When I was at school, one of my teachers also committed a pretty obvious mistake, and I couldn't help but to ridicule him telling him in front of everybody where he was wrong. Obviously he insisted he wasn't.
Next day I took with me evidence of him being wrong, and when I pointed it out, he admitted he was wrong. I had low marks for the rest of the semester.
I did learn my lessons:
1. If you need to correct somebody, try to do it privately.
2. It is pointless trying to win an argument against somebody who has power over you. Specially when the argument is absolutely inane.
3. Lower marks as a prize for ridiculing your teacher is way too high a price.
I think it was Ben Franklin who said something in the line of "winning discussions is pointless, because even when you can get victory sometimes, you never get any good will".
My point? Teachers may be not the brightest of them all, but it doesn't take a genius to educate a youngster. It is a blend of knowledge, psychology, maturity, and strong personality. I can forgive a blunder here and there, as long as the teacher helps my kids to become persons of worth. Regarding the letter that originated this thread, we don't know the forms the kid used to address the teacher about her mistake. The point maybe was "the kid was right, but even then I deemed appropiate to give him a lesson". It is not about my mistake, but about teaching the kid a lesson about how to deal with being right.
I have delivered training for some time -to adults-, and sometimes I have told mistakes, if only for not being thinking on what I was talking about. When somebody pointed a mistake out, I had a line that helped me to get out of the situation with my pride untouched: "yeah, that's right. I already knew it, I just wanted to know who of you were paying attention. Good job dude, now what were the rest of you thinking about?"
That used to work, along with a big smile all over my face.