What cultures are portrayed in the movie To Sir, with Love?

To Sir, with Love deals with the culture of working-class young people in 1960's London. The culture of teachers and the educational system is also a focus of the story line.
We see several clashes among these cultural elements in the story. Thackeray (Sidney Poitier) is an idealistic teacher who clashes with both the kids in his class and with his cynical, bigoted colleague Weston. The underlying issue, however, is at least partly a racial one. Because Thackeray is black, he must deal with a reflexive hostility from his almost entirely white class, as well as an unstated "skepticism," which is, at least unconsciously, felt about him by some of his colleagues. Though Thackeray was born in British Guiana, he's American-educated and basically identifies as American. This leads to the (mostly good-natured) rivalry of thought and attitude that typically occurs between American and British people. When Weston comments on an item in the news in which, he says, "the Americans have mucked it up as usual," Thackeray light-heartedly says, "Give them a chance. It takes time to learn how to lead the world." Weston then begins to say, "I should think that if anybody would agree with me on this, it would be you," obviously alluding to Thackeray's being a black man.
The culture-clash issues in the staff room (which we Americans usually call the faculty room, of course!) are not the main point of the film, however. Thackeray and his students are, unsurprisingly, at odds with each other at first. Initially he is put off by the somewhat crude behavior of some of the students—their insolence, bad language, and seeming unwillingness to learn. But eventually he wins them over and bonds with them. The backdrop of the film is, of course, "swinging" London in the 60s and the surrounding changes occurring in sexual attitudes, fashion, popular music, and so on. As expected, Thackeray has to deal with a crush one of the girls in his class Pamela (Judy Geeson) has on him. But in the end, all works out for the best, and at the graduation party, Thackeray and Pamela dance to the song "It's Getting Harder all the Time," done by minor pop group of the time The Mindbenders.
To Sir, with Love, like the novel by E.R. Braithwaite on which it's based, was pathbreaking in its time. It showed a realistic school without the violence of the earlier The Blackboard Jungle (in which Poitier had also starred, but as a student), and its message of empathy was one more nail in the coffin of prejudice and bigotry.