He lectured students on topics as diverse as poetry and the physics of time, generally using a chalk board to assist him. It was during one of his lectures when he first noticed the qualities of Bill Potts. (TV: The Pilot)
The Doctor lectured students, including Bill Potts and Heather, on Time And Relative Dimension In Space, which he said meant life. He referred to past days, future days, present days, little moments and big moments and hypothesised them all happening at once. (TV: The Pilot)
He later led a lecture on the vacuum of space, and the effects that exposure would have on the human body. He was questioned by a student as to what that had to do with crop rotation, which he dismissed as space was much more interesting. (TV: Oxygen)
Behind the scenes
Students attending the Doctor's lectures, aside from Bill and Heather, included Classmates' Caroline Lie and Joe (TARDISArchives), who can both be seen in multiple locations around the hall in The Pilot. The student who challenges the Doctor in Oxygen was played by Lauren Pate, one of Doctor Who series 10's assistant directors.