Educated for what?

THE NONMATHEMATICIANS I've talked to seem surprised to learn that the entire 1992-93 school-year crop of math PhDs in the United States, although up 14% from 1991-92, still amounted to just 1,202 souls. They are further surprised to learn that, as reported in the November 1993 issue of Notices of the American Mathematical Society, only 526 of these new PhDs were U.S. citizens, of whom in turn 145 were women. Each number seems like far too few.

But then I tell them that most of the new PhDs who were able to locate any kind of mathematical work went on to teach, at unexceptional salary levels (about $34,000 per annum for both men and women), and that their unemployment rate as of late September 1993 was the second highest ever (12.4%, down from 12.7% in September 1992). So in the end we're surprised to discover that the small PhD crop actually turned out to be too big.

A math education, it seems, raises the same question as one in philosophy or classical languages. Once you have it, what do you do with it?

Given all the ink spilt over our nation's need to train more people in higher math, the ordinary nonmathematician expects something more impressive. It doesn't have to be six-figure starting salaries or head-hunters enticing math whizzes into computer-lined hideouts to work for industrial or Wall-Street tycoons, but we'd certainly like to see something more promising than new and eager math PhDs sending out hundreds of unanswered resumes.

Professor A. W. Goodman of the University of South Florida, writing in the aforementioned mathematical Notices, put the current state of affairs this way:

If a person loves mathematics and wishes to devote his life to that study, he or she should be encouraged to do so. But in my opinion it is just plain stupid to try to entice people into that study.... Many students who devoted four or more years' training to do research in pure mathematics must now readjust to teaching arithmetic or basic algebra (assuming they can find such a job) or drop mathematics and retrain themselves for another type of work.

I don't know to what extent leaders of American industry and government grasp the present situation. Some of them may be bad at math. All, however, are presumably expert in interpreting signals produced by free-market economics. They should then have no trouble understanding the CEO of a computer-engineering-software company who wrote (in a private letter), "I think that the general handwringing over the lack of math and science ability in the American populace is misplaced. People who have the ability do not enjoy exceptional demand for their services."

One of the works that aggravated our anxiety over American math education was Everybody Counts: A Report to the Nation on the Future of Mathematics Education. As a math buff, I greeted this 1989 publication with enthusiasm. It sported an awesome list of collaborators and supporters: the National Research Council, the National Association of Science, the Mathematical Sciences Education Board, the Board on Mathematical Sciences, the Committee on the Mathematical Sciences in the Year 2000, etc., etc. The report's press was widespread and positive.

Thus it pained me all the more, on actually reading the report for sense, to find myself embarrassed by it. I even found myself obliged to go on record as a critic (as did Professor Wilbur Whitten of The University of Southwestern Louisiana, who in the May/June 1993 issue of Notices called it "self-righteous baloney"). I can only hope that some of the report's prestigious sponsors signed on board before reading it. The report may reasonably be described as math-industry puffery decked out in graphs, photos, and designer typography. Its main title, which had seemed at first so humanly apt, dissolved into an advertising ploy, a loaded pun for the mathematical lobby. Then the first sentence -- but I'll spare you the details, which I've covered elsewhere. (1)

I used to bury my embarrassments without ceremony. Now, in what I consider to be a more mature practice, I brave postmortem pains to discover the cherished and unchallenged cognitive habits and beliefs that I've learned are usually the trouble. "So why," I asked, "did I expect so much from a math-industry report that I was mortified to find it reading like a union pamphlet?"

What I discovered in myself was an overgeneralized and underanalyzed faith in math and mathematicians. Challenging the faith led to questions about the place of math, mathematicians, and math education in our society, which culminated in the title question of this piece, "Educated for What?"

The key answer I reached is one that until recently would have struck me as highly unlikely. Simply put, I no longer believe our country needs more math education. Just as we don't need more than a small stream of math PhDs, so we don't need a flood of math teachers or courses in trigonometry and calculus, let alone still more advanced math, and we don't need spectacular new techniques for teaching such courses.

What we need instead is a better integration of numbers and sense, what I call "mathsemantics." Some people will, of course, also take advanced math courses, whatever they happen to need for engineering or scientific work. The majority, however, won't take any advanced math, for the solid reason that they have no need for it.

I spent most of a vacation in 1950 studying the first 354 pages of R. G. D. Allen's Mathematical Analysis for Economists, because Professor Milton Friedman said I needed calculus to take his University of Chicago course in price theory. He let me in and I enjoyed the course. However, I've never had any occasion since then to use calculus. The suggestion that all educated people should know calculus strikes me as ludicrous. Educated for what?

It would seem to make sense to try to educate everyone to use numerical relationships comfortably in meaningful ways. We need as many people as we can get who are able to handle ordinary arithmetic, simple rates, the percentages that fill our media, and such useful multiplications of dimensions as "person-hours." We need people who recognize that counts of customers or patients aren't necessarily counts of people that averages and prejudice have a lot in common, and that accounting figures don't represent current market values. An education for such purposes, however, requires no advanced math at all. What it does require is basic mathsemantics. The Mathsemantic Monitor regards it as education for citizenship, for life in America.

An education for citizenship wouldn't stop anyone who needs more math from taking it. Quite the contrary. It would make advanced math courses more meaningful. Those who create or report statistics, for example, would know that a more challenging audience awaited their output. Their reports should improve, because good audiences help make good performers.

To shift our society's emphasis from math to mathsemantics, and thus to achieve an active integration of math and semantics, we must overcome barriers produced by some cherished and virtually unchallenged beliefs. In Korzybskian terms, we need to grapple with our assumptions. Overcoming them may be more than half the battle.

First, we need to grasp the total cultural oddness of our saying that math is a language but our then proceeding to insulate it -- both curricularly and psychologically -- from English and the normal questions we ask about language. We need a new outlook in which we can unselfconsciously ask what the numbers mean, not just whether they're accurate.

Second, we need to accept that normal childhood semantics simply can't handle numbers. Swiss child psychologist Jean Piaget and his followers have amply proved that most math meanings are beyond children. Our schools needn't produce any more mathematiphobes as confirmation.

Third, we need to accept that superimposing math instruction on ordinary childhood semantics forces children to disassociate math from ordinary life. One might even say that the typical American has a culturally-split personality, a math personality and a verbal personality.

Fourth, we need to accept that forcing advanced math on inadequately prepared students neither corrects their childhood semantic problems nor repairs the disassociation of math from their ordinary lives. It can, however, aggravate the problems of self-confidence that arise from bad early experiences with math.

Fifth, we need to accept the fact -- remarkable only for its being so ignored -- that almost all adults who get along well do so without advanced math. Indeed, some successful adults freely admit their antipathy to math.

Sixth, we need to accept that our most consequential mathematical shortcoming is not one of instruction in advanced mathematics, but that of understanding basic mathsemantics. We need to make ordinary numbers and numerical relations part of our everyday cognitive apparatus.

Seventh, we need to recognize that our basic mathsemantic failings foster serious social problems. Our failures to relate meaningfully to numerousness, to cumulative effects, and to the difference between numbers and real-world events, for example, continuously hamper efforts to reduce prejudice, pollution, and financial disasters.

Eighth, we need to accept that the task of improving mathsemantics must initially be one of adult education. Academe is hopelessly committed for now to the separation of math from English, of numbers from meaning. Even if it were not, we could hardly afford the nearly 100 years it would take for educators and their boards to agree first on what needed to be taught, and then to train a new cadre, who would in turn train the young teachers, who would gradually take over from the older teachers the task of teaching young students, some of whom might then eventually arrive at positions of leadership long after you and the Mathsemantic Monitor had left the scene.

So, to achieve anything in this lifetime, we need first to develop our own mathsemantics. We must understand how it has happened that childhood semantics has thwarted math instruction. We must reconcile math and general semantics, starting with the conflict between the math teacher's dictum that "one can't add apples and oranges" and our conclusion that, because "no two things are ever exactly the same," one never adds anything, so to speak, except apples and oranges. We must follow such leads until we've demolished the artificial barriers between math and semantics. It's up to us.

REFERENCE

1. See Mathsemantics: Making Numbers Talk Sense (Viking, 1994) for more regarding Everybody Counts and math education within a general-semantics perspective.