Lisa Bejarano’s post Two Kinds of Simplicity offers a useful idea about teaching complex fractions, but much more interesting to me are the three kinds of knowledge she puts to work in her class.

**Knowledge About Teaching**

Lisa has read widely from sources online and offline and has a great memory. So when she asks herself, “How am I going to teach [x]?” she can quickly summon up all kinds of helpful posts, essays, books – even the mental recording of previous classes she’s taught on [x].

**Knowledge About Students**

I stopped to think about how this would work with my class.

Lisa has taught long enough and knows her students well enough that she can test each of those resources out in her head, *all during the lunch break before class*. You can see her swiping right and left on each of them – “Yeah, maybe this idea. Definitely not that one.” – as she sees her students in her imagination. I’m sure Lisa is open to the possibility that her flesh-and-blood students will differ in surprising and awesome ways from her *mental model* of those students. I wouldn’t bet against her intuition, though.

**Knowledge About Math**

She ultimates decides to start her precalculus students with the elementary school analog of their lesson, turning an abstract fraction division problem into a more concrete one.

Then, as her students acquaint themselves again (or in some cases for the first time) with helpful models for that division, she builds back up to the abstract version of her task.

Lisa is only able to move up and down the ladder of abstraction like this because she *knows* a lot of math – specifically where it builds from and towards. If she *doesn’t* know that math, her options for helping her students basically shrink down to “let’s solve a few together.”

**Finally**

I don’t know if it’s possible to *practice* what Lisa is doing here. It’s *knowledge*, the tightly connected kind you get when you spend thousands of hours in math classes, reflect on those observations, write about them, talk with other people about them, and then use them to inform what you do in *another* math class.

It’s possible, even easy, to spend the same number of hours *without* acquiring that tightly connected knowledge.

It’s something special to see it all put to use.

**BTW**. My guess is a lot of those knowledge connections were tightened because Lisa is a dynamite blogger. On that theme, let me recommend The Positive Effects of Blogging on Teachers, an article which does a great job describing ten reasons why teachers should think about blogging.