I think women definitely tend to be more right brained, and men more left, but there are variations of course.
I’m a very right brain thinker myself–and I’m glad my parents pushed math (although I absolutely hated it), making me work so hard to get used to that kind of thinking (I was very unattentive, daydreaming, etc).
So I think I have a pretty good idea on how right-brained thinking goes.
The left front lobe is associated with language, speaking, organizing, planning, abstraction. Since the left side is always preferred when handling business (learning the language and integrating into that particular culture)–modern society values quick-thinking–as Carl Jung said “The United States is extraverted as hell.” It expects people to be integrated into the rules, and react accordingly (to be more extraverted–no introversion, no abstract questioning; stimulus-response.
Right brained thinking is often called lateral thinking–I visualize left brained thinking as a few connections–from one thought to another–which clearly stand out to the others. The two concepts are both so powerful, and so associated/conditioned with, that one tends to follow another. Right brain thinking I visualize as a far larger collection of connections of varying brightness. It doesn’t keep on track as well when tere are rules, but it is associated with creativity as you connect different concepts (with images, it is the visual thinking side) in different ways to try to arrive at what you think is the image of a solution, but you see several circus mirrors altering its reflection.
To try to train yourself in right brain thinking you might want to look at an ambiguous image somewhere, and try to think what it looks like, then think of a story about how it came to be there, why it decided to do so, etc.
How the child grows up says a lot about how they train their brain and habits of thinking. Girls are encouraged to talk, which keeps them holding together different elements of a story and how it all fits together (the task of trying ti understand how the other is thinking, learning how different people express themselves according to the individuals they are. Men tend to talk direct, they send messages, but the women also look for, and produce metamessages–a message behind what is literally said. “Is it cold in here?”-“Please close the window”. Deborah Tannen, in “That’s Not What I Meant!”. I read that in my first year of college and got a lot out of it, so you might like that one.
Boys tend to hang out doing some activity in which there are collectively accepted rules, and one does what they can according to those inarguable rules. Status comes out when certain people excell at certain activities–and their status is proof of the validity of their opinion.