In Gender Trouble, Judith Butler argues that our gender identity is entirely socially constructed. In other words, society has decided that a group of people it designates as females behave in a certain way and a group that it designates as males behave in a different way. To achieve the "proper" gender identity, parents, when they see a girl, doing something gender appropriate such as pretending to cook, praise that behavior and tell the child she is "all girl." Likewise, when a boy plays at an aggressive game, parents will laugh and say "boys will be boys." Conversely, if a girl is caught running around in aggressive play, parents will say "sit down like a little lady!" and boys who pretend to cook will be told "real boys don't do sissy games." Over time, as most children respond to the push and pull of positive and negative reinforcement from parents, they start doing what is considered appropriate for their gender. Then—and this is key in Butler's theory—when the children do what they have been conditioned to do, society says this behavior is "natural" for their gender when, of course, it is anything but natural.
According to Derrida's notion of deconstruction, the male/female gender identity split is a false binary established as a bricolage or kluge (patched-together construct) that society created to organize itself. The identification of woman with "nature" and male with "culture" is constructed, damaging, and easily deconstructed under scrutiny.
According to Marxists, capitalism uses gender ideologies to brainwash women into accepting and even welcoming their own oppression and to pit women against men so that the ruling class can more easily control both. Women are taught a false ideology that they are happiest when caring for their children in a nuclear home and when their energies and talents are directed towards consumption and care of largely decorative goods. Marxist theory would erase gender differences and liberate women from the false consciousness that leads them to over-value their unpaid domestic slavery. Russian communism emphasized shared housework and cooking in group living environments that were designed not to be "pretty" but easy to clean and maintain. They encouraged women to work in factories and other male dominated domains. Women fought side-by-side with men in World War II. Today, Marxists advocate for paid housework: it is labor and should be compensated as such, they argue.
In all these theories, gender identity has a profound effect on how females and males function in society and in all cases is far less "natural" than society would have us believe.