[B]y its very existence, [mathematics] poses a serious threat to their entire world view that there is no such thing as objective truth and what they have to say on any subject is just as valid as what anyone else says.
Imagine a baseball field with a ball resting on the ground. Two teams come onto the field, play a complete nine-inning game with the ball, and then depart, leaving the ball lying at rest on the field. It is an indisputable fact that there exists a straight line passing through the center of the ball and fixed in the ball, that now has exactly the same orientation in space that it did when the ball was resting on the ground prior to the start of the game. How can this not only be known for certain, but even be possible? The answer is that, in 1776, long before the invention of baseball, the Swiss mathematician Leonhard Euler proved the following theorem:
In whatever way a sphere is turned about its center, it is always possible to assign a diameter whose direction in the translated state agrees with that of the initial state.
The veracity of this statement, now known as Euler’s Theorem on Rotations, is, by virtue of Euler’s proof of it, beyond question. However, if the issue of whether the statement of the theorem is true or not were put to a vote of people with meager mathematical knowledge, it is likely that a—possibly vast—majority of such voters would deem it to be false. Many of these voters, upon finding out that they had chosen incorrectly, would become terrified and angry at this unequivocal demonstration of the undemocratic nature of mathematics. After all, democracy puts all voters on an equal footing—one person, one vote, but math pulls the democratic rug out from under them. This is not at all like arguing about the subjective issue of whether the University of Michigan or the University of Notre Dame has the better fight song. In mathematics, right and wrong are not matters of opinion. There is a metaphysically right answer—totally independent of people’s beliefs, desires, or cultures—and infinitely many wrong ones. And compounding the voters’ angst is the fact that a person’s ability to arrive at this indisputably right answer does not depend in any way upon that person’s census category, position in society, education in non-mathematical subjects, or debating ability; it depends solely on that person’s knowing how to do mathematics. This is why the theorems of mathematics are regarded as discovered rather than invented. They do not originate in the minds of men; they are “out there” waiting to be found. It is not surprising, therefore, that for people ignorant of mathematics, the subject, by its very existence, poses a serious threat to their entire world view that there is no such thing as objective truth and what they have to say on any subject is just as valid as what anyone else says.
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I submit that this is the cause of math anxiety, defined by Tobias and Weissbrod in their Harvard Educational Review essay “Anxiety and Mathematics: An Update,” as “the panic, helplessness, paralysis, and mental disorganization that arises among some people when they are required to solve a mathematical problem.” Math anxiety has been attributed by many “experts” to a variety of causes, such as math teachers who lack empathy, timed math tests, fear of public embarrassment, and so forth. But all of these rationalizations are unsatisfactory because they are superficial and fail to come to grips with the metaphysical and epistemological source of the psychological problem of math anxiety, namely, the destabilizing effect of mathematics on people whose self-image and ability to function depend on everything in their lives being subjective. But not everyone reacts this way to math.
For example, Professor Thomas Kane, in his graduate courses in dynamics at Stanford, often would bring to class some device comprising two or more rigid bodies and ask the class to vote on how it would move subsequent to a particular set of initial conditions. “We do everything democratically around here,” he would announce prior to calling for the vote. After recording the result of the poll on the blackboard, he immediately would launch into a derivation of the dynamical differential equations governing all possible motions of the system, and then present a solution of the equations subject to the particular set of initial conditions under consideration. Of course, the mathematical solution would never correspond to the motion the—usually significant—majority of the class had picked. Then Kane would subject the actual device to the same initial conditions employed in the solution of the equations and it would always move in accordance with the mathematical solution, never the class’s choice.
However, as far as one could tell from their reactions, all of the students, being engineering majors, invariably would find this state of affairs extremely satisfying on many levels. For one thing, it was yet one more demonstration of the supremacy of reason over opinion—especially majority opinion. It also was further confirmation of the rational and objective nature of the universe. Clearly the mathematical laws of mechanics did not wait around for the results of opinion polls and adjust themselves accordingly. But it is not difficult to imagine how different the reactions would have been had the audience for the lecture been postmodern humanities students rather than engineering students.
Viewed in this light, math anxiety is seen as a character flaw rather than as the popular view of it as a mental disorder. However, there is an additional contributor to math anxiety, one that works hand-in-hand with the aforementioned one: For the past four decades, the math curriculum in the public schools has been a Progressive one, carefully designed to frustrate students. It is structured to prevent them from gaining a step-by-step mastery of math—an intrinsically hierarchical subject—with the result that they are continually confronted by concepts with which they have been systematically rendered unable to cope. It is no wonder that “panic, helplessness, paralysis, and mental disorganization” is the students’ natural reaction. This problem is compounded by Progressive educators’ insistence on telling certain groups of students whom they suspect of having math anxiety that they should not be anxious about math, an action that has the predictable effect of instilling anxiety because the students understandably conclude that the issue wouldn’t be broached unless there was a compelling reason.
Math anxiety was not a significant problem in America two generations ago, and there is no good reason that it should be now.
Image: “Lots of math symbols and numbers” by Averater on Wikimedia Commons
