I am an incoming freshman at the University of Chicago and had a quesition about which math course to take my first year.
I've done my fair share of math during high school. I took two years of Calc (5 on the AP BC exam), some number theory, linear algebra, group theory, high-level geometry, algebraic topology (at SUMaC, if anyone knows of it). Although I wouldn't say I can regurgitate all that material right now, the point is that I've seen some college-level mathematics.
Now, as someone who wants to major in mathematics or mathematics w/ specialization in economics, I didn't know which course I should take. From what I gather, I have little to no chance of getting into Honors Analysis. (Even if I could, I don't think I would want the load.) So, that leaves me with either 16100-16200-16300 Honors Calc or (apparently new) 19900 Introduction to Analysis and Linear Algebra.
What do you think will be a better fit? I'm worried that taking a whole year of calculus again will leave me with no time during junior and senior years to take as many high-level math courses as I'd like. It'd be also cool if I could take Honors Analysis sometime--would one choice be better for this than the other? I'd really appreciate any advice from phuriku, mathgrad, and other math students at Chicago that I've seen on CC..
Parent of Chicago second-year math major here...
When you get on campus, go have a chat with Dianne Herrmann or John Boller. They are undergrad math advisors and are awesome folks, by every account I have ever heard.
The calc placement exam will give you a sense of where you should be -- but S1 had a math background similar to yours (but with a lot of proof work) and went directly to Analysis 203 IBL. Qualified for HA, chose the Inquiry-Based Learning section of analysis (where students derive the proofs and everything themselves) and loved it. He will be taking Honors Algebra this year.
The question in my mind re: placement is whether you have done proofs. If you want the option of taking HA as a second year, you must take the Honors Calc 16000 sequence and then apply for HA.
15000 Calc >>> 19900 >>> goes to Analysis 203
16000 Honors Calc >>> goes to Honors Analysis 207 or Analysis 203
19900 (Intro to Analysis) >>> goes to Analysis 203
20300 (regular or IBL Analysis) >>> goes to Honors Algebra or Algebra
20700 (Honors Analysis) >>> goes to Honors Algebra or Algebra
The Honors Calc 16000 sequence also has an IBL section. It is heavy on proofs and does Lin Alg at the same time. S spent a lot of time helping folks with 15000 and 16000 courses, and says the 16000 series is NOTHING like BC Calc. No regurgitation.
My recollection is that Phuriku took HA and mathgrad did 160s followed by HA, so if they chime in, you'll have three perspectives on ways to do these tracks.
While you are waiting on phuriku and mathgrad to chime in, I want to underline Counting Down's point about talking to Hermann or Boller. The University of Chicago math department has plenty of experience with educating math-centric undergraduates. They have absolutely no interest in "keeping you down"; there's no big overhang of bureaucratic inertia. In fact, they want you (and everyone else) to fly. They (and their placement test) will probably be able to figure out where you are on their map of math development in a matter of minutes, and what your best options are.
Rather than trying to figure things out completely on your own, you should probably listen to them.
Well, let me talk about Honors Analysis first, as well as some other higher-level courses here.
So Honors Analysis is extremely theoretical. You won't find anything of use there for economics, if that's what you plan to go into. It probably won't help your grades either, since it requires a lot of time dedication. If you want to do it for fun, then go for it, though. I know a few econ majors who did it as 2nd years and thoroughly enjoyed it AND did well in it.
With your background, you may be able to place into HA. Before coming to Chicago, I had only taken BC, AP Stat, Multivariable Calculus, Differential Equations, and Linear Algebra, and none of them were as theoretical as they should have been. I decided to get into Rudin's Principles of Mathematical Analysis during the summer, and that is what eventually led me into Honors Analysis. If you really know some group theory and algebraic topology, I'd say you have a pretty good chance at getting in.
Having taken Honors Analysis, I've also taken the following courses: Honors Algebra-1 and -2, Functional Analysis, Topology, Complex Analysis, and Partial Differential Equations. This is all after my 2nd year, so assuming you took Honors Calculus your first year, you could take all of these by your 3rd year, or any other courses in its place (I'm an analyst, so these are the only high-level courses that are appropriate for me, since I found that the Differentiable Manifolds course here is basically a Differential Topology course). Also, note that I have taken only 3 classes per quarter every quarter (although I had something like 10 credits coming in, so I haven't had to worry about the Core as much). This coming year, I am taking the Graduate Analysis sequence, 4th-year Japanese sequence, and Accelerated Latin sequence. Why the latter two? Because I've pretty much run out of math classes to take that I'm interested in. I could theoretically take Graduate Algebra or Graduate Geometry/Topology as well (assuming Sally would let me to take these simultaneously). This means that theoretically, even if you start out with Honors Calculus, you should be able to take all the high-level courses you want take by the end of your time here, including all 3 of the core graduate mathematics sequences (assuming you're willing to sacrifice some electives, of course). So there should be very little worrying on your part.
I should put out there that I'm heavily biased in favor of our Inquiry Based Learning courses and IBL in general, so you should probably take my advice (and, for that matter, everyone else's - we've all got our own biases and assumptions) with a grain of salt.
That said, with your background I think you may really enjoy IBL 160s: you build up the real number system from the ground up, do all your own proofs, and really *own* the material in a way I've not experienced since (I took this course back when they were still calling it "the experimental section" - clearly the experiment was a success, as they've expended to IBL Analysis and the occasional IBL topics course as well). There's very little repeat between this and your typical high school calculus course (I actually saw a lot more similarity to my point-set topology course third year than anything we did in AP BC Calc), and it's just a tonne of fun - albeit perhaps a bit more work than your typical calc class, definitely less than the infamous Honors Analysis, and (in my opinion) more interesting work seems easier to attack even if it's harder/takes more time. The real benefit for your situation, though, would be scheduling - it sounds like you'd be able to place into Analysis, and I'd completely agree with you that while some first years do take Honors Analysis, it's really something that's better left for later - let first year be about settling in to college, making friends, learning the city, managing roommates/dormmates, and whatnot, save the chaos of one of the hardest undergrad courses in the country for once you're a bit better established here. 199, meanwhile, is an awesome course, but takes only one quarter - if you're set on taking Honors Analysis, which with only one section begins in the fall, you either have to forgo two quarters of math or find other maths with low-level reqs to fill out your schedule. IBL 160s, meanwhile, firmly solidifies your proof style/background and takes all 3 quarters, very convenient for those hoping to take Honors Analysis. (I'll spare you my rant on why Honors Analysis is not worth it, because I know coming in to the school I wouldn't have listened to me, either - I was taking that course come hell or high water)
A year in 160s will definitely not keep you from the upper level courses: I feel I took plenty, and never felt held back from not taking Analysis first year. My math (ok, some are technically CS) courses:
1st year - IBL 160s
2nd year - Honors Analysis
3rd year - Honors Algebra, Point-Set Topology, Honors Combinatorics
4th year - Discrete Math, Algorithms, Logic 1&2, Representation Theory, Commutative Algebra, Algebraic Topology
(the lack of things like complex or functional analysis comes not from scheduling issues, rather my preference for the algebraic over the analytic - could have fit them in if I was interested)
I'll reiterate phuriku's point: Honors Analysis will not help you with Econ. It didn't even help me with Analysis - I'm starting grad school now, and I know all sorts of awesome things from that course about the p-adics and measure theory, but have never formally seen Stokes' Theorem: I'm basically starting from zero in analysis. It's much more of a topics-in-whatever-the-professor's-into course than an analysis one. Useful in its own regard, but know what you're getting into if you're set on taking it (or if you're set on Econ).
And one more time, just for good measure: IBL is pretty much the coolest thing on the planet, and a really unique opportunity - very few places have such a program in place. I wouldn't trade the experience I got from that course for anything, and the people I grew close to through it became some of my closest friends and favorite people in college and beyond. Definitely not your high school's calculus, nor even a typical college calc.
Given the OP's background, I agree that IBL 160's is probably a very good choice. Note that mathgrad is suggesting that it's obviously not impossible to take Honors Analysis after taking 199 (probably by begging and/or impressing Boller or Sally), but given the perhaps insufficient amount of preparation in 199 as well as the scheduling issues, this is not the route to take if you really want to take Honors Analysis sometime in the future.
As a prospective econ w/ spec major who took non-IBL 160's this past year, I will share my opinion on the sequence. First of all, it will not help you in econ, but that's not really a surprise is it? A minor digression: for the primary sequence in economics (200-203), the most important mathematical tool you will need is perhaps lagrange multipliers, which is an optimization tool in multivariable calculus. Essentially, this comes down to computation skills in an AP Calculus course. It is probably true, however, that theoretical mathematics will help in econ research and in this case, the pure math background makes a difference.
In general, regular 160's is not a difficult sequence if you are familiar with proof writing or if you devote the time to learning the basics in the beginning. My experience with two different professors for the sequence (one substituted for the other during Winter quarter) suggests that most of the problems assigned are only meant to increase your comfortability with the material at hand but not to really challenge you. Similarly, many of the test questions are very similar to problems you have encountered though there is usually one problem that is a bit trickier than the rest. The professor I had for winter quarter (who in my opinion happened to be a better teacher overall) even made it so that you only had to do 4 out of 5 problems on the midterms to score 100%. In general, grades are easy to obtain in the course, and you may not feel sufficiently challenged at times. Although neither of these professors are teaching the course this year, I have seen material from other UChicago profs teaching the course and I don't think the difficulty level varies much.
Having said that, the text for the course is superb. In fact, Michael Spivak's Calculus is used at many North American universities for a theoretical, proof-based calculus course, and I suspect the level of teaching for this particular course doesn't vary much from school to school. The text (re)introduces calculus as a subject that is about estimation and serves as a bridge to more advanced analysis. If there was anything in your AP Calculus course that you felt was not fully justified, Spivak will rectify that. Spivak provides a lot of problems, and some of them are pretty difficult. If you work on the problems that you find challenging or interesting, you will likely gain a lot from the course.
S1 likes Spivak and got it to read for fun the summer before heading to Chicago. He was also a Honors Analysis-come-hell-or-high-water guy, sat in on HA, regular analysis and IBL analysis the first week, and decided on IBL.
His HS math was Analysis I (BC Calc on steroids), MV, DiffEq, calc-based stat, LinAlg, Discrete, Complex Analysis, Origins of Math (proof-based) and a summer at HCSSiM. Also took lots of heavy math-based CS courses and mathematical physics.
So far at Chicago he's taken IBL Analysis, grad Discrete Math, grad algorithms, will take Honors Algebra, Honors Combinatorics and something else (some sort of formal logic, probably) this year. Most, if not all, of his non-math major courses will be on the theory/math side of the CS department.
If you check the link I posted previously, there are requirements for the Math w/Economics degree, and IIRC, you have to get through analysis. Boller and Herrmann will let you fly whichever route you take. S was very successful at negotiating a good placement with them. They are also familiar with teh various summer math programs so they'll have a good idea of your skill level in various areas from your SuMAC experience.
Wow, I didn't expect such detailed replies! Thank you all so much. You're fabulous. If everybody at Chicago were this helpful, I'd be all set for the next 4 years =)
I'm pretty much set on 160 w/ IBL... I mean with a HUM/CIV class, Intro Chem (which I believe is infamously difficult to do well in?), and another TBD Core class, I don't think HA will be a good idea even if I could make it. (And I sincerely doubt I could make it.)
If anyone has any other opinions, I'd appreciate it any time!
Hey! I'm hoping/set on 160 IBL this coming year, as well.
Anyways, the other reason I'm posting is because you sounded like you were hoping to take 5 classes this fall. At UChicago you can only take 4, unless you get a waiver (and from what I've heard from some of the upperclassmen is very rare and even less students end up enjoying). This is all to say if you don't count P.E.
S1 intended to take four courses first quarter, decided to drop one because he realized he didn't want to do HUM and SOSC in one year. Said it was an excellent decision, as it enabled him to get out and meet a lot of folks and get involved with things on campus, yet still get his coursework done and get excellent grades in the process.
He takes four courses every quarter now, does research on the side and will be a Junior Tutor this fall, in addition to other activities he's involved with. He's happy.