Thanks to a love of math and being an advanced special snowflake, I’ve taken a lot of math classes in my time, and I can expect to take a lot more if I go to grad school. Most of my math classes, even my self-directed course, were taught at a reasonable pace. When I didn’t understand a concept, I usually caught up before not understanding snowballed and there was no looking back.
I started only a couple of math courses ahead, though. If I went to Harvard and took the famed Math 55, I may have been crying for Godel within a week. Take a look at the 2009-2010 edition of the course for 55a and 55b. For those not familiar with the various fields of math, the course crams four math courses and then some into one year–four math courses that deserve a lot of time and attention. The students who take this course already have an extensive math background and probably are familiar with some of these topics, but the intensity of the course and problem sets, and the amount of independent study a student has to do in order to succeed make the course more difficult than a typical semester of abstract algebra.
“Survive” would be a more appropriate word. Math 55 is a course for the elite entering first-years in mathematics, and they’re often looked upon in awe. After the first few days, they see exactly what they’re getting into and start dropping like flies. Very few women complete the course, which has been noticed in research on gender and the sciences. The Harvard math department website warns students with these nuggets:
“This is probably the most difficult undergraduate math class in the country; a variety of advanced topics in mathematics are covered, and problem sets ask students to prove many fundamental theorems of analysis and linear algebra. Class meets three hours per week, plus one hour of section, and problem sets can take anywhere from 24 to 60 hours to complete[…The] class often contains former members of the International Math Olympiad teams, and in any event, it is designed for people with some years of university level mathematical experience. In order to challenge all students in the class, the professor can opt to make the class very, very difficult.
You should take [Math 55] if one or more of these describes you:
* You are fairly certain that you want to be a math concentrator and want to be challenged to your limit.
* You have a solid base in advanced mathematics and are very comfortable with proofs and rigorous arguments.
* You want math to be your most important class.”
After looking at the course site and reading about the course elsewhere, this description sounds about right.