You will notice that many of these themes appear in the form of questions. The History Boys seems to be more interested in making you think than in providing easy answers.
She got the boys through their A levels with factually accurate answers, but it is implied early in the play that this is not enough. She is the only devoted teacher.
Hector:
He inspires but educates with no particular aim in mind, it appears. He reveals towards the end of the play that he never meant to be a teacher, but only intended to do it for a short time. The different boys over the years have worn him out, and he has lost the drive he initially had. He tells Irwin not to remain in teaching, and we know that Irwin will not. Hector sees himself as a sort of clown as opposed to a teacher.
Irwin:
He is a hypocrite who challenges but does not offer a truthful alternative.