1 Corintians 11: 4 - 7
4 Every man who prays or prophesies with his head covered dishonors his head. 5 But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved. 6 For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head.
7 A man ought not to cover his head,[b] since he is the image and glory of God; but woman is the glory of man. 8
In roman (in contrary to greek) paganism the priest would have covered head during rituals (google capite velato). I don't know if and how much is it related to this, however.
EDIT: After checking some commentaries, it seems it's not this way; more like being uncovered/covered would mean being in/under authority, according to custom of that time and place.
My theory is that modern (post industrialization) schools fundamental purpose in society is not to educate but to prepare a workforce. They acclimate people into accepting very arbitrary rules that will be normative in the work place.
Traditionally, it's considered disrespectful to wear hats inside. Why that tradition exists, I'm not sure, but it's one of those etiquette things, like not putting your elbows on the table.
It could also be along the same lines as hoods - obscuring your face and making policing the school harder.
I suspect the tradition came from hats being used to protect one's head from the elements. Wearing one indoors suggests a lack of confidence in the building.
I've heard from a teacher in my high school over a decade ago that while it is disrespectful for men to wear hats indoors, traditionally women were exempt from this expectation.
1 Corintians 11: 4 - 7
4 Every man who prays or prophesies with his head covered dishonors his head. 5 But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved. 6 For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head.
7 A man ought not to cover his head,[b] since he is the image and glory of God; but woman is the glory of man. 8
Schools are all about conformity and enforcing arbitrary rules. Schools are designed to make you docile and get used to bullshit so you're less likely to revolt in your job.
Fuck I have no idea. Probably some kind of hangover from wierd victorian ideas about training the populace to accept authority and reduce individualism & identity idk. Fuck school.