Nash corrected the student's spelling in a tweet that was directed to her. And, looking at the exchange, that student needed some real help with his spelling. How could they fire a new teacher for going above and beyond? The student didn't take it personally and Nash did nothing inappropriate. This was idiotic.
If that's the only reason she was fired, then no. It could be that her district strictly prohibits any social media contact between teachers and students, but she's a web coordinator, so that doesn't make sense. Her tweet wasn't mean or mocking and the student wasn't upset at her correction of his spelling, so I think it's wrong.
I do not believe that Katie Nash should have been fired for her tweet. While some may have viewed the tweet as unprofessional and inappropriate for someone in her position, the tweet was not meant to be malicious, and was clearly meant as a joke as the student her tweet addressed was not upset by her tweet. I think that in a world where we are constantly bombarded with negativity and distress, people need to stop taking things quite so seriously and realize that there is nothing wrong with trying to find a little humor in things.
She was just having fun, and as she put it, being personable with the student. She wasn't being rude or hurtful to the student. It was a clever response, and one which I think people should have found a little more humorous. Sometimes we just try too hard to find fault with people and have to walk on egg shells so as not to offend anyone. I don't think this was that big a deal at all.