[joe_the_user]

More precisely, in order to decide whether it is a good idea, one should assess (i) how difficult it is to give an explanation of why some procedure works and (ii) how difficult it is to learn how to apply the procedure without understanding why it works.

Well, teaching basic math at a commuter college years ago, it felt like the issue of "teaching procedure" to "teaching understanding" was complex. The course I was teaching was close to the end of the math requirements for a significant percentage of the students. I was very attracted to teaching ideas but this group of students essentially had the attitude that they wanted a procedure to memorize rather than an explanation, not matter how complex the procedure. It had a certain logic - mathematical explanation would have touched a world they were happy to and committed to leaving forever soon after this. They'd suffered through this world up this point and thinking about it was more painful than simply acting.

Which is to say, I don't think there any easy answer for how to teach math. The failure of American "new math" years ago is something of a lesson in the push-pull of concepts versus concreteness as they can become ideologies in society at large.

[mncharity]

This is a widespread tradeoff. I'd a conversation with a first-tier college biology professor, about a way to give a more integrated, transferable understanding of a topic. He liked it, but observed, my students will shortly be taking the MCAT (high-stakes medical school entrance exam), and our time together is limited, and the MCAT doesn't test for understanding of the topic, only for something superficial and memorizable, so, I would be doing my students a disservice if I reallocated time to understanding.

Perhaps early primary school is an opportunity to escape this tension. With weaker test constraints, and more years of payoffs over which to amortize the costs of better understanding.

[kiba]

My understanding of learning is that it both takes a longer time to actually learn something useful and that we're inefficient in learning or retaining anything because we rushed from one topic to another in our courses.