<p>You have to test out of at least Math 1a and 1b (Calculus and Linear Algebra, respectively, but note that Ma 1a is not your typical high school calculus class, to be honest it’s closer to an analysis class than the typical AP Calculus class you might have taken in high school.) in order to test out of any math at all. (Otherwise you won’t be able to replace them with Ma 2a and b).</p>
<p>As to the advantages and disadvantages of testing out of Ma 1 and Ma 2, it really depends on your math background in high school, (e.g., how often did you have to do proofs?) and how much you plan on pursuing or caring about math at Caltech. IMO, Ma 1a doesn’t really teach you anything useful (assuming you know calculus pretty well) other than how to get used to proofs. Ma 1b and c on the other hand, are really quite useful, especially depending on your major and what track you take. The analytic track is a good deal more proof based and often approaches the course material in a different way, (esp. in linear algebra, where you start with vector spaces and move to matrices in the analytic track and vice versa in the practical track.) To be honest, if you can pass out of the class you should be fine, unless you were really struggling the with the test, and somehow only got the right answer by a random fluke of nature.</p>
<p>IMO, the most important subjects in core math are linear algebra and ordinary differential equations. (Everyone knows calculus, the multivariable calculus you need to know for most subjects is actually pretty easy, and probability and statistics you can usually learn as you go.) Most people will see differential equations again in ACM 95b, but for nearly everyone, Ma 1b will be the only class they take that is devoted to linear algebra, and the subject is pretty important, so if you feel that you don’t have a good background in the subject, I would advise against skipping out of Ma 1b.</p>