I've written before about how I feel like a concept we think our students get that they really don't get is the composition of certain functions, specifically trig functions and log functions. I made a vow to myself to emphasize compositions with my Pre-Calc students a lot this year, so that when they do get to calculus, the Chain Rule and u-substitutions will be two of their best friends, as opposed to worst enemies.
I was reminded of this vow when I asked a student to read an exercise from the book out loud. The exercise started like this:
And this is how she read it:
"Sin" [as in a transgression, not a trigonometric function] "times pi over two minus x."
I wanted to say, "When have you EVER heard anyone say it like that, girl?" But, I remained calm. I ignored the mispronunciation (we have bigger fish to fry here), and focused on the "times" part.
It seems like every time I have this conversation ("It's not 'f times x,' it's 'f of x,' guys."), I feel like the kids are just nodding to get me to shut up. I can't blame them. I did the same thing in grad school [way] more than once. As long as I make the prof think I understand what he's saying, all will be well.
But, inputs. They're kinda a big deal. What worries me is that it seems like students often view inputs as some kind of multiplication as opposed to actual arguments, which makes sense as the notation is very similar (parenthesis for both).
I continued to notice this was a problem as we were verifying trig identities. I don't know if the kids just got so into the proofs that they forgot a few fundamental things...like what sine and cosine are...or what was going through their heads exactly. But, let me tell you, I saw crap like following slide all. the. time. So, I made them figure it out:
I would not tell them what was wrong, but I did mention it was subtle. When they finally started figuring it out, we talked about why we need all those theta's! Our dear trig functions are meaningless without them!
It's a small step, but if it gets them to remember that these trig functions must have an angle at which they're to be evaluated, even if that angle is arbitrary, well, then, that's a good thing.