Watch an Expert Math Teacher Put Three Kinds of Knowledge to Work in the Same Class

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.”


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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Hey, Dan! I am giving a short talk this week to some Master’s students about what I learned from Dynamic Skill Theory and Dynamic Systems. I’m going to use this as a(nother) perfect example! Thanks for sharing the work of create educators.

  2. The three types of knowledge you reference here, knowledge of content, knowledge of pedagogy, and knowledge of students, form the basis for acceptance into and ongoing development of teachers in the New York State Master Teacher Program, which I believe adopted their model from Math for America. Here’s a link:

    Here are some pipe dream methods for developing teachers’ capacity in these areas:

    1. Spend lots of time teaching in a wide variety of grade levels. This variety will not only expose you to content as it develops over the years, but also to the different needs of students of different age levels.

    2. Spend lots of time writing your own learning materials from scratch. The process of crafting a learning experience from the ground up forces you to consider elements from all three of the domains in excruciating detail.

    3. Spend lots of time teaching with a wide variety of other teachers. Watching other teachers teach is extremely informative. Picking up on and mimicking things you actually watch another teacher do well live and in-person is a fast track toward improvement.

    The above experiences are not always easy to have, especially in large amounts, given teachers’ schedules and job requirements. But the more the better. Keep looking for opportunities.

  3. “Spend lots of time writing your own learning materials from scratch. ”

    That’s exactly what no one at the national level really understands, and if they do, it’s discouraged.

    I’m struck by how at odds her thought process is (a process I’ve gone through myself many times) with the national discourse on teaching math. Which textbook? How much technology?

    Rarely do you see a discussion of the typical life of a teacher in an at-risk community, where we struggle to teach content at something approximating the course title while moving the needle for the many kids who don’t have the skills for the course title.

    They talk about technology engaging students, or whether or not discovery curriculum will help students more than a more “traditional” textbook, or whether we need teachers with more skills or more diversity. Reality is about teaching rational expressions when a lot of your students add one half and one fifth to get two sevenths, or, happier story, can do fractions so long as there aren’t any of those pesky letters. Or teaching exponential functions when none of your students understand percentages. I recently wrote about a little discovery activity I did ( and why it wouldn’t have helped to have more technology and certainly wouldn’t have been possible with a textbook. I was trying to help my students become aware of the unpredictability of exploration by (paradoxically) giving them seemingly random instructions that seemed pretty logical. I had to plan the activity carefully because I had both very high achieving and motivated as well as extremely low-motivated, relatively low ability students. How do I keep the timing and discussion reasoanable for the range?

    This is very much related to your last post. That is, we are dealing with seniors in high school, in pre-calc, who aren’t comfortable with fractions. Then they are going off to college, where they originally were put in remedial math. Now some state university systems are declaring sure, put kids who don’t know fractions into precalc, so we can admit them, but we’ll be unsurprised when they can’t do fractions and put them in middle school math. For credit.

    And so then I wonder: we don’t see teachers discussing these challenge in stats. Does that mean stats is easier, or that teachers haven’t even begun to struggle with teaching unprepared students stats? I am really skeptical that they don’t exist. Make stats universal and see what happens.

    “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.”

    This is why I mentor, both officially and unofficially.

    Good post on her part, good catch on yours.

  4. Glad I never had to teach in the USA – whole career in Ontario, Canada – a world leader in education. If you want to know how to teach fractions here is the current state-of-the-art research results – based on work in Ontario and around the world. Stay tuned a section on x and ÷ soon to be added to the Fractions Learning Pathway. and Enjoy