
I don't know if it plays in Korea like other manhwas, but if so then it could be possible that MC/Teach was serving in the military and then got his teaching certificate (I don't know what the certificate/graduation exactly was).
Guwon probably can read certain/basic words or understands them.
After all we don't know the two guys that well and I'm really curious what will happen.

Reminder all Korean men over 30 have been in the military with only a few exceptions because they have 2 years of compulsory service. Might have been in the middle of his service before he got a chance to teach when the world went to hell. But hes been surviving in a dystopian world for I think 4 years. Who knows what he's been doing to survive
I’m 80% sure that the teacher is actually not a teacher but someone with a military background. The way he acts, his instincts and reflexes are too sharp for someone who taught kids.
Besides. How does guwon know that he’s a teacher if he can’t read? How did he understood the certificate?
All men do is lie.