by Arthur T. White
Professor Arthur T. White, a teacher of mathematics at Western Michigan University, discusses ongoing developments in mathematics education. Mathematics, like English and other disciplines in the humanities, has begun to feel the influence of postmodern pedagogies of the same type that have led, for example, to the declining standards of freshman composition.
I was attracted to mathematics over forty years ago, and have remained devoted to the discipline since, because of its qualities of truth, beauty, objectivity, and aloofness from matters mundane. Bertrand Russell said it better in The Study of Mathematics (1910):
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of stern perfection, such as only the greatest art can show. . . . The generations [of mathematicians] have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from dreary exile in the natural world.
But the academic movement known as postmodernism, which is present now in all disciplines, takes a different view. As Gertrude Himmelfarb alerts us in her essay on “Academic Advocates” in Commentary:
The animating spirit of postmodernism is a radical skepticism and relativism that rejects any idea of truth, knowledge, reason, or objectivity. More important, it refuses even to aspire to such ideas, on the ground that they are not only unattainable but undesirable—that they are, by their very nature, authoritarian and oppressive”
Postmodernism—with its origin in the writings of Derrida and Foucault, for example—took hold, in this country, in our departments of English and comparative literature. Traditional literary criticism, in which the text is primary and the reader is secondary, has been “deconstructed” and replaced by an array of approaches currently featuring “reader response” theory, where the reader is primary and the text is secondary. This inversion has the effect of increasing the self-esteem of the reader, while diminishing the legacy of the great thinkers and crafters of language of the past. The deconstruction of history has led to the National History Standards (vehemently rejected, in their first incarnation, by the United States Senate), to the Enola Gay exhibit at the Smithsonian Institute (similarly rejected), and to Afro-centrist history (currently being hotly debated). In each case, established fact is subordinated to the need for self-esteem, or for addressing grievances of various groups claiming victim status, or for replacing Western culture and tradition with multiculturalism.
The deconstruction of science has produced a distrust in the scientific method and in technology, and has heightened interest in alternative medicine, creationism, astrology, and the paranormal. The further deconstruction of patterns of Western thought has led to “emotion based reasoning,” which produces such phenomena as the advocacy of jury nullification in contexts of social engineering. With Russell, I had thought mathematics and mathematicians to be secure from all such social and political inroads. But are we? There is evidence that we are not.
A mathematician, I believe, is quite likely to be motivated by the Platonic view that mathematics is external to the human mind, that mathematical truth is discovered and—within a given system of axiomatic assumptions–that it has the desirable quality of being absolute. This traditional view is today being deconstructed by some mathematicians and by many mathematic educators. The notion of mathematics as objective and eternal is today being replaced, among mathematics educators, by the postmodernist notion of “social constructivism.” According to “social constructivism,” knowledge is subjective, not objective; rather than being found by careful investigation of an actually existing external world, it is “constructed” (i.e., created) by each individual, according to his unique needs and social setting. Absolutism is deliberately replaced by cultural relativism, as if 2 + 2 = 5 were correct as long as one’s personal situation or perspective required it to be correct.
The philosophical stance one takes on these issues would seem to have substantial impact on one’s pedagogy. The new pedagogy in mathematics, as represented by the “Curriculum Standards for School Mathematics” and the “Professional Standards for Teaching Mathematics” of the National Council of Teachers of Mathematics, and as embodied in the calculus reform movement, has much that even a mathematical Platonist like myself can find of value. For example, I have enjoyed using small-group guided discovery in my classes where appropriate, and for many years I have been stressing mathematic reasoning (as opposed to rote calculation), problem solving, connections, and writing experiences in the mathematical classroom. I also use a modified Moore method of instruction and a tutorial system in my “Mathematical Proofs” course. I am not, therefore, locked into a traditional lecture-style pedagogy exclusively.
But the aspects of the Standards and of calculus reform that trouble me are those that just might be more motivated by postmodernist egalitarian—perhaps even neo-Marxist—political, rather than pedagogical, considerations. And so I raise the following concerns.
(1) Mathematics education, through both the NCTM Standards and calculus reform, has the goal of making mathematics accessible to all students. But I wonder, after we pare away whatever of mathematics is not accessible to everyone, whether what remains will still be mathematics. If, in fact, we hope in vain that the masses can master mathematics, then perhaps calculus (or algebra) does have a legitimate role as a filter for some students, as well as being a pump for some others. If we make calculus (or algebra) easy enough for everybody, we might well find that we have so dumbed it down that we have dumbed it away.
(2) Reform trends indicate that group work (cooperative and/or collaborative learning) should be almost universally appropriate in the classroom. Students teaching each other material they don’t know, and perhaps have no affinity for, might be effective in a relativistic sense, but if we don’t want to discard millennia of the best that careful thought has produced, then I doubt that the pedagogy is universally effective in any rigorously objective sense. To whatever extent we assess by groups, then I fear that we are following the failed Marxist maxim: “From each according to his ability, to each according to his need.”
(3) One calculus reform text openly states its goal of replacing elegance, one of the traditional criteria of mathematical thinking, with brute force. This, perhaps, would make the material both less mathematically pleasing to some students, and more accessible to others, thus more nearly approximating the egalitrian equality of outcome. But this nation has long stood for equality of opportunity, not necessarily of outcome.
(4) The increased emphasis on technology and on practical applications, as with many other aspects of the reformed pedagogy, should be of benefit to true users of mathematics. But we should also concern ourselves with the future producers of mathematics. When much of the proposed pedagogy is driven by educators who are not themselves mathematicians, or who perhaps are seeking to politicize mathematics, then the inadequacy of the training for the next generation of mathematicians becomes suspect. I believe that realistic, but conceptually and numerically cumbersome, applications are better left to their specialty disciplines, as they hinder and even obscure the mathematical tools being developed in mathematics courses.
(5) The increasing emphasis on inductive reasoning, with the concomitant de-emphasis of deductive reasoning, might not be the best way to prepare careful thinkers. Instead, I detect here the specter of the postmodernist rejection of rational thought. The “definition-theorem-proof” format that has survived scrutiny since Euclid, and stands as the model of mankind’s intellectual potential and achievement, is now under such an attack that, without resistance from its supporters, it might soon vanish entirely from the high school and calculus curricula. “Writing to learn” and classroom discourse can be effective pedagogically, but if carried to excess, they threaten to distract from precision of thought. To what extent do the “rule of three” (numerical, graphic, and symbolic approaches) and redefining mathematics as a laboratory discipline make pedagogical sense? To what extent do they inject sociopolitics into our discipline?
(6) Reformers would have us avoid problems having just one correct solution. Surely this is postmodernist relativism asserting itself again.
(7) Granted, nationwide the results of calculus instruction are distressing. Some think that this should force a reform of the pedagogy. Others might prefer to reform the attitudes and abilities of students who take calculus. Perhaps the lecture system really is the worst system of instruction there is . . . except, of course, for all the others.
(8) To the multiculturalists, to the postmodern cultural relativists, to the selectors of the amazingly politically correct photographs for the NCTM’s Professional Standards for Teaching Mathematics,” we should simply say— “We welcome all who wish to join us in our glorious adventure; we will support and encourage you. But there is just one culture here: It is mathematics.”