Friday, July 25, 2008

Janet Hyde: boys = girls in math? Not really

Hundreds of media outlets - including the New York Times whose Tamar Lewin mentions Lawrence Summers in the very first sentence - uncritically report that "girls are equally good as boys" in math. This highly surprising statement is claimed to be based on a paper in Science magazine.
Janet Hyde et al.: Gender similarities characterize math performance
Janet Hyde herself, a self-described feminist psychologist, seems to be a prototypical 2nd wave feminist, too (click her picture or check her publication record). She thinks that "her biggest contribution to feminist sexology is her college sexuality textbook" (exact quote). She believes that the supposed higher rate of masturbation among boys and men has "enormous implications" which, she believes, was her second "major contribution" to feminist sexology. This difference between sexes goes in the "right" direction so these "scientists" choose to believe it and promote it. We can finally appreciate her third "major contribution". ;-)

All five co-authors are women who have been writing similar cargo cult scientific papers for quite some time.

It is enough to read a brief review in Science to see that the headlines don't follow from the paper at all. The review explains the main reason of the results: they have detected no signal because the tests were too simple. They didn't really test "g" or the ability to think mathematically but rather attainment. And the standards are dropping. David Malakoff writes:

The study's most disturbing finding, the authors say, is that neither boys nor girls get many tough math questions on state tests now required to measure a school district's progress under the 2002 federal No Child Left Behind law. Using a four-level rating scale, with level one being easiest, the authors said that they found no challenging level-three [strategic thinking] or [level]-four [extended thinking] questions on most state tests. The authors worry that means that teachers may start dropping harder math from their curriculums, because "more teachers are gearing their instruction to the test."
That's what the media should actually worry about - decreasing quality of American kids' math skills - but no one does. It's just too bad if there are no challenging problems. The tests they have processed had nothing to do with strategic thinking and extended thinking. It wasn't really about the math that matters in math-loaded occupations.

The last sentence says:
Among students she's observed, she says "the boys tend to be a little more idiosyncratic in solving problems, the girls more conservative in following what they've been taught."
Yes, that matches my experience, too. The latter approach is normally called "a lack (or shortage) of creativity." If you combine this observation about boys and girls with the statement that "more teachers are gearing their instruction to the test", you may guess whether it is the boys or the girls whose scores will be raised.

Check e.g. California's standardized math test for grade 7 to see how low-brow and mechanical these tests are.

Newspaper articles build their far-reaching statements on grades 2 through 11. The paper by Hyde et al. however did find that boys have an edge in the SAT tests before they enter the college. Hyde herself offers a conceivable but not straightforward interpretation of this fact - one that she chose not to apply in other cases. For a certain reason, this portion of the paper - which is much more important for our understanding of the development of the math talents - is not being reported or, to say the least, it doesn't influence the headlines.

The magnitude of variations

Most importantly, those five co-authors don't seem to discuss the variance of the distributions in the media. The larger second moment of the male distribution is what primarily decides about the small percentage of women in math-loaded occupations (especially the top ones), more than the central value does: see detailed calculations by La Griffe du Lion. Before those five spice girls made their bombshell statements about girls matching boys, they should have made sure that they could match boys (such as Rushton and Nymborg) in mathematics themselves. ;-)

Not to be unfair, their paper does mention that in the 99th percentile, they found the boys:girls ratio to be 2.06:1 (and for the 95th percentile, it was 1.45:1). Incidentally, these numbers roughly agree with Figure 2 in La Griffe du Lion's text about these matters. But Hyde et al. were very careful that this particular result didn't get into the media.
National Post (Canada) explains that these findings of the paper about the variations support e.g. Lawrence Summers' statements...
In the paper, they also say some "likable" stuff about the ratio being different for Asians: some people argue that this shows "complex cultural factors". That's of course complete rubbish. The ratio is closer to one for Asians simply because the Asian boys and girls have a higher central value, and the 99th percentile of the whole society therefore cuts their distributions closer to the bulk where the difference in variances doesn't play such a role.

Even if this effect were not enough to explain all the data, one should realize one more obvious thing: there is no reason why the magnitude of math skills gap between the sexes should be exactly universal for all races. Of course that different races have different innate aptitudes for maths and they can also have different differences between the sexes. This would show that the biological factors are more important, not less. In the (highly selective) case of the mathematical Olympiad discussed below, boys have a significant edge among the Asian nations, too.

It's been more than three years when Larry Summers introduced the width of the statistical distributions into the public debate but when you make a Google search, you will see that 99% of people still don't seem to get the point. Most people in the world are just stunningly stupid.

If you're one of them and you're disturbed by all these differences and you want to hear something encouraging about ordinary people, let me tell you that it can be calculated that if you pick a random man and a random woman, the woman will be g-smarter than the man in 45% of the cases. You need to calculate a somewhat tough integral to get this result. As Barbie correctly said, math class is tough. ;-)

Mathematical Olympiads

The girls tell us that the gap is now gone and everything is equal. The only problem is that no airplanes land and the percentage of girls in math-loaded occupations stays comparable to 10% or so and among the Fields medal winners, it is 0.0%.

In order to show you a much more meaningful and transparent measure of the real mathematical talent among American boys and girls, let us look at the winners of the U.S. Mathematical Olympiad. Every year, twelve winners are officially announced. We will start with the year 2007.

Click the picture to zoom in and read more about the 2007 ceremony (click). The guy on the left in an official and the remaining 12 high school seniors are the winners. The young lady in the light blue dress on the right side is the only female winner. Congratulations! She happens to be of East Asian descent. The East Asian people happen to have a higher IQ by about 10 points than the Caucasian people. At least, the PC people could be thrilled about one black guy kneeling in the middle. (Oops, it turns out that he is actually an Indian American, would you believe?) I am thrilled, too - and so am I by the remaining 11 winners.

But let's forget about geography for a while.

If the probabilities that girls and boys become winners were equal, the probability that there will be less than 2 female winners would be equal to 13/4096 = 0.003. That's approximately one part in 300. But yes, you might still say that it is a coincidence. Moreover, 2007 is not the most recent year. Why doesn't Mr Motl look at the 2008 ceremony (click)? It must have been an inconvenient one, right? Well, here it is.

Click the picture to zoom in. The background is a bit more complicated now :-) - there is one Einstein and 2+2 old officials at the bottom side of the picture. Nevertheless, there is not a single girl here. (There is no longer any guy who looks black here either, but we don't want to analyze geography in this text.) If you assume the 50:50 probability for them to become winners, the probability that there is no female winner equals 1/4096 = 0.000024.

Finally, you can combine the two sets of winners. If you assume that the a priori chances are 50:50, the probability that among 24 winners, there would be less than 2 female winners is equal to 25/2^24 = 0.0000015, roughly one part per million. That's already pretty much a five-sigma falsification of your hypothesis about "equality". Moreover, all these kids have spent all their lives in the atmosphere of political correctness so one would have to be really mad to argue that the small percentage of girls is due to the terror against female scientists. ;-)

At this moment, the egalitarian people have to claim that all the organizers of math Olympiads, including the old lady (Mary McKay) on the 2008 photograph, are despicable sexist pigs.

But because this result is so perfectly reproducible, I could add as many teams as I wanted, reducing the probability that such a thing can occur by chance virtually to zero. For example, among hundreds of known names of Czechoslovak participants of the International Math Olympiad (which I kind of screwed back in 1992, unlike the national level), you only find four female names: the probability is of order 1/2^200 = 10^{-60}. Analogously, there are 4 girls in the 1993-2008 list of the 96 Czech Republic participants, below 5 percent. (Feel free to check other countries.)

(Ms Jana Syrovátková whom I know very well earned a gold medal in 1993 which was amazing. If you care about these things, the last 1992 Czechoslovak team where I competed was the 13rd country which was one of the worst results in history. ;-) On the other hand, the 10th place of the first 1993 Czech team was fine; Slovakia was 12th in 1993. However, after 1993, both Czechia and Slovakia were always much worse than we were in 1992 and recently our teams are both around the 35th place in average. We simply can't compare to Hungary or Romania and I am not sure about the exact reason.)

But whether you like to hear it or not, the idea that there is as much math talent - and I am talking about the higher end - among high school boys and high school girls is utterly absurd. In the fast comments, additional statistics about the International Mathematical Olympiads are discussed. For example, the female percentage in the teams (10% in average in 2008) is heavily correlated with the success of the team. The winning teams are near 1% while the losing teams at the bottom approach up to 30% of girls.

This correlation can't be explained by any bias in education "before the olympiad": you would have to accuse the IMO graders of fraud. Instead, this correlation proves that the average global cutoff for the girls to attend IMO was somewhat lower. Despite this fact, the girls only made up 10% of the participants.

And the significant gap is not dropping in any sense - the two pictures above would indicate just the opposite trend but such a conclusion wouldn't be statistically significant. What may be dropping is the quality of the generic math education in the U.S. which is the thing that the journalists should worry about. But because most of them hate maths anyway, they prefer to focus on (and twist) different things.

The rate of female physics/chemistry Nobel prize winners has dropped, too (there's been none since the early 1960s), for exactly the opposite reason. Relatively speaking, it becomes tougher to earn a Nobel prize "by chance" (sorry, Marie Curie!). This fact doesn't mean that the new discoveries are more revolutionary than the old ones. On the contrary, they may be less revolutionary but they require a lot of systematic high-expertise work. Quite clearly, the more selective your math-related tests become, the lower percentage of females you will obtain (the boys have a greater variance of the distribution).

The high school math Olympiads are more selective than tests accepting undergraduates to colleges but less selective than Nobel prizes. The silly mechanical tests underlying the paper by Hyde et al. are not really selective and if you score well, it doesn't mean that you can do math-loaded things. It only shows that as a child, you were not quite left behind. ;-)

PC = -IQ

Incidentally, another new study found the political correctness of disciplines to be a highly decreasing function of their IQ. The dumber you are, the more politically correct you are. For example, as far as PC goes,
Psychology > Sociology > History > Political science > Biology > Mechanical engineering > Electrical engineering
As far as IQ goes, these disciplines are sorted exactly in the opposite order. Check the previous two links! ;-)

Click the IQ sticker below to see more articles about innate aptitudes and related stuff.


  1. I agree there is some difference in brain structure and how we handle problems and there will allways be. However I dont think the results of those Mathematical Olympiads are that meaningfull. Rember that 100-150 years ago women didnt have even the right to vote. if you look at the last 800 years of human history, I dont think that a woman who was good in science had much more oportunities to survive than a woman that was beautiful. On the countrary, during that time, males had the chance achieve prominent positions in society using their inteligence. As a result of that, women are culturally less oriented towards maths and maybe genetically inferior (due to pure evolutional reasons). But this also means that they have a big room for improvement. Lets see what the future brings.

  2. Dear ramon, I am not sure whether your remark is meant seriously or as a joke. Are you waiting for a genetic improvement in the future that you will "see"?

    Evolution of species normally takes a lot of time, often millions of years. And it is not even clear that there exist any pressures that would be leading men or women towards any improvements.

    Do you expect these "improvements" to materialize in 20 years instead of those millions of years? Or next week? ;-)

    I am also surprised by the comment about the "big room for improvement". Once you adopt the idea of further evolution, wouldn't you agree that men have a "big room for improvement", too? Why do you exactly seem to think that women should be getting "more rapidly improved" than men?

    Also, I have no idea how the right to vote can influence whether someone can solve problems from a mathematical olympiad. Do you really believe that there is a causal relationship and that you must vote before you solve it? Most of the math olympiads kids have never voted in their life!

    I am just baffled by this kind of discussion and unable to contribute "nicely" to it because you seem to reveal a complete ignorance of the evolution, psychology, society, time scales, causal relationships, and virtually everything else in the world. Sorry.

  3. Lumo, as to evolution, speciation is thought to generally take millions of years, but selection within a species has been observed in much shorter periods. In fact, in rapidly-reproducing species, shifts in the frequency of genetic traits- not mutations, mind you- have been observed in a matter of years, months, or even weeks. In bacteria and viruses, both of which reproduce at a very high rate, evolution is extraordinarily rapid, leading to superbugs like MRSA. Of course, in humans, the change would not likely be noticeable in a lifetime, but then again, "let's" in "let's see what the future brings" could refer to society as a whole, not to particular individuals. The whole phrase also does not necessarily refer to evolution, but also to social changes. Your attack on ramon seems almost ad hominem.

    What's more, ramon does point out the pressures that would cause selection, and the note on voting is clearly a mere example of cultural bias, not a statement that it carries a causal relationship with mathematical performance. However, it is noteworthy that in modern society, highly intelligent females are not any more likely to produce more offspring than unintelligent females. The reverse is true, I believe. Still, this does not mean that genetic changes are impossible. I'm not saying that I agree with ramon, but I am saying that I disagree with your comment.

    Why should women be more rapidly improved than men? If we follow ramon's suggestion, it is because men have been operating at or closer to peak capacity, while women have a social block that can be and is being removed. Again, I'm not saying that I believe in any of this, but I'll argue against an attack like yours.

    So, please, before you accuse someone of being ignorant of "the evolution [sic], psychology, society, time scales, causal relationships, and virtually everything else in the world," make sure you have your own facts straight.

  4. Dear zorkway, sorry, but you are a moron, too. Yes, this is an ad hominem comment because I am talking about you. Not sure how could I possibly say that you are stupid without being ad hominem.

    Evolution and natural selection are two parts of the very same process and this process, to achieve significant changes comparable to 100%, simply takes a lot of time - for large animals like us, it is comparable to millions of years. Whether it is faster for some bugs is completely irrelevant for this discussion, unless you confuse humans and insect.

    Moreover, there exists no pressure in the contemporary society for such improvements. It is not true that smarter women have more children than less smart women, to quote an example. Quite on the contrary, the society has eliminated most of the filters that were driving natural selection and stupid people have a rather clear reproductive edge today. The progress of the society is largely driven by the improvements of technologies, not improvements of the human race.

    Whether sex-dependent rules of voting are called "bias" or anything else has absolutely nothing to do with the discussion about the sex differences in math skills, because voting has nothing to do, and whoever thinks that it has something to do, is simply an imbecile. An imbecile who is ready to say the word "bias", and much like "Allah" for Islamic nuts, he assumes that the empty word explains the real world. It doesn't explain anything in the real world, certainly not anything we talk about here.

    Faster evolution of females... You say that it will occur because "men have been operating near their peak capacity". Is it a joke? Every generation is operating at a capacity that is tens of percent higher than the previous generation. Except that the character of our work and activity is changing as technologies are evolving. But there is no peak capacity, or at least not one that can be saturated in a foreseeable future.

    I find it obvious that women are roughly as far from their peak capacity as men. Your indication that you want to bury men is an example of your agenda-driven bias and bigotry. If you don't believe these utter stupidities, then why are you flooding my blog with them?

    I won't be approving further comments that I will consider as dumb as the comments by ramon and zorkway because I don't want this to become a discussion forum of idiots.

  5. Your commentary about the idiocy of the media and the poor research methods rife in the social sciences are well taken. Most academics in psychology and sociology seem to have a rudimentary or no understanding of the statistics used in the software packages that process their data.

    That being said, do you really think YOU know whether boys are better than girls in maths? I don't know about the rigor of these studies, but in "The Number Sense: How the Mind Creates Mathematics" Stanislas Dehaene implies that high-level math ability is correlated with synaptic complexity, an adaptive feature of the brain, not an immutable, genetic trait.

    Is it more likely for a boy or a girl to hear, from an very early age, that they could never be good in maths simply because of their gender? Are girls or boys more likely to have more responsi- bilities to family and in child rearing? Maybe most girls never put the time and effort into developing the complex synaptic array necessary to win a Fields Medal for these very reasons.

    It does not follow, eo ipso, that they are not capable of doing such work in high level maths, given the right environment. Furthermore, your very allusion to race is perplexing. A persons "race" is a purely social construct and has nothing to do with genetics. I would think a scientist, such as yourself, would be wary of using such ill-defined, loaded, engineered concepts.

  6. First of all, all the children doing math olympiad in 2008 have heard, throughout all of their lives, that girls can be equally god as boys in maths (even though it is not statistically true) from 99% of the people around. Nevertheless, 90% of the finalists are male.

    Your idea that you can create the mathematical talent by telling people some general cliches is truly ludicrous. You know, brain is an extremely fine gadget with billions of neurons connected and interacting in very subtle way.

    The idea that you can create a fine brain to solve a difficult mathematical problem by washing someone's skull with some dumb one-sentence encouragement or discouragement is utterly absurd.

    It's like arguing that you can transform an oven to a very powerful computer by pouring a hot or cold water on it.

  7. mthgk,

    "Stanislas Dehaene implies that high-level math ability is correlated with synaptic complexity, an adaptive feature of the brain"

    Correlation does not necessarily imply causation. It may well be that the synaptic complexity is an intermediate link that has a genetic parent or is associated with exposure to a high level of androgens in utero.

    Consider checking out the book "Brain Sex" by Anne Moir and David Jessel.

    It is possible but rare for women to have a male brain. FMRI is usually pretty good at distinguishing between male brains and female brains because both the structure and functionality of the average male brain is completely different from that of the average female brain.

    The implication that women are doing 'stereotypically female stuff' that prevents them from doing high level math is incorrect because it presumes that (a) that such women do not want to engage in typically feminine activities (b) these women do not perceive what they consider better alternatives for themselves and (c) one can do (in Kantian fashion) what one wants to do regardless of the limitations one's natural abilities. OR the natural limitations are small or non-existent

    The probability of bucking (a) and (b) and fulfilling c is small. Thus, it may very well be that some women are very good in math stuff but are equally adept in, say, biology or something to do with people, rather than abstract concepts.

    What evidence do you have that race is a purely a social construct? Consider reading the following (somewhat dry) articles [Steve Hsu and follow up at Gene Expression] which may change your views on the subject.

  8. While I agree in general I would add that some females have not gotten the Nobel prizes they deserved, probably foremost amongst them Lise Meitner. While there of course is males that have been overlooked as well (you might for example think that the relativity theory should have earned A Einstein another prize), the overlooked women would have had an impact since females are such a small portion to start with.

    That would change the stats somewhat but not the over all picture. And anyhow we don't know wether the stats reflects inability or indifference. Personally I find a lot of the things girls do for fun to be rather, ehh well, incomprehensible. ;-)

  9. "The idea that you can create a fine brain to solve a difficult mathematical problem by washing someone's skull with some dumb one-sentence encouragement or discouragement is utterly absurd."

    Quite aside from the discussion of boys vs. girls is the fascinating finding that one can in fact strongly influence development of intelligence (and other traits) through "dumb one-sentence encouragement or discouragement":

    As for the present (okay, past) discussion, I'm confused as to why anyone would think that genes on the sex chromosome should be able to affect some physical traits (genitalia being the least controversial, but also musculature) but not others (development of the brain, for instance). Mind you, I'm not claiming that they _do_, but I don't see why such a hypothesis is prima facie absurd.

    I can only imagine it has something in common with the reason why "your eyes are blue" is a less angering statement than "you are dumb." Some phenotypic characteristics we hold nearer and dearer than others.

  10. Dear Aditya,

    I thought you would be able to deduce without a help from others that my important statement also implies that the work of a Ms Carol Dweck is an utterly absurd pseudoscience rather than a "fascinating finding".

    Discouragement may convince someone not to *do* something but it cannot make someone *unable* to do something.