- This is part of the "Better teaching in any subject" series:
- Part 1: Problems with modern inquiry based methods
- Part 2: Meaning is use
- Part 3: Facilitate learning through the inner game of meaning as uses
- Part 4: Worries over losing deep conceptual knowledge (forthcoming)
Meaning is use
Meaning as use is a philosophical concept of meaning from Wittgenstein:
"For a large class of cases of the employment of the word 'meaning' --- though not for all --- this way can be explained in this way: the meaning of a word is its use in the language" (Philosophical Investigations).
Many traditional and folk understanding of meaning explains meaning in terms of mental representations, or idealized objects in some (sometimes mathematical) objective space, etc. --- i.e. stuff in people's heads or in some Platonic ideal space that has no practical significance for teachers in the classroom.
So while we may not necessarily agree that meaning is philosophically just its use in the language, it's certainly practical to see it that way for teaching! Because we can, as Wittgenstein urges, look and see the variety of cases in which a word is used, but we cannot look into the heads and minds of students --- and more importantly, nor can students look into the minds of teachers in learning what the teacher meant.
"So different is this new perspective that Wittgenstein repeats: 'Don't think, but look!' (PI 66); and such looking is done vis a vis particular cases, not generalizations." 
Meaning as discussed usually refers to meaning of words in a language, but math is no different. Math is itself a natural language, with a grammar and semantics that's evolved in the mathematical community, used to talk about things and their relationships. We need not look further than many science research papers wherein authors write mathematical notations and formulas, interweaved with English prose, to see how math is very much a language we can talk about things with.
If we accept that meaning (of words) comes from their use, then the meaning of a thing like a math formula is also built up from the use of it. The proper meaning of a math formula doesn't come from the instructor explaining it, and it doesn't even come from students discussing and talking about it. The meaning of a math formula comes from the proper use of it.