"Why do they call it `Introduction to Mathematics' when it's really `Goodbye'?"

My student meant, "My last mathematics course! Hurrah! Can you interest me? Can you give me a useful tool?"

Most college students find mathematics boring, irrelevant, and difficult. They do not know what Mathematics is, though they studied it for thirteen years. They hate the rose because they know only its fibrous stem and sharp thorns. They haven't seen its beauty.

Every field of study has its fibrous and sharp parts--its boring and demandingly precise parts. As a child I hated History because it was taught as a meaningless string of kings, conquests, and calendars. Even Mr. Downes, who made the Civil War live for us eighth graders, omitted its most important fact. For that I had to look as an adult through the eyes of a guest from Russia. She saw the central passion of the Civil War in a moment: brother against brother!

What is the central passion of mathematics? Mathematics is the science of patterns. Patterns-- surprising connections--are everywhere to be found. I too would hate mathematics if it were simply sums to calculate and theorems to memorize. It is not. Mathematics is music for the mind, just as music is mathematics for the ear. "Music is number made audible," says Claude Bragdon. Mathematics is both beautiful and relevant. I read George Gamow's paperback, One, Two, Three, Infinity, five times in high school, I was so taken by the beauty and relevance of the mathematics there. Only my New Testament was more shopworn.

Mathematics is beautiful. Fibonacci in the 12th century described a number sequence that measures the growth of rabbit populations. The same number sequence describes the arrangement of leaves on a stem to maximize the amount of light they receive, the design of the Parthenon, the shape of a seashell, the divisions of Beethoven's Fifth Symphony, and the most efficient way to sort data on a computer disk. Descartes in the 17th century described a formula for counting the parts of solids with many faces. The same formula describes maps on a plane, chemical isomers, and allowable ways of encoding data efficiently for computer transmission.

Mathematics is relevant. "A good theory is the most practical thing that there is," said psychologist Kurt Lewin. A mathematician secluded in a room with only pencil and paper can predict the future. Will it cost me more to tile a floor on the diagonal? Add a supercomputer to the room. What will tomorrow's gathering thunderclouds look like in 3D and in color? Mathematics can illuminate the past. Who authored the disputed Federalist papers? Mathematics can improve the present. What would a child missing two years ago look like today? How wonderful is our Creator that all of these questions can be answered using mind-amplifying mathematics!

I argued that mathematics is neither boring nor irrelevant. I did not argue that it is easy. Nothing that is rewarding is easy. As a mathematics major at MIT, I earned a D in the first semester of a mathematics course. I thought that my career was over. One of my professors sustained me with two pieces of advice.

"If you love mathematics as much as you say you do, you won't let a little thing like a D in a course stop you."

"But I don't understand the material," I protested.

"Sometimes you don't understand one step until you see how the next step builds on it."

Armed with his encouragement, I tackled the second semester and earned an A. You, too, even if your major is far-removed from mathematics, will be ready to give up when you hit the wall, when you find a seemingly impassible difficulty. Don't blame mathematics; it happens in track and field as well. Look for the meaningful connections.

But you picked up on my word "secluded," didn't you? Is mathematics an anti-social activity? I confess that in addition to being attracted to the beauty and usefulness of mathematics I was attracted to its sterile black-and-whiteness in a world of septic personal grays. I was a "nerd," a "geek." We're admired at cocktail parties. "Oh, you're a mathematician. You must be really bright." Then they always add, "I was never very good in mathematics." But we mathematicians are rarely the life of the party. We MIT men rarely dated MIT women, who were bright beyond belief, because they were rarely the life of the party. Yes, not everyone has the temperament for mathematics.

So to be honest I must tell you about the Spring of my first year of graduate study. That's when I turned from being a soloist to joining the orchestra. That's when mathematics for me stopped being a formalist's game like chess and started to become deep, meaningful connections.

The occasion is simple to describe. A qualifying examination in May hung over the head of every mathematics graduate student. It determined whether we would be welcome to aim for the prize of a Ph.D. or would be invited to settle for the consolation prize of a master's degree. Dick, George, Karen, Kalyan, Don, Harvey, Victor and I decided that we would meet every Sunday afternoon for an hour or two to prepare for the exam.

Group effort magnified our individual efforts. We honed our problem-solving skills; we reviewed all of the mathematics that we had supposedly learned. Two things particularly struck me for the first time. First, formal algebraic facts have motivating visual interpretations. Too many of my teachers had assumed that I could figure those out for myself without even telling me to figure them out. Some mentors, I later discovered, did not themselves know the motivation and meaning of the mathematics they marketed.

Second, mathematics vocabulary is everyday vocabulary. One's speech could be elliptical or hyperbolic. One can be an exponent for a cause. A discriminant allows me to decide among several factors. Mathematicians do not invent arcane language just to mystify; they merely make precise the language of general culture. No one had pointed out the connection.

I wish that I could say that I immediately began to teach in the collaborative way that I found so helpful that semester. But too often we teach as we have been taught. Only recently have I begun to trust the benefits of teamwork in the classroom. Only recently have I begun to focus not on what I can cover in the classroom but on what I can uncover.

"Why is it that only kindergarten students and graduate students get to experience mathematics as it is actually done by mathematicians?" asked educators Neil Postman and Charles Weingartner in their 1962 book, Teaching as a Subversive Activity. Why, indeed. Mathematics is too often taught as its results--a deductive and closed body of facts. Mathematics is too seldom taught as its process of discovery--an inductive, ever-changing body of deeply connected observations.

Now my Introduction to Mathematical Sciences has but one topic for the whole semester: interconnections. I want my students to be surprised by how interrelated all of their previous mathematical experiences have been with each other and with their academic and personal lives.

Jackie Carr, 1997 Messiah College graduate, says it best. "Logic and creativity. Who would have guessed that they could be related?"