>A woman should let a man teach her how to swim, but if she is already a swimmer she should be clever enough to pretend and let him teach her anyway.
Why?
Personally, I'd find that type of behaviour extremely repulsive and unattractive, why hide what you can and pretend to be "weak"? Why play some sort of theatrical act?
>I treat him like a man and he treats me like a woman
I've never understood that type of stereotypical thinking. For example, I wouldn't mind being together with a girl who is, say, physically stronger than me, and I don't consider myself less "manly", than my male friends, classmates and relatives.