One pairing that is particularly interesting to me is this:
Status inequality is acceptable for college teachers. Universities exist within a finely gradated status structure, with certain schools like Brown clearly more elite than other schools. University departments are carefully ranked and compete for superiority.
Status inequality is unacceptable for high school teachers. Teachers at this level strongly resist being ranked. It would be loathsome to have one’s department competing with other departments in nearby schools.
Brooks is only overstating a bit. Higher education does have many public rankings of schools, and even of the same discipline in different schools. However, disciplines within the same school generally adopt the polite fiction that they are on same level as one another, sharing the school's overall status. This is much like what Brooks says about high schools.
Nonetheless, I have noticed, comparing my life in higher education with my wife's work with elementary and secondary education (she is an education policy wonk) that the lower the age of students being taught, the more important it is to teachers to maintain that they are all at the same status level.
It is also true that the lower the age of students being taught, the more likely it is that the teachers are women.
And women, in general, are more likely to wish to treat all of their social relations as if everyone were on the same level, whereas men are more comfortable with the idea of hierarchy.