I haven't had kids of my own but I have spent a lot of time working with children and biology does have a strong influence. While these are generalisations, female brains mature emotionally much faster than male brains. Girls do learn to be quiet and focus on tasks earlier than boys do and I think society and teachers harness that, if only to get things done or some semblance of order out of chaotic kids. We often have to tell little boys that "big boys don't cry" because they frequently do cry and act out so much more than girls because they have a harder time containing their emotions at a comparative age than girls do. Boys are generally about 2 years behind girls in emotional brain development and it's wise to bear that in mind when trying to teach kids.
What bothers me is the societal construct of boys being taught that they are superior to girls and the natural inheritors of the earth, despite the fact that they frequently behave so badly. At puberty, it is true that female behaviour and skills can be very strongly influenced by hormones. For instance, prepubertal girls are very much boys' equals in science and mathematics but after puberty, many boys outstrip girls. Does this mean that all girls are crazy and irrational, as society likes to think? No, but there is an added hormonal emotional dimension to the female experience.
Boys on the other hand, tend to get into high-risk activities under the influence of testosterone. Society attempts to harness that testoterone by encouraging boys to win, often at all costs.
I believe that there are different mental abilities that do fall into general categories but it is a mistake to pigeonhole kids. I also think it is true that some men are more in touch with their emotions than others and it does depend on whether they were handled skillfully or brutally by parents and educators. The same goes for women when it comes to whether they accept traditional gender roles, at the expense of stifling their own intellect.
Society is a construct that humans use in order to live together. As such, the roles assigned always depend on the tasks of the time. It can only ever be a blunt instrument.