This Freshman Math Class at Harvard Looks Insane

[ActionPotential]

If this is the wrong forum for this, I apologize. I tried to narrow this question to the right section.

I somehow ended up on a Harvard undergraduate maths webpage for incoming freshman. The first two courses I read were pretty gnarly but I have read enough posts on this website throughout the years that I have become familiar with the kinds of people who would excel in these courses out of high school. However, the third course that I read sounded unreal.

I was curious if anyone has taken this course and what it was like? I can't fathom the type of people who are this advanced at mathematics by the time they are a freshman. Granted, I was never talented in maths (I always have had to work hard to do well) but I can abstract my experience far enough out that I can imagine how/why more talented people that I have met are better at solving problems.

Mathwonk: If you happen to read this post, let me know if this is the kind of course you took. I have read some of your anecdotes and distinctly recall you discussing how insanely difficult your freshman math course was (using Spivak I believe). I have always been curious.

http://www.math.harvard.edu/pamphlets/freshmenguide.html

"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.¹ This class is usually small and taught by a well-established and prominent member of the faculty whose teaching ability can vary from year to year. A thorough knowledge of multivariable calculus and linear algebra is almost absolutely required, and any other prior knowledge can only help. Students who benefit the most from this class have taken substantial amounts of advanced mathematics and are fairly fluent in the writing of proofs. Due to the necessity of working in groups and the extensive amount of time spent working together, students usually meet some of their best friends in this class. The difficulty of this class varies with the professor, but 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 this class 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."

¹This is what really stood out.

Another Question:
I have always been curious what the experience is like to be able to connect so many seemingly disconnected and generalized concepts. Are you able to follow the threads of objects and interactions and sort of watch it emerge into a connected, unified, network of inner-reality? I mean, when your mind is that advanced mathematically, is the mathematical world you experience in your mind far more interactive than the average person? Is it possible to even describe?

 

[WannabeNewton]

I had a couple of friends at a rival high school (Stuyvesant) who took point-set topology, real analysis etc. at nearby colleges while still in high school. It isn't as uncommon as you would think, that is kids who are very advanced as far as mathematics and physics goes by the time they enter college. Certainly it isn't the norm but it isn't uncommon. Also take a look at honors analysis at UMich and UChicago.

 

[ActionPotential]

Man, that is crazy to me! Did these kids still have varied interests or were they pretty dedicated to mathematics and physics for most of their life? I always wonder if the kids who are that advanced necessarily have a unique talent for mathematics or if perhaps some of them simply practiced a lot from a very early age and their interest, passion and hard-work remained consistent through their preteen and teen years.

I had no desire to learn mathematics in high school and so I never practiced and never really learned a strong foundation. I thought you just had to have a talent for it. After high school, as I grew older, I discovered a beauty for mathematics and realized that I had to work hard to learn Algebra and Trig the correct way and now I am learning Calculus and working through Spivak as best as I can in my spare time. I wish I could have realized how important practice was when I was young as I would be much further along but I also might not have developed the passion for it that I have now, so who knows. /ramble

Obviously some brains just seem to be constructed especially for mathematics while other's aren't quite as finely-wired.

EDIT: Forgot to ask - what kind of maths require 24-60 hours to solve the problem sets (in the context of this particular math course, not in general)?

 

[WannabeNewton]

Knowing them personally, they were pretty vested in mathematics but they still enjoyed non-academic extracurriculars like piano, violin etc. I don't want to get into the can of worms of talent vs. hard work but suffice to say they are very bright kids. The interest and passion to actually study advanced topics while still in high school also comes into play of course.

Forgetting math 55 for a second, there are problems in Spivak's calculus text itself (which is used in various freshman honors calc courses) that would take a very long time to solve; some problems in analysis simply take a while before they click. IIRC, the problem sets from previous math 55 courses can be seen online so you can judge for yourself.

 

[ActionPotential]

I wasn't intending to open a "can of worms" my man. I really kind of just went off on a stream of consciousness style tangent related to lower levels of mathematics. I have no experience nor opinion regarding anything beyond single-variable calculus which is probably why I am fascinated by these types of people.

 

[WannabeNewton]

Well certainly it would be nice to be one of those people but one does the best that one can do

 

[CubicFlunky77]

I will only speak for myself: If I call a subject hard, it makes it that much harder for me to get through it. If I allow my passion/interest to guide me I will pursue a certain subject for as long as my passion/enthusiasm dictates. Ever heard the cliche: You can do anything you put your mind to? I do not believe in frivolous words like hard,challenging, difficult or whatever 'intimidating' words people (mostly pedantic people with polished terminal degrees) use.

My advice to anyone would be: STOP thinking about what other people are doing. Do NOT try to be the best because you will fail miserably. Statistically speaking anyone person who is the 'best' at something has an extremely high chance of being 'out-bested' by someone else simply because there are so many darn people on Earth! Find something (it doesn't have to be math, physics or whatever) you are interested in and DO IT. Don't label it. Just do it.

 

[jgens]

I cannot speak for the courses or the people at Harvard, but I did go through the math program at UChicago, so I can add my two cents about that:

I went to a fairly ordinary high school and for my first three years there I had no real interest in mathematics. During my freshman and sophomore years I actually wanted to be a musician and I spent a huge amount of time working on that. Then sometime during my junior year I lost my passion for music and decided that I should pursue physics instead. Unfortunately this did not mean studying more physics on my own. Rather it just meant taking whatever physics-related courses my HS offered. Anyway it was only during my senior year that I discovered my passion for mathematics. During this year I did work through a decent portion of Calculus by Spivak, but that was the extent of my mathematical education. All in all my education was fairly balanced. I spent about as much time working on math and physics as I did working on English and history. I also did normal high school things like hanging out with friends, playing music, doing sports, etc.

So at this point I went off to the University of Chicago. One of the first things they have you do during O-Week is take a math placement test. The test itself was pretty simple: The first half was just easy single-variable calculus stuff, while the second half consisted of a few easy proofs. I did pretty well on this and got placed into the honors analysis course (Math 207-208-209). As a first year I was worried about taking such a demanding course and some of my peers in the class had fairly intimidating mathematical backgrounds compared to my brief affair with Spivak. But what I found is that none of this really mattered. If you put the work in, then everything turned out fine. A big part of undergraduate math education is learning how to think about mathematics, and while some kids certainly had a leg up initially, when you are putting in 40-50 hours a week just on math, that gap closes surprisingly quickly. I suppose the point of all this being that while you might feel behind now, it is really not so difficult to catch up.

Lastly to answer your question about what kinds of problem sets can take 20-60 hours to complete, the truth is that there are all sorts of ways this can happen. Some problem sets are just extremely long, while others have extremely difficult or open ended questions that require a lot of work to make headway.

 

[WannabeNewton]

^That's pretty awesome man, considering you're just going into senior year now and know so much math and physics already. Quite inspiring really :) cheers. Although, dare I say, you're being modest ?

Just out of curiosity, how'd you manage to learn so much by the time you were a sophomore/junior anyways e.g. differential topology and algebraic topology, given what you said about your mathematical background when you entered. Was it a lot of self-studying?

 

[jgens]

Just out of curiosity, how'd you manage to learn so much by the time you were a sophomore/junior anyways e.g. differential topology and algebraic topology, given what you said about your mathematical background when you entered. Was it a lot of self-studying?

A lot of self-study combined with a math-heavy course load. I mean as far as courses go I managed to exhaust most of the undergrad math classes by the end of my second year (the trade-off is that I am still taking core classes). The other thing that helped is the UChicago REU.

 

[WannabeNewton]

Haha exhausting most of the Uchicago UG math classes by the end of 2nd year, now that's really awesome :)

Thanks for the input!

 

[ActionPotential]

For clarification:
I am 27 years old (Undergrad Freshman). I have worked different jobs, lived in different parts of the country and pursued different interests. After these years, I finally decided I was ready for school. I am not feeling behind or confused about what I want to do or anything like that. I have actually been excelling far beyond what I ever thought I could do.

I have always just been interested and fascinated by people with exceptional talents in particular areas (athletics, music, arts, maths and sciences, literature, medicine, etc.) and since I have only recently been able to understand what differential calculus is, it made me start to appreciate the minds of those who are working on advanced maths.

JGens: I thoroughly enjoyed your anecdote. Great story!

I have been enjoying everyone's perspectives! :)

 

[mathwonk]

I went to Harvard in 1960, not having had any calculus at all in high school and very little math by the standards of top schools, just basic geometry and algebra, not even trig. In those days there was a freshman course before math 55, called math 11, and I took that. I was one of the better enrollees in terms of ability, but a hopelessly poor studier. I went from a B- to a D- the first year in math 11, attending only once a month the second semester.

After third semester I was kicked out, worked a year, then returned for my 4th semester, taking diff eq and linear algebra, and became motivated again to continue with math 55 which i then took as a junior. With this advantage, having had linear algebra, and having acquired some discipline (I had learned to attend class e.g.), I succeeded in the course, getting B+ and A- in the two semesters.

In those days the course was basically what is in Dieudonne's book Foundations of modern analysis, but the official text was by Wendell Fleming. The next year the professors, Loomis and Sternberg, wrote their own book.

A few years ago Wilfried Schmid taught it and my friend's son wrote up the notes. That young man had taken not only calculus before college but several advanced graduate classes as well, and was highly qualified to take the course as a freshman.