I hate the way math is taught, both in high school and in university.
Imagine, for a minute, that you want to teach a student about history. Imagine that you do it by picking a specific topic, like the cause of World War One, and giving him a page of history “problems”. Each problem requires that the student read a paragraph and write a paragraph in response. But each paragraph that he reads is almost identical to the previous one, differing only in emphasis. And each paragraph that he writes has to follow from the one that he read. Furthermore, the paragraphs are arranged so that there is only one right answer. If he writes the wrong thing, he has to do it again. It’s going to take him all evening repeating almost the same thing over and over again. And, for as long as he’s in your course, he’s going to get another page just like that one every couple of days, each on a slightly different topic.
How much is he going to hate history? A lot. How long is he going to stay in your course? Not long.
Yet, that’s exactly the way that math is taught in high school. The student is given a page of a couple of dozen simultaneous linear equations that increase in difficulty and is expected to solve one after another until the page is finished. And next week, it’ll be a page of quadratic equations that need solving. And the week after that, a page of binomials that need factoring.
In university, it’s even worse. The professor stands at the board and writes proofs of various theorems. Then the student is given a page of unproved theorems and told to get them proved by next class.
I groan just thinking about it. If any subject, from English to psychology, were taught the way that math is taught, students would hate that subject, too.
When I was a graduate student in psychology, I spent a year working on a project with a Ph.D. candidate from the math department. Watching him do math was a revelation.
Mathematicians don’t work the way they teach. At all.
First, they don’t waste their time solving the same kind of problem over and over again until they are ready to puke. They pick a single, challenging problem that interests them and work on that for weeks, months, or maybe their whole career. The problem they chose will take time but they work at it because it’s worth solving.
Second, they don’t waste their time working through each step of the proof of published theorems. They assume that, if the proof is published, the theorem is correct and move on.
They read a theorem that seems to bear on the problem that they are trying to solve, make up a simple example to make sure that they know how the theorem works, and then use it for their own purpose.
That’s what math really is: an exciting hunt for intellectual big game. It’s the ultimate collection of wonderful puzzles. Why do we teach it like it’s nothing but intellectual weight lifting? Every math lesson is more boring reps with heavier weights instead of another exciting challenge.
An incidental byproduct of math as professors do it is that it becomes a social activity. Mathematicians talk to each other about their problems. They exchange ideas. They don’t spend all their time sitting in a quiet room alone.
It’s true that a student should graduate with a common, basic understanding of the important concepts in algebra, geometry, trig, some calculus, and some statistics. But he doesn’t have to be taught every detail. He’s not going to remember the details anyway. What he really needs is to know that there is a known way to solve most of the problems that he will encounter in life, that he knows where to find the solutions, and that he knows how to apply them.
Let students work on bigger, more interesting, more challenging problems at the edge of their ability (which varies from student to student) so that they learn what math is about and how it works. That’s got to be better than the current state-of-math-education that leaves students with entirely the wrong idea about what math really is. And that teaches most of them to hate it.