Teacher Discussion Forums

In American English:
1. He teaches school. correct
2. He teaches high school. correct
3. He teaches college. correct
4. He teaches at a university. correct
5. He teaches university. not idiomatic
Why?
This is my biggest headache now.
Help me please.

I don't know why it's giving you a headache. (If I worried about every seeming exception in English I'd have a permanent migraine . ) And unfortunately, I don't really have an answer for your quest. However, in the future you might like to post questions like this to the "Applied Linguistics" thread where there is a lot more talk about what is or is not grammatically or lexically acceptable and why.