This is where I get to kick and scream and whine, without hurting anybody. I'll probably change this, or add to it, as I have more tantrums.
This is a fable about how we teach mathematics today, and what I think we're doing wrong.
Once upon a time, you join a chorus because you want to sing. At the first rehearsal, the conductor sits you all down, and you study the score. You analyze the harmonic structure of the piece, you learn the rules about the resolution of the the dominant seventh. You analyze the passing notes and suspensions.
At the second rehearsal, you study the rhythmic structure of the piece. You learn about strong beats and weak beats, and syncopation.
Then you analyze the phrasing. You mark your score at the approved places to take a breath. You practice breathing in unison.
The next week you practice your pronunciation. You do mouth exercises to limber up your lips and tongue. Finally, you get to sing! You sing scales, and arpeggios, and all sorts of warmup exercises.
Then it's the night of the concert. The doors of the concert hall are locked, and you are not allowed inside because you don't know the secret handshake. If you listen closely, you can just barely hear beautiful music coming from somewhere inside the building. Then the doorman tells you move along.
As you walk home, you say to yourself "I hate music!"
That's like saying "I hate breathing!" Music is built into us. We learn the rhythm of our mother's heart even before we are born. We hear lullabies before we learn language. The joy of music is part of what makes us human.
Back to the chorus. Yes, you all need to be singing in the same key, you need to breath at the right times. Yes, everyone in the choir has to be following the same beat. Yes, you practice your "S's" so you don't sound like a bunch of snakes. But you do that so you can finally get to the beautiful music. That's what makes all the practicing worth doing, to participate in the creation of something beautiful.
Now, back to mathematics. You go to math class. The instructor recites a definition, and tells you it must be memorized exactly, word for word. Then the instructor recites a theorem (that's the mathematical residue left behind after someone else has had all the fun), then proves the theorem, and then finally shows you a few examples of what you've been learning about. The answers to the odd-numbered problems are in the back of the book, and yes, this will be on the test. No wonder you think you hate math. You've been doing all the practicing, but you've never gotten to the good part.
Look at a musical score. It's covered with all sorts of strange symbols, and it takes some figuring out, but you're willing to do that because after you get past the symbols, you get to the music.
Look at a page in a math book. It's covered with all sorts of strange symbols, and it takes lots of figuring out. But once you figure it out, you turn the page, and there are more strange symbols to figure, and more and more. Finally, you get to the end of the book, and you still haven't gotten to the good part. You close the book and say "Enough of that math stuff."
Just like music, the joy of mathematics is part of what makes us human. But we mathematicians have conspired to keep that a secret. We save all the good parts for ourselves. Well, Spud the Masked Mathematician, is here today to break the rules, to teach you the secret handshake, and let you in on all the fun stuff that's been going on behind the curtain.
Comments on speaking English versus speaking Math
Have you ever watched Abbott and Costello do "Who's on First"? It's a famous, very funny, vaudeville skit that's based on Abbott and Costello misunderstanding each other. They think they're speaking the same language, but you can almost see their words flying past each other, never quite meeting in the middle. When this happens in mathematics, it's not nearly so funny.
Mathematicians have a terrible habit. (In fact, Spud has several, but that's another story.). All mathematicians share one particular bad habit. In the library you will learn that I'm saying that the intersection of the set of bad habits of each mathematician is non-empty.
-- "What is this potatohead ranting about now?" you might ask.
Hey, give me a minute, okay. By now you know I like to sneak up on important ideas gradually. Think about this. When the biologists start to talk about axial asymmetry among invertebrates, at least you know right away where all the tricky words are. You can ask about axial asymmetry, and invertebrates, and the biologists can explain about animals without backbones (invertebrates) that don't have matching left and right sides (axial asymmetry), and then they can point to the big claw and the little claw on a lobster, and you go "Aha!"
With the biologists, at least you know which ones are the tricky words. The technical vocabulary jumps out at you like it's a different language. The warning bells go off in your head, and you can stop and ask a question if you don't know what the words mean. Then the biologists can explain using plain words, and then point to an easy example.
Then the mathematician starts talking about a field. You don't even realize that you're in trouble! You're lost already, but it doesn't set off the warning bells. The baseball fan starts to wonder about left, center, or right, and the Field of Dreams with Kevin Costner. A pilot might think about Lindbergh Field, the airport in San Diego, or Lambert Field in St. Louis. A farmer might think about rows of corn and soybeans, and worry about next year's prices. And the mathematicians - it's too scary to say out loud. So, right off the bat, when the mathematicians talk about fields, you don't know what they're talking about, but you don't know that you don't know, because field sounds like a pretty ordinary word for most of us.
So, your alarm bells don't go off, and the mathematicians keep talking, until you eventually realize that your train of thought has jumped the track, and even though it sounds like English, you don't know what these math people are talking about.
So, you ask them what they mean when they say field. They tell you about "integral domains in which all non-zero elements are units".
--Gee, that helps alot. You want to run that by me one more time?
Then they "explain" that an integral domain is a commutative ring with no zero divisors. Even worse, some of the mathematicians start to sniff at you, looking down their pointy little noses, trying to make you feel dumb so they can feel superior. If you have any sense at all, it doesn't take you long to get out of Dodge, and move on to a "field" you can understand.
This inability to communicate about mathematics is a terrible problem, but the problem is not with you, it's with the way mathematicians use words. No matter what the reasons are, mathematicians are using language in a way that inevitably excludes most of us. Even though it sounds like English, it's a secret code designed to confuse everybody else. When mathematicians start talking about a field, a ring, normal, function, order, they are using precise, technical terms just like axial asymmetry and invertebrate. The problem is, most of us don't use words like axial asymmetry everyday, so when we run into them, we know they're special. In math, the ordinary sounding words invite you to skate out onto the thin ice until you fall in. Suddenly you're in over your head, and you don't know what went wrong. And it's very cold. And you learn to avoid mathematics.
Why should we, as a society, value mathematics if the mathematicians can't even explain what they're doing. Take a look at a stack of old newspapers or magazines. Count the articles about medicine, about astronomy, environmental biology, meteorology, etc. Then count all the articles about mathematics - it won't take you long. The last time I tried this experiment, I couldn't find a single mention of mathematics. This is not because mathematics is any less important, in fact the doctors, astronomers, biologists, and meteorologists couldn't do their jobs without advanced mathematics. No, it's because we mathematicians have been talking to each other in our secret code for so long that we have forgotten how to talk to anyone else.
When Abbott and Costello do "who's on first?" it's funny. It's not so funny when you're a student trying to learn mathematics. I have read about a math test that included this problem - show that a number divisible by 6 is divisible by 2. A student wrote down that "18 is a number, 18 is divisible by 6, and 18 is divisible by 2". The teacher gave the student a zero.
What's going on here? The student was reading in English and the teacher was writing in mathematics. When mathematicians say "show that a number..." we mean "show that any number..." and when we say "any number" we mean "every number".
--Then why didn't you say so?
I don't know. Third base.
Dunham Does It Right - How to Write About Math
In his review of William Dunham's fine book, Journey Through Genius, Allan Heydon has written "Mathematical truths possess a beauty quite unlike any other work known to man, and the ability to appreciate that beauty should not be limited to expert mathematicians. In the preface to his new book, Journey Through Genius, William Dunham notes that there are books on the great works of art, literature, and music, but that discussions on the mathematical "classics'' are scarce. His excellent, thoroughly enjoyable, and articulate book fills that void. Most importantly, the book is written for the layperson, requiring only a familiarity with high school geometry and algebra. "
You can read the full review here, and follow the links to reviews of William Dunham's other books, The Mathematical Universe and Euler: The Master of Us All.
Since I complain as much as I do about what mathematicians are doing wrong, I feel obligated to celebrate when someone gets it right. William Dunham knows how to communicate important mathematical ideas without getting lost in the jargon and notation of the professional mathematician. Go to the library and find his books, and read them.
Just do it.