What are different ways we can get women to study mathematics? In my own experience, the higher the math class, the less women in the class. Most women tend to go on the math education track and do not have to take the more advanced classes. These more advanced classes can seem like hostile places for women, especially if they are the only female in the class, or only of a few female students.
As an educator, how can we encourage women to study math and become mathematicians?
(I am a woman in my last year of a PhD in low-dimensional topology, I'll be a postdoc in the fall, and I have research aspirations. I'm mostly trying to articulate how I got to this stage in my life. In a real sense, my answer shouldn't matter very much since the many things that keep women away from mathematics clearly didn't work on me too well, or on the several other women that are currently in mathematics; but perhaps it is still valuable to see what worked for me.)
As a woman and an Indian in mathematics, the following is one of my 'favorite' phenomena: stereotype threat. Here is a more detailed non-Wikipedia source. Briefly, there have been studies involving Asian-American girls, where the group who were reminded that they were of Asian descent performed better than the control group on a given math test, and the group that was reminded that they were girls performed worse. It's unclear to me how we should respond to this phenomenon as educators. One possibility is to straight up mention it to students, assuming that awareness might dampen its effects. Another might be to be entirely indifferent to the genders of our students (and play up other similarities. A woman in your class might be the only woman there, but she presumably belongs to shared groups with some of the other students.)
Personally, I grew up in upper middle class urban India, in a part of the country where it simply never occurred to me that women and men could have (markedly) differing mathematical abilities - gender discrimination abounds in India, but at least in our socio-economic class and region, it wasn't pronounced in terms of intellectual ability; in particular, my elder brother suffered from math anxiety - I expect that my desire to compete with him had at least something to do with my interest in the subject. In any case, intellectual pursuits were strongly encouraged for everyone. We revered the academically successful students (class ranks were a matter of avid discussion) and the top positions seemed equally distributed (in my high school, there was an award for highest aggregate score in math and physics in our 12th grade (national) exam, and I recall the genders being equally represented. Nonetheless, there were various such pursuits where I was often the only female participant (inter-school quiz competitions, etc.). The OP mentions such environments as being perceived as hostile to girls and women, but I did not particularly feel this way -- while my fellow participants were not the same gender, we had bunches of other similarities, for example, interest in participating in the relevant activities.
Anyway, as a child I didn't know that one could do mathematics for a living. I like to plan ahead for things (and I believe this is correlated with gender), and my youthful planning for my life didn't include mathematics since I just didn't even consider it an option. My love for mathematics was kindled at a very early age by my grandfather, whose games with my toddler-self consisted of doing additions and subtractions using little circles. Coming to the US as an undergraduate I was excited to learn about the concept of 'double majors' and my decision to pick up a second major in mathematics was motivated in part due to my interest in it, but also due to being a typical over-achiever. My decision to switch to mathematics for graduate study was motivated by the following: math was more intellectually challenging than my primary major (biochemistry), mathematical results contain way more certainty than anything one would be doing in biochemistry, and I just liked my mathematics professors more. As an undergraduate I did attend the Nebraska Conference for Undergraduate Women in Mathematics which was a valuable experience just to know how many women there are in mathematics.
Outside of my own personal experience, I really like the idea of Sonia Kovalesky Days. I was recently involved in an iteration as a panelist, and it was really exciting to see so many young women participating and seemingly having a great time with mathematics. One of the young women asked the panelists to give them names of notable women mathematicians, and I think even knowing about women mathematicians (and women in other STEM fields) would be neat for them. On which note, I'm also a fan of the AWM essay contest.
tl:dr Here are the factors which I personally credit for my being in mathematics:
I attribute any success I might have to having excellent advisors, teachers, and mentors throughout my life; they were almost all male, which is unsurprising, since most mathematicians are.
Here are some possible steps one could take as an educator to interest more women in mathematics:
I think there is no clear answer, although there has been some research on this topic. I remember one study which focussed on gender differences of university math students:
Mischau, A., Blättel-Mink, B., Daniels, J., & Lehmann, J. (2004). Doing gender in mathematics: indications for more gender equality in German universities? Bielefeld: IFF.
The researchers found that women have partly different interests, e.g.
Roughly speaking: the purer the maths, the less women are interested in it. Presumably, even women who decided to study mathematics enter university with different interest profiles. Consequently, we should have a look on gender differences at school.
Research showed differences even in primary school. We have more boys than girls taking part in math competitions, although there is no (or hardly any) overall difference in mathematical ability (see Hanna, Hyde et al., Ma, Lindberg et al.). Differences can be found if we look at attributions (e.g. when they solved a task, girls might think "the task was easy" or "I was lucky" whereas boys would rather think "I am generally good at maths".), see Dickhäuser & Meyer. This might be due to behaviour of parents, teachers and peers (in short: the society). But that's a private speculation.
All in all, it looks like the phenomenon you described is just the tip of the iceberg - maybe the result of hundreds of situations in childhood. I have no clear answer, but maybe I could illustrate it's not so easy to get one.
First, we need to be aware of the ways that women are discouraged, and when those are not related to what mathematics is, we need to change the culture. Does math have to be approached as a competitive sport? No.
Related to this concern, please note that I am the only obvious woman in the top 40 users here. There are plenty of women blogging about math and math education. What about this culture alienates women? I think people do not try very hard to be diplomatic when they disagree with a poster: "This question does not belong.", "This is not an answer." I have a particular goal, and see myself as a central actor in the math education online discussion world (as a blogger at least), so I have persevered. But if I were less motivated, I might have decided the aggravation wasn't worth it.
My story may help others understand subtler issues which are not related to obvious sexism. I got my masters in a program that was easy, and enjoyed it immensely. I decided to go for a phd, and attended UCSD. It was very hard (as was U of Michigan, where I did my undergrad), and I didn't like it. I knew going in that that might happen, and quit very quickly. My interest was in teaching, so for me it wasn't a bad thing. But I am considered one of the casualties, since I didn't 'go on'. And I love math. But I also want to help others in the work I do. My interest in math education satisfies my passion for math, along with my interest in making a difference in the world. Is there a way to change the way we structure the study of higher math so that someone like me would want to continue? Maybe. I'm not sure yet what it would look like.
Two fundamental issues, in my observation: pathetically, quite a few (male) people subconsciously decide that, while they are not athletes or whatever, their machismo can be proven in ... mathematics. By math contests and being aggressive in class. Vehicle for ego. As a consequence, there's the "oop, it's not macho if chix can do it, too" unfortunate-riff...
And, the next point is that there is indeed a substantial tradition of exhorting kids to engage by invoking machismo.
Aaaand, there's the traditional "omuerta" of "not asking questions" (because it reveals weakness, which a good Spartan would never do, blah-blah-blah), which seems statistically less "reasonable" to females than males. And, srsly, what is the point of "classes" if asking the natural questions is belittled? Crazy! From an intellectual, pedagogical, and social perspective.
This ground presents serious difficulties for encouraging women (or any person not feeling at home in the nerdish pseudo-macho environment) to engage with mathematics.
An uncomfortable corollary is that the traditional quick-nerdy-combative kids have to be discouraged, or redirected. This is not easy, given the ambient culture that reinforeces this.
There are further subsequent problems... too many to explain in short. Perhaps people who're interested in my take on this should email me.
I see the original question as actually two questions asked here:
How do we engage women into studying mathematics?
How do we get women to become mathematicians?
Studying mathematics for the goal of becoming mathematicians (pure, applied, etc.) is a subset to the first question. For the first question, I have read about some of Jo Boaler's research in gender and mathematics education: http://gender.stanford.edu/news/2012/sugar-and-spice-and%E2%80%A6-math-under-achievement
Essentially, it comes down to how classes are being taught. I have the impression that mathematics education is traditionally set up in a competitive environment, which has the tendency to turn off the collaborative learning aspects that academia is supposed to foster. If everything in the learning environment is geared to working individually and getting the technically correct answers without having the space and time to develop deeper understanding and allowing a safe environment to compare and discuss answers, then it tends to turn off the girls (and possible future mathematicians). Content has less of a bearing to motivation than the delivery and environment of mathematics education.
One way to interest women in mathematics is to teach applications, including statistics.
Consider the following survey of 2012 doctorates in the U.S. in the mathematical sciences:
\begin{matrix} \text{Men} &\text{Women} &\text{% Women} &\text{Type of Program}\\ 653& 233 & 28\% &\text{public math programs}\\ 204& 62 & 23\% &\text{private math programs}\\ 105& 56 & 35\% &\text{applied mathematics}\\ 195& 118 & 38\% &\text{statistics}\\ 87& 85 & 49\% &\text{biostatistics}\\ \end{matrix}
So 45% of the women are getting PhDs in applied fields, while only 31% of the men are.
The choice of topics to teach is only one way of encouraging women to study mathematics, but it is an important way.
