So, I know this is an question that has been covered by a ton of people, but I just thought I'd add my own two cents' worth.
Can you teach writing? Yes and no.
Writing--good writing--comes from practice. It doesn't come from some magical mythical invisible fountain of awesome inside your head. Writing is really a learned skill, in my opinion. Even getting good plot ideas is learned, I think. As you look at the world around you and learn to see the beauty and the terribleness in it with an eye for story and character, you learn how to get inspiration, as strange as that sounds. You just really have to devote yourself to it. Let your inhibitions go, don't be afraid, and write like your life depends on it.
However, that being said, I don't think I'd be the writer I am today if I hadn't attended the California State Summer School for the Arts (studying creative writing) and the Iowa Young Writers' studio. In both those places, I attended workshops and class-type-things that were mostly discussion-based and interactive. By having my peers critique my work and critiquing the work of others, I learned how to view and dissect and truly understand my writing in a way I hadn't before. As I discussed in my other post, What it Means to Read as a Writer, going to these programs has taught me things I don't think I could have effectively learned on my own.
But, when you really get down to it, I'm not sure my experiences at CSSSA and the IYWS weren't exactly teaching so much as supplementing my practice. If I just went to those programs and I didn't practice, I would be a pretty crappy writer. You're not going to become a great athlete or musician by learning about it intensively for two weeks a year, or even six months of a year. Without exploring something on your own, you're going to miss out on a large portion of the learning process. Writing is the same as any sport or musical instrument in that way.
Basically, I believe that writing can't be taught, in the most literal sense of the word. No one can tell you exactly how to do it. There's no set of rules you can memorize, it just doesn't work that way. At the same time, critiquing work and having yours critiqued in turn are essential parts of the writing learning experience. This doesn't mean that you have to go to school for it--you can, if you want, I think it can be very useful--but it does mean that, if at all possible, you should seek out critique groups or summer programs or whatever you might want to do. That sort of thing has been absolutely invaluable to me.