<p>I think of numbers when I hear “engineering” or “applied math.”</p>
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<p>I’d say you’re right. My gut feeling is to take the undergraduate offerings this year, trying to do exceptionally well in them, and then to take graduate courses my remaining two years. </p>
<p>Surely two years of graduate courses as an undergraduate has got to count as something. I’m simply afraid it won’t count for enough to matter to top graduate schools.</p>
<p>I also know that many of the summer REUs expect an inordinate amount of graduate-level work. It makes it feel like I’m playing catch-up with people who had extraordinary opportunities in high school and were able to take analysis and algebra their first semester of college. The worst part of it is that it seems like there’s virtually no way to actually catch up without jumping ahead three years in the curriculum, a step that seems a bit too large for me.</p>
<p>OP, some of my peers took Master’s level Abstract Algebra their sophomore years. Then again, I go to a school with a ton of smart kids and these were the smartest of us all. If I were you, take the undergrad levels as a sophomore and take the Masters levels as a junior if you found them to be “easy”.</p>
<p>I know no one who did analysis or algebra their freshmen years. I’m pretty sure most universities would tell a freshman who has passed out of linear algebra and DiffEQ to take other upper level math courses that are slightly easier like number theory before jumping into analysis and algebra.</p>
<p>A few schools offer such courses to entering freshmen. The most famous is Harvard, which presently offers abstract algebra, abstract linear algebra, real analysis, and complex analysis to entering freshmen in a two-semester sequence known as “Math 55.” The exact content of this sequence seems to vary from year to year. The topics I listed seem to be a good description of its content last year (all the resources were posted online, including lecture notes, some of which I read since they were of high quality).</p>
<p>The material is at the undergraduate, not graduate, level. Still, students who complete this sequence (and equivalent sequences at select other schools) typically go on to take graduate courses only for the remaining three years. (This is understandable since that one sequence includes almost all of undergraduate mathematics!)</p>
<p>I’m so glad I left mathematics as a major. I got so much from it and I was pretty decent, but the study of it pained me so much. I read this stuff and I cringe. But I do cheer those that like it. (:</p>
<p>Here’s an idea. If you’re both concerned and self motivated, give an undergraduate text a shot this summer. If you make substantial progress, you’ve both prepared and proven yourself. I’ve heard good things about Pugh for analysis: not quite the standard that Rudin is, but supposedly it delivers a similar understanding and is a bit more illuminative for the self instructed.</p>
<p>I had this same idea and have already picked up a book from the library. I picked up the same book that the undergraduate analysis class here uses so I could get a feel for whether I’m familiar with the material already or not so much.</p>
<p>Thanks again to all the people who have written here.</p>
<p>how many people at Harvard take math 55? At caltech my friend took a really proof based calculus course his freshman year which probably is as tough as the analysis course I’m taking next year. I wouldn’t worry about “only” having 2 years of grad courses though. How many undergrads really have the opportunity to take 3 years of graduate level math outside of super prepared ones at tippy top universities.</p>
<p>I will give you the advice I always hear from my husband, who teaches in a top engineering department: college is not a race! It is far more important to get a solid foundation and excel in undergraduate courses than to skip material and enter grad classes before you’re ready. </p>
<p>I should add that in engineering, it is very common to take more than 4 years to graduate, so students don’t need to feel rushed to reach upper level classes. I don’t know if an extra semester is an option for you, but be aware that grad schools do not care how long it takes to complete all the courses you feel you need. It is very common here for undergrad students to take an extra semester of grad classes and then do a semester of research before applying to grad school.</p>
<p>You sound like a great student who is doing all the right things. I think it’s a good idea to discuss your future plans with professors in your department and ask if there are any opportunities for research at the undergraduate level. Get to know your professors and continue your excellent work so that you’ll be able to compete for a summer REU. Good luck!</p>
<p>10-13 take Math 55. It seems that 2-3 usually could have taken it if they had been given the opportunity to get a background in proofs. But not all of those people will move on in mathematics.</p>
<p>I think spending an extra semester or two could be a pretty good idea depending on how everything works out. After the discussion in this thread and some discussion I’ve had with others, I’m almost certain at this point I’m going to stick to undergraduate courses next semester.</p>
<p>lol ur making me fearful to be a math major. My school only offered AP Calculus AB so i’m going to have to take calculus 2 my first semester which definitely puts me behind the curve. This summer i’ve been going through some problem-solving books which cover topics usually present on the AIME and USAMO ( combinatorics, probability, elementary number theory, sequences and series, tough geometry, and of course proofs) hopefully this will help a bit.</p>
<p>all good advice given here so far. but I can’t stress enough, talk to your math advisor or seek advice from a professor that you feel comfortable with. First, they know all the ins and out of the program and can be an amazing resource. Second, once you find some one that you feel comfortable talking to, keep talking to them and keep them updated. This relationship will be real helpful in applying to grad schools, because many times it comes down to who do your professors know. Third, independant study is often a way to bridge the gap between undergraduate and graduate level work, as you can pace yourself and adjust accordingly. Based on the descriptions of the grad classes you gave, it might not hurt to drop an email to professor teaching the class. Tell them the classes you took and ask if it’s appropiate for you to take the class. If there’s an option to audit the class, you can do that to and drop out if you can’t handle the work.</p>
<p>ChaoticOrder, my high school actually didn’t offer AP Calculus (neither AB nor BC). In fact, it didn’t offer any AP classes at all.</p>
<p>In order to catch up, I took summer classes, so I ended up being able to finish the honors calculus sequence, linear algebra, and differential equations that year. Good luck. :D</p>
<p>Also, you shouldn’t let my posts inspire fear in you. I tend to be overly worrisome about academic matters. Don’t let that rub off on you. </p>
<p>Insane_Membrane, when you say independent study, do you mean a formal independent study program sponsored by the school or just simply studying in one’s spare time? My school has something like the former, but it’s only for graduate students and is intended to let them receive credit for doing research; I’m pretty sure there’s not something like that for undergraduates.</p>
<p>As I said earlier, find a professor to talk to and explain your situation. Even if the independent research is only an option for grad students, it doesn’t hurt to ask. At worst they’ll say no and suggest something else and at best they’ll make an exceptions for you. I’m not sure your university’s specifics, but if you want pm their website, I can check it out for you. Usually, this info is listed on the department website but can also be fund on the registrar’s site.</p>