Not good with numbers

I have not written much about “girls and math” or “women and math” here: there’s this, this, and that’s all I can recall. Well, you can run, but you can’t hide. Here are some answers to questions I’ve been asked here and in real life. Be warned that this is long and a bit rambling. Or just skip it if you wish. I’ll get over it somehow.

1. First things first. Gender imbalance in mathematics has much more to do with the place of women in the society than with their mathematical ability or lack thereof. That’s what I believe, and this is consistent with various studies such as this one. The full text is behind the pay wall, but this article summarizes the relevant points:

[...] the lead researchers Janet Hyde and Janet Mertz manage to show a significant correlation between the percentage of girls on a country’s International Mathematical Olympiad Team, and that country’s World Development Indicator Gender Gap Index. The emerging pattern is quite clear: The greater the gender parity in a country, the more girls go to the Math Olympiad; thus indicating a significant role – who could have doubted it – of social equality in girl’s performance on this (and other) indicators of mathematical achievement.

(See also here.)

Who could have doubted it, indeed? There is something unnerving about having to cite a published scientific study in support of something that should be self-evident enough.

2. What about innate differences in mental abilities between the sexes? This article mentions a few physical differences that might or might not be relevant: women’s brains are proportionally smaller, have a higher ratio of gray to white matter, show different activity patterns while engaged in problem-solving. What does this imply about our relative mental capabilities? We don’t know, the article tells us. But what I really would have liked it to say is: do you actually believe that a person’s mathematical aptitude could be directly related to the size of her brain? Then you might as well choose your next computer based on its size and weight – the bigger, the better – with the physical size of the CPU and the total count of various components also taken into account. That’s about the same level of sophistication.

3. But you can’t say that in a news article. This has to do with the distinction between different types of journalism. (I’ve seen good articles explaining this, but I can’t locate them just now. I’ll add a link later if I can find it.) To cut a long story short, in a news article – as opposed to an opinion piece, for example – you report the facts and do not comment on them. There’s also the general expectation that both sides of any dispute should be presented in order to avoid journalistic bias.

Suppose, then, that you’re writing a news article about a women-and-math study such as the one I just mentioned. You can’t just say outright that the innate differences theory is bonkers, or at least that it should be assumed false until some evidence is actually produced in its support. That would be an opinion. What you can report instead is that Dr. X said that the innate differences theory is bonkers. For balance, you should then also quote Dr. Y who believes that, while the study does debunk some of the popular myths, many questions are still left unanswered and the relevant biological differences may yet be discovered. Ironically, through its pursuit of objectivity and impartiality, news journalism ends up repeating what amounts to no more than hearsay and innuendo.

4. Why is there a problem with that? It casts suspicion on our intellectual abilities and professional qualifications without anything ever having to be proved. That’s how hearsay and innuendo work. When presented with both sides of an argument, especially on a subject that they don’t know much about, people will tend to assume that the truth lies somewhere in between. If innate differences are discussed so often, there must be something to it, right?

And there is no way to disprove it once and for all. That’s the thing about the yet undiscovered innate differences: they can conveniently remain undiscovered for as long as necessary. The other thing about them is their biological inevitability. No matter what we have already achieved, no matter how much evidence there is that we do in fact have mathematical talent, at some point a Fate will cut one of our strings and we will collapse on the office floor like rag dolls, weighed down by our biological limitations, unable to progress further. How encouraging is that.

5. So what is mathematical ability, anyway? I’ve had various people tell me that they have never been “good with numbers”. Guess what? I’m not necessarily good with numbers, either. I can’t add large numbers quickly in my head, let alone test them for primality, and my tax returns would invariably come back corrected when I used to do them by hand. Another time, a carpenter who was doing some work around my house told me that he wasn’t good at math. I pointed out that as a matter of fact his job involved a good deal of mathematics, especially geometry, in figuring out where to place things and how to make them fit. He responded that this was easy because it just made sense – the abstract stuff was hard, though.

Making sense – that’s quite possibly the best description of mathematical thinking that I could come up with. Mathematics is not only about adding numbers and solving equations. It’s also about identifying similarities, structures and patterns, making a logical argument, finding your way through a maze of converses and contrapositions. If you can build a tree house, or read through a Supreme Court verdict and understand its logical construction, or make a coherent argument in a blog post, chances are that you have – and make good use of – some of the same skills that professional mathematicians rely on every day.

6. OK, let’s be more specific. The sort of general mathematical skills that I have just described cannot really be analyzed in any quantitative way. Instead, any reasonable study must focus on some well defined group, from elementary and high school students to research faculty, and rely on quantifiable indicators of mathematical talent or accomplishment. The general patterns are consistent with what I said in my first point. But we also run into a new set of problems: how do we know that we are not conflating things that shouldn’t be conflated, or failing to make connections that need to be made? For instance, this article (see also here) points to a study showing a correlation between teachers’ math anxiety and the math performance of their female students; the male students were less affected. What else is there that we don’t know? Social factors in particular could be easy to miss: they’re hard to isolate and their influence could be delayed or indirect.

7. What about female faculty in math/science/engineering? There’s a book-length study funded by the NSF that documents the many ways in which gender bias affects our careers. I won’t try to reproduce it here – you really should read it if you have the time. I do think that specific manifestations of gender bias, such as when female faculty are paid less or have to wait longer to get promoted than their equally accomplished male colleagues, call for targeted corrective action. They do not call for a general discussion of how the female students somewhere in the U.S. tested on the SAT and what this might or might not mean in terms of our innate abilities. Please also spare us the arguments that we can’t possibly still be subject to bias because the number of female graduate students in the math department has been increasing (all the way to 20%, but never mind.) The time wasted on such arguments could be spent much more productively otherwise.

8. Why have there been no female Fields medalists? It should not be unreasonable to say that the absolute top level of accomplishment – in mathematics or any other area of science – requires a confluence of exceptional talent and circumstances favourable enough for such talent to flourish. Such circumstances might include encouragement from family and teachers, early access to high level education, acceptance and support of the academic community, and generally having control over one’s own life. This is doubly important in the case of awards where an early age limit does not leave much time for overcoming serious obstacles. Emmy Noether might have been a contender for a Fields medal if it had been established a few decades earlier. The Fields age limit is 40. Noether was only able to get a paid job at a university in 1919, at the age of 37.

That was only 90 years ago. Much has changed since then, but the playing level is still far from level. Give it another 200-300 years, raise the Fields age limit to 50 to allow for childbearing, include women on the Fields selection committee from time to time, and if there are still no female Fields medalists at the end of that period then that may be grounds for discussion. I’m not convinced that we’ll have to wait that long, though. We do see more and more women winning prizes and honours in mathematics, including those with age caps. For all I know, it could happen within the next 10 or 20 years.

9. To the extent that there is a power imbalance, we can’t just talk our way out of it. If a high-ranking administrator gives a speech to the effect that Gender Bias is a Bad Thing, that’s not enough to solve the problem. It’s a possible good start and that’s all it is. Also, we should act a little bit less like scientists sometimes. It’s tempting to think that if we just make our case and lay out all the arguments, our colleagues will have to see the light and mend their ways. I know because I’ve been guilty of that. That’s not how it works in politics, and that’s not how it works here, either.

10. I have hardly mentioned childbearing and family responsibilities at all. Not because it’s not important – it clearly is and plenty has been said about it. If that were the only issue, though, then those of us who have never had children would not have been affected by gender bias, and that’s not consistent with our experience.

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16 responses to “Not good with numbers

  1. David


    is a rather non-standard take on the issue. He basically says that under the current conditions – where anyone who is both smart and dedicated enough to get an academic hard sciences position can retire at the age of 50 – you’d have to be a (very smart) idiot to start an academic career; he also says that most very smart idiots are testosterone stuffed teen-age boys.

  2. Hmm. Just to begin with, not all of us were smart enough to be born in the right country…

  3. David

    @Izabella Laba – He addresses this point: if you are a western woman you should have (and usually does have) better judgment, and if you are a non western woman, chances are you will not immigrate (two reasons: the first is that most immigration waves start with young men, for the same reason – so I think he claims – they go into science; the second is that non-western countries tend to suppress woman more than western ones, so your chances of getting a good enough education are smaller).

  4. Does Eastern Europe count as “western”? If not (and for this purpose it shouldn’t), then (a) women did get a good enough education, (b) academic immigration was usually a rational decision motivated by the consideration of living standards, much like what Greenspun writes about choosing a professional career over an academic one, rather than some testosterone-fueled gung-ho adventure. At least that’s what it was in my case.

  5. David

    Now I’m intrigued: was the men/women ratio in hard sciences closer to one in eastern europe 20-30 years ago (I’m not sure that measuring it today is a good idea – a lot of eastern europe is rather western today) ?

  6. As far as academic jobs are concerned, the ratio was similar to what it is now in Canada and the U.S. Women’s education was not suppressed and we were free to pursue the same study programs that men did. On the other hand, most female university students in science would choose the “teaching stream” as opposed to the “research stream”. It’s a long story, probably the subject for another post.

  7. Anonymous

    Couldn’t the disparity could be explain, in part, by physiological factors other than innate ability? For example, maybe men are just more likely to be interested in a career in mathematics. Note that mechanics and firefighters are overwhelmingly male and nurses and elementary school teachers are overwhelmingly female. There are probably social factors that play into this, but it is conceivable that there are physiological factors as well. Of course, I’m not disputing that there are social forces at work — indeed I have seen first hand faculty members dismissive of female students.

    I think most of the mathematical community (there are always exceptions) agree that there is a problem. I think what is less clear, and more controversial, is how to address it. I know a senior (male) faculty member who feels passionately about this issue and uses all of his political capital to hire promising women when positions are available. There are always bright female applicants out there, and everyone he has hired are well deserving of their position. At the same time, however, I feel this is unfair to the male applicants. Essentially we have a system where a male applicant needs to be clearly superior to the entire pool of women applicants to win out. Of course, many might argue that since women have a harder time succeeding in math that this is fair. While this may be true on average, it says very little about any two specific cases that come before our hiring committee.

    Preferences like this also can create a backlash, or even harm top women in the field. In the above situation, it is sometimes the case that a woman is hired because she is (1) equally qualified to her male competitors, and (2) she is a woman. In other cases, a woman is hired because she is clearly the best candidate for the position. When the later happens, it is easy for someone critical of professor X, who knows that there is generally a preference for women hires, to say “she was given preference because of her gender.” This statement would be somewhat true and somewhat false and impossible to ever verify or disprove. Of course, no one will ever write on her reference letter “Yes we do typically have a preference for women, but even if we didn’t we still would have hired her.”

    I’d be interested to know if you (or anyone else) have ideas on how to help address the problem, without creating a host of new problems at the same time.

  8. There are clear physical reasons why firefighters are overwhelmingly male. On the other hand, the hypothetical physiological factors that determine math ability have never been identified and aren’t likely to be identified any time soon, and I’m tired of the shadowboxing. (Also, “more likely to be interested” is not a physiological factor.)

    Addressing that mess is hard. I have indeed seen examples where affirmative action procedures were threatening to go way off the rails and defeat their own purpose. On the other hand, as the power imbalance becomes less extreme, it gradually becomes easier to address gender bias in a more sensible way… oh, wait…

  9. NN

    Good on you for writing about this issue; my observation is that most people in Canada pretend (or believe) that discrimination based on sex/gender/sexuality has gone away. I think it will continue as long as people keep thinking of men and women as intrinsically and significantly different (other than in their reproductive roles), and, more generally, thinking of physical sex as being a discrete, black-and-white dichotomy, instead of a rich, splendid spectrum.

  10. Hany M. El-Hosseiny

    Look at a muslim country, where women are socially considered non-majors. A lot of young women study basic science, medicine and engineering (in science and medicine students the male/female ratio in recent years has become 2/3). However, still few women choose research. The good ones (I get frequently very good female students) seem to be less free than their male colleagues, they think more about marriage, they are not free to travel alone … etc.
    As in western countries, women seem to be more successful in Biology and Chemistry. This is intriguing, why the same pattern keeps appearing in societies so different? Is the direction of social pressure identical?

  11. Rather not the point, but: I find the phrase “manage to show” in the quoted paragraph in 1. strange. To show a positive correlation between two sets of data amounts to applying a standard statistical construction, right? Presumably nowadays it means inputting the data into something like a spreadsheet and clicking on one of several “correlation” buttons. If the Drs. Janet had to do something truly creative in order to acquire a positive correlation, wouldn’t that make the outcome less convincing?

  12. Much more seriously: yes, the Fields Medal, with its 40 year age limit, is unquestionably discriminatory towards women. It places a completely arbitrary deadline (which to my knowledge no one has seriously attempted to justify, beyond the anemic point that Fields’ intention was to use his award as an inducement to future work) up against a much more serious and meaningful biological deadline for women, but not for men.

    I realized this in the fall of 2006 when the Fields Medal was the talk of the (not just mathematical, for once) town. I even wrote a little essay about it, including pointing to a specific female mathematician whose unquestionably stratospheric mainstream results came soon after the age of 40, and presumably would have come before the age of 40 had she not had children.

    Interestingly though, most female mathematicians whom I have talked to about this agree with it but feel less strongly than I do. It did not take this argument to convince them that the Fields Medal is a little silly.

  13. Rather not the point, but: I find the phrase “manage to show” in the quoted paragraph in 1. strange.

    That could be played both ways, you know. It might be possible that the study authors had to do something creative. It might also be possible that the author’s turn of phrase reflects his unconscious bias in undermining the results of a perfectly legitimate study. Frankly, I don’t believe that either interpretation holds much water. The wording of a blog post is not usually subject to such close scrutiny; most likely there’s absolutely nothing to it.

  14. I looked at the paper:

    The statistical work seems quite straightforward. In fact, at the end:

    “Acknowledgments: We thank Jonathan Kane and Jennifer Petersen for help with making figures and performing statistical analyses…”

    It’s worth reading, I think.

    I also don’t think that it’s a matter of any unconscious bias against the article on the part of Daniel Hawes. I’m just being a picky mathematician who is interested in what research is like in other fields.

  15. Adam – Would you like to finish the sentence?