[avmich]

Yes, the first one looks like an important quote :) . When I studied math, I usually had trouble understanding or memorizing a rule unless I had at least a rough idea why it holds. In this case I'd suspect just remembering rules and then using them without understanding would - often - cause inconveniences or later errors, when a rule is remembered incorrectly. So maybe it's subjective - what should be taught first?

[winchling]

As philosopher Daniel Dennett puts it: competence comes before comprehension. It's totally possible to do something well, as animals do, without understanding what one is doing. But for comprehension one needs to have something in place to reason about and make connections.

And this isn't the full picture. Motivation comes before competence. One needs reasons to acquire skills: they have to address problems in one's mind if the mind is to fully engage. Which is why coercive education with its curricula, exams, etc, largely fails.