Demystifying Math 55
By Anastasia Yefremova
Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies in Algebra and Group Theory” and Mathematics 55B “Studies in Real and Complex Analysis,” it is technically an introductory level course. It is also a veritable legend among high schoolers and college students alike, renowned as — allegedly — the hardest undergraduate math class in the country. It has been mentioned in books and articles, has its own Wikipedia page, and has been the subject of countless social media posts and videos.
Most recently, Harvard junior Mahad Khan created a TikTok video dedicated to Math 55 that has received over 360,000 views to date. His is only one of many — his older brother created one, too — but it has the distinction of an insider’s perspective. “I thought it would be interesting if I cleared up the misconceptions about Math 55,” Khan said. While he hadn’t taken the course himself, he wanted to go beyond its reputation. “I wanted to get a real perspective by interviewing a former student and current course assistant.”
Over the years, perception of Math 55 has become based less on the reality of the course itself and more on a cumulative collection of lore and somewhat sensationalist rumors. It’s tempting to get swept up in the thrill of hearsay but while there might be kernels of truth to some of the stories, many of them are outdated and taken out of context. At the end of the day, however, Math 55 is a class like any other. Below, we take a stab at busting some of the more well known and persistent myths about the class. Or, at the very least, offering an extra layer of clarity.
Myth #1: Math 55 is only for high school math geniuses
Most articles or mentions of Math 55 refer to it as filled with math competition champions and genius-level wunderkinds. The class is supposedly legendary among high school math prodigies, who hear terrifying stories about it in their computer camps and at the International Math Olympiad. There are even rumors of a special test students have to take before they are even allowed into Math 55. But while familiarity with proof-based mathematics is considered a plus for those interested in the course, there is no prerequisite for competition or research experience.
In fact students whose only exposure to advanced math has been through olympiads and summer research programs can have a harder time adjusting. Their approach to the material tends to be understandably more solitary and that can be a disadvantage for the level of collaboration higher level mathematics require. “It has become a lot more open to people with different backgrounds,” said Professor Denis Auroux, who teaches Math 55,. “Our slogan is, if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.”
Also, there is no extra test to get into the class.
Myth #2: Just take a graduate class, instead
Math 55 is hard. Whether you’re just 55-curious, or a past or present student in the class, this is something everyone agrees on. The course condenses four years of math into two semesters, after all. “For the first semester, you work on linear and abstract algebra with a bit of representation theory,” said sophomore math concentrator Dora Woodruff. “The second semester is real and complex analysis, and a little bit of algebraic topology. That’s almost the whole undergraduate curriculum.” Woodruff — incidentally, the student Khan interviewed — took Math 55 as a freshman and returned her second year as a course assistant. She is intimately familiar with the course’s difficulty level.
So why not just take an upper level undergraduate course to begin with or even one at a graduate level, if you’re really looking for a challenge? What justifies the existence of a class with the difficulty level of Math 55? One argument is that the course helps structure and systemize the knowledge with which many students come to Harvard. It gives them a firm background in preparation for the rest of their math education. Math 55 is difficult and it is purposefully structured that way as it’s meant to help students mature as mathematicians rather than as simple course takers.
But more importantly, “it’s just not true that Math 55 is at the level of a graduate class,” Auroux said. “It goes through several upper division undergraduate math classes with maybe a bit more advanced digressions into material here and there, but it sticks very close to what is taught in 100-level classes. The difference is we go through it at a faster pace, maybe with more challenging homework, and ideally as a community of people bringing our heads together.”
A core goal of Math 55, according to Auroux, is to build a sense of community. Other schools might encourage advanced first-year students to take upper level undergraduate or even graduate classes, but Math 55 helps build a cohort of like-minded people who really like math, are good at it, and want to do a lot of it during their time at Harvard. That’s the experience Woodruff had, as well. “The community can be very strong,” she said. “You meet a lot of other people very interested in math and stay friends with them for the rest of college.”
Myth #3: Homework takes between 24 and 60 hours
Horror stories of endless homework are synonymous with the class. You’ll read or hear about “24 to 60 hours per week on homework” in almost every reference to Math 55. But one, there is a world of difference between 24 and 60 hours that is never explained, and two, this timeframe is quite misaligned with reality.
Auroux frequently sends out surveys to his students asking how long homework takes them and the average for most is closer to 15 hours a week. Those with more extensive prior math backgrounds can take as little as five to ten hours. The key factor is collaboration. “This class doesn’t lend itself to self-study,” Auroux stressed. Once they have thought about each problem set on their own, students are welcome and encouraged to talk to their friends and collaborate. “As soon as I see that something took over 30 hours I ask the student, do you know you’re supposed to be working with people and come ask me questions when you’re stuck?”
It is true that between reviewing lectures, digesting the material, and solving the problem sets, students usually end up devoting between 20 and 30 hours a week to the class. However, that includes the time dedicated to homework. So while students are discouraged from taking too many difficult classes and extracurriculars in the same semester as Math 55, they are also not expected to spend the time equivalent to a full-time job on their problem sets every week.
Myth #4: less than half of the class makes it to the second semester
Math 55 is just as infamous for its attrition rate as it is for its difficulty. Most sources like to cite the 1970 class, which began with 75 students and — between the advanced nature of the material and the time-constraints under which students had to work — ended with barely 20. Since then, the rumor has been that the Math 55 class shrinks by half its original size or more before the first semester is over. The reality is much less shocking and a bit more complicated.
Enrollment in this past fall semester’s Math 55A peaked at (ironically) 55 students. Well into the spring semester’s Math 55B, 47 students were still enrolled in the course. “On average, a drop of about 10-15 percent is much closer to what I would expect,” Auroux said. And those numbers become even more flexible if one takes into consideration the weeks math students have at the beginning of each semester to try out different classes and “shop” around before they have to commit to anything. This means students find their way in and out of Math 55 in a variety of ways over the course of the academic year.
According to Auroux, some students shop Math 55 in the fall and switch to the less intense Math 25 for the remainder of the semester. Others start out in Math 25 and, if not sufficiently challenged, switch to Math 55. Even people who end up in academia are not exempt from this. During his time as a student, our own Department of Mathematics’ Professor Emeritus Benedict Gross switched to the lower level Math 21 after two weeks in Math 55. In fact, those two weeks almost made him reconsider his desire to pursue mathematics. “By the beginning of sophomore year, I had decided to major in physics,” he recalled. “But during shopping period that fall, I walked past a math class taught by Andrew Gleason and stopped in to listen. It turned out to be Math 55.” He enrolled and by the end of the semester had found his vocation in mathematics.
All this means that Auroux sees student numbers vacillate up and down throughout the academic year. “There are about four or five students in this spring semester’s Math 55 that took Math 25 or even Math 22 in the fall, and they’re doing mostly fine,” he said. “It’s a lot of work, but I think they’re having a great time.”
Myth #5: 55-er culture is cult-y and exclusionary
Even though her experience with Math 55 was a positive one, Woodruff is very aware of the unhealthy culture the class has been rumored to cultivate. It’s easy for students to form exclusionary cliques that consist only of other Math 55 students, and some look down on anyone taking lower level math classes. But Woodruff also stressed that the instructors are very aware of this and actively take steps to curb that kind of toxic behavior. She said Auroux frequently brings up the importance of keeping the Math 55 community inclusive through Slack messages and lecture references.
Some students come to Harvard just for the opportunity to take Math 55. Some view enrolling in the class as proof of their mathematical gumption and competence. A Harvard Independent article called Math 55 the “premiere mathematical challenge for overachieving and…ridiculously mathy freshmen” and a piece in The Harvard Crimson referred to it as “a bit of a status thing as far as math majors here are concerned.” Over the years, the Harvard Department of Mathematics has taken steps to correct these assumptions.
For one thing, neither the Math 55A nor the Math 55B official course descriptions boast the dubious honor of referring to it as “probably the most difficult undergraduate math class in the country” (don’t trust everything you read on Wikipedia). For another, “we’re trying to emphasize that there’s no magic to Math 55,” Auroux said. “It contains the same material as some of the other classes we have. People who take it are not intrinsically better or smarter than the ones who don’t.”
Myth #6: You have to take Math 55 if you’re serious about going into academia
One reason math concentrators could feel pressured to enroll in Math 55 is because they view it as a prerequisite for a career in academia. It’s a sort of badge of honor and proof of their commitment to the field of mathematics. It is true that quite a few graduates of the course have gone on to pursue a career in mathematics. Woodruff herself believes that will be the most likely path for her, and several faculty members in our own Department of Mathematics took Math 55 during their days as Harvard freshmen.
“Several times in my research career when I understood something fundamental, I would realize that this was what Math 55 was trying to teach us,” Gross said. “It was an amazing introduction to the whole of mathematics and it was transformative for me.” In fact, Gross met Higgins Professor of Mathematics Joe Harris when they took the class together, forging a lifelong friendship. When they returned to Harvard as faculty, they took turns teaching Math 25 and Math 55.
However, Auroux is quick to point out that while many graduates of the course do end up in academia, most professional mathematicians have likely never even heard of Math 55. “I would like to think that it’s a success story if people end up doing math, because the goal of Math 55 is to show students how beautiful math can be,” he said. “If they love it enough to go to grad school and become mathematicians, that’s wonderful. And if they want to take that math knowledge and do something else with their life, that’s just as wonderful.”