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This question is meant to explore the intuition that mathematical thought does not most naturally proceed from writing in one's native language. The hackneyed and not entirely satisfying slogan that mathematics is a language is also relevant here, in that I am trying to tease out how "linguistic thought strategies" are central in the mathematical practice of successful mathematicians. Things like Manin's essay here are interesting if you have wondered along similar lines.

The basic observation I am interested in appears already in the learning of "math facts", which obey a substitution grammar that is very different from the one appearing in everyday language use. One might, following a Wittgenstein language game approach, regard a response of "5" to the question " What is 2+3?" as somehow analogous to a response of "Fine" to "How are you?" in English. In the sense that there is a linguistic command or stimulus that has a known, standard, response, the above two are similar.

In the above situation, there is an aspect of the learning of a foreign language, in that there is no "translation stage" in either case. This feels similar to a native English speaker learning Mandarin Chinese, where it is most efficient to directly learn the new language with very little translation. (Of course, this ignores the fact that in both cases, Chinese and English may point to the same physical objects or analogous human situations and mathematics may not so obviously do this.)

As a professor in a liberal arts college, I deeply believe that teaching students to write will inevitably teach them to think. The trouble is, I have a nagging feeling that having students write in their native language about mathematics in order to think about mathematics is very inefficient, and if Poincaré and Hadamard are correct, most mathematicians avoid thinking explicitly in language, perhaps because it may somehow emphasize serial reasoning over a holistic intuition. (The suppression of linguistic tendencies by mathematicians is reminiscent of Betty Edwards' Drawing on the Right Side of the Brain.) If emphasizing writing too much can impede the mathematical investigation in similar ways to how language may impede drawing ability, then we should be very wary of such emphasis.

It may be, though, that mathematicians are using their linguistic faculty in a way similar to learning foreign languages that share Wittgensteinian family resemblances. By learning solutions to problems in a subdomain, a mathematician stores, in a "language game"-like linguistic fashion, a set of promising responses to collections of problems in the subdomain. The responses are like various acceptable responses to "How are you?" in English. Some of these work out to lead to solutions to problems (the mathematician's language includes in its kernel the grammar of basic logic, so the proof is possible). It may even be that mathematicians are always including the use of their linguistic faculty to combine various symbols (here understood as formal entities that can be communicated, in principle...which may include pictures and various sounds) in potentially promising ways in response to problems. Any such mental manipulation of a mathematical object is somehow a "linguistic move" in that it involves a distinct transformation of something that can be communicated. It seems that solutions to problems provide patchwork conversations to learn the "foreign language moves" needed to learn mathematics.

As a person stockpiles enough of these responses at the level of mini foreign languages with family resemblances, he or she eventually becomes a what is called a mathematician.

I apologize for the above preamble but have not been able to distill it after thinking about it over some time. The point is, writing in one's native language, describing mathematics in order to try to learn it, seems similar to writing about playing the piano without actually training to learn to play...which is also learning something that resembles language. It may directly impede the learning of mathematics!!! It seems that one must learn the language of mathematics with as little translation as possible, and then try to write in that language, to learn mathematics.

Here is the question:

Question: In the education literature on writing to learn mathematics, what are some solid references that compare the above "foreign language acquisition" aspect of writing to learn with an approach of writing about mathematics primarily in one's native language?

I realize that this question is (as I am always guilty of) very dualistic and therefore kind of ridiculous. I cannot seem to settle the issue in my teaching, though, because attempts to be (quoting J.P. Serre) precise yet informal and describe mathematics using one's native language is a translation activity that may fortify human understanding, which is the ultimate goal of developing mathematics in the first place. Well-written mathematics very effectively bootstraps native language intuition without losing precision...and the thought of blind algorithmic manipulation is repugnant to any red-blooded mathematician if such manipulation is not being carried out to develop intuition (which may actually be related to fluency in the foreign language being acquired).

Another reason the question may be ridiculous is that it is dangerously close to being something we all have wondered about ("drill versus thought" approaches to math education). I ask, though, because it feels to me like there is something subtle here about which I'd like to read more...but I haven't been able to find a place to do so after searching the web several times.

This question is also related to nearly all of my others along these lines, particularly my "Calculation versus Writing" question and "Is Metacognition Ever Bad?". The distinction of the present question from these two is that here I am mostly interested in the intuitive similarity of feel of learning a foreign language without first translating and ALL mathematical activity, including activities that may not be initially interpreted as calculation. Drawing diagrams and manipulating vague mental entities have this quality of trying to almost combinatorially obtain fluency with something foreign with the desire to eventually succeed in a form of communication that is solving the problem. (This is analogous to the feeling of being able to program a computer. A solution of a mathematical problem is somehow having adequate fluency to be able to successfully communicate in a foreign tongue.) This question can be interpreted as unpacking the description of Israel Gelfand of mathematics as "adequate language" (although, not just for science). (EDIT: This essay of Ed Effros seems to be close to what I want...so more things like that would help, I think...)

EDIT: I think the intuition I am chasing here, and the one I hope the literature can help clarify, is related to the reference to mathematics as "the science of patterns". If a pattern is understood as Poincaré describes it as an "atom of perception", something that we attribute stability to by naming it and delineating it in either space or time (in the field of our senses or thought), then it is very much connected to language by this delineation. Native language plays with the perception of new patterns by the appearance of analogy (like seeing familiar shapes in clouds) and this may be something of the role of native language in mathematics, where patterns are identified and named. The foreign language aspect of this comes about by trying to develop an adequate and efficient language for the patterns of thought that emerge. Most importantly in these observations, the pattern requires delineation, a stable entity arising by a linguistically defined boundary that can be manipulated as a meaningful idea. This naming or encapsulating process, much like the philosophical process of assigning order via language to vague intuitions, yields mathematical ideas. The sense of mathematical beauty is reflected in the ability to effectively perform this delineation...and the "choices" made in such delineation are much like those of a poet. In mathematics, the desire for recombination and something of the foreign language aspect (necessitated by the desired leverage that the adequate language of mathematics, its body of organizing principles, provides) is more important than in philosophical investigation, which proceeds more closely to the native language.

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    $\begingroup$ I'll dare to offer one more interpretation: learning to do math. is akin to learning to play a complicated game with some clearly written and some unwritten rules (you learn the latter by trying to do something and seeing that it just doesn't work at all, or finding out a (meta)-trick that seems to result in a substantial advance again and again). So I would also look into how people learn to play games from simple ball once to advanced virtual reality ones in addition to learning languages, if you really want to figure out what's going on here. Just my two cents. $\endgroup$
    – fedja
    Apr 2, 2023 at 1:16
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    $\begingroup$ And, as much as good writing skills are very important for communicating mathematics, they do not seem to be strictly necessary for doing it. One of the brightest students of one of my colleagues was almost hopelessly dyslexic. (to the extent that when he served in some elite Israeli army unit, they just gave up and merely assigned another person to write his reports). As to "writing in foreign language",we learned programming that way (all commands were English-based). It is fun and does help as long as the vocabulary is not too extensive. Again, just my personal experience. $\endgroup$
    – fedja
    Apr 2, 2023 at 1:29
  • $\begingroup$ @fedja: The idea of looking into how people learn games is an especially interesting one. Thanks! I wonder where to look for this? $\endgroup$
    – Jon Bannon
    Apr 2, 2023 at 23:45

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Based on a quick Google Scholar search, it seems like there was a flurry of activity in this area ("education literature on writing to learn mathematics") circa 1990, with at least 3 book-sized collections on the subject:

Connolly, Paul, and Teresa Vilardi. Writing To Learn Mathematics and Science. Teachers College Press, 1234 Amsterdam Ave., New York, NY 10027, 1989.

Sterrett, Andrew. Using Writing To Teach Mathematics. MAA Notes, Number 16. Mathematical Association of America, 1529 18th Street NW, Washington, DC 20036., 1992.

Countryman, Joan. Writing To Learn Mathematics: Strategies That Work. Heinemann, 361 Hanover St., Portsmouth, NH 03801-3912., 1992.

I haven't read them, so I don't what or how solid the findings are. More can be found here:

https://scholar.google.com/scholar?hl=en&q=writing+in+mathematics+education

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  1. I would be very cautious about the idea that math training (strongly) enables general problem solving. If anything, I think training in rhetoric may be more generally useful across fields. See for example:

    https://classicalchristian.org/the-lost-tools-of-learning-dorothy-sayers/

    But I'm not advocating Sayers per se (although she was a right sharp lady), because I think in general "general problem solving" is oversold as a skill. Most problem solving is domain specific. We are like Ugg the caveman teaching his son how to make the best shaped stone tools...really we are...and even this is a marvelous invention...allows the formation and transfer of culture and technology.

    What training in math mostly teaches is training in math. And nothing wrong with that! There are a lot of important things to be done. Difficult things. Non trivial things. And a lot of balls not getting across the goal line (students not learning minimum standards) that we should concentrate on correcting.

  2. I would be cautious about asserting that math is better performed in a foreign language (is that even your point...kind of a word salad to weed through, but trying to find/address your main idea that you want contested. At a minimum, I would want solid, repeated, multiple controls study done to show it. And burden of proof on the proposition, not on the contrary. After all math is hard enough, let alone doing it in another language! All that said my sister did a year abroad (as an English speaker to German region) and math was her easiest subject (aside from English as a foreign language!) Next hardest was science. Then history. Then German. Then...the absolute hardest was trying to learn Latin IN German. Kind of pretty much what you would expect, I guess.

  3. I don't believe Chomsky natural language is applicable to math (or reading). Both of these are functions that are very recent in our evolutionary history. Oral language on the other hand is not...and has significant dedicated brain structure (and is easily learned by children). To the extent the Chomsky stuff is useful (and I'm pretty positive on it, but not an expert), I think that is in the context of oral language. I would be very hesitant to extend oral language models to other domains...that is the mistake the whole language reading advocates made and there have been study after study showing several reasons why they were mistaken...to the extent people have run from the term. Definitely Dewed-eyed dreaming there! The more we learn about rigorous applied psych testing (or even brain imaging), the more this Rousseau-ian stuff gets crushed.

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