How I Use It
There are two things I like to do with Twitter. The first is student outreach. Students ask if you're on Twitter, and if they follow you your account can be a valuable way to show them cool stuff in off hours.
There is also a very specific activity I like to do with Twitter and "Twisters," which are short (very short) stories that are written in 140 characters or less. Students really start to value the power of a few words and minimalism when they are so restricted in writing their stories. I usually show them a few examples (@arjunbasu, @veryshortstory, @gumballfiction) and then have them write a few of their own stories. It's a great way to get students to think about what they DON'T need.
Taking away sources of information from students is stupid. The knee-jerk reaction to Twitter in a school setting is that it must be bad because it's a social networking product and kids use it on their phones. A lot of school internet filters have it blocked. I like it because it's an efficient way of getting information: news breaks on Twitter now. What could be better than having a classroom full of students who are informed on current issues immediately?
Of course, this information transfer can create some problematic situations. It seems like every few months a celebrity death is erroneously reported because someone started a rumor on Twitter. This can be a great teaching tool, however, because it allows students to evaluate sources of information.