This is Math

"The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when one contemplates the mysteries of eternity, of life, of the marvelous structure of reality." --Albert Einstein


"Each problem that I solved became a rule which served afterwards to solve other problems." --Rene Descartes


Somewhere, in and between these two quotes, is what mathematics means to me. It's not the techniques and the procedural ways to solve problems. It's not even knowing theorem after theorem. It's in the exploration I do myself, and the discoveries I make.

In my last two years of high school, I decided I was going to be a math teacher. As far as I was aware, I was good at math. I mean, really good. My mom loves to remind me how my HS math teacher, Mrs. C, told her how I would act as a translator for other students in the class, re-explaining during work time what Mrs. C had meant. I had almost no trouble getting it the first time. Math was recreating what the teacher had shown us on a variety of problems. Sometimes there was a cool little trick you needed. I liked those. They made me feel smart when I figured out what the trick was.

My first semester of college, I got a wake-up call. Sure, most of my math classes were recreating what the teacher was showing us, but one class stood out. Everything was open ended. We'd get a question (the first one was about ants and how their tunnels connected - hooray for graph theory!), and then we'd play around, trying to find an answer of some sort. David, the instructor, had some rules about the format of our notebooks and everyone having to share something in class each day, but he didn't seem to do much. (Now that I'm on the other end of the classroom, I know what a lie that was!) We met in study groups (required), shared with each other, got excited for each other's brilliant ideas, and hoped some rule we came up with would be named after us. (David started that, too.)

One of my group mates spent hours drawing and looking at graph after graph, counting vertices, faces and edges, until he came up with an equation relating them all. Our study group was so proud of his result - it was HUGE. I was envious of his dedication and discovery. (To this day, I think of it as the Wilmot Equation, not Euler's Formula.) That was math. That was what it meant to do math.

I promise I'm not really straying from my quotes. Yes, problems were given to us, but they were only the starting point. David encouraged us to extend and expand the problems, go down a variety of different paths, and ask more questions. We became curious, and more curious, as we found more answers. And every problem built upon the one before. The definitions, rules and explanations from one problem guided us and pushed us on the next one. Sometimes something forgotten would pop back up. It was like finding those little tricks, but so much bigger, so much more meaningful. There's nothing like it to make you feel like a genius and an idiot all at once.

That, my friends, is real math. Not school math. REAL MATH. Mathematicians-turning-coffee-into-theorems math.

How do we get it into our schools?

Personally, I stole the research log from David. (Though he helped a bit - so is it just borrowing? I'm not giving it back!) I pose open questions to my students, and let them try stuff, play around, suggest ideas and shoot them down. I'm nowhere near as accomplished at "doing nothing" as David was, but I'm working on it.  I make up questions related to calculus, and ask them to play with them. I've gotten some beautiful definitions of tangent lines and explanations of how curves can have slope (without me introducing the topic first!), and I've gotten frustrated, stuck in the mud spinning the tires students. (It varies, depending on how willing they are to throw out how they've defined Math for all of these years...) I'm learning, slowly and painfully, how to scaffold the questions without making them obviously leading. I've got a loooong way to go on that. I need more calculus teachers in my circle of friends, so I can pick their brains.

Here's the notebook guidelines I give my students:

Research Log Guidelines
(Based on the syllabus of David Olson)