Stoops, there are a few studies around on the confidence issue, and demonstrating that attainment involves self-confidence as well as absolute ability
For example, Aspires (2013) groups those traditionally under-represented in post-16 physical sciences and mathematics (girls, working-class and certain minority ethnic pupils) tend to be less confident in their own abilities, and are less likely to identify themselves as being ‘good’ at science and/or mathematics – irrespective of their actual abilities and attainment.
In a survey of 23,000 12–15 year old girls who want to study physics beyond 16, more have lower confidence in their abilities than boys, despite tests revealing no difference in their actual conceptual abilities (UPMAP project, cited in TISME 2013.
Looking, I posted upthread about our grim % of women in engineering roles. I'm not terribly convinced by the 'it's because of apprenticeships' argument - why are those paths traditionally male, when they aren't elsewhere? Engineering UK has a good report on this - ie the reason girls don't pursue vocational engineering paths. And regardless, why are 85% of engineering graduates women?
Why do more girls study STEM when they're at all-girls schools, and then go on to do better?
Why are 76% of women in the UK who finish their education and are qualified in SET are not employed in the SET sector, compared to say 47% in Sweden?
You also spoke about male and female brains operating differently - I think there's evidence that this is much less than stereotypes, even within science, suggest and that there is evidence of stereotype bias - a Smith college study (Wraga et al in 2013) took 3 groups of females and got them to do a mental rotation task while in a brain scanner. This sort of task is one of the very few where there is some evidence of very small, group level differences between women and men. The group that was told about this difference performed worst and showed different patterns of brain activation, in those areas more often associated with emotional processes.
A group that was given positive messages performed best, and showed brain activation patterns consistent with visual processing and working memory. The study is worth a Google.
There's loads out there - I'll try to link to more later but I have to run now. The Wellcome Trust, Institute of Physics, IPPR and others have good evidence-based reports on this.