Many of us have heard the argument that you can't teach someone to write -- or that of course you CAN teach someone to write. One thing I've learned is that, no matter the subject, if there are extensive, reasoned arguments plus a lot of people on each side, it's usual (not universal, but usual) that each side has at least a piece of the truth and neither is completely wrong or right. Some kind of harmonizing of the views is in order to get the full picture.
I've long been on the side that says, "Of course you can teach writing. If you couldn't, why attempt to do so in schools? Writing may be art in a sense, but it is also craft, with specific skills in composition, grammar, and story, that can be learned and practiced. Raw talent, in any field, must be trained. If I didn't believe writing could be taught, why on Earth would I be teaching it?"
You know there's a "but" coming, right? BUT...when people say writing can't be taught, I do understand what they mean.
Editors and agents touch on it when they say, "I can help a writer with plot, but I can't help with voice." Teachers can help writers learn to plot, to use POV correctly, to show rather than tell, to use sensory detail, to flesh out characters, and to identify theme. We can teach grammar and sentence structure. But we can't teach a facility with language. Or a lyrical style. We can't help much with a constant tendency to choose almost the right word. There is such a thing as basically competent yet tone-deaf writers. That there comes a time when we can't bring them beyond their innate language talent level is what writers -- it's usually writers -- mean when they say writing can't be taught.
In this way, writing is like any of the arts. To teach any sort of artist, you take a person with a measure of inborn talent and set them a program in which they explore and practice different forms, media, and techniques, helping them improve weaknesses and identify strengths. You can teach writing, because having raw talent is no excuse for eschewing a proper course of training. But, you can't teach the talent, and no matter what we say about desire, hard work, and perseverance going a long way, and they do, the talent's got to be there.