The building blocks of understanding are memorization and repetition

You explicitly begin by mentioning that you think it's the similarity between the two topics that you think others are mistaken about -- that learning language and learning mathematics shouldn't be identified (as strongly). You now say that it's okay to (strongly) identify those two processes (as being the same), we need only talk about the relative amounts of subject versus language concepts needed, given an expected background for a student.

I wish you'd admit your phrasing could be misconstrued (at the very least).

> If you take the representation of Euler's equation, for instance, and consider how much conceptual depth underlies it, versus a string of 20 Cyrillic characters meaning "I went to the store," or whatever—you can see the difference. You'd be wasting your time trying to get a deep understanding of the Russian phrase, but there's a reason to do it with the mathematical phrase.

This statement, of course, only makes sense if you're comparing either just the language concepts of both or else the language and subject concepts of both.

Instead, your point here, as you last stated it, isn't that there's no reason to get a deep understanding of the Russian, but rather, that getting a deep understanding of the Russian is easier than getting a deep understanding of the mathematics because the deep understanding of the Russian is easier than the mathematical one to derive from your pre-existing understanding of English.

Okay, but is that really surprising? Or what you think you conveyed in that quote?

> So you always have language concepts only for a foreign language and language concepts + subject concepts for mathematics—practically speaking.

You actually always have both, but the bulk of learning a foreign language is language concepts, while in mathematics, you're exposed to more new subject concepts, which I think is what your point is.

Again, okay. But I think the language you used to describe that initially draws a needless (and ultimately, meaningless) distinction, and that we get a lot more utility out of talking about them as the same process for students, for which they've already done part in the general language case (and general arithmetic case!), since this automatically gives us a framework where we can meaningful discuss how learning Japanese, Russian, English, Category Theory, Set Theory, etc interrelate, the relative load of different concepts to learn in each case of learning a language, the process to go through, etc in very general terms.

So I don't think we actually disagree, except on whether your initial post accurately conveys what you seem to think.

It sounds like this is a lot of "perfect practice makes perfect" combined with "play begets understanding".

Memorization is a pre-requisite for perfect practice; and, if the core of what you're playing with isn't immediately at hand, how can you play with it? (If you have to look up every function every time, you'll have too many interruptions to grok much more).

I memorize in order to practive, and I repeatedly practice in order to understand.

Does that sound right?

By no means is memorization the end all be all. Oakley even writes "In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods". The techniques should be used hand and hand.

RangerScience said it best when he said "perfect practice makes perfect", I like to reword it so it reads "practice makes permanent". Obviously, practicing something wrong isn't going to help you advance.

Memorization should be "step one" for introducing a concept. I can explain how a for loop looks and show it to you, but forcing the student's hand to make a few loops before trying to implement it in a homework assignment can help build the neurological pathways (or motor engrams) so that they're no longer thinking about the syntax, just the chunk. "I need to type for(something;something;something)..." transitions to "I need a loop to go through this thing". The syntax is secondary as anyone who knows more than 1 programming language will tell you. Hell, I've build snippets into Sublime Text so I don't have to waste time with syntax either!

Some things just come from putting in the time and effort. No one gets their black belt after one class (otherwise I've been seriously doing it wrong for 10 years!)

Yes. Memorisation and repetition are useful. But of and by themselves, they are not enough, at least not for a deep understanding. That Ms. Oakley is a linguist makes the omission particularly glaring to me: it's finding a systemic understanding, that is, understanding the knowledge's grammar.

I've picked up a few skills over my life. Some music (poorly). A little bit of foreign language -- not much, but enough in a couple to get by as a tourist. Sport. A great deal of spatial knowledge. Some physics and economics, at uni. Programming and systems administration, some data management and analysis. More recently, synthesising numerous elements looking at questions of sustainability, collapse, or various modes of splitting the difference.

Some knowledge is almost mechanical. Music, sport, spelling, multiplication tables. X comes in, Y goes out. But simple repetition isn't fully sufficient -- this is what a good coach, in maths, music, or sport, offers. They know what you should be doing, see what you are doing, and then offer the cues necessary to get you to where you ought to be. The cues might themselves not make much sense overtly, but are the adjustment necessary to reach the desired result.

Practice without that intervention, and focusing on the right cues, only drills in the bad practices. And unlearning non-useful patterns is exceptionally difficult.

I'm tempted to say that easy learners are all alike. At the very least, none of them encounter the limits or barriers to learning (though it's possible each has some particular fast track to results). It's when learning comes hard that it's crucial to identify where and what the fault is, and to either correct it or bypass it.

In my current studies -- economics, political theory, ecology, energy, systems, and more -- what I'm finding most useful is to cover a great deal of ground, much of which is essentially circling a central problem sphere, but giving views on it from different directions. I'm quite literally finding myself re-acquainting myself with concepts, lessons, materials, and more, from the past 40+ years, and both dis-integrating and re-integrating them. I've described it as "refactoring my worldview" (mentioned on HN in a comment recently, also at https://dredmorbius.reddit.com), more to describe what the experience is like.

But the crucial element is not simply to repeatedly encounter facts until they're memorised, it is to create the structure into which they naturally fall. Or at least that's what I've found.

Some "systems" are less systemic than others. Virtually all have at least some path dependency, so history, law, politics, and literature will, in aggregate, at least follow some sort of path of low energy, if not an entirely logical route. In maths, logic, physics, chemistry, and electical engineering, the structure is more overt.

Again: Oakley approaches this concept, but never quite gets there. I found that disappointing.