> It's not hard to know what a derivative and integral is. There is mathematical depth to being able to do something interesting with that. That's a very good

> It's not hard to know what a derivative and integral is. There is mathematical depth to being able to do something interesting with that. That's a very good point. In some sense learning of the material at level N, is only done when you use and consolidate the knowledge at the subsequent level N+1, and maybe even N+2. Indeed, it would be fair to say that a student passing the final exam on topic X—even with a good grade—only truly understands 30-40% of the material. Only when the student has to apply this knowledge in later courses is the understanding complete. Perhaps the best strategy for the independent learner is to learn topic X and immediately follow up with applications of X. I mean that's what people recommend anyway, but I always thought of it as a suggestion or a nice to have, but in the light of your comment I'm thinking maybe it's a requirement.