i'll be attending harvard next year (deferred entry) and am planning to use this year to study calculus and linear algebra in preparation for either math 25 or 55. (See here for the various courses: http://www.math.harvard.edu/pamphlets/beyond.html ) I know math 23 will be too easy because I tried the problem sets and read some of Ross's Elementary Theory of Calculus and it seemed a tad too easy for me in that he explains too much and none of the problems are challenging.
However while going over the descriptions for 25 and 55 I noticed something contradictory which I hope someone can elaborate, especially if they have first-hand experience with either of the courses. The full description is:
"Math 25 and 55 are both full-year advanced courses designed for students with a very strong interest in theoretical mathematics. Each covers multivariable calculus, linear algebra, and some additional topics from a rigorous and advanced point of view. The students in these courses are frequently committed to concentrating in mathematics and are asked to put in extensive work outside the classroom. Many have had more than one year of college mathematics while in high school or have participated in various summer math programs. However, it is not necessary to have had multivariable calculus before taking 25 or 55. Although the syllabus of Math 25 is similar to that of Math 23, students will usually have had more preparation in math.
Math 55 is a faster paced course and covers topics more deeply. It is designed for students who arrive at Harvard with an extensive background in college level math. Math 25 and 55 differ from Math 23 in the level of outside work required: homework assignments in Math 25 and 55 are typically very time consuming. Math 23, 25 or 55 all provide an excellent foundation for further study of mathematics."
Notice that it says:
"However, it is not necessary to have had multivariable calculus before taking 25 or 55."
Then is goes on to say:
"It [Math 55] is designed for students who arrive at Harvard with an extensive background in college level math."
Now this is confuding; on the one hand, it says you can take 25 or 55 without multivariable calculus, which is really equivalent to "without college level math" (since most high schools cover everything below multivariable calc). On the other hand, it is saying that math 55 is designed for people with extensive background in college level math.
I don't think this was very smart of them because it leaves people very confused when trying to figure out which course suits them the most. Therefore, I am hoping someone can elaborate slightly on this. Also, my background in math is up to single variable computational (or "mechanical" as some people call it) calculus, but I have extensive (at least 2 years) first-hand experience with proofs from dabbling in competition math, and I have solved some very tough olympiad problems during my time (including several of modern IMO caliber), so I know the basics of some discrete math topics (mainly number theory and combinatorics - functional equations and inequalities as portrayed by competition math never appealed to me, having a sort of cheap plastic feel to them) as well as various proof strategies and general problem solving tricks.
This is probably why I found the course 23 not appealing: the course isn't challenging for me at all, but it has tons to offer with respect to knowledge, so it can probably teach me the most. On the other hand, this may back-fire because I'll find it boring due to the lack of challenge.
Any advice is highly appreciated!