Pershan's Pyramid of Teaching Greatness
It's all about generalization
You’ve probably seen Freytag’s Pyramid a few dozen times. It’s overused, abused, and has led to people making some very narrow and reductive claims about what constitutes a good story. Lincoln Michel, for example, has written about the need to think more broadly about story structure, and in my own less-prominent experience he’s correct.
Besides, it’s not even a pyramid! It’s a line with a bump. Come on, people.
My second, more serious point is this: I think in teaching there’s a similar sort of diagram that I often keep in mind, and it looks like this:
This is the form that I think a lot of learning—not all!—takes. You, as a teacher, are trying to start with specific knowledge that your student has. You are then trying to get them to learn a true and correct generalization from their knowledge. And then you’d like them to apply that knowledge to new situations.
Should we name the “rising” and “falling” action in this triangle? Can’t hurt. It would look like this:
It’s maybe easiest to see this at play when it comes to definitions:
Specifics: Here is a triangle. Here is another one. This is not a triangle. But this one is.
Generalization: Now, what is a triangle?
Specifics: And since you know what a triangle is, is this a triangle?
But this pattern is also handy for understanding how students learn procedures. Students learn by generalizing from solutions (e.g. a classmate’s idea, worked examples). But it’s not enough to hear or articulate an abstraction—to be sure that you understand it, you need to apply it to a specific case.
Learning a procedure can look something like this:
Specifics: A person solves 9 + 2 by counting on 9, 10, 11.
Generalization: “Oh wait! You can always count on from the larger number by the smaller number.”
Specifics: A person tries that out on 19 + 2.
I’m thinking about this because, in his newsletter, Dylan Kane wrote about a new study that found that self-explanations negatively impact learning from examples:
The authors reviewed 43 articles studying worked examples in math. They found (probably not surprising) that worked examples had a positive effect on learning. They also found (probably more surprising) that worked examples worked better when they only included correct examples, and also that including self-explanation prompts had a negative effect on learning.
And while I agree that it’s a surprising finding, I’m not that surprised. Because I’ve seen a lot of things called “self-explanation prompts” that don’t fit this pattern of instruction. In particular, they don’t do anything to move students up the knock-off Freytag Pyramid up from the specifics they are familiar with, up to some sort of generalization.
“Explain your reasoning” is a standard prompt that may induce some students to make generalization. Or it might not.
I should admit at this point that I stole this specifics/generalization/specifics thing from a lovely piece that I’ve written about before by researchers Alexander Renkl and Alexander Eitel. They have a busier version of what I’m talking about that has really influenced my teaching.
Once I saw generalization as at the core of so much of learning, it changed how I thought about my work. I started being much less relaxed about who was leading the abstracting, generalizating, explaining process. It made me realize that, if there was a moment to pick for me to step in, this was it.
My goal is to help students make those generalizations and get back to problem solving, and sometimes the quickest way to do it—the way that gets kids back into independent thinking the fastest—is to articulate the principle myself.
It doesn’t need to be a lot of talking, but if I’m going to talk, this is often where I’m going to do it.
I think a lot about writing, and I think a lot about teaching. And here’s a weird thought: if you tilt your head enough, “show don’t tell,” the popular creative writing admonition, is pretty similar to the teaching mantra “never say anything a kid can say.”
OK, fine, they’re not exactly parallel! But they’re both trying to warn you, as someone trying to engage others, that simply articulating your views isn’t enough. And that’s true. You need to help people find their own way into your world. How is this done?
One way might be like this: start with specifics, guide people to some new place—some generalization—and then set them free to try it out on their own.
I love this idea! Here's my my mind went: how does this apply to a few generalizations I teach? What I'm thinking is that there is a relationship between the complexity of the specifics and the complexity of the generalizations.
Example one, unit rates. A really helpful generalization in 7th grade math is "find the unit rate and use that to solve the problem." Students often have a pretty good intuition for these. "You go 10 miles in 2 hours, how many miles do you go in an hour?" is a problem lots of kids just know how to solve or can solve with some pretty minimal nudging. Same with "you are biking at a speed of 15 miles per hour, how far will you go in 3 hours?" The generalization is more complex -- unit rates are hard to explain, though with enough examples they start to make sense. So you can do lots of small specific examples, and then try to climb that ladder to generalization. Specifics are simple, generalization is complex.
Example two, two-step equations. The generalization here is something about inverse operations or undoing things or doing the same thing to both sides. But the specifics are hard! It is hard to understand how to solve 3x+2=14. I love tape diagrams as does the IM curriculum so we spend a lot of time on those. Here the generalization is still complex but so are the specifics. This generalization is always tough for me, we spend so much mental energy on the specifics it feels tough to really generalize. I still think your framework is helpful here! Just different, because of the complexity of those specific cases. Specifics are complex, generalization is complex.
Example three, complementary angles. Here I think both the specifics and the generalization are pretty simple, and not that different from each other. This topic relies on some prior knowledge and really benefits from students who have good intuition for angle size in general. But we do some specific examples and the generalization is mostly just recognizing right angles in places where they're not obvious and applying complementary angles in problems that include other stuff as well. Specifics are simple, generalization is simple.
All of those are my subjective judgments based on my experience. Maybe I'm trying to shoehorn this stuff into a box it doesn't fit into. (Also, are there topics where the specifics are complex and the generalization is simple?) But when I read this post I was thinking "whoa this applies so well to some things I teach but not to other things, why is that?" and this is what I came up with as a first stab at trying to understand.
How do you handle the tension between the implications of this framework and the desire to validate students’ own approaches, when those approaches may be less “principled”?
When I ask an open-ended question, a few kids solve the problem in the simplest “principled” way, but most will hack their way through using common sense or something. It feels disrespectful to highlight the “principled group’s work and then try to demonstrate that the other groups were using a disguised (less-efficient) version of principled method. Doing so creates a hierarchy of methods and communicates, “you got the answer, BUT…” to the “common-sense groups.
How do you recommend handling this?
Maybe it’s okay to just have straight-up conversations about efficiency, and maybe I avoid those too much.