So I was in a colleague's classroom today, and I wrote down what my notes would look like if I had been a student who took good notes in her class that day. Then I wrote down what my notes would look like if I'd been in a class I taught, covering the same material. Here they are (in order -- although first scan was cut off for some reason.).I think the contrast is quite telling.

On the one hand, there's a shitload more insight in the class as I would have taught it. On the other, if you were a student looking back over your notes to try to figure out how to do a problem, the notes you'd taken in my class wouldn't do you a whole lot of good -- or rather, they would, but only if you had the patience to wade through a lot of conceptual shit in order to get there.

This brings up two questions:

1. These two teaching styles are clearly addressed towards different ways of thinking about mathematics, and different types of learning goals. (Roughly, we could describe those ways of thinking as "how mathematicians think about math" vs. "how people who are not mathematicians, but are pretty good at routine mathematical tasks, think about math.") Which of these reflects how we should be teaching our kids?

2. Is there, beyond the intrinsic value of each of these teaching styles, a value either to diversity or to consistency of teaching styles? Are students advantaged by having experience with both of these? Or do students who are used to one way of teaching/thinking freeze up when they're exposed to another way?

Briefly, I think we do want to be teaching our kids to think about math the way mathematicians think about math -- the understanding is a lot deeper and is rewarding on its own merits, but I think these ways of thinking also enable you to learn new mathematical concepts (and relearn forgotten concepts) much more easily -- because if you focus on understanding the underlying structure of mathematics, you can intuit the definitions and processes that would make sense in the context of that structure.

Unfortunately, my experience has been that there is an advantage to consistency -- and that students often react with frustration to the confusion introduced by exposure to new ways of thinking about concepts they already (at least partly) understand. I think I'm doing more harm than good by trying to teach mathematics the way I see it: by upending the expectations that kids have developed during years of math classes, I take what was once safe and predictable and render it frustrating and alienating. And I think, ultimately, that it's more important for students to have positive relationships with mathematics and with academics more broadly than it is for them to see it in particular ways, no matter how visionary or appealing those ways of seeing may be.

So I think teaching mathematics in shitty high schools is exactly the wrong place for me.

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I think given that you admit that on the onset you would be a fine teacher. Hell, I wish I had math teachers who focused on the theory, maybe I would of been interested in it during High School and college.

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