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The author is arguing that 'the latest wave of educational reform in mathematics' (in the U.S.) is overemphasizing conceptual understanding, and that doing so w

The author is arguing that 'the latest wave of educational reform in mathematics' (in the U.S.) is overemphasizing conceptual understanding, and that doing so will be damaging to students. The author advocates instead using a memorization and repetition based approach. I can see where she's coming from, but the argument she is making is flawed. She describes a process for developing fluency both in foreign language and mathematics, wherein she experiments and plays with the constructs being learned—this is a great suggestion, but it's not in opposition to understanding: the two complement each other. I think the error arises from making an overly strong identification between learning a foreign language and learning mathematics. 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.

johnnyhillbilly 10y ago
Didn't this turn out a bit of a straw man? As far as I understand, she advocates an understanding of the concepts complemented by some appropriate type of repetition. There is also a third concept: That of imagining. She speaks specifically of one equation in classical mechanics, which she played around with in her mind until it became second nature. Of course, this is based on her experience. It appeals to me based on my own experience, yet it isn't a showcasing of scientific rigour within pedagogics.
- throwaway729 10y ago
  > yet it isn't a showcasing of scientific rigour within pedagogics. I think this is exactly the correct criticism of this and most other critiques of mathematics curricula. It's ironic to see a huge conversation about mathematics pedagogy that never goes and looks at the data.
SomeStupidPoint 10y ago
> 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. I think you're simply wrong here. You're not talking about the amount of conceptual depth underlying it, you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning. On the whole, the Russian phrase has greater conceptual depth; however, Russian is closer to English than mathematics is, and thus you get greater conceptual transfer between your knowledge of English and Russian than your knowledge of English and mathematics, so more of your past experience remains relevant. In essence, I have something like 36-72k hours considering semantically equivalent statements to the Russian one in English, while I have something like 5-10k hours of studying mathematical statements at all. Your comment just amounts to saying that having 6x as much practice at task A as task B makes task A relatively easier, which, while true, isn't perhaps as profound as you'd meant it. Consider this: the entire field of academic inquiry containing Euler's identity is fundamentally an effort to fully understand "I went to the store".
- westoncb 10y ago
  > you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the subject the language is used to talk about. To be clear, I'll call the first category of concepts 'subject concepts' and the second 'language concepts.' When you learn a new foreign language you must learn new language concepts, but not new subject concepts. You aren't re-learning what it means for one to visit a store, you are only learning a new scheme for representing that concept, which isn't nearly as deep as the initial concept itself. In learning mathematics you're having to pick up the language concepts and the subject concepts, and the subject concepts (as in the example I gave) can be quite deep.
  - SomeStupidPoint 10y ago
    You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while knowing no languages, since it's purely acquisition of language rather than subject concepts. (This is actually untrue -- which is why it's easier to say, switch to romance languages than going to an Asian one from English; there's a bit of subject conceptualization in the nature of the language concepts.) It's not comparing apples to apples, which sort of reduces the point about the relative complexity of statements and the way that you learn the underlying concepts -- both language and subject. You're comparing the complexity differential of two encodings on the one hand and the total complexity involved in the other. Nonsense comparison. If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? Because the language differential between spoken languages (eg, you do learn new subject concepts if you learn Japanese versus English) is comparable to the difference in mathematical underpinnings, eg, the switch from set theory to category theory.
erroneousfunk 10y ago
Interesting that you should use the phrase "I went to the store"... I took Russian for two years in college, and, while I can get around Moscow alright, ask for directions, and exchange a few sentences of banter with Russian speakers I meet at parties, I'm hardly fluent. But the phrase "I went to the store" could be constructed in a HUGE number of ways in Russian, as there is no word in Russian that simply means "went" or "go" like we have in English. Did you go to the store and come back? Was it a unidirectional journey? Do you go to the store often? Did you drive to the store? Did you walk to the store? Did you go inside the store? Did you obtain the endpoint of store as a goal? The phrase "<foo> went to <bar>" can be (and, indeed must be -- no easy outs) conjugated and declined in so many freaking ways in Russian, with subtle meanings encoded in those conjugations and declensions, it's nuts. I used to be great at the verbs of motion a long time ago, but, because I haven't been religiously practicing and thinking about them, I just avoid them altogether at this point. (The last time I tried saying "Are you driving to Boston?" I accidentally said "Are you wandering around aimlessly in Boston by car?") My point though, is that I would argue that there is a very good reason to think deeply about the sentence "I went to the store" in Russian, and think deeply about all of the ways you can say it, modify it, and replace its component parts to achieve different meanings, and how those meanings relate to the meanings of other sentences containing the same component parts. I certainly wish I had done that more often!

johnnyhillbilly 10y ago

Didn't this turn out a bit of a straw man? As far as I understand, she advocates an understanding of the concepts complemented by some appropriate type of repetition. There is also a third concept: That of imagining. She speaks specifically of one equation in classical mechanics, which she played around with in her mind until it became second nature. Of course, this is based on her experience. It appeals to me based on my own experience, yet it isn't a showcasing of scientific rigour within pedagogics.

throwaway729 10y ago
> yet it isn't a showcasing of scientific rigour within pedagogics. I think this is exactly the correct criticism of this and most other critiques of mathematics curricula. It's ironic to see a huge conversation about mathematics pedagogy that never goes and looks at the data.

SomeStupidPoint 10y ago

> 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. I think you're simply wrong here. You're not talking about the amount of conceptual depth underlying it, you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning. On the whole, the Russian phrase has greater conceptual depth; however, Russian is closer to English than mathematics is, and thus you get greater conceptual transfer between your knowledge of English and Russian than your knowledge of English and mathematics, so more of your past experience remains relevant. In essence, I have something like 36-72k hours considering semantically equivalent statements to the Russian one in English, while I have something like 5-10k hours of studying mathematical statements at all. Your comment just amounts to saying that having 6x as much practice at task A as task B makes task A relatively easier, which, while true, isn't perhaps as profound as you'd meant it. Consider this: the entire field of academic inquiry containing Euler's identity is fundamentally an effort to fully understand "I went to the store".

westoncb 10y ago
> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the subject the language is used to talk about. To be clear, I'll call the first category of concepts 'subject concepts' and the second 'language concepts.' When you learn a new foreign language you must learn new language concepts, but not new subject concepts. You aren't re-learning what it means for one to visit a store, you are only learning a new scheme for representing that concept, which isn't nearly as deep as the initial concept itself. In learning mathematics you're having to pick up the language concepts and the subject concepts, and the subject concepts (as in the example I gave) can be quite deep.
- SomeStupidPoint 10y ago
  You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while knowing no languages, since it's purely acquisition of language rather than subject concepts. (This is actually untrue -- which is why it's easier to say, switch to romance languages than going to an Asian one from English; there's a bit of subject conceptualization in the nature of the language concepts.) It's not comparing apples to apples, which sort of reduces the point about the relative complexity of statements and the way that you learn the underlying concepts -- both language and subject. You're comparing the complexity differential of two encodings on the one hand and the total complexity involved in the other. Nonsense comparison. If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? Because the language differential between spoken languages (eg, you do learn new subject concepts if you learn Japanese versus English) is comparable to the difference in mathematical underpinnings, eg, the switch from set theory to category theory.

erroneousfunk 10y ago

Interesting that you should use the phrase "I went to the store"... I took Russian for two years in college, and, while I can get around Moscow alright, ask for directions, and exchange a few sentences of banter with Russian speakers I meet at parties, I'm hardly fluent. But the phrase "I went to the store" could be constructed in a HUGE number of ways in Russian, as there is no word in Russian that simply means "went" or "go" like we have in English. Did you go to the store and come back? Was it a unidirectional journey? Do you go to the store often? Did you drive to the store? Did you walk to the store? Did you go inside the store? Did you obtain the endpoint of store as a goal? The phrase "<foo> went to <bar>" can be (and, indeed must be -- no easy outs) conjugated and declined in so many freaking ways in Russian, with subtle meanings encoded in those conjugations and declensions, it's nuts. I used to be great at the verbs of motion a long time ago, but, because I haven't been religiously practicing and thinking about them, I just avoid them altogether at this point. (The last time I tried saying "Are you driving to Boston?" I accidentally said "Are you wandering around aimlessly in Boston by car?") My point though, is that I would argue that there is a very good reason to think deeply about the sentence "I went to the store" in Russian, and think deeply about all of the ways you can say it, modify it, and replace its component parts to achieve different meanings, and how those meanings relate to the meanings of other sentences containing the same component parts. I certainly wish I had done that more often!