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That's right, talking calculus. Calculus is, after all, a language--the language of change. It is used to calculate, of course. But students of chemistry, biology, economics or government, as well as doctors, options traders and secretaries of defense, all read, hear and talk calculus. Mathematics in general is a language, with an alphabet, a vocabulary, a grammar. Every equation is a sentence. Mathematics is used in the writing of weather reports, great theories of the universe, State of the Union addresses and the side panels on Cheerios boxes. There are dialects of mathematics. There is mathematical slang. Mathematics borrows from other languages and they in turn borrow from mathematics. You might argue that real languages evolve over time, that they aren't invented. But mathematics did evolve. Some of its terms were invented, as were such English words as "elevator." But much mathematical notation has come down to us greatly modified over time. "Yes," you might say, "but mathematics is logical. It proceeds step by step." But languages are logical too. And while mathematics may be taught step by step, that's not how it's learned. Many a student finishes her first semester of calculus feeling that she's missing something. She gets by on a mechanical level--"If I see this, I know I should do that"--but still feels at sea. That's quite normal. She'll understand that first semester's work much better after a second semester of study, and even better after a third. (Ultimately, true knowledge may be beyond the grasp of all of us. In the words of mathematician John Von Neumann, "In mathematics we don't understand things, we just get used to them.") "But," you counter, "where is the literature of mathematics, its works of beauty, power and grace?" There are masterpieces of mathematics--Euclid's Elements, for example, which until recently had been published in greater numbers than any other book. Great mathematics is almost always beautiful. Superior poets and mathematicians alike have acknowledged that poetry and mathematics share the potential to express profound truths with brevity and elegance. "The mathematician who is not somewhat of a poet will never be a perfect mathematician," noted 19th-century mathematician Karl Theodor Wilhelm Weierstrass. And 17th-century poet John Dryden wrote that one "should have a reasonable, philosophical, and in some measure a mathematical head to be a complete and excellent poet." You may argue that mathematics is more than a language, that it is in fact a great body of universal truths. So it is. Like any language, though, it can express falsehoods as well as truths. Scientists don't believe their equations are universally true. Equations are descriptions, and no description is complete. Science is metaphor. The physicist doesn't say the earth's orbit around the sun is an ellipse, she says it's like an ellipse. The demographer says population growth is like an exponential curve. As biologist James Bullock said a few years ago, "It is just as metaphorical to call the world a sphere as it is to call it a stage." Yet another objection is sometimes raised to seeing mathematics as a language: "Anyone can learn a language, but not everyone can learn mathematics." The idea that some of us have "math brains" and some don't is pernicious. It's sad when someone all but boasts that she can't do mathematics, that no one in her family can. Would she claim that no one in her family is capable of learning to read or of learning his or her native language? All of us can learn mathematics, just as we learned to speak. "So you say," you may retort, "but I can't learn calculus!" Yes you can-and to prove it we're going to give you two quick lessons in the language of calculus. The first introduces a symbol, the prime, which looks like this: ´ If x represents something, then x´ ("x prime") represents how fast that thing is changing. Population´ is the rate of population growth. Distance´ is the rate of change of distance (also known as velocity). [Water in the boat]´ is the rate at which the boat is taking in water. When you read in the paper that the national debt is $3 trillion, you can express it like this:
When you read that this year the country will post a budget surplus of $70 billion, you know the national debt will shrink at the rate of $70 billion this year. You can express that this way:
That's the first lesson. You're already talking calculus. The second lesson introduces pictures. If we read in English, "The number of applicants to Smith has been increasing steadily at the rate of 200 per year," we can translate that to:
When we feed this equation to a computer, it can show us a picture of the rise in applications:
--a straight line. If we read "The number of Smith applicants has been increasing steadily at the rate of 6 percent per year," we can translate that to:
This time, the computer shows us a curve
and we see the difference between a constant growth rate and a constant percentage growth rate. That's enough for now. A third lesson would explore where those graphs come from and how to draw them. The point is that it's not hard to say things--meaningful things--in the language of calculus. High school students often have trouble seeing and believing in the ultimate meaningfulness of mathematics. As algebraic rules rain down on them, students can lose any sense of the "rightness" of algebra. One rule looks like another. Take fractions: are either of the following correct?
One is, one isn't. Many students suspect that the rules that make this so have no logical basis, no intrinsic meaning. But they do. This lack of faith in mathematics has serious consequences, for individuals and for society. Consider the recent debate over the census in the year 2000. It was suggested that we use statistical sampling techniques to augment the traditional head count. This raises two questions. One is political: If sampling is used, what will be the consequences for the configuring of congressional districts and the distribution of government funds? The other question is mathematical: Will sampling make the census more accurate? The problem is that some people maliciously confuse the questions. In fact, sampling would almost certainly result in a better tally. But one side doesn't want you to believe mathematics and argues that the idea of using sampling is nothing but a political maneuver. It may be acceptable (though unstatesmanlike) to oppose sampling because it will cost your side seats in the House of Representatives. But it's inexcusable to argue, as some do, that sampling is inaccurate-and, sadly, it's all too easy to fool a public distrustful of mathematics. In our calculus courses we emphasize the linguistic nature of the subject. Our students read and write about weather, disease, finance and arms races-all in the language of calculus. They talk calculus. They work in groups, discussing, questioning, guessing, testing, persuading. They talk extensively with native speakers (the computer is a native speaker of calculus). The course is designed so that in a 13-week semester students learn enough to discuss their major disciplines in the language of calculus. The mathematics department is also experimenting with a 12-credit intensive course, like those in French, Italian and Japanese. It starts with students hearing and speaking whole sentences of calculus--jumping in with both feet. Students feel at first as if they are in a strange new land. But like students in a Spanish immersion class, they quickly adapt. Our calculus students study a language. They become literate. Sometimes they write term papers in lieu of taking exams. This past year they wrote on toxic waste, Third World development, retirement planning, cancer cell growth, automobile skidding, the psychology of baseball and other topics. Galileo said: "The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics." And anyone can learn it. ............................ David Cohen and Jim Henle are professors in Smith's Department of Mathematics. Their book Conversational Calculus is published by Addison-Wesley. |
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