“The next theorem could be extracted from this construction with very little extra effort.” “Trivial.” “Easily verified.” “The theorem follows.” “The obvious inequalities prove (c).” “The proof is quite simple.”
These quotes are all taken from a math textbook, the one used by students in their first semester of one of the math major tracks at Princeton. Walter Rudin’s Principles of Mathematical Analysis is revered by some and despised by others, but whether you take his word as gospel or curse the day you bought his little blue book at Labyrinth, understanding every word is the surest path to success in the course.
Odd as his turns of phrase might seem, Rudin’s language actually fits in quite well with the rest of the math world, as does the mathematics department at Princeton. It’s not uncommon to hear words like “obvious” or “trivial” used to describe the results proven in class or on problem sets, and textbook authors frequently leave proofs as exercises for “the clever reader.” I remember being taken aback by this teaching methodology, which struck me as conceited and unwelcoming toward beginning students, but, like most of my peers, I decided it was endearing and quirky and learned to ignore it.
The language of the textbook was something I could get past, but I was more concerned when I began to see Rudin’s brevity and condescension reflected in the classroom dynamic. I remember my first-semester math professor stopping in the middle of a lecture to ask if he needed to finish the proof, or if the result was obvious.
What? At first, I was shocked—I mean, how do you answer that? If you raise your hand and ask him to finish the problem, what does that say about your ability as a math student? But what if you stay silent and miss out on important material? I watched from one of the back rows as my classmates cautiously scanned each other’s faces for signs of understanding, confusion—something to help them figure out what to do. After an uncomfortable silence, someone spoke up and our professor finished the proof. But in a learning environment in which this is the norm, what’s a budding mathematician to do?
In the long term, the answer might just be leave the department. Only thirty-five members of the class of 2017 are pursuing a mathematics concentration, followed by another thirty-five in the class of 2018. That the department is this small is already noteworthy, but even more perplexing is the fact that of these seventy students, only thirteen are female—that’s less than 20%.
To figure out why this gender gap exists, I talked to two juniors and one sophomore in the department (all of whom asked to remain anonymous), and to Dr. Christine Taylor, a professor and researcher at Princeton. We talked about their experiences learning and teaching math, both at Princeton and elsewhere, and what might be causing this gender disparity.
According to one sophomore in the department, it could have something to do with differences in the way men and women respond to challenges. “When women face a challenge, like a hard test, they tend to internalize it and say ‘I must not be good enough at this,’” she said. “Whereas men will say, ‘It was hard’—they externalize it. So it’s not their fault that they did poorly. It more has to do with the circumstances of the test.” The idea that men and women make different attributions about challenges (well-supported by research in psychology) could “play into the way [male students] interpret language like ‘of course’ or ‘obviously.’” While women might perceive this language as questioning their intelligence or capability, men may have the added defense mechanism of the external attribution—the ability to say, “That was a difficult test” instead of, “I must be a bad student.”
But the gender disparity doesn’t start in Princeton’s math department—it’s long before they are faced with Rudin’s talk of triviality that girls are discouraged from entering the field. As Liza Milov ‘20 wrote in the recent Nass article “Women in Philosophy,” research by Dr. Sarah-Jane Leslie has indicated that girls as young as five and six years old are already inclined to avoid games believed to be for “smart people.” Leslie has also shown how this perception continues to affect women throughout their academic careers: majors that tend to be associated with a “brilliance requirement,” like mathematics and philosophy, tend to be predominantly male, whereas those that require “hard work” attract more women.
The issue, then, is that in the minds of female students, math becomes a game for “smart people”—and over time, fewer and fewer girls retain the confidence to overcome that perception. This might be related to the differences in attributions that I mentioned earlier—women tend to internalize their failures, while men tend to internalize their successes. According to research in psychology, teachers might play a role in the development of these attributional styles: teachers tend to praise boys for their intelligence when they succeed, and girls for their effort.
There may also be more overt differences between girls’ educational experiences and those of boys. The sophomore I spoke with theorized that girls might arrive on campus with less robust math backgrounds than their male peers, and that this inequality is a result of the outreach efforts of math camps. “It’s all about opportunity,” she argued. “A lot of people never had the opportunity to go to math camp, because maybe they didn’t know about it. Even if [girls] are doing math in school, and excelling at it, they might not be actively pushed in that direction. Or pursuing mathematics might not even cross their minds, owing to social factors dating as far back as early childhood.”
She also noted that even in math camps for high school students, the gender gap is alarmingly wide. A junior in the department, reflecting on her awareness of the gender divide in math, shared that she hadn’t really noticed an underrepresentation of women in the field until going to a math camp, which was about one-third female.
While math camps may seem like purely extracurricular summer activities, they actually carry significant weight in terms of preparation for Princeton’s rigorous introductory math sequence—or at least that’s how some students perceive them. The sophomore math concentrator’s impression is that “many of the students in the 216/218 sequence, as opposed to the standard 215/217, enter college with particularly strong advantages, such as intense summer math camps or mathematically focused high schools.” And since the accelerated MAT 216/218 sequence seems to yield more math majors than MAT 215/217, female students’ underrepresentation in math camps (and subsequent disadvantage in the development of a background in math) could contribute to the gender disparity we see in the department here.
Of course, these anecdotes merely reflect students’ perceptions of the way the department works—that math camps are virtually prerequisites, and that the accelerated sequence is a more surefire path to entering the department. Taylor, who taught at Harvard and MIT before coming to Princeton, expressed some frustration with this perception, saying instead that “there is no reason why anyone can’t be a math major who started in 103.” She recalled a successful mathematician who started in the equivalent of 103 at Harvard and mentioned one of her own former students at Princeton who started in 201; he went on to win a Math Department prize for “outstanding accomplishments in mathematics by juniors” and was nominated for the Rhodes Scholarship by the University. But these examples aren’t readily available to undergraduates interested in becoming math majors, and “just the fact that you know there are people taking 215, or 216, puts you off.”
In order to encourage aspiring math majors, Taylor told me, she tries to reach out to individual students who perform exceptionally well. She recalled several students who had planned to enter other departments but were convinced to major in math because of individualized outreach. For talented students who might not recognize their potential as mathematicians, this outreach offers an incredible opportunity to connect with the department and has the potential to change students’ lives.
But for the majority of students, who don’t receive this kind of recognition and who are already experiencing stereotype threat, perceptions about math—regardless of their veracity—have a heightened impact on their level of comfort in the department. It’s easier to make someone question their belongingness in a field if they’re already questioning it themselves.
Men in the department are not immune to discouraging perceptions like these—one male junior I spoke to described his experience with what he described as a “big macho vibe” in the department. “People want to be very masculine by showing other people that they’re better, or that they have more knowledge. So you’ll be asking a question and someone else will jump in… and attempt to make you feel worse,” he said. “I think that’s kind of a toxic vibe in the department.”
Although we’re aware of these gendered differences in educational experience and attributional style, and perhaps of some of the ways in which stereotype threat might affect women in math, it remains unclear exactly what causes such a dramatic gender imbalance in Princeton’s math department. Do fewer women arrive on campus with an interest in pursuing math than men—that is, does the gender difference already exist at the start of the first semester in the introductory prerequisite sequence? Or is there something specific about the culture within Princeton’s math department that combines with preexisting forms of discouragement to keep women from pursuing a math concentration?
The answer is probably some of both—yes, there are fewer women than men at the start of the first semester of 215, and yes, this decreases further with time—both undergraduate women I spoke to had taken classes in which they were the only female student. Of course, since math is a difficult concentration for any student, we can expect to see decreases in the number of women (and men) in math classes over time. But how does the first week of 215—during which the class was about one-third female—devolve into a senior class with only three women concentrating in math? Are women discouraged by the department more quickly than men?
If, like Taylor, the department has an interest in a more even gender breakdown, perhaps this question deserves consideration. The sophomore math concentrator I spoke to contended that while the department is “not overtly unwelcoming toward women, they make limited attempts to advocate for increased representation of women in the department. This type of neutrality is precisely the problem.” And I agree with her. By buying into the culture of the math world—calling some results “trivial” and neglecting to explain others—the department isn’t necessarily doing something wrong. To change the environment and to attract more female concentrators, however, there has to be active encouragement. In ignoring the problem, argued the sophomore I spoke to, “you’re not doing anything to counteract subtle forms of discouragement. And then the issue just perpetuates.”
In this respect, the math department is definitely lacking. Compared to departments like computer science, in which organizations like Princeton Women in Computer Science support and recruit potential new concentrators, the math department lags behind in its support for its undergraduate women. Princeton’s math department does have an organization for female members, the Noetherian Ring, but its sole sponsored activities in recent years have been lunch events held twice annually (not to mention that its website hasn’t been updated since 2014).
Comparable institutions seem to have more active organizations for women in mathematics: at Harvard, the organization Gender Inclusivity in Mathematics (GIIM) is “dedicated to creating a community of mathematicians particularly welcoming to women interested in math and reducing the gender gap in Harvard’s math department.” Their website features upcoming events like math socials and pset nights, as well as discussion series and information about guest speakers. At Stanford, the organization Stanford Women in Math Mentoring (SWIMM) offers tutoring, peer help hours, and mentoring resources for women considering majoring in math.
The gender imbalance in math is not a problem exclusive to Princeton: according to the American Physical Society, just over 40% of mathematics and statistics B.A.s go to women, yet women make up just below 60% of all bachelor’s degree recipients. The numbers are worse in graduate programs—in 2014, the Annual Survey of the Mathematical Sciences in the U.S. reported that less than 30% of math Ph.D. recipients were women. But although the problem exists outside of Princeton, our math department seems hesitant to address it, perhaps more so than at other universities.
If Princeton does want to close the gender gap in the math department, something has to change. Whether it’s subtle condescension in the classroom, outdated textbooks that refer to the reader as “he” (like the one used in 217, part of math concentrators’ first-year experience with the department), or lack of direct access to mentorship and role models within the department, there is clear opportunity for improvement—and with only three female concentrators graduating in the spring, a clear need for it as well.