- 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.
That means for students to look and see the meaning of a math formula, they have to see it being used properly --- used as parts of explanations and descriptions of the world, used to solve problems in the world, and used in a variety of particular cases, not generalizations --- and the students have to similarly use it themselves properly too in order to exercise the use of the meanings.
Math is very much a utilitarian thing, created and valued for its usefulness (as much as its beauty), so it shouldn't be a surprise that its meaning is built up from its being used.
But notice that I said the proper meaning of a piece of math comes from its being used properly. If teachers and students see and use various pieces of math wrongly, then for those students, the mentioned pieces of math (formulas, terms, or whatever) will naturally pick up their meaning from the wrong usages! If meaning is use, then students seeing and imitating wrong usages will naturally learn the wrong meanings.
That's why all the experiencing and reflecting that students do in today's experiential, discovery, inquiry, or constructivist style classrooms is terrible if they are not shown first the proper way of use, be it in math, English, etc.
That doesn't mean students should be taught with traditional back-to-basics methods though.
Meaning comes from a variety of particular uses, not from being told a single general way to do things
Traditional back-to-basics methods usually imply students being told a single general way of doing things. As in:
Here's the step-by-step you must take and show on paper when told to add two two-digit numbers together: line up the numbers digit-by-digit vertically, work right to left, add digits column by vertical column, if there's a carry then write it down in the next column of digits on top, etc.But that is not the meaning of "add two two-digit numbers, 12 and 13, together".
At this point, constructivist, discovery, inquiry, and experiential style proponents might eagerly raise their hands and suggest things like tiles, interlocking cubes, strips of paper for number lines, reflection journals on strategies they discovered playing with those math manipulatives, etc.
Even if a student figures out how to use the interlocking cubes properly, that is still not the meaning of "add two two-digit numbers, 12 and 13, together".
That's because all of those are just individual, particular uses of "add two two-digit numbers, 12 and 13, together". Even the traditional addition algorithm or strategy described above is just another particular use.
But a student who sees many varieties of particular uses of "add two two-digit numbers, 12 and 13, together" can begin to learn the meaning of that phrase. The student must look and see a variety of particular uses, including but not limited to, tiles, cubes, paper strips, traditional adding algorithm, etc., all being used properly to represent and solve "add two two-digit numbers, 12 and 13, together".
"Meaning is use" means more accurately that meaning comes from a variety of particular uses, and students need to look and see while teachers show the varieties of uses properly in order for students to learn the proper meanings from their proper uses.
As they say, "show, don't tell". But that's not enough.
 Biletzki, Anat and Matar, Anat, "Ludwig Wittgenstein", The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.), URL =