Quantumblue
I have always been passionate about avoiding girls becoming disengaged with Maths and Science. In my university degree, I was part of a very small minority of girls studying abstract Maths and physics, and do want to change this. It is not necessarily about pushing girls into STEM subjects regardless of dreams, but making sure they know they are capable of it and can follow their dreams.
With regards to specific strategies, it is not any one that works and it is not something that I consciously focus on. To begin with, I am a role model to show that it is possible; seeing a woman with these qualifications is one indication that it is possible. There are so many regional and rural schools which have no female Maths or Science teachers, and girls do not always see that it is a potential path. I have had past students go into science or maths degrees at university because they have seen that if I could, they could. I know this seems like boasting, but I have received emails or Facebook messages to this effect.
I have very high expectations of my students; male and female and will not accept it when they give up. That is is easier said than done, but my classes are generally small enough that I can pay a lot of one to one attention. I have had girls attempt to do poorly to not show up "weaker" boys before, which can be a very interesting discussion.
Keeping them engaged is about continuing to make the girls interested in Maths or Science and this is different for every student. It might start out by discussing financial maths, which most (sadly not all) students can see a definite need for in their lives, and then extend this to more theoretical concepts.
I have also run Maths related quiz night to encourage engagement with Maths on a different level, and have noticed that this has improved attitudes somewhat.
It is very important to stop peacocking in the classroom and to immediately halt instances of attempting to impress the opposite gender by stuffing around or deliberately doing poorly.
Unfortunately there is no definite answer to keeping girls interested in science and Maths. Ultimately for me it is the relationship I have with my kids; when I was at my last school I managed to increase the number of girls studying year 12 Maths by a significant amount to the point of extra classes being offered.