<p>Hi. I'm currently registered as a Mthematics/Actuarial Studies Major, but I'm right at the beginning of my college "career"- I'm taking Calc I now, next semester I'm slated for Calc II, Probability Theory, and Theory of Interest. </p>
<p>I was wondering- what exactly is math really about, at the higher level? Is it a continuation of the math stuff I've been learning all my life? Or does it morph into something different? Does it get much more difficult at the higher level courses? Is it so much more difficult than other majors? </p>
<p>I'd appreciate any information you can give me, but please do include some info about yourself and your credibility- are you a math major/grad, and what did you study?</p>
<p>Hello. In my opinion, math does get easier high up–perhaps because you will start to find greater meaning in it, or start seeing it around you. I think at the higher levels is more about concepts (it will de emphasize rote tasks). </p>
<p>Well, one branch of higher math deals with a 3 axis (3D) coordinate system, and plotting 3D parabolas and spheres, etc.</p>
<p>At most colleges courses up through multivariable calculus (calc 3), linear algebra and differential equations are very reminiscent of the way math was approached in high school. You will learn new concepts and theorems in class, and then spend a lot of time with computational examples, both in class and on the homework. As leolibby said, you will probably spend less time with rote tasks, but the focus is still on doing computations.</p>
<p>Then there’s what feels like a big disconnect between these classes and upper-level courses like real analysis and abstract algebra. The focus of your classes will suddenly shift from doing computations to writing proofs. (The theorems you learn and prove are still useful for computations, but you will rarely be asked to actually compute something.) It also gets a lot more abstract. Instead of thinking about 3 dimensions, you will be studying n-dimensional spaces (where n could be 3 or 10 or 1843 or infinitely many). Instead of doing arithmetic with real numbers, you study other systems you could do arithmetic in (e.g. systems with only finitely many numbers). Instead of studying polygons and curves and surfaces in 2- or 3-dimensional space, you will think about “objects” that don’t quite fit into our 3-dimensional world at all - objects which you can define in abstracto and say quite a lot about, but which are too crazy to build any sort of physical model of.</p>
<p>At this point it is hard to say if you would enjoy abstract math or not. Many students hit real analysis or abstract algebra and decide that they are more interested in a quantitative field than pure math. Others don’t really find their passion for math until they get to the abstract courses. Everyone I have ever met agrees that abstract math courses are harder than the computational math courses. Above all, they require a tremendous amount of dedication. It might take you several hours to solve a single problem. Most people find this process frustrating rather than enlightening. </p>
<p>But I wouldn’t worry about it at this point. Just enjoy the classes you are taking now!</p>
<p>b@r!um- thank you very much for your reply, I appreciate it. Question: You said many students find they’re more interested in a quantitative field than pure math. Is applied math included in quant. fields? Is transitioning from pure to quant at that point simple/easy/reasonable? How does one go about that? I just want to know if I should do anything now to prepare for that time, and if majoring in math is a good way to do that. Thank you!</p>
<p>P.S. More responses are always welcome. Lurkers, I know you’re there! Come out and say what you’re thinking, FWIW!</p>
That depends very much on how it is taught. Applied math can feel very much like pure math, in the sense that you can spend most of your time writing proofs. That might be more common on the graduate than the undergraduate level though. Most undergraduate programs in applied math seem to emphasize how to use the models and methods without going into too much detail about why they work. </p>
<p>Generally speaking, I think there might be two major differences between pure and applied mathematics. First, pure math tends to work with ideal conditions, while an applied mathematician has deal with a lot of variables beyond his control (imprecision of measurements, etc). That’s why pure math produces much stronger results than applied math. Secondly, a pure mathematician is often only interested in the existence of a solution, while an applied mathematician is interested in algorithms to find it (and their complexity, numerical stability, etc).</p>
<p>Judging from the people I know and their career paths, it seems to be much more common to switch from pure math to applied math than vice versa. “Cutting-edge” applied math actually uses quite sophisticated tools from pure math. A solid background in the traditional areas of pure math will help you a lot if you decide to pursue a graduate degree in applied mathematics or a related field.</p>
<p>For example, the robotics lab at Penn sends their graduate students to take differential geometry and differential topology classes otherwise taken by pure math PhD students. Apparently they use these concepts in robot motion planning. Or look at economics graduate school admissions. The selective economics programs expect their applicants to have quite a bit of training in pure math, including real analysis, topology and PDEs. </p>
<p>On the other hand, if you are more interested in a lucrative job straight out of college, a more applied undergraduate program will probably serve you better.</p>
<p>There definitely is a shift towards theorem proving. I remember in my calc 3 class, my professor would spend the day doing example calculations from the book, now I’m in his linear algebra class where a normal class is : Theorem, Proof, Theorem, Proof, Theorem, Proof, example, Theorem, Proof. Also, the focus shifts to be more on the general structure of what you are studying, like in analysis we only ever studied the properties of integration and how they are proved and the difference between antidifferentiation vs. integration with the Riemann integral. Right now in my linear algebra class we’re studying the various spaces isomorphic to the Hom vector space and Sylvesters theorem along with spectral theorem for operators. There are almost no calculations in this section and it is expected that homework will include a few problems which will be proof oriented and quite challenging. However the de-emphasis is also a blessing as a professor can truly evaluate you for your work and your thought process, as there is no final answer, and there are always multiple ways to go about a problem.</p>
<p>I personally think you should enjoy and be good at both proofs and computation for an applied math major, as you will undoubtedly see hundreds of proofs in applied math classes as well as in homework and tests. But more importantly, you really can’t expect to retain any applications if you don’t at least understand the main idea of the proof. Otherwise you’re just memorizing recipes, so when you’re presented with a problem outside of class, not only will you forget the exact procedure, you won’t even remember which theorem to look up. In other words, if you find that you’re not interested in pure math, the odds of you getting a very useful education out of an applied curriculum are small.</p>