> I think the error arises from making an overly strong identification between learning a foreign language and learning mathematics.
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.
Without getting as precise as using formal mathematical notation about all of this, there is going to be sufficient ambiguity that, if you desire, you can construe it to be taking one or another subtly different stance on several points.
Interestingly, those ambiguities are pretty readily solved without the formal notation if both parties are actually interested in communicating, rather than demonstrating that an initial statement said 'X' rather than 'Y'.
You are clearly in the second camp here, so the ambiguities weren't easily resolved. shrug
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.