False Equations

Overcoming Math Anxiety.
by Sheila Tobias.
Revised edition. Norton. 260 pp. $23.00.

The reasons for our widespread and embarrassing ignorance of mathematics were aptly described by Jacques Barzun in Teacher in America some 50 years ago:

Early in life, people come to think of themselves as having or not having that mysterious [mathematical] “mind,” and until recently I do not believe that anyone dared to dispute its existence. But the belief is a superstition, and one that is largely unproductive. . . . [T]he feeling is given that the whole system dropped down ready-made from the skies, to be used only by born jugglers. This is what paralyzes—with few exceptions—the infant, the adolescent, or the adult who is not a juggler himself.

Barzun elaborated his diagnosis and common-sense prescriptions in eight characteristically brisk pages, and urged us to get on with the practical tasks of teaching. My own experience of teaching mathematics suggests that he is right: there is good reason to expect success when mathematics is properly taught—at the right pace and connected with students’ other experience—and when (as one of my colleagues used to say) it is studied with the expectation of pleasure.

Sheila Tobias’s Overcoming Math Anxiety, first published in 1978, and now reissued in a revised form, puts a contemporary spin on the problem that Barzun addressed a half-century ago. It is a self-help book, of which part is how-to and the rest remakes Barzun’s main points with a feminist and therapeutic gloss.

The occasion for the gloss is the persisting gender gap that shows up in classrooms and on standardized tests commonly thought to measure mathematical aptitude. Tobias seems to experience those vexing test scores as an affront. The very existence of a research group that “continues to find . . . that . . . the top males on average outperform the top females” (emphasis in original) is cause for complaint. Her replies to such provocations pad out the ideological sections of this book with arguments that sometimes lamentably recall Wolfgang Pauli’s lapidary dismissal of bad science: “Not even false.”

“Since not all math underachievers are women and not all women avoid mathematics,” reasons Tobias,

it is not very likely on the face of it that poor performance in math can result from some genetic or hormonal difference between the sexes. Moreover, no amount of speculation so far has unearthed a “mathematical competency” in some tangible, measurable substance in the body.

The first of these assertions appeals to a principle whose soundness can be tested by applying it to the matter of height. Since not all short people are women and not all women are short, it would be implausible, according to this principle, to attribute differences in average height to genetic or hormonal difference between the sexes. Note that Tobias uses the phrase, “on the face of it,” as a kind of rhetorical shoplifting: she walks away from a bad argument but still pockets the goods.

The second assertion has force only if one resorts to caricature, distorting the proposition, “x has some biological basis,” to mean “there exists some special x-causing chemical that possessors of x have in unusual abundance.” Most of us would agree that Mozart was different from ordinary mortals, and was born that way; but not even the crudest materialist would thereby be committed to believing that Mozart’s genius was the result of some special music-chemical or musician-gland, or that his career could be fully explained by his physiological inheritance.

Having insisted that the gender gap in mathematical aptitude cannot be fully explained by “nature”—something no one seriously disputes—Tobias seems to admit only one other contrary possibility: that it can be fully explained by (maladroit) “nurture.” Girls, she reports, receive, on average, less encouragement about mathematics and more discouragement than boys do. If that observation seems a little frail to carry the load of indictment with which she freights it, one can always resort, as she does, to rounding up the usual suspects: “Why have you (or most of the men you know) never heard of . . . Sonya Kovalevskaya, who took 19th-century mathematics by storm. . . ?” Alas, Tobias’s question could just as well be asked about Hilbert or Gauss.

Finally, Tobias declares an upset victory for what is little more than a truism: “Apart from general intelligence, . . . the most important elements in predicting success at learning math are motivation, temperament, attitude, and interest.” But she has already insinuated a great deal more than this, and her explanation continues into the realm of psychopathology. It seems that the origin of “mathophobia” is a culturally conditioned “math avoidance,” which can blossom into a syndrome whose “victim” becomes a “math-anxious person,” needing “math therapy,” possibly through the “intervention” of a “math counselor” or other “expert” in the problems of the “math-disabled.”

The discovery of widespread but hitherto unrecognized psycho-pathologies is, of course, a growth industry, and one would like to think that such claims incur a large burden of proof. In its stead, Tobias documents, at length, some commonplaces of teaching: students profit from discussing their work with friends (and girls often lack “math friends”); students who attribute their failings to lack of effort do better than those who attribute their failings to lack of ability; and so on. It is a slim yield for so much effort.

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The how-to sections of the book are a mixed bag. They reteach simple, if often mystifying, subjects such as fractions and negative numbers, and illustrate doing mathematics by walking through some word problems. Tobias seems unaware of George Polya’s classic introduction to mathematical thinking, How to Solve It (1945, and still in print). Moreover, her understanding of calculus is unfortunately marginal, and her idiosyncratic take on that subject is unlikely to kindle understanding in many readers.

The most serious failings in these sections of the book are, for want of a better word, cultural. Take, for instance, this howler:

Socrates’s best-known treatise on government . . . starts with an idea, the notion of justice. . . . We might not agree with the definition of justice as it finally emerges from the Dialogue, but we have learned a lot about government from the progress of the debate.

Here Tobias misidentifies both the author and the title of what is arguably the most famous and influential philosophical work in the history of Western thought. More depressing, if possible, is that no one seems to have noticed—not the acquaintances of the author who read the manuscript (we know the names of some, from their endorsements on the jacket and contributions to the text), or the editors at Norton (who let it pass in the first edition, too), or a decadeand-a-half’s worth of presumably grateful readers.

The problem goes deeper than ignorance or complacency. Thus, in a motivational pep talk, Tobias recommends mathematics solely as a booster to one’s prospects for success in the world of affairs, showing no grasp of what it might mean to take possession of so remarkable an intellectual instrument: a body of deductively organized knowledge by which significant, remote, and surprising consequences can be rigorously developed from simple principles.

The wonder inspired by the power of mathematical knowledge has played a remarkable role in the Western tradition—think of Plato, Descartes, Galileo, Newton, Kant. Jacques Barzun recalls the event that marked Thomas Hobbes’s “philosophical awakening”: a chance look at a copy of Euclid that astonished the middle-aged Hobbes into exclaiming, “This can’t be true!”

Our students, of any age, are entitled to that astonishment. But who will teach the teachers? Not all know the value of what they claim to teach, or of the tradition it comes from, and the damage from those failings cannot be measured by standardized tests alone.