Sexism

Last modified: Thu Apr 9 14:39:25 PDT

Note (9 April, 98): The following article has been written over several years, and some of the data is now a few years old. It was originally submitted to the Notices of the AMS, which declined to accept it unless I removed the anecdote concerning Jenny Harrison; since anecdotes are an important part of the subject, and are usually unwritten and hard to verify, I did not wish to discard this one. The Newsletter of the AWM did not have the space, (but suggested a short letter referring to this article), so electronic publication in this form seemed to be the best alternative.

I encourage replies, and will append them if the authors permit.

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SEXISM IN MATHEMATICS???

by Rob Kirby

"The persistence of rampant sexism seems almost unique to mathematics among scientific disciplines today.", [Paul Selvin, Science, 255, 1992, 13 March, page 1383].

This is the most extreme charge of sexism that I have run across. But lesser charges are common; for example, they are sprinkled through the pages of the Notices in September, 1991, and sometimes appear in issues of the Newsletter of the Association for Women in Mathematics. Often, it seems to me, an assumption of sexism in math stands behind calls for various forms of affirmative action.

It is the purpose of this essay to discuss the charge of sexism within the math community at the elite, research oriented institutions. The charge of sexism rests on the quality of evidence (or lack thereof) for sexism. I wrote in this Newsletter (vol 25, no 6, page 22) that "there is, in print, nothing remotely close to evidence or argument that women are discriminated against in the math community". Out of context, this statement looks a bit extreme, and it is not surprising that it generated opposition; in particular Mary Beth Ruskai wrote: "I'm not sure what standard Kirby is applying here -- mathematical proof? guilt beyond reasonable doubt? preponderance of evidence? a videotape of a meeting in which someone says, "we won't hire any women"?" (AWM Newsletter, vol 26, no 2).

To understand what is meant by evidence, it is useful to first discuss the evidence for the proposition that women can't do math as well as men; the standards for evidence in this example should be good standards for evidence of sexism in math.

There are a variety of scientific articles, some published in Science over the years, which describe experiments which purport to show that women's brains are biologically different than men's, or studies about left handedness and the degree of specialization of the two sides of the brain which have some relevance to gender, or tests of electronic activity in different parts of the brain when it is active in different tasks which are sometimes sex related, and so on.

More specific to math are the variety of tests such as SATs, GREs, or the Putnam Exam which often show a distinct difference in the performance of men and women.

And more pertinent yet is the fact that many more men than women have been performing at a high level in math throughout history; there are many ways to package such statistics, but men essentially always come out well ahead.

That's a sample of three different sorts of evidence; what do we think of it? The first category of biological articles tends to consist of articles which fade over time. It is important to remember that such research has many ugly episodes in its past, although that does not imply anything about current research. But my impression is that such research has not stood up well in the past, and given its tenuous connection with math, it should not be taken too seriously.

Tests are not all that well correlated with mathematical achievement, and certainly are affected by social factors, e.g. only a subset of math students are the kind that want to get heavily involved in Putnam type training and competition, and if women are less likely to be interested in such tests, is that good or bad? We would not want all bright math students to get seriously involved in the Putnam Exam.

And the greater achievements of men may be entirely due to social factors, particularly including the much more limited opportunities for women to do math in the past.

Furthermore, one has to ask what is meant by the phrase, "women can't do math as well as men". If it means that men do more math than women, then it is true and has as much value as saying more men play football. If it means that at birth, a girl on the average has a disadvantage over a boy at becoming a good mathematician, then it might mean that she is biologically disadvantaged or that she will be inevitably socialized in a way less conducive to math, or some combination of of the two. Most likely the phrase usually means that at any given time, speaking statistically, a woman who chooses to do math is less likely to do it as well as some statistical comparison group of men.

The phrase, with all its vagueness, clearly has some evidence (often uncertain or faulty) in its favor, and is a reasonable topic at least for the sophmore midnight bull session. However, I think that most of us believe it is not nearly strong enough for us to act on it; e.g. when considering candidates for admission, or an award, or a job, one has data about the candidates which far outweighs the sex of the candidate (when choosing tall people and picking between a 5'9" man and a 5'10" woman, one does not pick the man because men are on the average taller). And even more so, most of us are agreed that one should not make public policy (favoring men) on the basis of the evidence described above.

People who say that women can't do math as well as men are often called sexist, but it is worth remembering that some evidence exists and the topic is a legimate one, although Miss Manners might not endorse it.

To summarize, I think there is a consensus (which I fully agree with) that the evidence that "women can't do math as well as men" is sufficient to discuss the topic with care and sensitivity (but, usually, fruitlessly), but is not sufficient to show that it is true or to change one's behavior towards women, and is very clearly not sufficient to justify changes in public policy.

When I wrote the quote above: ". . nothing remotely close to evidence. . .", I had in mind that the evidence that does exist is much weaker than the evidence that women can't do math as well as men, which is evidence that we generally disregard to the extent that it is socially unacceptable to claim that women can't do math. On the other hand, it appears to be quite acceptable to write (not just say) that male mathematicans are sexist.

After this lengthy preamble, I hope that I have now outlined rough ground rules for a discussion of the charges of sexism in math. Evidence should be treated with the same respect or skepticism as similar evidence above, and one should be equally careful about advocating actions or public policy.

I will begin with some statistics.

1. The percentage of PhD's in the US who are female has climbed from 11% in 1974 to 20% in 1983 to the mid 20's in the 1990's.

2. Every four years a number of distinguished mathematicians are invited to speak at the International Congress of Mathematicians, from 12 to 20 as plenary (hour) speakers, and the remainder to speak (for 45 minutes) in one of the different (currently 18) fields of math. Here is a table of the women who have spoken at past Congresses, followed by the number of speakers of both types since 1954 (corrections, please, if I have missed someone).

Female speakers at ICMs:

1932 Emmy Noether (plenary).

1936 ---

1954 Mary L. Cartwright

1958 ---

1962 ---

1966 Ol'ga A. Ladyzhenskaya.

1970 Yvonne Choquet-Bruhat, Zofia Krygowska, Nina N. Ural'tseva.

1974 Jacqueline Lelong-Ferrand, Mary Ellen Rudin.

1978 ---

1983 Nancy Kopell, Ol'ga A. Ladyzhenskaya, Karen K. Uhlenbeck, Michelle Vergne.

1986 Izabella Grigorievna Bashmakova, Sun-Yung Alice Chang, Judith Grabiner, Caroline Series, Nina N. Ural'tseva.

1990 Karen K. Uhlenbeck (plenary), Lenore Blum, Shafi Goldwasser, Dusa McDuff, Colette Moeglin, Mary Rees, Eva Tardos.

1994 Ingrid Daubechies (plenary), Marina Ratner (plenary), Fan Chung, Deborah Hughes-Hallett, Frances Kirwan, Joyce MacLaughlin, R. Parimala, Karen Parshall, Bernadette Perrin-Riou, Claire Voisin, Lai-Sang Young.

Year 1954 1958 1962 1966 1970 1974 1978 1983 1986 1990

Speakers

Plenary 20 19 16 17 16 17 17 12 16 15

45 minute 42 40 57 54 240 140 119 123 148 139

3. Each year the NSF awards over 1000 grants of the traditional type which include some summer salary, travel, perhaps support for graduate students, and perhaps support for computing or other items. In 1992, about 7 to 9% of these NSF grants (in pure math) went to women and this did not vary too much according to field (there were roughly 70 women and 800 men). I obtained these percentages just by counting names in the Summary of Awards in the Division of Mathematical Sciences of the NSF (this document has not been published since 1992, and a modest attempt to obtain better and more recent figures from the NSF was not successful).

4. Each year the members of the AMS vote to elect 5 members of the Council of the AMS for 3-year terms. The nominating committee nominates about 7 candidates, waits to see if more are nominated by petition, and then, if necessary, increases the number of candidates to a slate of 10. In the 24 years from 1971 to 1994, there has always been one female candidate, but never more than three, with a total of 37 women in 20 years (statistics for four years between 1971 and 1994 seem to be missing from the AMS archives).

In 14 out of the 21 years, a woman has been the top vote getter. The average vote for a woman has been 1856 and the average for a man has been 1448, a 400 vote differential. Moreover, women were doing approximately as well over twenty years ago as they have been recently. The number of voters ranges between 2000 and 4000, so a relatively small fraction of the AMS membership votes. Staff at the AMS have kindly provided me with these statistics.

5. Here are some statistics at various levels at a particular university that I am familiar with, the University of California at Berkeley.

A. Over the past 25 years, Berkeley has made about 10% of its tenure or tenure track offers to women, with a higher acceptance rate by women.

B. It is not so easy to determine what fraction of post-doc type offers have gone to women because of the changing nature of hiring at Berkeley over the years. But beginning in 1972, assistant professorships became very scarce (the math department lost 13 FTE due to budget cuts in 1972), and all post doc hiring was done with 2-year "lectureships". Procedures changed little until 1984 when the lecturers unionized and negotiated a 6 course per year standard. Thus the math department, which could not be competitive with that teaching load, had to hire under a variety of different titles, and statistics are harder to sort out. But from 1972-73 to 1983-84, Berkeley hired, as lecturer, 51 men and 20 women, so about 28% were women.

C. For nearly 30 years, the department has set aside 10% of its admissions (to graduate school) and 10% of its teaching assistantships for members of underrepresented groups. (Note that everyone is eligible for the other 90%, and then the 10% goes to those from the underrepresented groups who were not already admitted or supported.) Incidentally, this procedure will not continue in this form, for the Regents of UC and the statewide Proposition 209 have banned such preferences in admissions (it may be that this particular form of affirmative action was already disallowed by the Bakke case).

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If a statistical charge of sexism is going to be made, it may be easist to make in the two "contests", invitations to the ICM and NSF summer grants, which are easier to examine than other examples such as hiring in individual departments. In the case of the ICM, the pool of candidates is clear, namely everyone. The prize, an invitation to speak, is also clear. The criteria for selection are not so clear, in that selection is usually based on dramatic work in recent years, but lifetime work, lecturing ability, mathematical and geographical diversity can also be considered; nonetheless the criteria are clearer than in many other competitions.

There are about 18 committees in the different specialties of math who are charged with coming up with a short list of names to be invited to speak at an ICM. I was on the committee in topology in 1986, and I imagine my experience was typical. We produced a list of 7 names with two alternates. I had to defer to the other members of the committee in those parts of topology, i.e. algebraic topology, where I am not competent to make subtle distinctions between the best people. Even in my field, geometric topology, there was debate and I was not entirely convinced of all of my choices. In other words, it was not easy to choose people even in an area as small as, say, 2% of mathematics.

I can easily imagine that five disjoint committees in topology, each picking 7 names, would list at least 15 different people. Would the other eight (other than the actual 7 picked) have grounds for complaining of unfairness? Only if they had a strong mathematical case that the actual committee had erred, and who is to judge the merits of this case if not the actual committee who was picked for its wisdom in these matters. Note that the arguments here are usually over male candidates and have nothing to do with sexism.

So although the ICM appears to be a good example to examine for sexism, it would not be easy to make a case unless the sexism is widespread. Probably the charge of sexism at ICMs really began, and rests, with the examples of ICMs before 1980 which had so few women speakers; it would have seemed inconceivable that so few women deserved an invitation.

Similar comments can be made about NSF grants. The heads of the programs sometimes compare notes about the level of the applicants who either just do, or just don't, get a grant. I'd guess that care was taken that women were being fairly treated at this level. The place for bias to creep in is with the peer reviews from the mostly male peers. But how can one test for subtle bias by these peers? Who is better better informed about the quality of applicants than they? It is worth noting that the NSF (and other government agencies) has brought great pressure to bear on institutions like MSRI to enhance the opportunities for women, and it would be ironic if they themselves did not treat women fairly when it came to summer NSF grants.

But some will claim that the low percentage of NSF grants as compared to the number of PhD's is a prima facie case of discrimination, that the truth is somewhere in between, that the percentage of NSF grants that would go to women if there was no sexism would be somewhere in the teens. Two points: suppose the number of NSF grants going to women in a bias free world would increase by 50%, say from 8% to 12%. This would mean that every year, in pure math, around 35 women would be denied NSF grants that they should get. Is this plausible? Would we not have been noticing this? And if there was enough bias to cause this, how is such bias consistent with other statistics (such as voting or hiring at Berkeley or some other schools)which would still be favorable to women even if 12% accurately described the research by women? Second, one has to be careful with claiming that 7-9% must imply sexism; for example, the fact that about 75% of the National Basketball Association players are black does not mean that the NBA discriminates against whites (if anything, the opposite is mildly true).

I think that if a case is going to be made for more than occasional incidents of bias, then the ICM or NSF is the place to look; the game is quite well defined and is not tainted by harder to measure questions of collegiality and personality that can affect hiring in a department. Anyone wishing to claim sexism in math should first put themselves to the test of making an argument of sexism at the ICM or NSF.

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The various elections to positions in the AMS provide further tests for bias. In an election to the Council, for example, a voter has a secret ballot in which to vent whatever noble or ignoble feelings he/she has.

What can one conclude from these voting records? It may be that in the minds of some voters, women represented change and this is what they wanted to vote for; in other words they may have been gender blind but pro change so they voted more heavily for women. It seems likely that the extra vote for women was partially due to the desire of women to see more women on the Council; in fact it is conceivable that men were slightly biased against women but that this bias was far outweighed by women voting for women (which we do not call bias). Still, these statistics give absolutely no support to the notion that males are sexist, at least those who vote (who are also more likely to be active in organizing and running the math community).

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The statistics in 5B on the hiring of lecturers at Berkeley in 1972-1984 are interesting. During those years, the average percentage of US PhD's (citizens) that were women was 14.5%. Thus, Berkeley was hiring at a rate twice that of the pool of women with PhD's. In fact, since the percentage of women who get NSF grants is half to a third of the percentage that gets PhDs, and since Berkeley is presumably trying to hire at the level of the very best PhD's (i.e. those that have a good chance at getting NSF grants), it looks like Berkeley was hiring women as lecturers at a rate three (conceivably four) times the rate at which they would have hired women purely on merit.

However it may be that Berkeley received unusually many applications and acceptances from particularly strong women, and that Berkeley was simply hiring on merit. To get some idea of the likelihood of this explanation, I looked up the number of publications for each of the lecturers in the MathSci database. The average number of papers for the men was 26 and the average for the women was 12. 36 of the 51 men averaged at least one paper per year, which is some sort of indication of a successful career, whereas 5 of the 20 women averaged one per year. Several women and men left academic life and are known to me to be quite successful in other jobs not involving publication of math papers. Six women and four men essentially did not publish beyond their theses, but given the bad job market of the 70's, these may have been highly talented people who saw better opportunities elsewhere. Nonetheless, these numbers (imperfect as they are) support the notion that Berkeley was eager to hire women and not hiring purely on mathematical merit.

My perception of the Berkeley department (during the 70's and 80's) was that most of my colleagues were strongly in favor of doing well by women. The statistic in 5B does not surprise me although I had never looked up the data. But apparently no one had any idea that the percentage, 28%, was so high. The hiring was done by many different hiring committees under at least 6 different chairs. More to the point, however, is that the critics of Berkeley never bothered to check up on one of the simplest and most useful of statistics, nor, curiously, did the defenders of Berkeley.

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Much of the "evidence" about sexism is anecdotal: in his Science article, Selvin gives several examples; the Notices issue on Women in Math (volume 38, Sept. 1991) is liberally sprinkled with examples; they occur rather rarely in the Newsletter of the AWM (which seems to be quite careful about such matters). But most charges of discrimination against individuals only exist at the oral level; a written charge is usually not made publicly, so it is hard to pin down and evaluate the charge.

So I'd like to discuss four examples of an anecdotal sort which have appeared in print. The first and most famous concerns Julia Robinson. The San Francisco Chronicle on 10 Dec. 1987 carried a story including the following paragraphs:

"The late Julia Robinson . . was denied a post as professor at Berkeley in 1948 because her husband was already a mathematics professor there. . . .

Even after her husband retired in 1973, she still did not get the appointment--until she was elected the first woman in the National Academy of Sciences in 1976."

In a later editorial, (8 Nov. 1993), referring to discrimination, almost exactly the same words appeared. This time Constance Reid, Robinson's sister and noted author, wrote to the Chronicle explaining that because of her health Robinson had never wanted a position at Berkeley, and that the story that she was "denied" a position "is simply not true" (22 Nov., 1993).

The Chronicle writer then wrote to Reid, explaining:

"Since no one wrote or called to correct the original article, I have had the unwarranted belief for the last several years that what I wrote was correct, and I felt free to draw upon it in the recent editorial. As nearly as I can recall, my source was one of your sister's former fellow woman faculty members."

This story about Julia Robinson, often in more accusatory terms, has appeared in many places, but the version above is interesting because the nature of the source is explicit and implies that the writer could put a damaging story in the paper on the basis of only one source without checking its veracity. It also shows the importance of diligence in correcting the media.

A second example begins with Paul Selvin again in the same Science article referred to at the start of this essay. Selvin presents a bar graph with a caption stating: "In a 1983 study both male and female mathematicians rated math papers they thought were written by a man ("John T. McKay") more highly than those they thought were written by a woman ("Joan T. McKay")."

However, Selvin's caption was far wrong, and Science printed a correction (volume 256, 19 June 1992, page 1615) which I partially quote: ". . a caption . .. was incorrect. The data did not show how male and female mathematicians evaluated mathematical articles. In fact the articles in the study that was discussed [M. A. Palardi and W. D. Bauer, Sex Roles 9, 787 (1983)] were about politics, the psychology of women or education; the subjects were not mathematicians but male and female college students." One wonders how Selvin could have obtained a photo of the bar graph and then gotten the caption so wrong.

Unfortunately, the example doesn't end there because the Economist in June 1996 in an article in their Science and Technology section about women and science repeated Selvin's caption. When I emailed them to inform them of the mistake, they apologized, but declined to print a correction. So one again wonders how many readers of these serious and widely read magazines, not knowing much about mathematicians and their ways, carry around the factoid that it has been demonstrated that they discriminate against women.

A third example is taken from a letter by Lenore Blum that appeared in the Berkeley Independent-Gazette on 25 April, 1975. This letter might have been long forgotten except that it has recently been reprinted on page 72 of the delightful book, "Julia", by her sister Constance Reid. I suspect this will be an inspiring gift to many girls, who, if they carefully read page 72, will come away with a poor impression of Berkeley's math dept.

For Blum writes of the department's "long history of contempt for, and discrimination of, (American) women mathematicians", and backs her remarks with four examples. One concerns Julia Robinson and is discussed above. Of the remaining three examples, the one with the most substance is that "during last year's procedures, women were asked to apply for positions that the Math Department had already committed (in writing) to two men" (this refers to spring 1974).

Blum's statement is accurate, but there is more to the story. Berkeley's math dept conducted a reasonable search to fill two assistant professorships, and ended up choosing two males, X and Y. They got letters from the dept offering them jobs, but were told that the offers were not binding until they went through the administration, and that the administration rarely if ever overruled the department. X and Y accepted and turned down at least one other job.

Then a campus committee in charge of affirmative action decided that the search for women and minorities had not been adequate, and that the offers could not be made unless a further search was done. The dept hiring committee had done its usual search, and in addition the dept had a committee W whose sole purpose was to search for women. But presumably the campus committee did not realize how unlikely it was that a woman as strong as X and Y would not have been aware that she could apply to Berkeley, or that such a woman was not already known to Berkeley mathematicians.

The additional search went forward, as Blum charges. If a woman as strong as X and Y had applied, then it is almost certain that the administration would have given the dept an extra position for her, rather than refusing to approve the dept's offers to X and Y. This actually happened a year later in somewhat different circumstances.

There is no evidence here of any "contempt for, or discrimination" against women by the math dept. The extra search essentially happened because of an overly zealous campus committee, and could have resulted in a woman being hired.

A fourth example is on the one hand rather trivial, yet is illustrative, and keeps surfacing. The 7 July 1996 edition of the San Jose Mercury News carried a front page article about obtaining court records about various cases of alleged sex discrimination at the University of California. The article began with the "most dramatic story" of Jenny Harrison and included the following: "Another professor, Robion Kirby, said: 'Women belong at home in the kitchen caring for their families,' Harrison claimed in court papers. Kirby has denied this remark."

Harrison's accusation has appeared in several California newspapers, is in the Congressional Record, and has been made in various forums (I have heard audiences react audibly to this charge). It resonates with many non-mathematicians, conforming to their beliefs about how women are treated. When Harrison filed suit against Berkeley in 1988, I obtained a summary of her charges against me. The first is:

"In October 1977 Harrison had a long conversation with you. During that conversation you explained that you were a libertarian and believed that women belonged at home, in the kitchen, caring for their family."

Normally, examples such as these four are based on something, and in this case, I recall a conversation which took place in my car as a few of us returned from a topology seminar. I could have said what I believed then and now, that raising children is an admirable occupation and Americans should put more time and effort into this (I was a single parent for a number of years). I might have said that, given society as it is, more of the task of raising children would fall upon women. But libertarians are not so likely to tell people where they belong. I can only guess that Harrison did not understand my remarks too well, and that over 11 years, they evolved into the allegations she seems to firmly believe. But I do not and did not believe what she attributes to me.

This illustrates the point that it is remarkably hard to get accurate quotes. How well do you remember a conversation 11 years ago? or even yesterday? Do you make a habit of immediately writing down things you hear, as reporters often do (and look how often they get things wrong). (The Associated Press put out a version based on the San Jose Mercury News article, and turned the quote into ". . Kirby told his class that 'women belong at home . . .'", an error which adds weight to the accusation because it implies there are witnesses who might confirm it; the error arose because the AP picked out the phrase "told his class" from a sentence about someone else later in the same paragraph.)

As scientists, we would naturally check with the speaker to see if the speaker had indeed said (or meant to say) what we thought, but it is awkward and difficult to check such things without being confrontory. So, it is not surprising that these charges of sexist statements are rarely verified and hence rarely to be relied on. And it is unfortunate that we usually do not act like scientists when dealing with issues of sexism.

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What about all the books and articles written about women in science? I have browsed through and read parts of many of these. Almost all of them are about science with math forming an unclear and small subset of the cases. These books are sometimes interesting, but I believe the evidence they offer is far weaker than the kinds of evidence which we disregard when talking about "women can't do math", as I discussed earlier in this essay. I have two comments which apply to many of these works.

Showing sexism in science does not show it in math, for the cultures of math and of the other sciences, which consist mostly of experimental laboratory work done in teams under some supervision from a senior scientist, are surprisingly different. (For example, the fraud one has been hearing about in science has occurred mainly in labs in the biological sciences, and there is far less opportunity for such fraud in math.)

Second, some articles about sexism in math seem to rely on the assumption that the male and female PhD's from elite university X are statistically equivalent. This may be true, but there is reason to be suspicious and not allow such an assumption to be crucial in an argument for sexism. If one considers the strongest half of the PhD's from X, one cannot assume that the fraction of women is the same as for the weaker half. Each time one restricts the pool of mathematicians to a more selective group (e.g. high school grads to college grads to PhD's to NSF grantees to ICM speakers) the percentage of women falls off. This phenomenon may be happening with X's PhD's.

Is there any study in print which offers good evidence for sexism in math, given the standards that I have set out in this article? Those who have been claiming sexism should have some ready examples.

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I've been told that the best evidence for discrimination against women is the large number of mathematicians that believe it is so, that with all that smoke, there must be fire. I am skeptical of this because the cases that I have been able to investigate, some of them famous, have turned out to be utterly without merit. Our society is focused towards paying attention to (and believing??) charges of sexism against women, (but not towards examples of men treating men badly or treating women particularly well). And one should remember the amazing capacity for humans to believe things for which there is little or no evidence.

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My view of the math profession is that it is generally fair. We have pretty good habits because it is relatively easy to measure our worth as research mathematicians, and because it is hard to make unfairness pay off. For example, we have little temptation to fudge our experiments to make the data look better for we rarely experiment; and we don't know how to fudge a proof to make it better. It appears to me that mathematicians would genuinely prefer to see women do better in math; certainly the establishment pays lip service to this ideal (see their election statements) and why should we be skeptical? Our administrations and the NSF, among others, have been exerting considerable pressure for more action to increase the participation by women in math.

Yet my view is, apparently, a minority one. The view that we often hear is represented by Cora Sadosky in the January, 1994, AWM Newsletter where she writes: . . . I learned of a very able . . woman . . whose tenure was denied. As in many other cases, a perception of gender discrimination is not an overreaction. What is to be done? How can we force fairness in the tenure and promotion process?"

I believe such statements are unfair, and the only way I know to deal with them is to present the kind of arguments and statistics that I've given above. I apologize for them, for they are likely to be disheartening to all those who wish to encourage women. In my view, the smaller number of women in math is not due to discrimination by men nor to any inherent inferiority in women, but rather is due to the simple fact that more men than women choose to enter mathematics. I agree that these choices are due to many factors, including the environment that girls and boys grow up in, but little of this has anything to do with our mathematical community, which strikes me as fair as any I know.