my first thought was 'arrogant', too. Not sure I'd like a teacher saying it to a child in quite that way, though, but it might have been that she had a point - some children can be really smart-alecky, get one over on the teacher, prove them wrong, make them look stupid, etc, and they can have a certain tone in their voice that conveys that. Not that your daughter necessarily did, of course, but maybe something made the teacher think that she might, just in the way she said it, or based on some history of their interactions, or poor word choice on your daughter's part, or whatever. Or the teacher had had loads of people doing similar already, and one more potential interaction like that was just enough to tip her over the edge?
I once had a teacher who was talking about the word 'miscellaneous' and how we could write it on our jotters, though she would have to think about how to spell it. I piped up "I'll tell you how to spell it" - when what I meant was 'I'll go look it up in the dictionary for you, in order to be helpful'. I got seriously snapped at, with the teacher clearly thinking I was trying to show I was cleverer than she was, etc. I wasn't intending that at all!! As it happens, I was an excellent speller for my age, and probably could have spelled some word she didn't know, but I really truly wasn't trying to show off at all. I can see now, though, how it sounded like it - and in several other situations, where I might well have known more than someone, but didn't express it in a way that made it clear I was still deferring to them in some way. That can be an important lesson for children to learn, especially those who do know lots of weird/random things that someone else might now - how to put that across without pissing the other person off, however 'right' you might be. Learning to let someone 'save face', if you will, or just respecting their feelings. I was too enthusiastic about wanting to share things I knew!