DISCLAIMER: I am a software developer with a molecular biology degree. I am probably unqualified to make judgments about gender and gender roles. Imma do it anyway, because this is the internet, yo. Plus, I lived with an Anthropology major for three years, and sometimes I believe this qualifies me to participate in these sorts of discussions.
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I think somewhere in here, people are getting that gender is a cultural construct, and that its not the same thing everywhere, but forgetting that cultural constructs as widespread and fundamental as assigning a societal role based on biological sex usually exist for a very good reason. These reasons probably have a lot less to do with brain waves and a lot more to do with muscle mass and uteruses.
ON AVERAGE, men are taller and stronger than women. It is the woman who has to carry the child and then nurse the child. Even if those were the *only* things that biologically separated male humans and from female humans, those two things would be real reasons to construct the institution we call gender.
And we can say things like "but now we understand better, and the notion gender is antiquated! We should throw the whole damn thing out!" But I can think of a bunch of ways that would be bad for people born biologically female--in a totally modern context.
For example:
I played high school sports. When you're little, its easy to say "everyone play together!" because the differences in athleticism between little boys and little girls are pretty much nil. The little-kid soccer league I played for was coed until the under 10 age group, and probably could have been until under 12.
Once you hit high school, if we eliminated gender altogether, 90% of the people who got to play sports would be the ones who were born with penises instead of vaginas. Because they are going to be the ones who are, on average, bigger, faster, taller, and stronger.
Say you decide that's okay, and we should just have the really athletic people (who will mostly, but not exclusively, be male) play competitive sports, but not the other people. Okay. The other people will go do other things, like school plays or yearbook club. But then you're still ordering people--you're just doing it in a less intuitive way.
And like it or not, human brains are hardwired to make snap assumptions about people and situations, and gender is a visible, real way to start the foundation of your impression of someone. That's not a bad thing--as long as you follow up that immediate impression with your slow brain checks on whether or not your assumptions were true.
I think its a red herring to discuss the value of gender based institutions in terms of brain function. Of course girls can do math. Women can learn computer programming. Of course boys experience empathy. Men can learn to sew. When you say "but these stereotypes exist because of gender! Let's get rid of them", I think you're throwing the baby out with the bathwater. It's better to teach kids that some--or even lots of--boys like Barbies and girls like Matchbox Cars than it is to aspire to tear down the whole institution and say that the words "man" and "woman" don't qualify as words. Because you're never going get people to stop making snap judgements based on biological sex. Fast brain/slow brain. It's always going to happen. We just have to remember not to discount the slow brain part.