Consider:
-holders of advanced degrees and credentials (gatekeeping)
-assigns labour (literally called worksheets!) to pupils (proles)
-gives out sanctions, extra work, and bad reviews to punish bad work behaviour
-literally trained in discipline
-sharing of knowledge is forbidden, calls it cheating
-insists that you not work directly with your best friend, assigns you pairwork with a gender you're not yet comfortable with
-according to most media, teachers are the most liberal profession
-creates quarterly performance reports for workers, stressing constant need to improvement (report cards)
-have direct access to whiteboard markers and notebooks, the means of instruction!
Obviously not, but there are still interesting things to be said about the often unwilling role that teachers play in reinforcing class society. Some of our first experiences with capitalist propaganda are handled by teachers, who have to since it's in the curriculum.
Homework is literally classism, kids whose parents / paid tutors have time and knowledge to help will be better off.