On justice and mathematics

What does it mean to study in a field defined by inequalities?

Tasmin Chu, Features Editor

There is a passage in Plato’s Meno that goes something like this: The well-born Meno asks for proof of Socrates’ claim that no one is ever taught anything, and instead they recollect things they already know. Socrates calls over one of Meno’s enslaved attendants and asks the boy, who has no mathematical experience, to solve a geometry problem. With Socrates’ guidance, the boy discovers how to double the area of a square, and Socrates suggests to Meno that what appears to be learning, then, was merely recollection: “These notions have just been stirred up in him, as in a dream.”

It’s a strange and interesting thought experiment, one that neatly crystallizes a belief agreed upon by most mathematicians—that math is a priori, meaning that mathematical truths come from theoretical deduction rather than experience. But in recent years, the more I think about this anecdote, the more the social context stands out to me rather than the philosophical argument. After the dialogue ends, the boy’s chance to engage in mathematics is over. He goes back to serving his master. What he is intrinsically capable of is philosophically interesting, but realizing his mathematical abilities is never considered.

In some ways, math can feel like the most apolitical subject of all. Its theorems proceed from axioms, not empirical data; it’s easy for, say, a pure mathematician to feel insulated from the world. But the field’s demographics reflect the stark inequalities of the society we live in. Despite an influx of initiatives in the past few years, women remain underrepresented in STEM—and the problem is particularly severe for fields like mine. While women have made significant gains in some areas of science, like psychology and life science, math-intensive fields remain behind in increasing female faculty representation. Racial minorities face the same problem: Data from the United States’ National Science Foundation reveal that only 4.5 per cent of mathematics PhD recipients in 2017 were Hispanic or Latino, and only 2.8 per cent were Black. Census data from the same year suggests that Black people accounted for 12.3 per cent of the whole population, while Hispanic people accounted for 18.1 per cent.

Unfortunately, not everyone believes that there’s an obligation to change this state of affairs. University of California, Berkeley’s Rob Kirby has maintained that the mathematical community is “generally fair” to women and minorities. On his website, he wrote, “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).” In 2019, the topologist Abigail Thompson condemned what she called “mandatory diversity statements” in a piece published in Notices of the American Mathematical Society. She was referring to the statements about “contributions to diversity” that some universities solicit from job applicants—a hiring practice which she compares to a McCarthy-era loyalty test. Recently, she and Kirby, among others, founded a new organization for promoting mathematics called the Association for Mathematical Research (AMR).

The AMR’s website is frustratingly vague; little distinguishes its purpose from pre-existing associations, like the American Mathematical Society (AMS) and the Mathematical Association of America (MAA). The number theorist Michael Harris, who was invited to join, saw the organization as a possible reaction to the AMS’s increasing focus on equity. As the AMR’s letter of invitation to him read, “Though individual members may be active in educational, social, or political issues related to the profession, the AMR intends to focus exclusively on matters of research and scholarship.”

The AMR has been met with backlash —including by Louigi Addario-Berry, a math professor at McGill. Addario-Berry has criticized the organisation on his blog, writing, “The mathematical community is impoverished by its lack of diversity! A professional society that doesn’t share that view is not one that I see a great value in.”

When I talked with Addario-Berry, I asked him why there was so much backlash to diversity initiatives. He told me that empathy tends to be asymmetrical. For example, while I might think a lot about why white tenured academics find diversity initiatives unfair or censorious, they probably don’t think about someone like me before they pen their next op-ed.

“I think a lot of AMR signatories, or at least the founders, are really people who fundamentally haven't spent very much time building empathy or putting themselves in the shoes of people who don't feel welcome by the mathematical community,” he told me. “If you have never given serious credence to the idea that the mathematical community is unwelcoming and discriminatory, and you really do think that it's a level playing field, then you’re going to be resentful of programs that are designed to level that playing field.”

In truth, Kirby or Thompson’s statements don’t particularly offend me. But I do find what they say stupid. How can math be apolitical? The great mistake I see in the AMR is the unconsidered assumption that certain human pursuits can be isolated from the material and social conditions of the society in which we live. All human activities are bound up, inevitably, with the normative. Even the surge of research into machine learning, for example, reflects the fact that companies are invested in using these tools for profit. And more generally, racial and gender minorities who endure discrimination outside the classroom are not instantly insulated from those experiences within it.

At the undergraduate level, gender representation in math is relatively decent at McGill. According to McGill’s official enrollment statistics, there are currently 308 undergraduate female students and 508 undergraduate male students in math. (No other category for gender identities is listed, so it’s possible some of these students are non-binary or genderqueer as well.) But things change as you go up the ladder. This year, only nine of McGill’s 45 master’s students in mathematics are women, and the gender breakdown of the PhD students is even worse. In Fall 2020, only 11 per cent of the math department’s 61 PhD students were women. Even more startling is the fact that this percentage represents a drop from recent years. In 2016, 24 per cent of PhD students were female, but by 2018, it was down to 15 per cent. Despite a surge in equity initiatives in recent years, the gender gap has only widened. The lack of female representation is certainly something that Shereen Elaidi, a master’s student in mathematics, notices in her program.

“You walk into the grad lounge,” Elaidi said. “And I wouldn't think about this [normally], but you realize at some point: ‘I'm the only female here in the grad lounge.’”

It’s a depressing thought. I’m often grateful, at the undergraduate level, that so many of my classmates—and a handful of my professors—are women. At the same time, though, resolving inequalities has to go beyond diversity training and increased representation. Cost is a huge barrier for graduate studies at McGill. In order to maintain full-time studies, graduate students are only allowed to work 180 hours per term, or 12 hours a week, while completing their degree—something that can leave self-supported students with few solutions. For instance, Elaidi receives an $18k stipend, which also comes with the obligation that she works as a teaching assistant for two classes. At the same time, though, she pays $20k in tuition as an international student, meaning she studies in Montreal at cost.

“I've had to work so many jobs just to pay,” Elaidi said. “It sucks. It’s mentally exhausting.” She added, “The funding kind of assumes that you have another source of income to help you live.”

While graduate students will probably remain overworked for a long time, McGill could at least give international students a livable stipend. Cost is just one reason that the pipeline is leaky, a metaphor for that way that women and racialized minorities gradually disappear from STEM the higher up you go. But I also wonder if earlier interventions, like better undergraduate teaching, could draw a greater diversity of people into math. After all, bad teaching, especially in math, can end up testing for academic background, rather than ability—and a student’s high school background in math will obviously intersect with race and class. Reaching people with less mathematical maturity is worth the struggle.

“Giving a talk where you get across the interesting and new ideas from a subfield in a way that gives some inkling of what they're about to a broader audience is a real challenge,” Addario-Berry said. “Frankly, a lot of mathematicians don't like to put in the time, and I understand it. We were almost all, in some sense, chosen for this, in the sense of having succeeded in getting a job based on a very narrow set of skills, which is almost exclusively the ability to write papers that get into good journals.”

Maybe it’s time to broaden what we look for in the mathematical community. It’s always struck me as somewhat perverse that in a field where it is unusually hard to distill and transmit information from one person to another, we still don't seem to care that much about good teaching. And while I understand the idea behind making academics teach—giving back to the scientific community, putting students in touch with current researchers—I also wonder if it’s time to separate these professions more fully, at least for introductory classes.

As Gavin Barill, a PhD student in the mathematics department, put it to me succinctly: “If you're not investing in teaching, then you are using undergrads for their tuition.” He would know. Barill himself was turned off from math in undergrad by what he described as a “gatekeep-y” first-year analysis course; he ended up majoring in computer science. Speaking to him reminded me of all the people I know who are driven away from mathematics by courses that are rigorous but, frankly, taught poorly. Who gets excluded by this kind of pedagogy?

In many ways, I have an unusual level of privilege when it comes to mathematics. My father is a category theorist by training, and growing up, he would show me the odd proof here and there, demonstrating that √2 was irrational or that an infinite series converged to 2. Usually, I didn’t understand these proofs, and I would feel frustrated and mystified. At school, I excelled in math, which unfortunately meant I was forced to write math contests. But there were upsides, too; in seventh and eighth grade, I was placed in a small, collaborative math class. It felt like a class where I did puzzles all day with my friends.

By high school, though, things became more computational. Deep down, I often felt like an imposter: One who could easily take a derivative but lacked the creativity necessary to do real mathematics. So I started my degree in biology, and later, philosophy (with a minor in math). My image of a real mathematician was that of Carl Friedrich Gauss, who derived a beautiful summation formula as a child—or maybe it was Terence Tao, the youngest person to ever win a medal in the International Math Olympiad. Personally, I found mathematics contests stressful. As soon as I could avoid them, I did.

“We have explicitly and implicitly quite narrow ideas about who counts as a mathematician and what counts as mathematics,” Addario-Berry said. “On the spectrum, competitive problem solving is kind of the epitome of that, right? If you can solve tricky mathematical questions quickly, then you're good at math. Other kinds of thoughts that are slow and involve a lot of analogy, which is super important for advanced math—that's very much not selected for reward at the primary, secondary, or university levels.”

In my second-year algebra course, though, I got lucky. During the pandemic, I worked through the details of rings and groups with the help of a supportive TA. Alone in my bedroom, I began to wonder if it had been a mistake to give up on math. And I had the startling realization that I was good at math—or maybe good enough at math. I knew I wasn’t exceptionally talented. But I didn’t need exceptional talent to keep doing math.

The imposter syndrome I had was insidious—but in many ways, it was also something that was culturally reinforced. The stereotype of a mathematician is still just “a lone man” in his ivory tower, as Elaidi put it pithily. Like me, she came from a humanities background first—something that can make you particularly vulnerable to feelings of not belonging.

“This is something I've noticed about the math department compared to other departments at McGill: Effortless talent is kind of rewarded,” Elaidi told me. “That culture thing made me dread it. Because none of this comes easy to me.”

Yet she stuck with math. During her undergrad, Elaidi participated in the math department’s Directed Reading Program (DRP), which pairs undergraduates with graduate student mentors. With the guidance of her mentor, she researched special relativity and differential geometry. “That was literally what made me think I want to do math research,” she recalled.

Now Elaidi helps organize the DRP along with the graduate student who founded it, Peter Xu. The DRP gives students an opportunity to explore research and topics of their interest with a mentor. It’s a breath of fresh air compared to the NSERC or SURA research awards, which are highly dependent on GPA. The application for the DRP doesn’t take into account your transcript—a choice which Elaidi explained to me was intentional.

Talking with her reminded me that making the field more equitable doesn’t only look like diversity training. That’s an important part, to be sure, but increasing equity can be as simple as good pedagogy, dropping GPA requirements, and increasing the accessibility of research projects. Another thing that instructors could model is a growth mindset—something Agnes Totschnig emphasized to me. Totschnig is one of several math students who founded Diversity in Math, a student group that aims to inspire people from all backgrounds to discover mathematics. “If you see math as something that you’re good at or not and everything comes easily to you, the first time you get stuck, it can be really scary,” Totschnig said.

So far, Diversity in Math has held workshops on mental health and imposter syndrome, as well as a panel demystifying the process of finding research projects. In many ways, Diversity in Math owes itself to the work of Rosalie Bélanger-Rioux, a faculty lecturer who has done enormous work for equity in the math department. In 2020, Bélanger-Rioux began the Math Equity Reading Group, giving faculty and students a chance to discuss issues of equity in the field and academia more generally. Right now, she’s thinking about organizing a training session for TA’s to get them thinking about pedagogical techniques and equity.

“The two of them actually really mesh together,” she explained to me. In fact, the same pedagogical techniques that have been shown to be good for underrepresented minorities are also “good for everyone, basically.” It was a refreshing thing to hear at a time when so often the needs of marginalized groups are pitted against the needs of everyone else. In the same vein, Bélanger-Rioux hopes that accommodations necessitated by the pandemic—generous grading schemes, optional midterms—might become more commonplace in the future.

“Whether or not COVID is over or almost over or whatever, bad stuff happens all the time,” Bélanger-Rioux said. “Yes, there was more bad stuff happening for everybody. But bad stuff happens all the time. Being more accommodating to students doesn’t mean being easy on them or giving them higher grades. It just means giving them a better opportunity to show us that they do know the stuff after all.”

These days, as I write my proofs and correct my notation, I think about my great-aunt, Diana Yun-dee Wei. She wrote her PhD thesis on torsion theory at McGill in the late 60s, a fact that was mundane as a child but became more extraordinary as I grew up. I can’t imagine earning a degree at that time and place with the background she had, but nevertheless, she survived difficult supervisors and grueling courses. After her was my father, who immigrated here from Taiwan when he was 14. He didn’t speak English before he came; as he told me once, “Math was the only subject I was good at.” But a bachelor’s became a master’s, and a master’s became a PhD. As for myself, I still wonder whether I can contribute at all to this field—if I have the ability or the discipline. But I’m reminded that improving the state of mathematics can be so much more broad than doing research. Doing math can also look like teaching my friends about the Cantor space and remaining critical of the status quo. The academy is not paradise, as bell hooks wrote in Teaching to Transgress in 1994. But, she went on, “Learning is a place where paradise can be created. The classroom, with all its limitations, remains a location of possibility.”

Illustrations by Xiaotian Wang, Design Editor