Why is it wrong to say 'why to correct students' instead of 'why correct students'?
Because "to correct" is only the base form of the verb when it is listed in translation dictionaries. to read, to listen, etc. When you actually say or write it, you would say "I read," "I listen," "I correct." You only add the to if you are saying you "want to" do something, like "I want to correct my students," or "I want to drive the car." If you are actually doing it, then you drop the to. Saying why to correct students is asking why and addressing that question toward students that are correct or who have been correct in the past. Saying why correct students could be a third person reference instead of a 2nd person address like in the first case.