I do freelance tutoring; the math-related ones are mostly 1-2 hour one-on-one sessions for high school and college students taking calculus classes. Often I find myself scrambling for words - I don't want to stare blankly at the student for half a minute planning what phrasing to use. Unfortunately, the aforementioned scrambling means that sometimes terminology which I know is incorrect slips in. (Indeed, every single bad phrase from this post has come out of my mouth during one tutoring session or another!)
An example from a quick conversation in recent session [paraphrased]:
Me: Do you know how to find the derivative of $f$?
Student: Wait, that ("that" is $\frac{d}{dx}[f(x)]$) means to take the derivative?
Me: Yes, it's equivalent notation.
Student: I need to find $f$ prime then.
Me: Yep, just take the prime. [internal facepalm]
Student: Hmm... [starts scribbling work down]
Should I have done anything in particular after I accidentally used incorrect terminology? I don't want to distract too much from the student getting to work on the problem and practice doing calculus, but I'd also hate to contribute to what I know is widespread misuse of basic calculus terminology, if for no other reason than I like fancy terms. Is it worth interrupting the student's work to correct myself, probably stumbling over more words in the process of explaining why my initial phrasing was wrong? Or is there another remedy I can employ?
I should clarify that this is exclusively verbal terminology - I'm a stickler about how I write down mathematical work due to a few strict teachers who managed to drill in, for example, that you never stop writing the $\lim_{x \to 0}$ during the intermediate steps if it's still there. I will also clarify that I get both students who just want to get through their homework, and students who want to learn more broadly e.g. to prepare for a test or get caught up with their class, so it would be helpful if answers could consider both those cases separately or together.
