MacTutor
Sofia Kovalevskaya
Tom Burslem
Appendix
An analysis of some of the problems faced by Sofia, comparing them to the problems faced by other female mathematicians, both past and present.
The very first sign of any problems to come in Sofia's life was her parents' disappointment at the birth of a girl. In many countries around the world it is still the case that a son is preferred, and a daughter will often live her life being resented by her parents. There are stories of baby daughters being killed where they are not considered as valuable as boys. There are obviously many different cultures around the world and women have different roles in them. In many parts of the world women are forced to work harder than men, receive less schooling than men and are even denied equal access to medical care. Worldwide approximately 500 million children start primary school, but around 100 million of these drop out before completing four years and of this 100 million around two thirds are girls.
A problem that Sofia encountered as a girl was common at the time and many would argue is still evident today. That is the influence of society on young girls, the idea that they should be pretty and graceful above all else, with the ultimate goal of marriage in mind. This is illustrated by Sofia's rejection by her parents as a young girl when she did not match up to her elegant elder sister, and was not presented to guests as her sister was. It could be argued that society still plays a large part in pushing girls towards certain stereotypes. Toys marketed towards girls are usually things like dolls that "cry and wet" or household items or make up sets etc. Do these toys point young girls towards the roles of mother, housemaker or beauty? Or is it simply that these are the toys that young girls prefer and we are reading far too much into it all?
In the media, women are generally portrayed in one of three categories: mother, sex object or girl. Certainly the perceived "nerdiness" of mathematics would dissuade a young girl under pressure to appear attractive and ultimately find a husband, from studying mathematics. But then it could easily be argued that such an image might put anyone off applying themself to the subject. Is there any evidence that women are put off studying maths at school or university level? Looking at some figures you could easily believe that too much is made of any gender gap. The percentage of women receiving bachelors in mathematics in America has steadily risen since the sexual revolution in the sixties, from 38.2% in 1971 to 46.5% by 1988. However looking at the percentage of doctorates in mathematical sciences rewarded to women we see a clear gap. This figure rose from 7.8% in 1971 to just 16.2% by 1988, and if we look to Europe the indicators are slightly more worrying. Of students of maths and computer science in higher education in 1994 only around 26% of them were women . We cannot directly compare this figure to those recorded in America as it lumps together all forms of higher education and also includes computer science. There were also some impressive anomalies, in Italy and Portugal the figure approached 50%, whereas in Luxembourg and the Netherlands it was only 10%. But the survey certainly gives us a clear indication of quite a large gap between men and women in mathematics. Is this evidence that women are dissuaded from studying mathematics though?
In the past there have certainly been many cases of women who really did want to study mathematics but then found that they were met by many obstacles. There are examples of this in the life of Sofia Kovalevskaya, her father forbidding her to study, her inability to travel without a chaperone and the universities that rejected her, all on account of her being a woman. This is also illustrated in the lives of many other female mathematicians; Sophie Germain (1776-1831) is one such example. Her family took away her fire, her light and even her clothes in an effort to drag her away from mathematics, but still she managed to study by the light of contraband candles and eventually persuaded her family to allow her to continue. Even so, she still had to face the problem that there were no places available for her once she had prepared herself for higher education. There were many new seats of learning being founded, but none of these were open to women. There have certainly been advances since then, the University of Göttingen is one notable institution that has helped to further women's education, also the creation of colleges for the higher education of women such as Girton near Cambridge.
Girton College was quite a step forward in England, founded in 1869 it was a college set up for the higher education of women. In 1873 they moved to just three miles from Cambridge and women were allowed to attend many of the lectures at Cambridge University, with the permission of the professors. The only drawback with this situation was that Cambridge University also maintained the Spinning House, a prison for prostitutes and suspected prostitutes. At that time you could be suspected of being a prostitute just for wandering around unchaperoned, which presented problems just trying to go to and from lectures! And although you could receive a higher education there were almost no postgraduate opportunities available. There was certainly no overnight change, and in fact it wasn't until 1948 that women could actually officially receive degrees from Cambridge University.
One woman who was perhaps partly responsible in bringing about change was Charlotte Angas Scott (1858-1931), her situation was different from Kovalevskaya's and Germain's in that she received a great deal of support from her family right from the very start. Her brother was the principle of the newly formed Lancashire Independent College and managed to provide her with tutors, a great help when very few secondary schools were open to girls. In 1876 she won a scholarship to Girton College having never attended a secondary school. When Scott took the Tripos exams, something that she could not take for granted as not all women were permitted, word escaped that she had scored the eighth highest mark. However, because she was a woman she could not be present at the award ceremony or even have her name read out. There are stories that when the names of the men were being read out it reached the eighth and the room cheered "Scott of Girton!". All the publicity from a woman succeeding in a man's subject put pressure on Cambridge to change and permit female students to take the exam as a point of policy rather than exception. And on February 24th 1881 this was finally passed. However women still had to look to elsewhere, like the University of London, to receive their degree externally having taken these exams at Cambridge.
These are all examples of women who were not so much dissuaded from studying mathematics as simply found that there were few options for them if they wanted to. I think that another factor remains to be discussed, one that is perhaps partly due to the influence of society. It is now worth looking at the motivation to study mathematics to a higher level. One reason might be to look towards an academic career, perhaps lecturing or research. Another might simply be in order to get a job in industry where mathematical knowledge could be applied. Again we look to the past and notice the obstacles met when trying to pursue these goals, the attitudes of the university faculties and society in general. In the life of Kovalevskaya we saw the trouble women faced being accepted in the academic world as colleges and universities turned down her applications to teach, despite her qualifications. The only option open to Kovalevskaya in the academic world was teaching at girls' schools. With the difficulties of raising children and pursuing a career, again illustrated in Kovalevskaya's life, there seems to have been little left to motivate women to pursue higher education if serious academic goals removed, in fact there seems much to advise them against it.
A commonly held medical view in the nineteenth century was that women's brains were smaller than men's because women required extra energy to maintain their ovaries. To pursue further study would take energy away from the ovaries, causing them to shrivel, hence infertility would occur. This seems ridiculous in today's world, but many people still hold the view that women simply do not have the "right kind of minds" to progress in mathematics. In Anne Fausto-Stirling's book "Myths of Gender: Biological Theories about Women and Men" she refers to an article which appeared in the New York Times and Science magazine. The article talked of the discovery that men were genetically stronger than women at mathematics and gave the impression to some that women would be better off if they recognised their limits and didn't even bother trying to aim for things that were obviously beyond their grasp.
Even today, there is the argument that women are not as mathematically minded as men as they lack the same spatial skills, their brains working in a different way to men's. One woman who dealt with these attitudes in an interesting manner was Grace Chisholm Young (1868-1944), another woman who went to Girton College after a home education. She finished by scoring a first class mark in the Tripos Part I exam, but then had nowhere in England to continue her studies so left for Göttingen to pursue a Ph.D. There she studied under Felix Klein and earned her Ph.D. at the age of 27. The following year she married William Henry Young, who had been her tutor at Girton, and they began a strange partnership. Knowing that she could not publish work under her own name, she worked with her husband and together they published many works under his name. The explanation for this is shown in this letter from William to Grace:
The above quotation illustrates very well the feeling towards women academics in the late nineteenth and early twentieth century. The lack of acceptance and the idea that they could not have both a family and a career. But what of women with similar goals in the modern world? Looking at various accounts from women in mathematics published in the Notices of the American Mathematical Society in 1991 it would appear that attitudes still play a large part in preventing women from progressing in their field. Vera Pless is one of the women who gives an account of her path towards becoming a mathematician. When growing up she had no ambition to become a mathematician as "girls at that time aspired to be wives (of successful men) and mothers". The main goal seemed to be to settle down and you were some how considered unfortunate if you ended up with a career instead of becoming a wife. Vera Pless did marry, and in some respects it was her marriage that helped her to take the first steps towards becoming a mathematician.
While waiting for her husband to complete his Ph.D. she embarked upon one herself and when he completed his they moved to Cambridge. She was fortunate enough to be able to complete her Ph.D. by correspondence, and two weeks before her first baby was due she defended her thesis. However she had no plans to work after this, instead choosing to look after her family. There was never any idea at the time that men should help in the home and her husband continued to work long hours for little money. Short of money, a few years after her second child was born she decided to find work at a university near the nursery. Her prime concern being her children she just taught at the university, having no time for research, and although she taught as many classes as a full-time faculty member she was only paid a graduate's salary.
When her children became old enough she decided to take up a full time post. However, when applying for ones in the area she struggled to gain a post. Not just because it was so long since she had done any research, but there was also a very negative response from some institutions on account of her being a woman. She was lucky enough to find a job at an Air Force research laboratory nearby and take part in research in a new field of mathematics, error-correcting codes. While there she talked to other women and helped to set up WISE, Women in Science and Engineering, an organisation formed to help others who were meeting the same problems. But after five years of separation from her husband they divorced and the children moved across to join her. Vera Pless's story shows us how women can be pulled in all different directions by their commitment to the home, the pressures of raising a family and the need to work. At least it is an improvement from the situation for people like Kovalevskaya and Chisholm-Young in that higher education was readily available, but there is still a cause for concern in the response of institutions.
Another woman, Deborah Haimo of the University of Missouri at St. Louis describes how there were few options open to her after university. She considered teaching but found that most public schools would not allow her to marry and remain a teacher. This seems to be a common problem; the idea that women are useless once married as they will have children and be unable to teach for a time. Susan Landau describes how she married while she was still a graduate student, but was then warned "Don't have children until you finish", by the wife of a history professor when she was 25, then "Don't have children until you get a tenure", when she was 28. Unfortunately this could leave her waiting until she was 35 to start a family, a daunting prospect in the face of the complications that can arise in pregnancies later in life. Today childcare is steadily improving and though not anything near ideal yet, it is helping to prevent some women from having to make a choice between a career and a family.
Although University faculties are still dominated by men, I feel that it would be foolish to expect this not to be the case. It will take a long time for the balance to be levelled off in the classes themselves, and it will take a while for the number of pupils to filter through to the stage where they might want to take up academic posts. Until then universities should be responsible in making sure that when these women graduate the facilities are in place to make sure that a life in academia is a valid option. The ensurance of sufficient maternity leave and childcare along with an encouragement to society that parents can share the work of raising children may help to eventually level the balance. I believe that simply trying to fill posts with women in order to make the numbers balance would actually be detrimental to trying to raise women's status in the field. Until there is a large base of quality candidates available there is no point in filling places with people who are less suitable simply to better represent them. In fact it would give them a worse representation if there were people in roles that they were not properly qualified to fill. Everyone should be treated equally and decisions based on merit, rather than trying to create differences and only this way will a true balance be achieved.
The indicators for women are looking promising, in 1998 a study of students from 13 different countries showed that boys were significantly outperforming girls in mathematics in all the countries surveyed except for South Africa. However in an article in the Irish Independent in May 2000 it was suggested that this had been knocked on the head by the Chief Examiner's Report for maths in the junior cert. Generally girls were found to outperform boys, except at higher level where boys were found to receive a higher percentage of A grades. And in a report in the Sunday Times in March of 2001, it was suggested that "Girls have boys beat by the age of 3". The report claimed that "girls' basic reading, maths, general understanding and communication skills are far superior to those of boys from the age of 3". These results were echoed in similar studies among four and five year olds. In fact there are signs that women have overtaken boys so much that boys need help to be put on an equal footing with girls. In a recent effort to help boys catch up with girls, study guides aimed at 7-11 year olds and based around football teams were brought out in March of this year hoping this would get boys interested in the subject.
Are boys falling behind, or are girls overtaking them? Perhaps it is a combination of both of these factors. Increasingly the arrival such things as games consoles, a particularly male pastime in general, have brought with them distractions from other activities. Meanwhile women are receiving more encouragement to pursue their dreams, with new role models appearing in the media and the recent phenomenon of "girl power". So young girls are realising that their goals are no longer unattainable and they are being encouraged to aim high and fulfil their potential. Perhaps in a few years time people will be writing reports into how women are dominating science and industry, while men end up being left at home as industry has turned its back on them. An interesting thought, but I think we have a long way to go before that much change is apparent in the attitudes of our society.
The very first sign of any problems to come in Sofia's life was her parents' disappointment at the birth of a girl. In many countries around the world it is still the case that a son is preferred, and a daughter will often live her life being resented by her parents. There are stories of baby daughters being killed where they are not considered as valuable as boys. There are obviously many different cultures around the world and women have different roles in them. In many parts of the world women are forced to work harder than men, receive less schooling than men and are even denied equal access to medical care. Worldwide approximately 500 million children start primary school, but around 100 million of these drop out before completing four years and of this 100 million around two thirds are girls.
A problem that Sofia encountered as a girl was common at the time and many would argue is still evident today. That is the influence of society on young girls, the idea that they should be pretty and graceful above all else, with the ultimate goal of marriage in mind. This is illustrated by Sofia's rejection by her parents as a young girl when she did not match up to her elegant elder sister, and was not presented to guests as her sister was. It could be argued that society still plays a large part in pushing girls towards certain stereotypes. Toys marketed towards girls are usually things like dolls that "cry and wet" or household items or make up sets etc. Do these toys point young girls towards the roles of mother, housemaker or beauty? Or is it simply that these are the toys that young girls prefer and we are reading far too much into it all?
In the media, women are generally portrayed in one of three categories: mother, sex object or girl. Certainly the perceived "nerdiness" of mathematics would dissuade a young girl under pressure to appear attractive and ultimately find a husband, from studying mathematics. But then it could easily be argued that such an image might put anyone off applying themself to the subject. Is there any evidence that women are put off studying maths at school or university level? Looking at some figures you could easily believe that too much is made of any gender gap. The percentage of women receiving bachelors in mathematics in America has steadily risen since the sexual revolution in the sixties, from 38.2% in 1971 to 46.5% by 1988. However looking at the percentage of doctorates in mathematical sciences rewarded to women we see a clear gap. This figure rose from 7.8% in 1971 to just 16.2% by 1988, and if we look to Europe the indicators are slightly more worrying. Of students of maths and computer science in higher education in 1994 only around 26% of them were women . We cannot directly compare this figure to those recorded in America as it lumps together all forms of higher education and also includes computer science. There were also some impressive anomalies, in Italy and Portugal the figure approached 50%, whereas in Luxembourg and the Netherlands it was only 10%. But the survey certainly gives us a clear indication of quite a large gap between men and women in mathematics. Is this evidence that women are dissuaded from studying mathematics though?
In the past there have certainly been many cases of women who really did want to study mathematics but then found that they were met by many obstacles. There are examples of this in the life of Sofia Kovalevskaya, her father forbidding her to study, her inability to travel without a chaperone and the universities that rejected her, all on account of her being a woman. This is also illustrated in the lives of many other female mathematicians; Sophie Germain (1776-1831) is one such example. Her family took away her fire, her light and even her clothes in an effort to drag her away from mathematics, but still she managed to study by the light of contraband candles and eventually persuaded her family to allow her to continue. Even so, she still had to face the problem that there were no places available for her once she had prepared herself for higher education. There were many new seats of learning being founded, but none of these were open to women. There have certainly been advances since then, the University of Göttingen is one notable institution that has helped to further women's education, also the creation of colleges for the higher education of women such as Girton near Cambridge.
Girton College was quite a step forward in England, founded in 1869 it was a college set up for the higher education of women. In 1873 they moved to just three miles from Cambridge and women were allowed to attend many of the lectures at Cambridge University, with the permission of the professors. The only drawback with this situation was that Cambridge University also maintained the Spinning House, a prison for prostitutes and suspected prostitutes. At that time you could be suspected of being a prostitute just for wandering around unchaperoned, which presented problems just trying to go to and from lectures! And although you could receive a higher education there were almost no postgraduate opportunities available. There was certainly no overnight change, and in fact it wasn't until 1948 that women could actually officially receive degrees from Cambridge University.
One woman who was perhaps partly responsible in bringing about change was Charlotte Angas Scott (1858-1931), her situation was different from Kovalevskaya's and Germain's in that she received a great deal of support from her family right from the very start. Her brother was the principle of the newly formed Lancashire Independent College and managed to provide her with tutors, a great help when very few secondary schools were open to girls. In 1876 she won a scholarship to Girton College having never attended a secondary school. When Scott took the Tripos exams, something that she could not take for granted as not all women were permitted, word escaped that she had scored the eighth highest mark. However, because she was a woman she could not be present at the award ceremony or even have her name read out. There are stories that when the names of the men were being read out it reached the eighth and the room cheered "Scott of Girton!". All the publicity from a woman succeeding in a man's subject put pressure on Cambridge to change and permit female students to take the exam as a point of policy rather than exception. And on February 24th 1881 this was finally passed. However women still had to look to elsewhere, like the University of London, to receive their degree externally having taken these exams at Cambridge.
These are all examples of women who were not so much dissuaded from studying mathematics as simply found that there were few options for them if they wanted to. I think that another factor remains to be discussed, one that is perhaps partly due to the influence of society. It is now worth looking at the motivation to study mathematics to a higher level. One reason might be to look towards an academic career, perhaps lecturing or research. Another might simply be in order to get a job in industry where mathematical knowledge could be applied. Again we look to the past and notice the obstacles met when trying to pursue these goals, the attitudes of the university faculties and society in general. In the life of Kovalevskaya we saw the trouble women faced being accepted in the academic world as colleges and universities turned down her applications to teach, despite her qualifications. The only option open to Kovalevskaya in the academic world was teaching at girls' schools. With the difficulties of raising children and pursuing a career, again illustrated in Kovalevskaya's life, there seems to have been little left to motivate women to pursue higher education if serious academic goals removed, in fact there seems much to advise them against it.
A commonly held medical view in the nineteenth century was that women's brains were smaller than men's because women required extra energy to maintain their ovaries. To pursue further study would take energy away from the ovaries, causing them to shrivel, hence infertility would occur. This seems ridiculous in today's world, but many people still hold the view that women simply do not have the "right kind of minds" to progress in mathematics. In Anne Fausto-Stirling's book "Myths of Gender: Biological Theories about Women and Men" she refers to an article which appeared in the New York Times and Science magazine. The article talked of the discovery that men were genetically stronger than women at mathematics and gave the impression to some that women would be better off if they recognised their limits and didn't even bother trying to aim for things that were obviously beyond their grasp.
Even today, there is the argument that women are not as mathematically minded as men as they lack the same spatial skills, their brains working in a different way to men's. One woman who dealt with these attitudes in an interesting manner was Grace Chisholm Young (1868-1944), another woman who went to Girton College after a home education. She finished by scoring a first class mark in the Tripos Part I exam, but then had nowhere in England to continue her studies so left for Göttingen to pursue a Ph.D. There she studied under Felix Klein and earned her Ph.D. at the age of 27. The following year she married William Henry Young, who had been her tutor at Girton, and they began a strange partnership. Knowing that she could not publish work under her own name, she worked with her husband and together they published many works under his name. The explanation for this is shown in this letter from William to Grace:
The fact is that our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. No. Mine the laurels now and the knowledge. Yours the knowledge only. Everything under my name now, and later when the loaves and fishes are no more procurable in that way, everything or much under your name. There is my programme. At present you can't undertake a public career. You have your children. I can and do.
The above quotation illustrates very well the feeling towards women academics in the late nineteenth and early twentieth century. The lack of acceptance and the idea that they could not have both a family and a career. But what of women with similar goals in the modern world? Looking at various accounts from women in mathematics published in the Notices of the American Mathematical Society in 1991 it would appear that attitudes still play a large part in preventing women from progressing in their field. Vera Pless is one of the women who gives an account of her path towards becoming a mathematician. When growing up she had no ambition to become a mathematician as "girls at that time aspired to be wives (of successful men) and mothers". The main goal seemed to be to settle down and you were some how considered unfortunate if you ended up with a career instead of becoming a wife. Vera Pless did marry, and in some respects it was her marriage that helped her to take the first steps towards becoming a mathematician.
While waiting for her husband to complete his Ph.D. she embarked upon one herself and when he completed his they moved to Cambridge. She was fortunate enough to be able to complete her Ph.D. by correspondence, and two weeks before her first baby was due she defended her thesis. However she had no plans to work after this, instead choosing to look after her family. There was never any idea at the time that men should help in the home and her husband continued to work long hours for little money. Short of money, a few years after her second child was born she decided to find work at a university near the nursery. Her prime concern being her children she just taught at the university, having no time for research, and although she taught as many classes as a full-time faculty member she was only paid a graduate's salary.
When her children became old enough she decided to take up a full time post. However, when applying for ones in the area she struggled to gain a post. Not just because it was so long since she had done any research, but there was also a very negative response from some institutions on account of her being a woman. She was lucky enough to find a job at an Air Force research laboratory nearby and take part in research in a new field of mathematics, error-correcting codes. While there she talked to other women and helped to set up WISE, Women in Science and Engineering, an organisation formed to help others who were meeting the same problems. But after five years of separation from her husband they divorced and the children moved across to join her. Vera Pless's story shows us how women can be pulled in all different directions by their commitment to the home, the pressures of raising a family and the need to work. At least it is an improvement from the situation for people like Kovalevskaya and Chisholm-Young in that higher education was readily available, but there is still a cause for concern in the response of institutions.
Another woman, Deborah Haimo of the University of Missouri at St. Louis describes how there were few options open to her after university. She considered teaching but found that most public schools would not allow her to marry and remain a teacher. This seems to be a common problem; the idea that women are useless once married as they will have children and be unable to teach for a time. Susan Landau describes how she married while she was still a graduate student, but was then warned "Don't have children until you finish", by the wife of a history professor when she was 25, then "Don't have children until you get a tenure", when she was 28. Unfortunately this could leave her waiting until she was 35 to start a family, a daunting prospect in the face of the complications that can arise in pregnancies later in life. Today childcare is steadily improving and though not anything near ideal yet, it is helping to prevent some women from having to make a choice between a career and a family.
Although University faculties are still dominated by men, I feel that it would be foolish to expect this not to be the case. It will take a long time for the balance to be levelled off in the classes themselves, and it will take a while for the number of pupils to filter through to the stage where they might want to take up academic posts. Until then universities should be responsible in making sure that when these women graduate the facilities are in place to make sure that a life in academia is a valid option. The ensurance of sufficient maternity leave and childcare along with an encouragement to society that parents can share the work of raising children may help to eventually level the balance. I believe that simply trying to fill posts with women in order to make the numbers balance would actually be detrimental to trying to raise women's status in the field. Until there is a large base of quality candidates available there is no point in filling places with people who are less suitable simply to better represent them. In fact it would give them a worse representation if there were people in roles that they were not properly qualified to fill. Everyone should be treated equally and decisions based on merit, rather than trying to create differences and only this way will a true balance be achieved.
The indicators for women are looking promising, in 1998 a study of students from 13 different countries showed that boys were significantly outperforming girls in mathematics in all the countries surveyed except for South Africa. However in an article in the Irish Independent in May 2000 it was suggested that this had been knocked on the head by the Chief Examiner's Report for maths in the junior cert. Generally girls were found to outperform boys, except at higher level where boys were found to receive a higher percentage of A grades. And in a report in the Sunday Times in March of 2001, it was suggested that "Girls have boys beat by the age of 3". The report claimed that "girls' basic reading, maths, general understanding and communication skills are far superior to those of boys from the age of 3". These results were echoed in similar studies among four and five year olds. In fact there are signs that women have overtaken boys so much that boys need help to be put on an equal footing with girls. In a recent effort to help boys catch up with girls, study guides aimed at 7-11 year olds and based around football teams were brought out in March of this year hoping this would get boys interested in the subject.
Are boys falling behind, or are girls overtaking them? Perhaps it is a combination of both of these factors. Increasingly the arrival such things as games consoles, a particularly male pastime in general, have brought with them distractions from other activities. Meanwhile women are receiving more encouragement to pursue their dreams, with new role models appearing in the media and the recent phenomenon of "girl power". So young girls are realising that their goals are no longer unattainable and they are being encouraged to aim high and fulfil their potential. Perhaps in a few years time people will be writing reports into how women are dominating science and industry, while men end up being left at home as industry has turned its back on them. An interesting thought, but I think we have a long way to go before that much change is apparent in the attitudes of our society.