Math 55: Harvard's Most Challenging Undergraduate Mathematics Course
For decades, Harvard University's Department of Mathematics has maintained a reputation for academic rigor, epitomized by Math 55. Once described by the department as probably the most difficult undergraduate math class in the country, the course is designed for students who possess a deep love for mathematics and the significant time commitment required to master its advanced content.
While the course is legendary for its intensity, modern instructors offer a more nuanced perspective. In 2022, Professor Denis Auroux noted that for students who are reasonably proficient in math and can devote ample time to their studies, the course is entirely manageable.
Key Facts
- Reputation: Historically regarded as one of the most demanding undergraduate math courses in the United States.
- Time Commitment: Modern students average 20 to 30 hours per week, though historical estimates ranged from 24 to 60 hours.
- Coursework: Students typically use LaTeX (a high-quality typesetting system) to document their work, effectively creating their own textbooks.
- Assessment: The course concludes with a comprehensive take-home final exam.
- Prerequisites: Even students with Advanced Placement Calculus credits or USA Mathematical Olympiad experience may find the course challenging.
Course Structure and Enrollment
The path into Math 55 has evolved over time. Previously, students began their journey in Math 25, a lower-level version of the course created in 1983. After three weeks of studying point-set topology (the study of topological properties of point sets) and special topics—such as p-adic analysis taught by Wilfried Schmid in 1994—students took a quiz to determine their placement.
Since 2012, the process has become more flexible. Students can now choose to enroll in either Math 25 or Math 55 from the start. They are encouraged to "shop" both courses for five weeks before making a final decision on which level is appropriate for them.
Depending on the instructor, a diagnostic exam may still be administered after three weeks. A historical example from 1994 illustrates how these results were used: out of 89 students, only 15 scoring above 50% entered Math 55, 55 students scoring between 10% and 50% entered Math 25 (Theoretical Linear Algebra and Real Analysis), and 19 students scoring below 10% were advised to take Math 21 (Multivariable Calculus).
[ไม่มีภาพประกอบ]The Challenge of Retention
The attrition rate of Math 55 is a testament to its difficulty. In 1970, Richard Stallman estimated that the course condensed nearly four years of departmental coursework into just two semesters. Of the 75 students who initially enrolled that year, only 20 completed the course.
This trend of high attrition continued in later years. David Harbater, a student in the 1974 section, recalled that while 70 students started, only 20 finished, and only 10 truly understood the material. More recently, in 2005, student Scott D. Kominers logged a similar decline: starting with 51 students on the first day, the number dropped to 21 by the fifth week, coinciding with the deadline to transfer to Math 25.
These drops are attributed to both the extreme nature of the material and the common undergraduate practice of "shopping around" for courses at the start of the semester.
| Year/Period | Initial Enrollment | Completion/Retention | Notes |
|---|---|---|---|
| 1970 | 75 | 20 | Covered ~4 years of coursework in 2 semesters |
| 1974 | 70 | 20 | Only 10 students reported full understanding |
| 2005 | 51 | 21 | Retention measured at the 5th-week deadline |
| 2022 | N/A | N/A | Average workload: 20-30 hours per week |
Frequently Asked Questions
How much time should a student dedicate to Math 55?
While historical claims suggest workloads of 24 to 60 hours per week, 2022 data indicates that students typically spend between 20 and 30 hours per week on the course, including homework.
What is the difference between Math 25 and Math 55?
Math 25 was created as a lower-level alternative to Math 55. While Math 55 is the most advanced track, Math 25 focuses on Theoretical Linear Algebra and Real Analysis for students who may not be ready for the intensity of Math 55.
How do students submit their assignments?
Students typically typeset their homework using LaTeX, which results in a professional-grade document that essentially serves as their own personal textbook for the course.
Is it possible to take other difficult courses alongside Math 55?
It is generally ill-advised to take many other challenging courses or engage in heavy extracurricular activities during the same semester as Math 55 due to the significant time commitment required.
Can students switch between Math 25 and Math 55?
Yes. As of 2012, students have a five-week "shopping" period to decide between the two courses, and diagnostic exams are sometimes used to help guide this decision.