The left right brain idea is not only a myth as u/BlackHoleHalibut points out, but evidence has shown pretty paltry (if applicable) difference between women and men skillwise (in practice) and biology-wise. Things that were accepted as the norm back then (ie women being bad at math) were shown in later years as more women got into university were found to be negligible. Environment plays a great role in the manifestation of certain traits - socialization (it's generally accepted or expected women to be more social or emotional, while boys are taught to sublimate their emotions in different ways), something called the self-fulfilling prophecy or an expectation coming true cause the people it pertains to make it happen in part cause they are convinced it's inevitable. That isn't to say there aren't cases where gender presentation is different. Schizophrenia tends to manifest less severely in women and later in life - though in part due to the protective factor (something that helps manage a trauma or disease) that is estrogen. Women have been documented more at risk of developing Alzheimer's but the reasoning is still being researched.