I was exactly the same. I was good at integrating but had no idea why we were doing it. Then we did a physics-based module (called mechanics) and it was like yo

I was exactly the same. I was good at integrating but had no idea why we were doing it. Then we did a physics-based module (called mechanics) and it was like you: a hundred light bulbs went on. Nowadays I "see" numbers in a totally different way. If I see a time axis I can't help but see the rates of change etc. I wonder if it could be taught in a more top down way: start thinking about rates of change in some discontinuous functions, like finance. How growth of assets over time relates to income. Then move to continuous with distance, speed, acceleration. Then do the thing where you differentiate symbolically.