I have a friend who says no, and a wife who says no. But I disagree.
It should go without saying that no one person and no university course can teach you (or me) how to get ideas. But they can suggest places to look. It might be just a case of pointing across the street and asking the student to imagine what that person in the smart business suit is doing with that old lady who is hitting him with her walking stick – and then to think beyond the obvious. Just teaching a concept of thought as simple as that one may be all that's required to get someone writing – that can be taught.
All stories, in whatever format (verbal, novel, script, etc.) follow the pattern of having a beginning. middle and end. There's a bit more to it than that, of course, and there are cases where the temptation to mix these up has been too great for some writers and directors, but I shan't confuse the issue by mentioning them. Let's accept that structure is important and many would-be writers start off without any planning or, indeed, having any knowledge of the form required to lay out a story – but that can be taught.
A little knowledge of the history of the English language (much of it lifted from other languages) can help in carefully choosing words to provide emphasis. You can learn that.
We were taught punctuation at junior school, yet now many 16-year-olds leave school without a clue about correct use of apostrophes and even simple commas – that can be taught.
These are just a few instances of teachable aspects used in writing.
"Ah but," you may say, "this is not teaching writing per se, is it?"
I agree that writers need to feel almost a primal urge to write – rather instinctively like breathing, eating, procreating... and not just so they can see their name in print or on the rolling credits (which, on TV, are getting faster, have you noticed?) I believe that wanting to write should not be to satisfy vanity. Quite simply, writers must want to tell stories.
And that cannot be taught.