Asilomar #3: Green Knowledge

Session Title

“Red Knowledge / Green Knowledge: A Different Way to Think about Teaching”


Tom Sallee. Professor of Mathematics/Researcher, UC Davis.


Man’s a titan. He was when I took him for Non-Euclidean Geometry at UC Davis and he was in the Sanderling Room today where he kept an audience rapt with only his measured cadence and an old-school overhead projector. Ninety minutes passed in what felt like ten.

He spoke of red knowledge and green knowledge, the red stuff being procedural/factual and the green stuff being conceptual. He distinguished them by saying that, with red knowledge, “you know whether or not you know it.”

“What’s 8 – 7?” vs. “How do you multiply two-digit numbers?”

“You’ll never hear me say conceptual knowledge and procedural knowledge aren’t both important,” he said. “That’s insane.” But he said he preferred conceptual knowledge because it gave him “a fighting chance” with a problem he’d never seen before.

“We tend to hide our best stuff from our students,” he said, “and I don’t know why.” He was referring to problem solving, honed intuition, creativity, etc.

I reckon this distinction is on the mind of anyone with a blog and of most people reading this. Implementation remains kinda fuzzy for all of us, though.

So the basics are these: during lecture he focuses almost exclusively on green knowledge and he puts it almost exclusively on them to pick up red knowledge independently, through the text.

He focuses each lecture on what he perceives are the largest ideas in calculus (for non-majors). For example:

To three decimal places find a number b such that:

He’ll give that to kids who don’t know L’Hôpital’s Rule, kids who barely know the definition of a limit and draw them around to the idea that you can drop that equation into Excel and get a decimal answer accurate to three, four, five, or fifty decimal places.

He did this for a semester and compared survey results with a willing red-knowledge colleague. The results were descriptive and I wish I could recall the exact wording. My strongest take-away was that, while both classes evinced intellectual laziness that kinda comes with the college freshman territory (laziness that’s kinda our fault in high school) Sallee’s kids were much less intimidated by a problem they didn’t know how to solve and more often felt like class time had been well spent.

Presentation Notes

Acetane transparencies never looked so fine. He’d scribble on a couple, leave ’em up for a few minutes, and then put a new one up. Maybe because it was the second half of the day, maybe because I’ve got a nerd crush on the guy, I don’t know, but I welcomed the lo-fi approach.


  • Prefaced his talk with a two-minute summary so that anyone who wanted to bail could bail early rather than late. Classy.
  • Recommended reading: The Structure of Scientific Revolutions, Thomas Kuhn.
  • Rick West: “If you copy an answer, you’ve rented knowledge, you haven’t bought it.”
  • He’s retiring this next year, which lent all kinds of urgency to his talk.
  • The best assignment he ever received at CalTech: a sheet with 100 problems about which the instructor said, “Find all the problems you can do and don’t do them. Find a problem you can’t do and do it.”

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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. Nice post, Dan.

    This distinction, and its importance, is THE main problem I have with the over-reliance on standardized tests. Typical multiple-choice testing does little to nothing to encourage (to say nothing of “help to assess”) whether any “green knowledge” has been mastered by the student.

    This distinction is also a helpful doorway, I think, for bringing writing and reflection into the math classroom. (Same could be said for other disciplines that are not traditionally writing-centered.)

  2. Are any of these sessions videotaped?

    Anyway, thanks for writing these great summaries! I may need to look up some writings by this Mr. Sallee.

  3. Eric, strong second point. Introspection is the bridge between your world and mine.

    I’m conflicted on your first point, though. It is possible to test green knowledge through multiple choice examinations. California does.

    I think the gravitational pull of standardized testing though is toward red knowledge, which is unfortunate. I think if I taught more green knowledge my students would do better on these exams ’cause they wouldn’t shut down on problems whose solutions weren’t immediately apparent.

    H., thanks for the positive feedback. When Richardson says that he doesn’t read people who post too much, I get all self-conscious about my post volume. And sorry, haven’t seen a camera yet.

  4. While I agree that “green knowledge” can be tested on multiple-choice tests, it too often is not and is not really designed to do so (so why force it?).

    Also, not all types of “green knowledge” can be tested this way. You cannot design a multiple-choice test that can effectively evaluate a student’s ability to write. For that matter, a multiple-choice test will never tell you whether or not a student “knows” how to swim … only the lake will prove that.

    If we make a distinction between “knowing” and “doing” (i.e., knowledge and skills), then I think we just push the issue further back. If I answer all the right questions about swimming, but can’t actually do it, do I “know” it? And which part of that is red vs. green? Isn’t it more effective to just ask me to swim? And how do you get at my cognitive abilities if all I have to do is select someone else’s cognitive conclusions from a list?

    But I agree with your last point: more focus on “green knowledge” can’t help but spill over into improved testing of “red knowledge.”

    [And, to repeat what I’ve said elsewhere, I’m not against red knowledge or the testing of it through multiple choice tests. I’m against seeing both as the definitive assessment of a student’s education and of the educational system as a whole … which is the generic public view.]

  5. Dan, I have given up on reading some high-volume blogs, but those had texts where it was necessary to read a paragraph or two to find out whether the rest was worth looking at, or where the ratio of ideas to text was low. Not an issue here.

  6. Dan,

    I just wrote a post concerning two activities we did in class today. In my mind, both leaned to the “green knowlege” side of the spectrum. What absolutely shocked me in the two activities was the reaction of two of my most intelligent students. They HATED the activities. They weren’t comfortable with the trial and error involved in the two games.

    Now, I understand high achieving students like being successful. Most of the time this success comes easy to the intelligent students. Today, it took a little work. This took some being wrong. Has our emphasis on grades (even in elementary school where grades are more or less determined on whether or not a student turned in their work) ruined the students of even attempting to answer a problem where they might be wrong?

  7. I don’t know if I can wring my hands too much over our grading culture, but I do agree that students who have made smartness and quickness a part of their self-assessment do not tolerate challenges to those beliefs. They don’t appreciate a difficult problem that takes several wrong approaches before the right one reveals itself.

    The only solution, I think, is to keep issuing those challenges until, by the second term, no one thinks any less of them and everyone is comfortable with failure, even the kids who never fail.

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