You know how the monks life would be like. They renounce the world and practise asceticism for many years in mountains or temples, so that they can attain elightenment after many many years of that practice. In the process of it, they would go through many temptation, negative feelings and so on. However, they should tolerate and get over all those things to enter into Nirvana (In the Hindu and Buddhist religions, Nirvana is the highest spiritual state that can possibly be achieved) '도를 닦다' means that kind of practice. We use that experession in daily life to express that we've been eduring some unpleasnt situations againist our will or goals, et cetera.. but we cannot just give up on it for some reasons. For example, I have a mean boss, but I cannot quit my job because I have to pay my rent, but whenever I work with him, 도를 닦는 기분이야.

ohhhhh, you're a genius jamie.Thank you and Mojave

So, it's like he is really pissed off, but he cannot express that right away. He has to stand it and be nice to the girl because he is a tutor. Whenever he teaches the student, he feels like he is a monk or something.

I see. So it means to like morally come clean or spiritual enlightenment. But my friend said it like this with a video: "과외한번 할때마다 도를닦는느낌이다.. 이러다가 열반할것같다.... 하...." In the video she is trying to record her student but the student quickly grabs the phone and blocks the camera. And she says "Okay okay let go so I can turn it off" I have no clue lol

도를 닦다 means something like "cultivate the moral sense". If you look it up in daum, there are several examples that might help.