Twitter in the classroom can be a fun way to learn the curriculum. This past assignment has showed me that twitter can be used for more than just telling people how your day was, or what they saw. Who knew that important things could actually be tweeted too? I enjoy this because it's the first time I've gotten to use technology like twitter. We never got to experiment with things like this in high school and it's a cool new experience.
It can be used to discuss amongst the class about the material that you are teaching. You can administrate assignments to all of your students by just tweeting to your students.
There are also downsides that come with using twitter. First off not everyone has a twitter and I'm sure there are some people that are unwilling to get one. When you post something to do on twitter you are trusting your students to log onto their twitter account and to participate. That may not be that easy so in many ways it would be better to just hand out an assignment and then maybe just use twitter from time to time. Perhaps use it primarily as an extra credit activity rather than something mandatory.