Let me show you what's actually happening when someone explains something to you — the difference between the real thing and the performance.
What you're looking at
Two parallel columns showing the internal structure of an explanation — one that works (left, green) and one that's broken (right, red). Each flows from a question down through the machinery of understanding, through a test, to an outcome. The working side branches into a picture and math that feed into each other before converging on a predictive test. The broken side is linear: question to vocabulary to exam to conditional success.
Why I drew it this way
I wanted the branching and rejoining on the left to be visible — that's the key difference. Real understanding isn't a straight pipe from question to answer; it's a picture that generates math, math that sharpens the picture, both of them answering a question you actually have. The right side had to be a single narrow channel because that's what memorization is: one damn thing after another, no cross-bracing. I almost put the broken side in a literal pipe shape but decided the parallel structure would make the contrast sharper.
What it argues
That explanation quality is a structural property, not a content property. It's not about how advanced the vocabulary is — it's whether the pieces lock together. The picture has to constrain the math, the math has to illuminate the picture, and both have to answer a real question. When students fail, it's usually because we gave them the right-hand column and told them it was understanding.
What I left out
I left out the teacher. In a fuller version there'd be a third column showing what the explainer is doing — whether they're checking that you have the picture, or just listening for the magic words. But that would've made this about pedagogy, and I wanted it to be about the thing itself: the anatomy of a working explanation, which you can build even if nobody's teaching you. You can always ask yourself, "Do I have the picture? Does the math follow? Can I predict a new case?" Those questions don't need a teacher.