Also, understanding is not the same as doing, which is a distinction you fail to make. Understanding arithmetic multiplication is one thing. Performing it is something else, hence the distinction between mathematics and calculation. Calculation is very useful in mathematics, it's what K-12 and most college math is, but it's only one part of what mathematics concerns itself with. A physicist may be good at doing physics, but he doesn't necessarily possess an understanding of physics per se until he begins reflecting on it. Furthermore, there are many things you must learn before you can understand them because they require that you known things in aggregate and from a certain mature perspective. You must know the "what" before you can explain the "why" or the "how".
Furthermore, there is difference between comprehension and understanding. Most of what you know in the empirical order is comprehension of someone else's claims. You verify very little of those claims yourself because it is impossible for anyone to verify everything they learn. You take it on a practical kind of faith and trust that those who make these claims have done their jobs. If you end up working in the field, you may verify some of those claims, but in practice, science is a body of culture that is transmitted and only when anomalies begin to crop up do certain theories begin to undergo examination, verification, and reformulation by some scientists.
There is something to be said about teaching subjects with more context and coherence and being guided along a logical progression. Frankly, most teachers are awful, and compared to the classically trained, are a miserably uneducated bunch with little pedagogical skill. That and the idea of universal education is very problematic because it doesn't take seriously differences in students and the families they come from.
Well, to me piano is like sport, you get good to it by leveraging muscle memory, and when I was saying "with conscious enabled repetition" I was merely saying "no brainless repetition". I see no contradiction in our point of view here.
> Understanding arithmetic multiplication is one thing. Performing it is something else, hence the distinction between mathematics and calculation.
Multiplication was a maybe too simple example to give. Take bezier curve as a more advanced example: can you refute that you really get it threw visualization rather than by reapplying the formula on a series of homework assignation?
You know where repetition is really important? In sport. Muscle memory really work that way, with a "conscious enabled" repetition.
But for learning? Frankly you don't learn multiplication by repeating the tables, you really learn it by visualizing the repeated additions, thus by understanding it.
That being said, homework can be smart if it promotes self-discovery of a specific domain.