<p>I'm not sure where I should put this, but you guys are generally very wise. :D </p>
<p>What is a typical math sequence that many math majors in college follow? How about non-math majors? List a few, if possible. </p>
<p>Are college math credits taken in high school usually transferred? What if you're well into the graduate curriculum by the time you graduate high school? Assuming that the student has studied these topics with the rigor of a college math class [and has taken many of these classes from a college itself], what usually happens when the student enters college? </p>
<p>There are several entry points into a math major in college. In some colleges this is Calc 1 & 2 (the equivalent of AP-Calculus BC). Most prospective math majors in top colleges, however, already have AP-Calc, so they begin with Multivariable Calculus and Linear Algebra. At some colleges, there are different levels of these two courses, some are plug and chug, some are more proof-based. Prospective math usually majors go into the proof-based courses. If they switch into the major after taking non-proof-based MV Calc and LA courses, they usually take a course that is designed to teach them how to do proofs.</p>
<p>If you have already covered some college level courses, you will place into more advanced levels; some colleges give you credit for college courses taken in high school; others will not. You need to check the credit policy of each college you are interested in. At my S's college, no credit is given for college courses already taken, so advanced students have to take graduate courses in certain fields as there are distribution requirements within the math major (algebra, analysis, geometry) and the number of courses required for the major is the same no matter how advanced a student is.</p>
<p>In some colleges, students eligible for Advanced Standing may opt for a four years AB/AM degree in math. That can be limiting in terms of elective courses as this means another 8 math courses on top of those required for the AB.</p>
<p>Just a couple additional notes to marite's excellent summary:</p>
<p>Most non-math majors don't go beyond basic Calculus, or maybe Multivariable Calculus. The exceptions are students who have very math-based majors, like Economics or Physics. If that's what you are interested in, you should look at some colleges' catalogs to see the degree requirements for majors in those subjects -- that will give you a sense of how many/which math courses they take (although some will take more). </p>
<p>What I observe (at a considerable distance) is that math as a field, beyond the required-Calculus level, is generally (a) not that crowded, and (b) pretty achievement-based. It is usually pretty easy to tell whether a particular student is prepared for a particular course, whether she's a self-taught 12-year-old or a college junior who has gotten all the right tickets punched in the right order. Of course, in the latter case, the transcript suffices, but in the former case a short test or interview, or even a review of some work, will do fine. Math departments tend to be both familiar with the fact that some students have done a lot of math before college and open to placing those students in the courses for which they are prepared. However, it's also often the case that even advanced students are not fully fluent in proof-based math, and so may overestimate their ability to handle high-level college courses.</p>
<p>math tends to be the subject (atleast at the schools that I'm familiar with) which has a ton of different sequences and classes for students of all abilities, interests, and majors</p>
<p>There's the normal version of multi/linear, there's a harder, but still mostly application sequence, and then there's the proof based one geared towards math majors. There is also a combined multi/linear course for econ majors. It is possible to become a math major from any of the non-combined sequences, but the students who are going to become math majors tend to gravitate towards the proof based courses. </p>
<p>If a student enters having taken lots of classes, they can usually place out of the earlier requirements and take more upper level classes. As marite said, they still probably have to fulfill the requirements of the department in algebra/geometry/analysis, but that can be done with upper level and grad classes. </p>
<p>I disagree with JHS about it being an achievement based field. At the very upper levels of math its a very aptitude based field, it is not enough to work very hard in the course, or to have taken all of the right courses and worked hard in them. The upper level classes, atleast at my school, have a very bimodal distribution of grades, with a set of the class getting extremely high grades and a set hovering around mediocre.</p>
<p>DS1's experiences in looking at colleges is that the ability to write proofs is the linchpin to higher level work in math. If you've gotten through MV/DiffEq and Lin Alg, but have not done much work in proofs, that will likely be your first stop. A number of colleges DS has considered offered a proof-writing class for kids in that position. Other schools, once students have gotten through the basic Calc/DiifEq/LA sequence, throw kids straight into Analysis with proof. In any event, the opportunity to get placement beyond what BC Calc would get you seems to be standard operating procedure, and we have found this even if the courses were taken at a high school rather than a college.</p>
<p>Look at the online college catalog for the schools you are interested in. They will tell you required/recommended courses for each major, with suggested timelines. Different schools do/don't accept AP credits, may combine topics differently in courses, have different reqs. You can't always tell content by course title, either. Which courses to take/place out of would be the sort of individual thing to discuss with the dept. For example, an Honors sequence of 3 semesters of calc+linear algebra, highly theoretical, may be better than getting the AP credits for HS calc, doing 2nd and 3rd sem calc + lin. alg, plus a later theory course...</p>