How important is enforcing standard mathematical language and/or notation?
Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using nonstandard mathematical language. The example we discussed was correcting a student who says “x two” to indicate $x^2$ (instead of “x to the power of two” or “x squared”). I said that my primary role is to first determine what a student means, and if they mean $x\times x$, then I will almost always correct the language as saying such things may be misinterpreted as $x \times 2$.
This question came to me in the context of a discussion with other faculty about inclusive teaching practices, where a method shared involved allowing classes some freedom in creating and using their own language for items/principles/etc discussed in the topic. The perception was that allowing (and even encouraging) students to do this would build community in the classroom and let students attach some personal meaning to the items discussed. The instructors speaking favorably about this were from certain Humanities departments (writing, film studies, history). I was the only instructor from math or science in this conversation.
This is of interest to me because while my students must be able to communicate mathematics with others once they leave my class, I wonder if allowing some creativity here might do something positive for their overall experience studying math.
I have definitely had moments in certain classes where a student will (usually as a joke) propose a name for something — a solution technique, a common pitfall, a type of function — and the class adopts it. This happens when I don’t know of any standard terminology for it, so this class name may stick around for the term. I’m actually really happy when this happens organically. What I haven’t done is encourage playfulness with things that already have names.
So — is there any place in mathematics for student-created terminology and/or notation? How have you (or would you) allow it?
Finally, I understand what a huge setup for failure this would be if we didn’t enforce any language or notation — students would go to some other class and be completely lost, not recognizing basic things as typically presented. If you don’t like this idea in principle, that is totally fine. I am just wondering if there’s any place in modern mathematics for this kind of freedom for students.