As a teacher, I think it's important to remember that you're learning from the students just as much as they are learning from you. We are all learning even until the day we die. If you can't handle being proved wrong, then you're not setting a good example to the students. What's important is to learn from your mistakes and not repeat it again.
However, there should also be a limit to where a student should challenge a teacher's authority. Just because you're right doesn't mean you can be rude and shame the person in public.
As for screwing up in school, you don't really see the consequences until it's too late. I can't believe how important High School was for me to get into a good studying habit, which got me the marks to get into a decent University. From there, I worked extra hard, never took summer holidays in University to work in internships to earn tuition money + gain real work experience. Now I'm debt free + working overseas straight out of grad while everyone is still looking for work or working in retail. Everything does add up, so if I slacked off in HS, I definitely wouldn't be where I am now.