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.