I used to think calculus was about formulas. Then I had to teach it.
What you're looking at
Two panels split by a vertical line. Left side: what I thought integration was when I learned it at MIT — a symbol-manipulation game, formulas leading to formulas. Right side: what I understood later when I had to teach freshmen — it's counting, it's area, it's the sum of infinitely many contributions. The left side is red (wrong path), the right is green (the real thing). Arrows point down to what each view produces: a function vs. a number.
Why I drew it this way
The vertical divider is hard — no gradual transition, because the shift wasn't gradual. It happened the first time a student asked me "but what IS an integral?" and I realized I'd been doing symbol-shuffling for years without the picture in my head. I put the formula in a box on the left because that's how it felt: closed, rigid, something you either got right or wrong. The right side gets an ellipse because the concept is round, continuous — it flows. The annotations at the bottom are the punchline: procedure vs. meaning.
What it argues
That you can be technically fluent and conceptually blind. I could integrate anything you handed me in 1935, but I didn't know what I was doing until I had to explain it to someone who'd never seen it before. The diagram argues that teaching is where you discover whether you actually understand your own tools, or whether you're just good at following recipes.
What I left out
I didn't draw the middle part — the years of using integrals in quantum mechanics, path integrals, all the machinery I built on top of a foundation I didn't really see. That would've been a third panel, and it would've muddied the point. The before/after is sharper without the "during." The shame is in the gap, not in the journey across it.