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If 'linguistic knowledge is largely subconscious'3, why isn't math?

Most math instructors sermonize solving exercises and problems. But a student challenged why students need practice — because most students and adults don't need to know, or practice, any linguistics or socio-pragmatics to speak their L1 (native language)!

        Modern linguistic research carried out within the Chomskyan framework reveals the complex system of linguistic knowledge (grammatical competence) that enters into actual language use to be almost totally tacit or unconscious. Only a small fraction of that knowledge is accessible to conscious thought. In other words, the set of rules which native speakers employ to represent internally an utterance of any one of the infinite number of sentences of his/her language constitutes a system of unconscious knowledge; thus, no native speaker is capable of describing those complex formal rules — syntactic, semantic, morphological, phonological, and phonetic rules — which native speakers employ in their day-to-day communication. Only a fraction of this knowledge is conscious. The conscious aspect of linguistic knowledge is restricted largely to the native speaker's ability to offer grammaticality judgments about an utterance.
        Although linguistic (grammatical) competence is the central element in language use, knowledge of linguistic form must be integrated with socio-pragmatic competence/ knowledge in order for an utterance to be socially appropriate/ acceptable. Like linguistic competence, socio-pragmatic knowledge is also largely unconscious. Consider, for example, the rules of turn-taking in a conversation. It is remarkable how complex the factors are that enable speakers to determine unconsciously when and how to take or yield their turn in order to ensure the smooth operation and success of a conversation (see Green 1990 for the markers of cooperation in a drug case; Crystal 2011: 126—7 for the index of criminal intent through turn-taking in conversation). In contrast, social evaluation of speech, i.e., the notion of language and speaker categorization (e.g., 'good' vs. 'bad' language, 'intelligent' vs. 'stupid' speaker) and prescriptive rules which are learnt by means of formal instruction (prohibition of double negatives in a formal context) represent a conscious dimension of socio-pragmatic competence. Figure 27.1 sums up the conscious and unconscious aspects of the two knowledge-based systems which interact with each other to ensure socially accepted language use.1

To wit, why is math unlike "most of our knowledge of the complex system of English grammar [that] is unconscious — which is very fortunate considering the rate at which we produce and comprehend language in normal conversations.[?] Frankly, the quantity of information that we have to know to use a language is astounding, and we generally only become aware of the depth of that knowledge when we have to learn a second language. Then we begin to see that language learning is so much more than just memorizing new words and sounds! All the world's languages have complex rules, though they differ greatly from English. The speakers of other languages are equally unaware of the vast majority of their grammar rules as well."2

They "know their grammar" but do not necessarily know much about it, i.e. they are often not able to formulate the rules governing their linguistic choices.4

Why can't students rely on 'native speaker intuition' for math?

'Even though most speakers claim not to know the rules of grammar of their language, they can tell you if a sentence is grammatical or not. [. . .] Although native speakers could say that the sentence sounds funny and declare what sounds better, most could not provide the complete rule for the order of multiple adjectives in a sentence even though they have been apply- ing this rule correctly, though unconsciously, most of their lives. We may not even think about this rule until we take a foreign language and then have to overtly learn the adjective order rules for that language — which are most likely different from the rules of our own.'5

Native language users process grammar and vocabulary unconsciously, i.e., they use language automatically with no clear consciousness of controlling grammar (Bialystok and Shamood Smith 1985). In our daily lives, for instance, we may drive a car while talking with friends, which means that we can easily perform the two actions of driving and talking in parallel, suggesting that we can manage both actions unconsciously (or automatically). Driving would become very dangerous if intentional control were required in the process of driving and/or talking. Automatic and unconscious processing makes it relatively easy for people to perform different tasks in parallel.6

1 Tej K. Bhatia, The Handbook of Bilingualism and Multilingualism, p. 675.
2 Elizabeth Winkler, Understanding language A Basic Course in Linguistics, p. 13.
3 Markus Bieswanger, Introduction to English Linguistics, 2017, p. 122.
4 Bieswanger, p. 122.
5 Winkler, p. 12.

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    $\begingroup$ But you practice your language skills for hours every day. If you do maths as often as you speak, listen, read, write or even think, it'll come just as natural, and indeed, there are plenty of people like that. It's just that for many people, maths are something to be avoided, not sought after :) $\endgroup$
    – Luaan
    Oct 3, 2022 at 5:36
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    $\begingroup$ Many immigrants who don't speak their native language anymore become less and less proficient in it after a decade or two. Use it or lose it. It's just that the average kid spends thousands of hours a year practicing their first language without ever considering that "learning" or "practice". $\endgroup$ Oct 3, 2022 at 10:31
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    $\begingroup$ Who says they don't? People easily detect failures of syllogism intuitively, even when they cannot articulate why. Similarly with geometric intuition. But most mathematical questions are exactly about qualifying your intuition with proof and arguments, which is nothing like the knowledge people have about their native language. $\endgroup$ Oct 3, 2022 at 14:26
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    $\begingroup$ I question your premise. What's your evidence that people don't have native mathematical intuition? I'm pretty sure I do. $\endgroup$ Oct 3, 2022 at 19:34
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    $\begingroup$ Why compare math to the native language, instead of a foreign language? That being said, if you go long stretches of time talking only a foreign language, it very much does happen that you become less comfortable with your native language (for a time, until you practice that again). $\endgroup$
    – kutschkem
    Oct 4, 2022 at 11:45

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Probably evolution. Our brains are not purely, perhaps not even mostly programmable computers. You can learn to walk naturally. Doing a kip in gymnastics takes practice. Similarly, your brain is specialized to have massive innate control of your tongue, but not of individual smaller toes.

Language has probably been a part of human society for tens or hundreds of thousands of years. Math has not. Especially at a general level. Consider now, if you lost the ability to understand math or language, which would make your life worse.

The brain does have some flexibility and can learn to do kips, play piano, type, and solve equations. But these require practice because we are not evolved to need them.

In contrast, facial recognition is an innate talent. One which programmable computers have only recently achieved.

Or consider the differences in muscular control of your fingers versus your toes. Or your fingers versus a dog's paw digits.

Or your sense of smell versus sight. And part of that is in the sensing organ, but a lot is in the brain. Look at the size of your optical lobe on a diagram of the brain.

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    $\begingroup$ This seems to imply that language, in contrast to mathematics, does not require practice. But it takes years of language development for a child to go from being able to say their first word to being able to form grammatical sentences. The fact that people develop an unconscious intuition for language doesn't imply that it doesn't require practice to get there, it is the result of practice. Developing knowledge/skill to the point where it is deep-seated enough to become unconscious intuition doesn't mean it's innate. $\endgroup$ Oct 5, 2022 at 14:39
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L1 language competence requires thousands and thousands of hours of practice to reach. We don't realise it only because much of it happens so early in life. Nonetheless huge amounts of early schooling time is put into mastering language skills.

It's so fantastically difficult to achieve this level of fluency that it is generally reckoned to be impossible for adults to reach it in a second language.

Many people receive sufficient mathematical training for simple arithmetic to seem innate but we simply don't spend the time to achieve unconscious mastery of more complex areas.

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    $\begingroup$ Most of language learning in school is about reading and writing. Spoken language, including vocabulary and basic grammar, is learned naturally through exposure. $\endgroup$
    – Barmar
    Oct 3, 2022 at 13:22
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    $\begingroup$ @Barmar: That "naturally through exposure" is the thousands upon thousands of hours of practice. $\endgroup$ Oct 3, 2022 at 13:45
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    $\begingroup$ I was responding to your second sentence about earlyi schooling time. By the time you enter school you've already mastered the language basics. $\endgroup$
    – Barmar
    Oct 3, 2022 at 13:47
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    $\begingroup$ @JackAidley Language acquisition is much more than practice. Infants prefer language over other sounds within 2 days of birth and can differentiate between French and Dutch within 4 days of birth. That's not practice. $\endgroup$
    – Passer By
    Oct 5, 2022 at 14:33
  • $\begingroup$ When parents help toddlers to learn, they ask them a lot of questions, and they correct them (by reformulating the toddlers' incorrect sentences for example). That looks a lot like an exercise session. Also, they use a lot of "exercise books" (image books for example) and "teaching materials" (toys with a lot of colors, sounds, images, etc). $\endgroup$
    – Taladris
    Oct 5, 2022 at 15:48
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I believe there are people who develop mathematical competency without being taught but they are in the minority. No one taught me algebra, but I placed out of algebra by "intuition". I am sure there are people who calculate without being taught how.

Most likely everyone develops language because we use it all day from a young age. I do not have the same competency in Hebrew, a language that I started learning at age 6 and I often am confused about what's grammatically correct. Sometimes if I stop and think about it, I notice my mistake and sometimes I don't. My granddaughter who is a native Hebrew speaker, catches all my grammatical mistakes without effort. Of course, I have no ability in Japanese because I was never exposed to it.

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    $\begingroup$ Exactly this. Language is something we practice, all the time, it’s just not something most people consciously practice. In a similar vein, I’ve spent so long playing shooter games that target-lead calculations are intuitive to me, even though I would have to spend a lot of time ‘relearning’ the actual math to be able to explain it to someone any more concretely than just ‘aiming where the target will be when the projectile gets there’. $\endgroup$ Oct 2, 2022 at 17:22
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    $\begingroup$ Agreement: Speaking, listening and reading is practicing language. If people spent as much time performing computations and reading proofs as they did speaking and listening, they would have extraordinary mathematical capabilities. But these activities are less engrained in our daily social lives compared to the corresponding linguistic activities. $\endgroup$
    – Opal E
    Oct 2, 2022 at 18:00
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Why can't students rely on 'native speaker intuition' for math?

Because we don't usually get to teach students mathematics; we teach them number problems. To quote Paul Lockhart:

“The area of a triangle is equal to one-half its base times its height.” Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time— there is nothing left for the student to do.

Some students manage to figure out the underlying mathematics, and those students can often coast on their 'native speaker intuition' – until they can't, because practising is actually important for developing that mathematical intuition.

If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box! […]

Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason. There is something so wonderfully transformational about this art form. The relationship between the triangle and the rectangle was a mystery, and then that one little line made it obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is all about?

Consider two students.

  • One learnt about algebra, figured out how to solve quadratic simultaneous equations, and was bored for the next year of maths lessons, barely engaging with the work except when explicitly encouraged.
  • The other treated maths lessons as a game of "spot and apply the algorithm", without really realising there was anything deeper, but consistently producing well-written answers to textbook questions.

Who's going to do better when faced with the trigonometry unit? The student who has been learning to spot patterns, identify algorithms, and execute them in a clear and correct manner? Or the student who's barely done any mathematical thinking for a whole year, because they solved the general case two weeks in, and doesn't remember how to multiply fractions?

Neither of these students is mathematically-literate. The engaged one is at least numerate – but you don't become literate by reading Biff, Chip and Kipper 30 000 times. You do it by having conversations about all sorts of topics, listening to people from all sorts of backgrounds, and writing in all sorts of ways. When was the last time you saw somebody having a mathematical conversation in a classroom more sophisticated than “how do I do question 21b?”? When was the last time you read a piece of mathematical literature in the classroom? (Do your students even know that mathematical literature exists?)

A mathematically literate person doesn't need to memorise the quadratic formula, because they understand what it means, and so (with a little thinking) they can derive it. But they're likely to have memorised it anyway, or near enough as makes no odds, because quadratics come up a lot and it's useful for studying certain properties of them.

A mathematically non-literate person might memorise the quadratic formula, and be able to apply it really well to problems of the form ax² + bx + c = 0, but as soon as they forget it, that skill is permanently lost ­– at least, until they re-learn it from a textbook. Without an understanding of what that computation is actually doing, they might as well buy a calculator and familiarise themselves with its instruction manual.

Maybe if mathematics exams were more like language exams, there might be more scope to teach mathematics in classrooms, and more students might pick it up from immersion and exposure (and chocolate).

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Frankly, I'm surprised no one else has mentioned this already:
Speaking a language and doing math are two very different skills.

When you speak a language, you express some ideas of yours in a more or less precise way, in a more or less elaborate manner, in a more or less eloquate tone, etc. Every single category you apply to language is imprecise. There are some hard and fast rules in the grammar, but do you make your grammar teacher happy as you speak? You try to write texts with good grammar, but I have yet to meet a person who always talks in full sentences... Nevertheless, we typically have no problem understanding the intended meaning of the imperfectly chosen words that are bound together with broken grammar which some person utters at us. And the grammar is the most precise thing about language to begin with.

By contrast, mathematics requires perfect(!) precision. As a math student, you must learn to apply text transformations without error. Now, you may get most of the points for doing the right steps in an exam where you replaced a 42 with a 24 by accident, but whatever your final result is, it's simply wrong. End of the story. Worse than worthless. Dangerous.

That means, that it is plain impossible to learn mathematics by trial and error. You'd only produce errors. And guess what, that's the experience of all those math haters out there. They tried to learn math the same way they'd learn a language, and they failed abysmally. Learning math requires a totally different approach: The approach of knowing the rules. And only applying the rules that are known to be applicable. And do so 100% correctly. And, as any math lover can attest, this radical "stick to the rules!" provides those that can apply it the remarkably blissful experience of: "Stick to the rules, and you'll never get a wrong result. You might fail to find the solution, but you'll never get a wrong one!" This "stick to the rules" is not what we are born to do. We have no intuition about it. We need to understand its necessity, to practice it, and to exercise it until it's a learned reflex to stick to the rules whenever we use math.

This is the key difference between learning the exact sciences and the "softer" subjects like languages.

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  • $\begingroup$ This is a correct but schoolchild view of math. In it's higher ranges math is all about reifying imprecision. Topology is real analysis approximated. In a different direction, likewise numerical analysis is approx real analysis. And asymptotics is all about approximation. $O(2n +7)$ is approximated by $O(2n)$ by $O(n)$. And one generally considers the rougher the better! $\endgroup$
    – Rushi
    Oct 20, 2022 at 12:31
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    $\begingroup$ @Rusi Sorry to disagree, but there is nothing imprecise in asymptotics, for instance. The lim operator is precisely defined, and either has a precise value, or none at all. When I say "1/x approaches zero", I'm really saying: "For any $\epsilon > 0$ there is an $x_0$ such that $-\epsilon < 1/x < \epsilon$ for any $x > x_0$." The former is only a colloquial shorthand for the later, but if you apply scrutiny, you will find every mathematical term precisely defined. The $O()$ notation simply defines equivalence classes for functions and the relation $O(f(x)) = O(g(x))$ is precisely defined. $\endgroup$ Oct 20, 2022 at 14:56
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    $\begingroup$ @Rusi In short: Math allows us to precisely define what we mean with "a is an approximation of b", it allows us to talk about approximations. But it is not an approximation itself, nor are its results imprecise in any way. There is no ambiguity in the statement $O(n) \neq O(n^2)$. $\endgroup$ Oct 20, 2022 at 15:05
  • $\begingroup$ See en.wikipedia.org/wiki/Big_O_notation#Equals_sign : "according to Bachmann $O(x) = O(x^2)$ but not the other way round. Knuth calls these one way equalities". It doesn't help that CS-ists use $O$ when they mean $\Theta$ $\endgroup$
    – Rushi
    Oct 21, 2022 at 3:54
  • $\begingroup$ @Rusi I didn't say that this statement was true ;-) But I concede, big O notation is fraught with ambiguity. Especially since there is indeed the confusion about which definition is actually used. And, indeed, it's mainly a tool used by computer scientists for a very broad classification of complexities, and as a language to communicate them. Nevertheless, once I define mathematically, what $O(f(x))$ actually means, the ambiguity falls away. If I use the upper bound only definition, it follows that $O(n)\Rightarrow O(n^2)$, but not the other way round. $\endgroup$ Oct 21, 2022 at 6:58
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Maths is two things. One is a sort of "language" as you describe - which is much more unambiguous than "natural" human languages, but also more limited in scope. The other is a practice, a study. Think about the difference between speaking English, as you refer to, and carrying out debate or rhetoric in English effectively. Which one do you think will take more practice?

Unfortunately, the two are mashed together, and many don't even get to know the "grammatical" parts of the mathematical "language" until very late, if at all typically unless going into a more specialized track in college or even graduate school and if not that, then only by self-study, or even really just get an informal idea for what the various pieces of "language" actually mean. For example, how many would, if you ask, know that an expression like "2 + 2 = 4" is actually a term composed of two sub-terms, namely "2 + 2" and "4", where the "=" follows a grammatical rule that " = " is itself a term? Moreover, each term has a type (namely the term formed by "=" you might say type "Boolean", but its subterms may not, and the subterms on "=" must agree for type. And that's just one particular grammatical system that I just happen to like (look up "type theory" for the "type" idea, and even then that's far from a complete one and I've not found most fully decked-out ones entirely satisfactory); there are many others possible.

I would suggest you can learn mathematical "language", in this regard, without too much difficulty indeed, and you should also be able to understand individual mathematical concepts, i.e. the definitions of words, by themselves if you are conveyed the underlying intended intuition and semantics instead of a banal formalism (for example, "+" is a function that is constructed the way it is so that it can be used to represent quantifying aggregation with conservation). The real skill is in the mathematical problem solving, i.e. actually using them - e.g. determining whether "2 + 2" stands for the same number as "1 + 3" and thus whether the assertion "2 + 2 = 1 + 3" is true or false. This takes practice for the same reason it takes practice to do any other form of investigation or inquiry, and you have to learn the relevant methods and tools as well as develop the relevant facility in that regard.

Finally, even regarding the language part, maths' language is always, due to its more specialized expressive power, in the position of learning that a "second language" would occupy. And those always take more effort - it's only your first language that comes naturally and "effortlessly" when very young, and even then that's not entirely so since while you can get "workable facility" in it effortlessly, but it still takes practice to get great at it.

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Everything we learn, we learn as patterns. Our brain works by pattern matching and pattern prediction.

Our brain has an enormous flexibility, e.g. the same area can be used for spoken language as well as for sign language (if you are deaf). People without hands learn to do everything with their toes.

The main factor for learning language - and everything else - is exposure. From day one, we spend multiple hours per day learning and practicing language. In a multilingual environment, we can even learn several languages at the same time.

It's really the same with lots of other skills we acquire. We learn to drive a bicycle and a car on a subconscious level - even if, in theory, there is a lot of physics and mathematics involved. We learn to subconsciously apply what is best described in quadratic and differential equations.

We do not spend the same amount of time with mathematics, as it is taught at school and often used at work. We manage to learn the multiplication tables on a subconscious level, not much more.

Also, we do not want to do it subconsciously. We want to write it down explicitly, so that we, and the teacher, and the tax auditor can check what we have done. In many exams (at least in Europe), you get points for writing down all the steps, not just for the result - much to the chagrin of the gifted student who solves the task instantly in his head.

While we do get some gut feeling over time and move some of our problem solving skills to the subconscious, students always learn lots of new things and simply cannot spend that much time on any topic.


(For reference, I would like to point to the TED talks of Jeff Hawkins and Lýdia Machová, and their other works. You might also want to ask the question on Psychology & Neuroscience StackExchange.)

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Because as Otto Jespersen shows in his Philosophy of Language, we mostly speak based on patterns. So a child learns "John hit me" and from that has "X hit Y" to cover all future hittings. But though you have 3x + 4x = 14 as a pattern , it is a pattern of a setup but not of how to solve. If you form a sentence you know what you mean, but if you form a word problem you might have nary a clue to solve it.

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