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!
The intended purpose of public schooling is to prepare children for a job ie sell their labor to a capitalist. It's not some sort of secret; every public school will tell you this. Public grade school is designed to simulate working at a job. Public school teachers simulate the management of a job.
School teachers are not bourgeois.