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Women in math

February 6, 2012

This is crossposted from Naked Capitalism.

A study recently came out which was entitled “Can stereotype threat explain the gender gap in mathematics performance and achievement?”. One of the authors created and posted a video describing the paper, which you can view here.

As a preview, there seem to be four main points of the paper and the video:

  1. The papers on stereotype threat normalize with respect to SAT scores which is bad.
  2. Evidence for stereotype threat is therefore weak.
  3. We should therefore stop putting all of our resources into combating stereotype threat.
  4. We should instead do something easy like combating stereotypes themselves.

Before we go into the details of the paper, we need a bit of context. For that reason, this post is split into three parts. The first addresses a meta-issue, namely that of the “null hypothesis” in this discussion. A frustration that I have, and that I think is shared by many of the women I know in math, is that the (often unspoken) working hypothesis is that in fact women are just not as talented, and it is somehow up to us women to prove this otherwise, presumably by convincing men that we’re geniuses.

The authors of the above paper fall prey to this disingenuous line of thought, by proclaiming stereotype threat is an insufficient explanation but not offering any alternative explanations. This sets up a kind of implied false dichotomy: if it isn’t explained by such and such, it must mean girls are dumb.

Not only does this undermine serious intellectual debate, but it often turns people off from entering the debate in the first place, because they sense the manipulative nature of the discussion. But that’s a pity, since, with the correct assumption, namely that women and men have equal talents but things are holding back women, we could probably make lots of progress on what those things are.

The second part is directly related not to the paper but to the blog post which referenced the paper, which changed the conversation from “math performance gap” to the question of “why there are no women math geniuses”. This is an interesting twist, and in my opinion warrants addressing separately.

In the third part I argue directly against the paper and its conclusions.

1. The Null Hypothesis

Needless to say, I think the onus is on the scientific community to prove that women aren’t as mathematically talented as men. In other words, I do not accept the defensive position that I need to prove we are as smart: the null hypothesis is that a series of effects, one of them stereotype threat, explains any perceived difference in talent.

In his now famous lecture at NBER in 2005, Larry Summers putatively discusses the issue of why there are fewer tenured women in science and math departments at top universities. However, if you read the transcript, you will note that, when he gets to the “different availability of aptitude at the high end” part, he does us a favor of sorts by admitting what his underlying working hypothesis is: that girls aren’t as good at math. His argument using standard deviations of test scores is ridiculous, especially if you consider 1) how differently women do versus men on the same test in different conditions, 2) how much that difference has itself changed over time, and of course 3) the question of what the tests themselves are measuring.

To test why this null hypothesis is so damaging, my friend Catherine Good suggested the following thought experiment: imagine if he’d gone up to the podium and, instead of saying that women aren’t all that good at math and it was partly explained by when he’d given boyish toys to his twin girls that they took care of them instead of constructed things, he had instead substituted gender with race. Here’s the passage:

There may also be elements, by the way, of differing, there is some, particularly in some attributes, that bear on engineering, there is reasonably strong evidence of taste differences between little girls and little boys that are not easy to attribute to socialization. I just returned from Israel, where we had the opportunity to visit a kibbutz, and to spend some time talking about the history of the kibbutz movement, and it is really very striking to hear how the movement started with an absolute commitment, of a kind one doesn’t encounter in other places, that everybody was going to do the same jobs. Sometimes the women were going to fix the tractors, and the men were going to work in the nurseries, sometimes the men were going to fix the tractors and the women were going to work in the nurseries, and just under the pressure of what everyone wanted, in a hundred different kibbutzes, each one of which evolved, it all moved in the same direction. So, I think, while I would prefer to believe otherwise, I guess my experience with my two and a half year old twin daughters who were not given dolls and who were given trucks, and found themselves saying to each other, look, daddy truck is carrying the baby truck, tells me something. And I think it’s just something that you probably have to recognize.

It begs the question, why did the women in kibbutz quit working on tractors? The way Larry tells his story, he makes it clear he thinks that it’s because the women wanted it that way (thus his story about the twins). But surely it is as plausible that: 1) Men, having a vested interest in proving their manhood (which they do and in cultures around the world leads to certain types of work being seen as “manly”) weren’t keen about day care duty and/or 2) women were hesitant to cross the lines of gender stereotype (it might lead them to be perceived as being masculine, or even worse, emasculating). And it also isn’t hard to imagine that parents ooh and ahh more when small children play with what are perceived to be gender-appropriate toys and are quietly or even vocally uncomfortable when boys play with dolls and girls play with trucks.

One last word about the null hypothesis and why I’m so devoted to this issue: when I and two other girls (and, as it happens, no boys) in the 6th grade did well enough to go into a special, advanced 7th grade algebra class, my (female) teacher brought us up to the front of the room and told the three of us “I don’t see why you would challenge yourselves like this anyway since you are girls, and you won’t be needing math when you grow up.” I was the only one of the three of us to actually choose that class, and I was the only girl in the algebra class. One of my friends was one of two women in a class of 45 students studying artificial intelligence at Yale. She was expecting praise for being one of only two students to get a program to work on a particularly tough assignment. Instead, she was accused by the professor of stealing the code from her male classmate. She left the major. Until stories like this become rare, or even uncommon, I will assume that there’s too much cultural influence to figure out the real story.

Going back to Larry Summers, his lecture did two things: 1) it breathed new life into the age-old stereotype that women aren’t as good at math as men, and 2) it attributed that difference to an underlying innate ability difference- that is, he conveyed a “fixed ability mindset” regarding math (more on mindsets below). As the leader of an educational institution he introduced the two ideas that together are like a powder keg: they can undermine women’s feelings of belonging in math, which in turn informs their mathematics achievement and intrinsic motivation to remain in math.

Now more about Catherine Good. She talked at that same conference where Larry Summers put his foot in his mouth; in fact she was the speaker after Larry at that conference, and she was talking about her paper that gives evidence that the above “powder keg” message tends to push women out of math (but Larry didn’t stick around long enough to hear her talk, unfortunately). She is also an expert on stereotype threat and helped me look at the study. More on her thoughts below, but I still want to talk about the concept of “genius.”

2. Women and the concept of genius

Let’s define, as one of the commenters does from the blog, a “genius woman in math” to be any woman who has won a Fields Medal. Since there are no women who have won Fields Medals (versus 52 men), this is a pretty tight definition. I would argue, and I might in another post, that even without the above definition, the concept of “genius” is a social construct which is rarely if ever applied to women, except perhaps after they’re dead. Please comment with counterexamples if you know of any.

So here’s what I think. There are lots of reasons that women don’t win Fields Medals. I will name a few.

  • Fields Medals are awarded to mathematicians under the age of 40, for some reason, and women mathematicians typically do good work into their retirement age, whereas men usually do their best work young (this also explains why Harvard has so much trouble hiring women- by the time they are convinced the woman is a genius, she’s 55 and has grandchildren and frankly probably sees the offer as tokenism).
  • The commenter who defined a “math genius” as a Fields Medalist said that it would be an objective measure. But Fields Medals are awarded by a bunch of guys who decide what’s important and who’s responsible for the important results. In other words it’s a political process.
  • Women don’t care as much about winning Fields Medals. This matters, because I know of men who explicitly worked on problems in order to win the Fields Medal (you know who you are). It’s a serious and bizarre case of narrow focus.
  • Why is math genius defined so narrowly? I would personally define it more broadly (a topic for another post), and there’d be plenty of women geniuses. With my definition, though, I’d guess that women who are geniuses have lots of options and they often choose something they consider more personally rewarding than an academic job.
  • Women’s intelligence may also manifest in different ways: note that most of the assholes on Wall Street are men. This kind of makes sense since women are typically not as driven by testosterone and competitiveness. This doesn’t mean they aren’t geniuses or that they couldn’t have done the work the men on Wall Street did (my experience proves that).
  • The Fields Medal distorts the mathematical process itself, by implying that there’s a single superstar who swoops in and solves the problem that all the other people were incapable of doing. In fact mathematics as a field is an enormous collaboration, a scientific project, where everyone depends on the community around them for coming up with questions, defining the “interestingness” of questions, and giving context to results. The idea that there’s one winner out of all of this, or even one metric by which we could measure such a winner, is silly. See this post from Quomodocumque.
  • Another point about genius (in any domain): research is showing that to truly express one’s genius takes thousands of hours of practice. So genius may be a latent trait but will never be expressed without many hours of hard work. This point is very often lost and is related to women in that their apparent geniusness depends to a large extent on how supportive their environment is for all that investment of time.

3. The paper against stereotype threat

I am finally ready to address (with Catherine’s help) the issues of the paper in question, which I will repeat:

  1. The papers on stereotype threat normalize with respect to SAT scores which is bad

In fact the author “discards” a bunch of stereotype threat studies on these grounds. However, it is totally standard to normalize with respect to some other metric (would you rather we didn’t normalize to anything?), and in fact it essentially penalizes the studies, since it has been shown that stereotype threat is in play even for the SATs. On the other hand, the standard for normalizing (this is called “including a covariate”) is that the groups being compared should not differ significantly in the covariate, presumably because it’s harder to argue that your are in fact correcting for that aspect. Because men and women sometimes do differ significantly in SAT scores, including them as covariates could be a technical violation of the rules of conducting a so-called ANCOVA.

Is this what the author is complaining about specifically? Did he, for example, check to see if the samples in the “discarded” studies actually differ in the covariate? It seems he’s making the assumption that they did, but it’s not clearly stated that they did. It’s certainly not a given that the men and women in these studies did differ in the covariate, and he needs to make that precise. If they did not, then there’s no valid argument against using SAT scores.

  1. Evidence for stereotype threat is therefore weak.

There is ample evidence that stereotype threat is very real. Keep in mind that the authors of this study have not shown evidence against stereotype threat, but have simply complained that they don’t like the existing studies for it. And their standard for what “replicates” the original study is overly stringent- they only wanted to include studies that found significant interactions between gender and condition. Interactions are easiest to find when you have a “crossover effect” (e.g. males are higher in condition A but lower in condition B), but often we find “span effects” in which the males and females may be equal in condition A but differ in condition B. This can also be an example of stereotype threat. For example, in a paper written by Catherine, she didn’t find a significant interaction (males and females performed equally in condition A) but when the stereotype threat was reduced, women outperformed men. To discount this and other studies as not providing evidence of stereotype threat simply because an “interaction” wasn’t found is playing games with statistics.

  1. We should therefore stop putting all of our resources into combating stereotype threat.

Nobody who studies stereotype threat claims it explains everything. It is part of a larger picture. The good news is that there are interventions for it (described below).

  1. We should instead do something easy like combating stereotypes themselves.

The idea that it’s “easy” to combat stereotypes is completely naive. There are tons of ways that stereotyping is understood to be very difficult, if not impossible, to get rid of. Some of them have to do with an evolutionary need to simplify first impressions of people (i.e. categorize) so that we can tell if they are an immediate threat to our safety. This may be the most baffling part of the whole thing, because the authors should really know better.

I want to end on a positive note, because the news is actually pretty good. There is a way to combat stereotype threat, and I’ve tried it and it works. To understand it, it helps to think about the way people think about intelligence itself. As a simplification, people either think that intelligence is fixed and rigid (you’re either born with it or you’re not) or they think that intelligence is malleable and can be learned and practiced.

It turns out that if someone believes the latter “malleable intelligence” view, then they work hard and are hopeful and stereotype threat is to a large extent alleviated. Whereas if they’re convinced of the former mindset for intelligence, the effect of stereotype threat is more pronounced. In situations where the stereotype is salient (“girls are bad at math” is salient when taking a math test), the situation itself can convey a mindset of fixed ability and all the hallmark responses that go along with that mindset then follow. To encourage a malleable view of intelligence can help combat that fixed view and thus the threat of the stereotype.

The way I used this information was as follows. I started a class in teaching proof techniques at Barnard College (there were both Barnard students and Columbia students in the class). At the beginning of every class for the first two weeks I described how mathematicians aren’t born knowing how to prove things, but rather they learn techniques, and practice them until they are proficient. Note I wasn’t directly confronting or addressing stereotypes, but rather setting up the mindset where the studies have shown stereotypes have less negative power.

The class went great, and is still going on. I will post soon about my experiences starting that class and others like it.

  1. LindaCO
    February 6, 2012 at 8:19 am

    Good stuff! Found your blog through NC.


  2. Steve Smith
    February 6, 2012 at 9:25 am

    I think Marie-Sophie Germain is an example of a female genius who was unable to make a career out of mathematics because of gender bias. What evidence do we have that the societal issues and biases present then are not present today?


  3. JSE
    February 6, 2012 at 10:45 am

    “and just under the pressure of what everyone wanted”

    This is a great example of the sneaky rhetorical use of the word “just”:



  4. Luis Miguel Solarte
    February 6, 2012 at 11:20 am

    Matilde Marcolli don’t win Fields Medal for political reason, her post deserve a good read:
    http://listeningtogolem.blogspot.com/2011/12/eminence-and-demise.html .


    • plm
      February 7, 2012 at 6:35 pm

      Thank you so much, Luis Miguel. I have to read more of her blog, I did not know of it(!). Do you know where to find more information about the situation of Matilde? I am not sure who she has problems with of her past collaborators, do you know?

      I hope this gets sorted out, though I guess I am too optimistic.

      I once wondered on Le Bruyn’s blog why Matilde did not receive more recognition. I am not sure her work should have earned her a Fields medal, but certainly more recognition than it has received so far.

      She has many students, but I wonder just how well they can popularize her thoughts, perspective.

      And finally, even if she is admired I wonder how much of an example she should be (to other women), I think she would admit she is rather unhappy -beware this is only my guess.


  5. Bridget
    February 6, 2012 at 11:20 am

    One thing I noticed recently that the gender ratio is much better among the ranks of winners of awards/grants that allow self-nomination than it is among the ranks of winners of awards/grants that require peer-nomination.


    • February 6, 2012 at 11:22 am


      Interesting! I’d be super into helping you make that a statistical argument if you have some numbers.



  6. lawrence castiglione
    February 6, 2012 at 11:37 am

    Old Emeritus Sez: Stereotypes pass through statistical screening like air through a screen door. Culture is all about us, what we choose to express in dichotomous variables, cues transmitted without conscious awareness, cognitive lenses through which we view our experience and seldom yields even to sophisticated research design.


  7. February 6, 2012 at 2:13 pm

    I appreciate posts like this on the situation of women in math (and in many other male-dominated fields in which I have firsthand lopsided dynamics, such as computer science and chess), especially since my sister is a mathematician (actually, I believe you knew her at one point), and one thing that is extremely important and effective, I believe, is women speaking up both about their experiences and refuting superficial attempts to say that there is no problem. Actually, I think that in itself is discouraged in various ways (by those trying to portray such individuals as troublemakers).

    By the way, last week I nominated you for a Liebster award: http://franklinchen.com/blog/2012/01/29/liebster-blog-award-5-blogs-you-should-check-out/


    • February 6, 2012 at 2:20 pm

      Cool, thanks! And yes I agree there are plenty of cultural issues surrounding whether it makes sense to talk about this stuff. Luckily I am a loudmouth.

      I will need to look around for some blogs to nominate. It’s a cool idea.


    • George Hagstrom
      February 6, 2012 at 7:02 pm

      My reply is only of limited relevance, and you might already be aware of it. If you want a country where chess is female dominated look at Georgia. Perhaps it would be an interesting case study, to look at the differences in how female chess players are described/how their talents are portrayed inside Georgia and outside out it.


  8. February 6, 2012 at 5:10 pm

    I have said this in comments before, but Emmy Noether (abstract algebra) and Florence Nightingale (statistics) were both proper Maths geniuses. If you want males of similar stature to the female ones mentioned, there are people like Euler. However I suspect most Fields medal winners have had significantly less impact on the world.

    I am 51, so things may have changed, but when I was being educated, the gender of Newton, Liebniz, Gauss etc were mentioned but Noether’s first name was glossed over and Nightingale was described as a nurse!

    If we stopped erasing women’s contribution from our tales of mathematical history, we might find that we ceased preventing their aspiration to the mathematical future.


    • February 6, 2012 at 6:33 pm

      That, and Noether was 37 when she first got a proper paid job at a university, over the objections of many faculty. The Fields age limit is 40.


      • lawrence castiglione
        February 6, 2012 at 9:40 pm

        So noted.


  9. kris
    February 6, 2012 at 9:43 pm


    There was no fields medal when Emmy Noether was alive. If I am not wrong, the first fields medal was handed out sometime around 1960. Anyways, I think someone like her generally would have added to the reputation of an award, rather than the other way around.


    • February 7, 2012 at 11:47 am

      Yes, I know that the Fields medal was established later. My point was that women tend to come into their accomplishments later in life, often for reasons of social expectations and prejudice rather than childbearing or whatever.

      If “math genius” means doing stellar work by the age of 25, then most women miss the mark because (a) it takes them some time to figure out for themselves that they’re actually good at it, (b) then they have to convince others, (c) even if it’s established early on that they’re good at math (as it was in my case), they’re often left adrift in early career stages, expected to be everyone else’s little helpers, often discouraged from forging their own research programs because how would *they* know what’s important or not.


  10. brownian
    February 6, 2012 at 9:48 pm

    I am a woman in math. Making introspection, I never thought that women are not as smart as men, but sometimes I think that learning a lot of math, writing the coolest papers on hot topics and making herself a social environment from academic achievements is not a easily maintainable and lasting status for women (at least for me) while there is a more down to earth life and social environment they can be part of using their natural social skills, out there. But I should also add that staying in academia can be painful if you are not going to do it like a man. I am sorry.
    It’s also likely that women and men may be comprehending math in different ways. Assuming my brain is a good representative of a female brain, I like working and thinking with internalized objects than reading math as a super formal language. No need to say internalizing process takes time.
    At the end some of those who are predicted(!) to not be successful in math can not take more of the pitying looks and go to a place where things go more smoothly at a faster pace. I still do not know which model to adapt for myself.


  11. Constantine Costes
    February 7, 2012 at 3:13 am

    This talk by Prof. Kathryn Johnston was mentioned in a comment on the naked capitalism post; you might find it interesting: http://www.cfa.harvard.edu/cfawis/kathryn_johnston.pdf


  12. Dan L
    February 9, 2012 at 10:48 am

    I think it’s useful to partially categorize the People Who Think There is No Problem, because there are really different mentalities involved, and different arguments to be deployed:

    First, there are the people who think that males have some sort of biological advantage over females, OR at least suspect that this may be the case. Frankly, I find this view outright sexist and totally unsupported by any kind of rational, scientific reasoning, but there are many respectable people who fit in this category, including Larry Summers.

    Second, there are people who think that females and males are equal in intellectual capabilities, but that females have a “natural” inclination to choose not to reach that potential. (That is, they naturally prefer family life over career ambition, yadda yadda. They like to play with dolls rather than trucks, etc.) This is almost the same as the first mentality, but it seems a bit less sexist to me. It’s a sort of cultural relativism, saying that if women don’t want to be mathematicians, that’s not a sign of their inferiority.

    Third, there are people who think females and males are equal in intellectual capabilities, and may even concede that society causes much of the disparity in outcomes, but as long as there is no serious overt discrimination going on, this should not be considered a “problem” to be “fixed.” Or at least, the only real problem to be fought is discrimination, which these people typically believe has been mostly stamped out. This person would say it is wrong to discourage a bright young girl in mathematics, but also dislikes the idea of giving this child special encouragement because she is a girl. They might say stereotype doesn’t exist, or even if it does exist, it’s not discrimination so who cares. This is sort of the anti-affirmative action mentality, and I think it is the most difficult to combat.

    As for the Fields Medal, I feel confident that a woman will win the Fields Medal in my lifetime. There are so many high-quality young female mathematicians already that I feel it is just a matter of time. It hardly matters though. People will still ask, “But why are there *so few* female Fields Medalists?” These people will completely miss the significance of the fact that the question itself changed.


  13. Jay
    February 10, 2012 at 2:30 pm

    It took me no more than an instant to come up with two examples of living female geniuses (IMO):

    In my own field is the mostly unappreciated (because poorly understood) Bernadette Perrin-Riou. Her insights from the late 1980s and early 1990s are still ahead of their time.

    Not in mathematics but science/engineering is Angela Belcher. OMG just watch this:


  14. Joanna Bujes
    February 18, 2012 at 10:08 pm

    As for why women are bad at math: try this:

    Q. “Do you know why women are bad at math?”
    A. “no”
    Q. “Because all their lives they’ve been told that this [hold thumb and index finger a couple of inches apart] is eight inches.”


  15. goldgarf
    March 16, 2012 at 12:49 pm

    When your post is filled with statements like “There is ample evidence that stereotype threat is very real,” and “Women don’t care as much about winning Fields Medals” and “research is showing that to truly express one’s genius takes thousands of hours of practice” you need to link to all the evidence butressing your position. Instead, we are left only with your explanation of what the “evidence” says.


  16. Piper
    July 20, 2012 at 3:32 pm

    I’m so glad I came across your blog. I’m currently writing a paper on women in STEM fields and I’ve come across many studies RE: Stereotype threat. It’s for a gen ed class, but I’m definitely one of those women who always thought I was bad at math (no one ever told me I wasn’t. In fact my mom used to tell me she thought I was good at it) and this paper is personal on many levels. I thought my friends who were good at math were just born with intelligence, because they would solve problems quicker and get better grades, etc.

    Further, I recently changed my major to Computer Science because I became interested in programming just a couple years ago. The higher math and physics scared me, but I was ready to be challenged. My last Calc II class I failed…reinforcing (for me) my inability to do math and I almost changed back to Computer Systems (less math/physics). I took it again though and the 2nd time around I got a tutor and fell into an awesome study group. Even though I was the only girl in the group, I realized that my struggle with specific concepts weren’t uncommon. Maybe it’s a “duh” moment to others, but knowing I was keeping up with math majors and bio chem majors and realizing we were all struggling was a relief. I felt like I’d just discovered something everyone else knew..the people around me that were truly succeeding were the ones that worked the hardest. Sure, some of the guys were whizzes at doing calculations in their heads, but that didn’t make understanding when to use the disk, shell, or washer method any easier.

    I passed this time around, but I also realized, for me, learning math in the past was not about understanding concepts. I was just *memorizing* rules and not understanding where they came from or how they connected to other aspects of math and other fields. For example in this last math class, I got why mastering algebra and geometry was essential in solving more complex problems. For whatever reason, I didn’t understand that math (to me) was built on foundation after foundation and getting good at one layer (algebra and geometry) helped to strengthen other layers (calculus). In any case, I’m in the process of really *learning* math now and kicking myself for seeing the beauty in it so late in my life (I’m 31).

    In any case, I’m very happy to be reading your posts. I’m discovering so much about women mathematicians and women in other fields of STEM. It’s very heartening.


  17. Scott Adler
    July 23, 2012 at 5:20 am

    Perhaps I might add something to this debate-complaint. Perhaps there is another factor at play.
    Not long ago, I wrote an article on intellectual property law, during which I casually asked one senior attorney in the field, “What is the percentage of women who obtain copyrights?” He answered, “About fifty percent.”
    “And patents?”
    “One to two percent. I can go for years without seeing a single female inventor.”
    I went back to my other interviewees, male and female, and all said the same thing.
    I’ll add another example from my personal experience as one of the founders of the PC software business. The number of women at my level of creativity (and I was strictly third tier) was precisely nil. Zero.
    At least a million women had more training and skill than I did, yet not one came up with a single innovation.
    Women aren’t dumb, certainly my wife is a whole lot smarter than me, but very few of them pick at a problem all day and all night, obsessively study a real world or theoretical problem until a light goes on.
    The problem may be that women are just too normal. Most of us who invent things have a touch of Asperger’s, at the very least. Some are utterly disabled by it. Relatively few women have Asperger’s.
    I have Asperger’s. I find it hard to read faces, or make eye contact, or tell when people are lying to me. Consider yourselves lucky.
    Okay, there are no female Fields recipients, and there are more female hackers on TV than in real life. From my experience, you should consider yourselves lucky.


    • July 23, 2012 at 6:33 am

      Here’s the thing. The most creative inventor I’ve ever met is a woman, but she’s never gotten a patent. This is mostly because she’s a woman, and as such isn’t recognized as a potential genius.

      I’ll agree with you though that men are more likely to be functional and have asperger’s and be inventive, but it’s partly because the world has a path ready made for men like that, but not for women like that.


  18. namae nanka
    September 6, 2012 at 9:55 pm

    stereotype threat doesn’t explain the normal difference, but even worse performances than what would normally be expected. So the 1STDV gap between black-white remains present in its absence. The original study found as much.


    as for men and women,



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