Twitter can be a valuable tool to a teacher. It can provide the teacher with a connection to other teachers around the world, a place where they can collaborate and share. It can serve as on-going professional development. It can also serve as a place where students can connect to their teachers.
I recently discovered the Twitter feed of NCTM - the National Council of the Teachers of Mathematics. They put out a math "problem of the day" - every day. These problems are great - they would be perfect problems to start off a class:
If you haven't already, give Twitter a try. I guarantee you will find something valuable there.