I don't quite get it... she pranked him? Or maybe she just wanted an excuse to share an umbrella with the other girl?
At first I thought she had some special feelings for the teacher, but I'm not really sure that is the case here since she let the chance to share an umbrella with him pass. Anyway, I'd use an interesting-looking umbrella like that. He's just way too uptight and too conscious of his image or something.
At the end it says 変わった 傘ですなー "what a weird umbrella" which I assume was an offhand remark of the bald teacher.