As a first remark, I think one of the most subtle and troubling effects of sexism in developed countries is that many women fail to really develop a sense of personal authority. It's hard to say precisely what I mean but Dave in post  or Mia in post touch on it.

Here's an attempt. Think of great male heroes in your professional field, in your life, in the general culture. Imagine some kind of important situation: how you feel when this person is in the room. They immediately size things up, understand what is crucial, what is at stake. They count as a witness.  If something happens in the world and they see it, then it has not been in vain, it will not be lost.  E.g. Foucault's epitaph, that one thinks of the many problematic people who for all the years of his life kept silent knowing Foucault was there to respond.

Suppose one doesn't want to allow the vacuum of "you are not a boy" to exist. How important is it to fill this with positive stereotypes? Even ones which are slight misreadings or spins of current ideas. E.g. of course you would want to be a scientist as everyone knows women have the capability for those all-important intuitive flashes which all great scientists need, but can't be taught. Or depending on your family -- e.g. for expats, Well honey, of course you're the best in your class in physics, Newton was English, or The women in our tradition/culture/city have always had special gifts, or even, Yes I can tell she takes after her grandmother, she's going to be an outstanding diplomat. Anything to latch onto.

It's complicated, of course, but I think this is one of the major problems with the modern philosophy of equality in everything. When it is layered onto deep-set societal beliefs that men are attributed the positive qualities and women get the reflected light, it is not enough to give most women a positive sense of self and an appreciation of their gifts as being uniquely theirs (not as women, of course, but as human beings).

I came across my copy of The Second Sex today... Here is what Simone de Beauvoir wrote in 1952:

"Even so, it is very seldom that woman fully assumes the anguished tete-a-tete with the given world. The constraints that surround her and the whole tradition that weighs her down prevent her from feeling responsible for the universe, and that is the deep-seated reason for her mediocrity.

The men that we call great are those who -- in one way or another -- have taken the weight of the world upon their shoulders; they have done better or worse, they have succeeded in re-creating it or they have gone down; but first they have assumed that enormous burden. This is what no woman has ever done, what none has ever been able to do. To regard the universe as one's own, to consider oneself to blame for its faults and to glory in its progress, one must belong to the caste of the privileged; it is for those alone who are in command to justify the universe by changing it, by thinking about it, by revealing it; they alone can recognize themselves in it and endeavor to make their mark upon it. It is in man and not in woman that it has hitherto been possible for Man to be incarnated. For the individuals who seem to us most outstanding, who are honored with the name of genius, are those who have proposed to enact the fate of all humanity in their personal existences, and no woman has believed herself authorized to do this."

--The Second Sex, chapter XXV.