

#### by Alan Kay

The article about "math" by Donald G. McNeil, Jr., "The Last Time You Used Algebra Was ...", misses many important points about mathematics, not all of which can be addressed in this short reply. The only worthwhile opinion in the piece was unfortunately at the end, by Miss Collins, McNeil's daughter's mathematics teacher. She pointed out that children don't study poetry to become poets (nor is the reason for them learning to read and write to become professional writers and readers), and that many areas of education are not about direct vocational training but about developing "a habit of mind".

McNeil thinks that mathematics is what is taught in school and called "math", but this is generally not at all the case. Much of "school math" is not mathematics at all but attempts to train children in various kinds of calculation using patterns and recipes. Mathematics is actually about representing and thinking clearly about ideas. Science goes further: to try to come up with plausible ideas about the universe that are worthwhile thinking clearly about.

An interesting example of unclear thinking is the second question from an international math test in 2003 quoted in McNeil's article It has geometric figures and descriptions, and asks the student to find a match. To a clear thinker it should be glaringly apparent that there is no correct answer to the question as posed. It is worthwhile looking at it for a few minutes to see why. From the actual information given, there is no indication in the diagrams or text that any of the larger or smaller triangles are right triangles with one 90 degree angle (the triangles as drawn could have their largest angle be 89.9 degrees and we couldn't tell the difference). This is why the language of geometry indicates a right triangle by a square sign placed on the angle. Similarly, there is no indication that N and M are actually midpoints (there are special signs to indicate these as well).

Why is this an issue? Because in clear thinking in mathematics or science the mere appearance of something is not enough to indicate there is enough information to come to a conclusion. A large part of this kind of clear reasoning is looking for all the cases and knowing that one has all the cases. The diagrams as given allow for the large angles not to be 90 and M and N not to be midpoints, so a clear thinking child will answer "none of the above", and presumably be marked wrong since the multiple choice test (no such thing in real mathematics) has no provision for explaining the reasoning (in spite of the fact that explaining reasoning is the whole point of real mathematics).

One has to ask: why was a self confessed non-clear-thinker chosen as the author of an article supposedly about mathematics? And this leads to larger and more important questions: why isn't teaching children "how to think clearly using representations and reasoning" a major goal in the US? Perhaps not too surprisingly, the real thing is a lot of fun for most boys and girls, it's the junk ideas vended by non clear thinking adults that are hard for children to swallow.