<p>I’m a 3rd year this coming year, and may be of assistance, given I’ve seen enough of the non-lower-division coursework.</p>
<p>The difference in upper division work is that you’re assumed to be there to learn the math theory. Many lower division classes aren’t run explicitly like math courses, where formal mathematical communication and reasoning are only for “the math majors” – well, in the upper division, everyone is treated like a math major. The goal of the pure math classes is for you to learn the major theorems and framework behind a refined theory which is considered a basic prerequisite to thinking about more complicated things. For instance, the rough aim of Math 110 will be to classify as much as you can about linear operators on vector spaces – you start off with axioms defining the domains of these functions, and then develop the theory of special linear operators which naturally show up in a diverse array of fields outside of and within mathematics. </p>
<p>In a course like Math 104, you work with objects that are no longer discrete, and thus must develop a formal way of doing any reasoning at all with them. The axioms here will be more intuitive in a sense, because they involve things like distance, which you can truly use your physical intuition to picture. Though, the challenge is translating physical intuition into clean mathematical writing. </p>
<p>As you go on after that, there are courses which use the theory of these objects that commonly show up in mathematics to do more sophisticated things. The goal, it must be stated, is not to write proofs, but to understand a theory, and be able to communicate in its language, which <em>does</em> involve communicating in a formal language. If you would rather use the math theory others have come up with to see applications to something you can more tangibly see, the math major may not be for you, even the applied math one, given a lot of pure math classes are required. </p>
<p>As mruncleramos says, a little ability and a little effort (which usually means some enthusiasm) will get you a long way. </p>
<p>As you go up in the math classes, actually, your homework becomes more and more important, because fewer things can be covered effectively on an exam. You’re no longer developing skills – you’re developing a working knowledge and ability to think about the main ideas of a refined theory. Thus, spending lots of time working at home and reading and writing sometimes lengthy proofs becomes necessary. Different professors will put different degrees of emphasis on different aspects. Some professors in upper division classes will make it as much as 40% credit for your homework. Though, of course this means you really can’t slack off on your weekly work. This is obviously unheard of in Math 1A-54, and it’d not make sense, given if they’re testing skills rather than knowledge of a theory, an exam makes more sense. Makes less sense as you go on and on. </p>
<p>Overall, I agree with posters who say to try it out and drop if you need to. Usually you’ll know by your first upper division class if you’re built for it.</p>
<p>And with that, I’ll unfortunately not be immediately around if anyone has something to ask, but if you PM me, I probably will get back in a few weeks. I probably can answer most questions about the math department that’d come up here.</p>