Tanya Khovanova's Math Blog

China Girls Math Olympiad is becoming an international math Olympiad for girls. When I first heard about this competition I felt very sad. I need to explain myself here.

For many years I felt very proud that math Olympiads do not separate the genders. Most Olympic sports, like running or swimming, have separate competitions for men and women. I felt that joint competitions for math demonstrated the spirit of equality in our math community. I felt that insofar as gender didn’t matter, mathematics was more democratic than other sports.

At the same time I do understand how people might assume for the following reasons that a math competition among only girls would be useful:

• They promote the idea of math to girls.
• They can help girls who are into math to feel less lonely.
• They generate additional resources and training for girls.
• They might be less stressful for some girls, than mixed math competitions.
• They help promote the image of female mathematicians to society
• They provide further opportunities for girls to earn prizes and improve their resumes.

See also the article: First US Team to Compete in the China Girls Mathematical Olympiad.

On the other hand, this development scares me. If we have a separate girls Olympiad, will that soon lead us to have two Olympiads, one for boys and one for girls? Two separate Olympiads would be a defeat for women mathematicians. Or, maybe I shouldn’t be scared. The percentage of girls at the most prestigious mathematics competition, the International Mathematical Olympiad, is so small that it can be viewed as virtually boys-only.

Mathematics is becoming similar to chess. There is a World Chess Championship where both men and women are allowed to compete, and there is a separate Women’s World Chess Championship. The interesting part is that Judith Polgar, by far the strongest female chess player in history, never competed in the Women’s World Chess Championship. I suspect that I understand Judith. She probably feels that women-only competitions diminish her, or that chess is about chess, not about gender. In any case, I hope that one day the separate girls Olympiad will not be needed.

I am looking forward to the 2009 Women and Math program at Princeton. The irony is that I lived in Princeton for seven years and the only time I visited this program was for the lecture course on wavelets by Ingrid Daubechies.

I felt that mathematics should be genderless and pure; that the only basis for a program should be mathematics itself. I tried to ignore the problems of women mathematicians by pretending they didn’t exist. By the time I realized that I might very well love to hang out with a large group of female mathematicians, I left Princeton.

Can you imagine how glad I was when I got a call inviting me to join the organizing committee for the Women and Mathematics Program last year? I was so eager that I arranged a math party at the Program and gave my own talk about Topology in Art.

What can I tell you? I loved the program. For the first time in my life I didn’t feel like a loner, but rather that I belonged to a group. I also felt envious, because when I was a student we didn’t have anything like this in Russia.

I am going to be on the program this year too. The subject is Geometric PDE. I am so looking forward to it that I’m already planning another math party.

When you talk over the phone with an adult stranger, you can generally determine if this person is male or female. From this, I conclude that the voice characteristics are often noticeably different for males and females. There are many other characteristics that have a different distribution by gender — for example, height.

My question is: “Can we find a trait such that the distributions are the same for both genders?”

Trying to find the answer, I remembered what we learned in high school about the genetics of eye color. I checked the Internet on the subject and discovered that the story is somewhat more complicated than what I studied 30 years ago, but still we can say that eye color is defined by several genes, which are located on non-sex chromosomes. That means, your eye color depends on the genes your parents have and doesn’t depend on your sex. A boy and a girl from the same parents have the same chances for any particular eye color.

Since eye coloring has nothing to do with gender, women and men are equal in the eyes of eye colors.

Does that mean that if we check the distribution of eye color for the world population, the distribution histogram will be the same for men and women? That sounds like a logical conclusion, right? I would argue that this is not necessarily the case.

Let me remind you that the distribution of eye color depends on the country. China has an unprecedented gender imbalance, with 6% more men than women in its population. As the eye color of Chinese people is mostly dark brown, this creates an extra pool for a randomly chosen man in the world to have a darker eye color than for a randomly chosen woman. If we exclude China from consideration, we can still have different distributions. For example, in Russia the life expectancy for women is 15 years longer than the life expectancy for men. Consequently, Russia has 14% less men than women, while globally the male/female sex ratio is 1.01. Therefore, eye colors common in Russia will contribute to female eye colors more than those of male.

What if we consider only one country? Let us look at the US. Immigrants to the US are mostly males. If the distribution of eye color for immigrants is different than the distribution for non-immigrants, then male immigrants contribute more to the eye color distribution than female immigrants.

There are so many factors impacting eye color distribution, that it isn’t clear whether it’s possible to find a group of people other than siblings in which the distribution of eye color would be the same for women and men.

We see that eye color distribution, which theoretically doesn’t depend on gender, when measured in a large population can produce different distributions for men and women.

Recently I wrote a theoretical essay titled “Math Career Predictor”, where I assumed that the distribution of math ability is different for men and women. In reality, there is no good way to measure math ability, hence we do not have enough data to draw a complete picture. For the purposes of this discussion let us assume that we can measure the math ability and that Nature is fair and gave girls and boys the same math ability. My example with eye color shows that if we start measuring we might still see different distributions in math ability in boys and girls.

My conclusion is that if we measure some ability and the distribution is different for boys and girls, or for any other groups for that matter, we can’t just conclude that boys and girls are different in that ability. For some distributions, like voice, we probably can prove that the difference is significant, but for other characteristics, different distribution graphs are not enough; we need to understand the bigger picture before drawing conclusions.

In this essay I would like to explain why I am not yet a professor of mathematics.

Today at 49 I am still in search of my dream job. My gender is not the main reason that I don’t have an academic position or another job I like. My biggest problem is myself. My low self-esteem and my over-emotional reactions in the past were the things that most affected my career.

I remember how I came to Israel Gelfand’s seminar in Moscow when I was 15. He told me that I was too old to start serious mathematics, but that he would give me a chance. He said that at first I might not understand a thing at his seminar, but that every good student of his comes to understand everything in a year and a half. The year and a half passed and I wasn’t even close to understanding everything. Because of this I was devastated for a long time.

I had always had problems with my self-esteem and being a student of Gelfand just added to them. My emotional reactions, while they impacted my work in mathematics, were not exclusively related to mathematics. When my second divorce started, not only did I drop my research, I quit functioning in many other capacities for two years.

I was extremely shy in my early teenage years. By working with myself, I overcame it. When I moved to the US, my shyness came back in a strange way. I was fine with Russians, but behaved like my teenage self with Americans. For two years of my NSF postdoc at MIT, I never initiated a conversation with a non-Russian.

For the second time I overcame my shyness. Now, if you met me in person, you wouldn’t believe that I was ever shy.

I became much happier in the US, than I ever was in Russia, but still my emotions were interfering with my research. Because it was so difficult to find an academic job here, I felt tremendous pressure every time I sat down with a piece of paper to work on my research. My mind would start flying around in panic at the thought that I wouldn’t find a job, instead of thinking about quantum groups.

Over all, I think that my inability to control my emotions, together with my low self-esteem, might have impacted my career much more than the fact that I am a woman, per se. Being a mathematician is not easy; being a female mathematician is even more difficult. Still, in my own life, I know I can only blame myself.

The good news is that I have changed a lot, after many years of self-repair. This is why I have made the risky decision, at the age of 49, to try to get back to academia. And this time I have a great supporter — my new self.

This story happened at a colloquium that was conducted during the Women and Math program at Princeton last year. The room was full of women waiting for the colloquium to start. A young man appeared at the door. He looked around in complete surprise. I watched the fear fade from his face when he must have decided that he had the wrong room. He disappeared, but reappeared at the door very soon. He had obviously checked the schedule and had realized that, in fact, he had come to the right room in the first place. His face started changing colors. He was terrified. A few minutes later, he left.

I sat there thinking: women have to deal with this type of situation every week. He could afford to skip just one lecture, but if a girl wants to do math she has to be courageous almost every seminar. I mentally applauded the girls around me for being that courageous.

Wait a minute! I am a woman myself. I went to seminars where I was the only girl hundreds of times. How did I feel? Actually I think my mind never registered that I was the only girl. I never cared. The first time I really thought that the gender of people in a room might be an issue was during that colloquium last year.

I started wondering why it had never bothered me. Could it be that the Soviets did a good job of teaching me not to pay attention to people’s gender? Could be. But wondering back in time, I remembered something else too.

When I was a child I wasn’t a girly girl. I was not interested in dolls; I preferred cars. I didn’t play house or doctor; I played war. To tell the whole truth, I actually did have a doll that I loved, but I never played with it. I liked having it. The doll was a gift from my father’s second wife and it was way beyond my mother’s price range. I think I had an admiration for the quality and the beauty of this toy.

So, while appreciating the courage that the other girls might have needed to do math, I was sitting there pondering my own indifference to the gender of people at seminars and my relative comfort with large groups of men. But this comfort had its own price. I felt comfortable with the group I wasn’t a part of, while I felt different from the group I was a part of. My price of being comfortable at math seminars was loneliness.

Suppose you are a woman living in the US and you would like to be a mathematician and work in academia. Suppose also that you would like to have children and spend some time with them. Let us say that you want two or three children and you would like to be with them at home for at least their first two or three years. That is, in total you need to devote 5 or 6 years to this endeavor. When would be a good time for you to have children? American women mathematicians are commonly advised to postpone having children until tenure.

Let us look at the situation more closely. Is it a good idea to have children before you earn your PhD? There are many non-mathematical reasons not to have children too early:

• You do not know yourself well enough to be able to choose a good husband.
• You might not have met the man you love yet (see my essay The Mathematical Path to the Right Husband).
• You might not have enough money to support yourself and your children.
• If you have not yet definitely decided to become a professional mathematician, you might still be clueless that your future career contradicts your plans for children.

There are also mathematical reasons not to have children too early. My former adviser Israel Gelfand liked to tell everyone that mathematicians generate all their best ideas before the age of 23. His views may be extreme. After all, we know of mathematicians who made great discoveries later in life, but it could be that they did this using their mathematical wisdom rather than the processing power of their brains. It may well be beneficial to start your first research early in life. I am not sure about math creativity, but I swear that it is much more difficult to learn new things as I age.

In addition, you might need to relocate frequently to maintain your math career and it is more difficult to do that with children. It’s hard on the children too. Besides, having children early means that while working on your tenure, you will be distracted by your kids and their problems, leaving you less time for your research.

You might think that the closer you get to your PhD, the more the situation improves. In reality, there is a very important reason not to have children while you are in grad school. When you start working on your first topic, it is very important not to be interrupted. If you take a big break someone might finish what you started. When you come back, your topic might be resolved and you would have to start all over again.

The situation after graduate school is even worse. When you apply for jobs, employers are likely to count how many papers you have published per year after your PhD. So you need that number to be high. You can’t afford to dilute your paper count per year by a several-year break. Besides, if you have an interruption in your research it might be considered as a weakness and you might lose in comparison with other applicants. Let us break down the time between PhD and tenure into three periods: postdoc, visiting positions, and tenure track. For each period there are extra reasons not to have kids:

• Right after the PhD you begin looking for postdoc positions. All of them have formal time constraints: you are not allowed to apply for these positions if you are more than 3 (or maybe 5) years past your PhD. To have children during the postdoc period is really a bad idea.
• After the postdoc you might have several visiting positions. They usually require yearly relocations and you need to produce something new every year, so that your current curriculum vitae is different from the one you sent to the same place a year ago. At this stage of your life you are much better off without a husband — forget about children.
• When you have tenure track, you are so close to tenure that you might not want to put your dream at risk after so many years of sacrifices.

For all of these reasons, advice to wait until tenure makes sense. There is one big problem with it though: you usually get tenure in your late thirties. It might be too late for children. You might not want to risk that.

You can always compromise by having one child instead of three. Or you can suppress your desire to spend a lot of time with your children by having hired help, which means that you will miss a good deal of your child’s development, and your child will miss a lot of your love. You can compromise your academic goals by taking a more stable, but less research-oriented, technical job in industry. Or you might get lucky and marry a househusband.

In short, a math career is very kids-unfriendly — there is no right time. If you’re a woman mathematician who wants to spend time with your kids, prepare for pain and disappointment.

But here is an unconventional idea you might consider.

After you finish working on your PhD, postpone your actual defense by 5 years, and have your kids in between. This way all your PhD results will be published and no one can interfere with them. At the same time, the clock that counts your publications per year after your PhD will start 5 years later.

My idea is not a good solution — you will still have many problems — but it might be better than waiting ’till tenure. I do wish there were a better way.

I am interested in a career in mathematics. How hard is it to be a woman mathematician?

Let us look at some numbers from the American Mathematical Society Survey Reports for the year 2005:

• There were 40% of women among graduating math majors.
• There were 30% of women among Math PhDs granted.
• There were 11% of women among full-time tenured or tenure-track positions.

You can’t just say that women do not like math — 40% of those choosing math as a major is quite a large number, after all.

On the other hand, the downward trend of these percentages is striking. If women’s opportunities and abilities are the same as men’s, these percentages should grow with every age step, since, as we know, the percentage of women in the population increases with age due to men dying earlier.

But the numbers go down and very fast. There are many potential explanations for this, but today we’re going to look at one of them:

Women have less ability for high-level mathematics.

Was Larry Summers right when in his speech that cost him his Harvard presidency he compared math ability to height and to the propensity for criminality, and suggested that the distribution, especially standard deviation, of math ability differs for men and women?

To answer this question, I wanted to find some other data that correlates gender with math abilities. I took the results of the American Mathematical Competitions (AMC 12) for the year 2008. Among 120,000 students who participated, 43% were females. Here are some results:

• Among students scoring 72 points or higher there were 40% of girls.
• Among students scoring 98 points or higher there were 30% of girls.
• Among students scoring 134.5 points or higher there were 11% of girls.

This picture is similar to that of the academic career: the closer you climb to the top, the smaller percentage of girls you see there. Of course, winning a competition is very different from getting tenure. People who win competitions are smart and competitive — smart and competitive enough to go for money, rather than academia. On the other hand, people who are interested in mathematics often are not interested in anything else. Why would they waste their time in competitions when the Riemann Hypothesis is still waiting to be solved?

But still, both achieving tenure and winning math competitions represent mathematical ability in some sense. If Larry Summers was right and the distribution of math ability is different among males and females, then by looking around you at the percentage of females at your level, you should be able to assess how close you are to the top of the math field.

I propose the following math career predictor: Take your results in AMC 12. If among kids who did better than you, the percentage of girls is more than 11%, you do not have a chance at tenure. If the percentage of girls is more than 30%, do not waste your time working on a math PhD. If the percentage of girls is more than 40% maybe math majoring is not for you.

I hate my math career predictor. I hate it not only because it has so many flaws that it might just deserve the Ig Nobel Prize, but because it doesn’t take people’s effort into account. You really have to work very hard to be a math professor, whether you were a winner or a loser in math competitions.

You might ask why I created a math career predictor that is so flawed. My mathematician friends, those who are more honest than polite, tell me that I have no chance at getting back to academia. On the other hand, I had the second best result at the 1976 IMO, which means I have the ability. My predictor may be my only hope.

Ken Fan is running a math club for girls in Cambridge, MA, called “Girls’ Angle”. When I heard about it, my first reaction was surprise. When I was a girl, I would never have been interested in a girls-only math club.

Am I prejudiced? When I was growing up in Russia, there were not very many girls who were really good at mathematics. I would have expected that a girls’ math club would be less challenging than just a math club.

What if someone organized a boys-only math club? I would have been furious. I would have felt it was discriminatory. Should I then feel an equivalent fury about the girls’ club? But I do not feel furious, and I wonder why. Is it because deep down I think no boy would bother being interested in joining a girls’ math club? Is it because I still think a girls’ club would be weaker than a general club? I do not know.

At the same time I agree with Ken, for there are a variety of reasons why girls might prefer a girls-only club. For example, shy girls might feel more comfortable with girls or some girls might feel better able to concentrate without the distractions of boys. In some cases, the parents might have made the decision.

Obviously, since the club has students, there is a demand for it. If there is a demand, there should be a supply. I will support anything that works and helps improve American math education. I even volunteered to give a guest lecture at Girls’ Angle.

Was my lecture at the girls’ club different from my other lectures? Yes, in a way. I asked the girls to help me to finish a sequence. I started writing 1, 1, 2, 3, 5, 8, on the board and no one was shouting the next number. In great surprise I turned back to face the class and saw a forest of raised hands. They patiently waited for my permission to speak. Yes, it felt different.

I decided to check the pricing. It appears that the girls club is twice as cheap as other math clubs, like The Math Circle or S.M.Art School. I can’t help but wonder if the girls are signing up at the Girls’ Angle not because they want to study in the girls-only group, but because it is cheaper.

I am glad that Ken Fan is good at finding sponsors and that there are so many people sympathizing with his cause. However, this situation does seem unfair to boys. Should I be furious that boys are not allowed in this very affordable math club? I do not feel furious, but I decided not to give any more lectures at Girls’ Angle for free. At least not until I give a free lecture to a mixed-gender math club. I want to be fair.