### Learn only what I teach you, and teach me not.

One of my favourite classes to teach at Mathcamp is a bit of performance art I call “Calculus Without Calculus”. I cribbed the curriculum from a high school calculus textbook - take your pick - and from a class I took as a camper eight years ago, under the instruction of the inimitable Loren Larson. In my class, we solve a handful of calculus problems without defining a single function, evaluating a single limit, or taking a single derivative. My campers have eaten up the alternative methods, and often they return to camp the following year to tell me of the many calculus test questions they solved, for instance, by reflecting a line in the x-axis and doing three lines of simple calculations rather than setting a derivative equal to zero and doing a page or two of computation.

“I got a zero on the test for that question,” one camper reported, “because I didn’t use calculus.”

Another camper, who had two weeks of projective geometry but no calculus under his belt, used the former to find a tangent to a parabola. “I got it right,” he told me, “but my teacher didn’t know what I was doing and wouldn’t give me any credit for it.”

I can count on the fingers of one hand the number of times a university student of mine used anything but the method I’d taught them to correctly solve a problem. And I remember my reaction each of those times: I was *ecstatic*. I am pleased when students are able to apply what I have taught them, but I am *delighted* when they are able to think beyond my lessons and use prior knowledge to attack the problems I assign them. A handful of times at Mathcamp, a student will show me a solution that I’d never have thought of myself. It’s amazing to watch them assimilate my lessons and expose them to branches of mathematics about which they know more than I do.

Anyone who feels differently shouldn’t be teaching math.

Again:

ANYONE WHO FEELS DIFFERENTLY SHOULDN’T BE TEACHING MATH.

I think about the lessons that some of my campers’ teachers, who scrawl big X’s through their students’ *perfectly correct* work, are teaching their students, and they appal me:

- Mathematics isn’t about logic, or reasoning, or thinking creatively. It is about
*following instructions.* - If the teacher has never before seen what you’re doing, then it’s not worth any credit.

My campers know math, and love it, and they know that they’re good at it. Other students could learn to like it if they were encouraged sufficiently. And I can guarantee that these lessons about what is and is not acceptable in a math class will stay with *those* students far longer than the lessons about factoring quadratics or computing derivatives or plotting ellipses.

So often I hear adults dismiss their high school math education with the comment that what they learned wasn’t useful, that it was irrelevant to their lives and their work. And in a way, they’re right: there is not a single job in the world which a person will be asked to *optimize a solution by defining a real-valued function and setting its first derivative equal to zero*, which, if their teachers are anything like the ones my campers describe, is what comprised their high school math education. There are, however, plenty of jobs, and plenty of everyday tasks, in which a person will be asked to figure something out from some numerical data. The method won’t be given - part of that person’s job is to figure out which method or methods should be applied. If you’re insisting that your students use the method that you taught them last week, and ONLY the method you taught them last week, to solve Question #5 on Test #3, then how on earth do you *expect* them to react when they encounter a mathematical problem that’s not quite like anything they’ve seen before, a year after they took your course? And if you dismiss a solution that doesn’t look like anything you’ve seen before, rather than ask your student for insight and clarification, what message does that send them? I’ll tell you what message it sends them: it sends them the message that they must work within the static confines of what you know. It tells them that in mathematics, *the teacher, rather than the subject matter*, decides what is legitimate. *It discourages them from experimenting*.

What a sorry way to do math. What a sorry way to do *anything*.

Working with tremendously talented students, I am aware that it’s not easy to stay on top of the workings of all of one’s pupils’ minds. By allowing students to give solutions other than the ones that I have in mind, I relinquish some measure of control over my curriculum, and that’s difficult. Often I’ll ask a question in class, and get a response that’s wildly different from the one I had thought of. Sometimes it’s tempting to nod and quietly dismiss the student’s answer, and give mine - *the* correct one - as an alternative. But I make a point of not doing that; instead, I’ll work through my student’s reasoning, on the blackboard if necessary, and after class if the complexity of the response demands it. Sometimes my student is wrong. But often, very often, they’re right, if in a roundabout way. Other times, they’ve come up with an explanation of a proof or a justification of a theorem that’s better than the one I had, but one that I had to think about. And I’m awed.

If I ever stop feeling that way, I hope that someone has the guts to tell me that I have outlived my usefulness as a math teacher.