In Fire Emblem Three Houses your main job (when you’re not out fighting battles) is to teach the students in your house. As a professor, you’ll lecture and teach your students the skills that they’ll take out into the battlefield.
Focus on these activities to raise your professor level
Alongside your different combat abilities you’ll also be working towards your professor level as well. This level determines how much of a monthly allowance you get from the Monastery, your number of activity points to spend on free days, and the ability to unlock master classes. Raising your professor level is definitely important as you can always use more cash to purchase or replenish battalions, purchase items, repair or purchase weapons, and many other things that can prepare you for battles.
Increasing your professor level requires that you spend your free time doing activities in the Monastery. Completing activities like fishing or gardening will increase your professor level. Participating in activities with your students like tournaments, meals, choir practice, and giving advice will also raise your professor level. Finally, faculty training will also increase your professor level as well.
Everything sort of feeds into making your students stronger. As a higher level professor the money that you get and the activity points that you have will allow you to complete more and more activities during each free day allowing you to make your team stronger and stronger. The higher you raise your professor level the more quickly you’ll be able to unlock important Master Classes for your character.