Twitter in the classroom

I have been thinking about ways that twitter would be useful in my classroom, and I think that some reasons are obvious. Students can tweet about assignment and questions they may have as they work on them. But also they can run ideas past each other in a very direct forum. The advantage of twitter is that it is very assessable from your phone. Text messaging is becoming ubiquitous with students these days and twittering is a great way for them to be in contact with one another. They could ask a specific question in relation to an aspect of an assignment they are working on. They can post events that are happening in their lives personal or otherwise.
As their teacher I could assign specific twitters that they could follow, maybe in current events and then we could discuss the tweets in class. Maybe for an art project we could print out all the tweets from the week from all the class members and put them into a collage that would make a statement about our public/private lives?