course


May 2014 
This is absolutely a worthwhile course to take if you have the necessary prerequisites and are thinking of becoming a math or sciences major. Don't be put off by NSOP's awkward avoidance of the Honors sequence  if you're interested in proofbased math and up for a bit of a challenge, try it on for size. Like several students in Honors Math, I took both linear algebra and multivariable calculus in high school, so I very much appreciated a differentlymotivated and more rigorous perspective. And this course will prepare you for upperechelon math courses like no other. For example, MuTao starts the first semester defining fields and finitedimensional vector spaces, definitions traditionally seen first in abstract algebra, and often uses functional notation rather than matrix notation for linear transformations. In the second semester, Prof. Wang introduces you to analysis with the completeness of the real number system, and throws quite a few bones to topology along the way. As a previous reviewer noted, MuTao is an excellent, excellent lecturer. He structures and organizes effortlessly, he tries to develop intuition (especially geometrically; as a analytic geometer he's also an accomplished sketch artist), he proves almost everything he states in a way that's easy to understand  and he responds well to questions. He's also very friendly and helpful in office hours. Homework assignments are due every class (the assignments due after the weekend are generally longer, 2~4 hours instead of 1~3). In the first semester, most of the problems are MuTao's own, but a couple of weeks into the second, all of them come from Apostol. With that shift comes a shift away from rigorous proofs and towards problemsolving. Additionally, the cramming of all of Calc III & IV into the second semester may at times feel a bit rushed. But all in all, this is not a difficult course to follow if you put your mind to it. Exams are all inclass and can be tough to complete at times, but Prof. Wang understands this and grades generously. As the previous reviewer noted, there's a good mix of problems, so you can play to your strengths, but there's not a lot of correlation between the material on problem sets and the material tested on the exams. That said, do the homework anyway because it'll enhance your understanding of the material from in class. Then when an exam comes around, do the practice exams, go to the review sessions (one before each exam in addtion to regular weekly recitation), and reread your notes  you'll be good.
May 2013 
Overall, I walked away pretty happy with this course. Based on reviews of previous iterations of the course, it sounds like this one wasn't nearly difficult was the killer courses with Savin and Friedman. It follows the second volume of Apostol's textbook very closely, and served as coverage of linear algebra and multivariable calculus, and basic proofs. Since multivariable calculus is covered in two separate courses, calculus 3 and calculus 4, taking this class is a good idea if you want get the material out of the way in a single semester. Professor Wang was an excellent lecturer who covered material very thoroughly in class. He moved quickly, but he was very thorough and wrote down everything that was important on the blackboard. He did tend to focus on long proofs that were never to be seen again, but he also did a great job conveying the intuition behind the material. Homework was assigned twice a week every week, and the length of the homework varied a bit, but it usually didn't take too long. The assignments were almost always out of the book, and this was probably the biggest weakness of the course, because the problems in the book were generally not all that great. During the first semester, when we did linear algebra, they were generally decent. During the second semester, when we covered multivariable calculus, some of the problems required excessive tedious algebra without having much value or relating to what we were emphasizing in class. There were three times when the professor wrote his own assignments that were not from the book, and honestly these were the most interesting and relevant assignments that we had. It would've been nice to have more proofs assigned during the second semester, where the homework assignments were almost entirely computational. The exams were very fair. Usually 40% of the points came from true/false questions and the remaining 60% of the points came from longer problems. There was usually a good mix of computations, conceptual problems, and short proofs. Usually the average was between 70 and 80. He clearly didn't design the exams with the intention of them being brutal and the averages being very low, as he expressed disappointment when the average was in the low 70s. He is also very good at writing problems that simplify very neatly, and is not interested in making the students crank through a lot of tedious algebra on the exams. He also does not penalize students multiple times for the same mistake, and if a problem has multiple parts, missing one does not imply you won't be able to do the later parts. The most relevant preparation for the exams was the practice exams that he provided beforehand, for which solutions were provided, and he or the TA worked them out in a review session the night before the exam. Seriously, these were golden. The exams weren't copies of these, but some of the questions on exams did emphasize similar points (that might not be seen in the homework) and got you thinking in the right direction. Comparatively, the homework was not very helpful for exam preparation.
May 2012 
Well, I just survived a year of Honors Math, and now that I've gone home and thought about it, here's my review of the course. To begin, Prof. Savin is a savant (look him up online and be amazed at all his accomplishments, which includes a perfect score at the IMO) who can effortlessly teach without notes and solve most challenging proof problems on the spot. Really, the greatest impediment to the teaching is his wonderful but sometimes incomprehensible accent, but his notes on the board and his office hours more than make up for this minor shortcoming. If you truly enjoy the intricacies of math, I highly recommend you take this class, which serves at the alternative to the laughable Calc III sequence. (The class started with 90 but eventually whittled down to about 30 by the end of second semester.) It's sufficiently challenging enough to get you wellacquainted with the higherlevel math courses in the department, and despite the incredibly low test averages (see below), there is a very generous curve that will save your grade at the end of the semester. Plus, the second semester of the course is much easier than the first; perhaps at that point, Savin has successfully conditioned his students to expect the worst on his exams and they react accordingly. Lastly, the only way to study for Savin's tests are to make sure your definitions and ability to do computations are solid, because you're going to get destroyed on the proofs he gives out regardless.
May 2012 
Savin is a brilliant guy, but brilliance doesn't necessarily come from other brilliant people. This is not his first time teaching a freshman course, but he clearly did have much higher expectations at the beginning than we were capable of fulfilling. Much of this was due to the lack of transition from computational math to proofs. His proofbased homeworks were very insightful, or at least they would have been if anyone knew how to do them, and those who claim they did can't really be sure since the TA was clearly not thorough. It takes a lot to teach this class, and Savin did a great job. However, there is a stern line between teaching how to get from point A to point B and just telling a class it should be at point B by next Monday. This was particularly true once multivariate integration came back and the class had absolutely no clue how to handle parametrization let alone surface integrals, divergence theorem, etc. Savin had little time, and that's probably the Achille's heel of this class, so he has to send his students to the book for practice. Unfortunately, Apostol (our textbook) sucks at establishing a basic understanding of the subjects it teaches. It teaches from a prooforiented perspective and then throws a bunch of poorly worded and arbitrary calculation based problems in the exercises, but I do have to admit the grueling differentiation exercises were good. All and all, the nature of this course is much like reviewers for other professors have stated, so I can't really blame him for the content. The curve is alright but nothing to keep your hopes up for. I think Savin curved the mean to a B+, and if you screwed up any of the actually doable problems you were thrown back far. If you did the impossible, you were thrown ahead. Studying helps to a point, but intuition is where its at. Savin's personality is very unique. I don't think I could do it much more justice than has been done below, especially the amazon one. He's an impersonal but warm fellow who is always looking to help when he's teaching (he really does scan peoples expressions to see if they understand what's up), but he won't go out of his way to help you or show you a drop of mercy. The TA was good for the most part, but he quit giving homework solutions after they were turned in, which would have helped a lot for some problem sets. Overall I'm glad I took this course. It was a good introduction to higher mathematics and whats along the future. Would honors linear algebra and Calc 3 and 4 have been better? GPAwise probably. Proof learningwise probably. Stresswise probably. Experiencewise who knows... probably. Savin's great (not that he'll even be teaching it again), but the course is too vast to give more than a wishywashy understanding of all it entails and the grade distribution promises to hurt at least half the class.
Jan 2012 
This is Prof. Savin's first year teaching a freshman course, and it showed: he assumed that most freshman students actually knew what mathematics meant. But we didn't. We thought "oh we get to look at curves and surfaces and put them in our calculator, or write down their equations, and compute facts about them! After all, how could our high school give us the wrong idea about Math?" I think about 1/2 to 3/4 of the way through the year most people begin to come to the realization of what we should actually be aiming for in the homework sets (in Savin's words: "Think of the picture, then the proof is obvious, I don't need to write it"). Starting with single variable calculus, which in its standard form as required by every other science/engineering/econ major is horrendously mundane, Savin gave us "interesting" problems for the homework and tests, much to our naive horror. Looking back, those first few homework problem sets (well not the first one, which was rigorous proof in the real number system) were the most interesting, and difficult, part of the class. Unfortunately, by the time most of us had got our math bearings, other members of the Math department (or perhaps even Savin himself) had smelled the freshman blood in the water and the class became easier tenfold. Which is a shame, as all of the homework sets for Linear Algebra were directly from the book, which compared to Savin's sets are akin to a coloring book in difficulty (but a lot less fun). However, most of these obstacles aren't faults with his teaching, they are faults with the way Math is taught in the US all the way up through high school. It is not surprising that many of us did so poorly (and thus were saved by the curve) on Savin's first midterm: for most, this is the first real Math course ever taken. Math is an art, and Savin does try to help us starryeyed first year students understand that (without directly stating it, of course). Go into this class with that clear in your mind, and 1) the homework and tests will be easier. Especially the tests: Savin will take an elegant ideadriven proof over an algebraic plugandchug any day, 2) his digressions into interesting subtopics and counterexamples of Calculus I/II and Linear Algebra (Fourier, anyone?) will be the highlight of the course, not the bane of your existence, and 3) when Savin says his now (in)famous "go home and think about it", you will actually look forward to it. Don't take this class just because you were told you were "good at math" in high school Calculus (unless you were lucky enough to have received a real Math education, and were saying "duh" throughout this whole review); take this class because you are curious and actually want to play and struggle with the implications of our notion of derivative, integral, and vector space. Also, Savin is a really nice guy: if you do take this class, don't make the mistake of skipping out on his office hours. He directly helps you with the homework and is always excited to help you gain a foothold on understanding the sometimes abstract solution process of his more difficult problems.
Dec 2011 
Savin really knows his stuff, but sometimes it seems like he doesn't really teach so much as just illustrate proofs in class. Which is fine, especially since he's got this wonderful Romanian accent, but you're definitely left to struggle with the material yourself some moreâ€”his response to more basic questions is usually 'go home and think about it'. That said, the class will teach you more than any other math class you've taken, since you take things more or less from the ground up. Although it's proofbased math, there wasn't much of an introduction to proof, and at times it seemed like we were both learning something for the first time and reviewing it as though we'd been taught it ages ago. The lecture hall was full the first few days of class, but it thinned to about half by the end. On the plus side, nothing gives more camaraderie than knowing that all of you are struggling togetherâ€”the average on our first midterm was 30/100. The homework is hard, the tests are impossible, the grades are curved at the very end, but I'd definitely take this class knowing all that.
May 2011 
Professor Friedman is awesome. Having had absolutely no background in higher math, and a lot less experience than some of the class, I still truly enjoyed both semesters. That said, I'm not one to be demoralized by seemingly impossible problem sets, or less than fantastic grades, and didn't ever find that either of those things interfered with my enjoyment of the class. For those who aren't this way, be wary, but if you are willing to put in the work, you will come out of Friedman's class with a tremendous improvement and a truly deep understanding of what you've learned. Though all of my classmates that I've talked to have varying opinions on the class itself, we all agreed on that. Often Friedman wont answer questions immediately because he carefully plans out the structure of his classes, and likes to stick to this. This never bothered me because his lesson plan is brilliant and incredibly well thought out. I also really liked his teaching style and thought he was very clear, but it may take some getting used to. He wrote our class notes, which totaled a few hundred pages, himself. The problem sets are difficult, and existentialcrisisinducing if you decide to start them 12 hours before they are due. However, they are meant to be done in groups, and they often took my friends and I even longer to finish because they inspired conversations that gave us a deeper understanding of the material. Ultimately, I loved this class and was so glad I took it. Friedman is fantastic, and though the class is incredibly challenging, it is completely worth it.
Apr 2011 
I liked the class. Friedman knows his stuff and the course is well thought out, though he doesn't follow the "texts" at all, they're more like supplementary reading. His notes basically cover his lectures and are pretty complete and intuitive. I do agree that the notation is a bit confusing at times, but that can be easily clarified by just asking him or the TA what's up with it. The course is curved, but very little, no big deal. Just go to class, ask questions (not during class, friedman's got lotsa stuff to cover...), start your homework before Monday morning when it's due. It's not an easy course but being a good student should be enough for you to do well... On a side note, the guy does actually have a sense of humor and is pretty friendly.
Jan 2011 
To begin with, I recommend that you avoid taking this class with Friedman if at all possible. I realize that this is the math class for freshmen math majors and more often than not you will not be able to avoid it and this fact only makes all the more terrible that they assigned Friedman to teach it. We spent most of the semester on Linear Algebra and the last 2 weeks or so on properties of real numbers. All the calculus is presumably covered in Honors Math B. The biggest issues with this class are as follows: 1. You spend hours and hours doing difficult problem sets every weekend. They were assigned on Wednesday nights and due on Mondays. Obviously no one got a chance to look at them till the weekend and hence there was practically no help for the homeworks other than a Friday morning recitation which no one went to because no one started the homework till Friday evening. His office hours, being on Tuesdays(?) and Thursdays are pretty much useless for getting homework help. If you try to write halfassed proofs, you will lose lots of points and do badly in the class so you pretty much have waste your whole weekend on this class. 2. THERE IS NO TEXTBOOK FOR THIS CLASS. Friedman assigned a textbook because he was obliged to but never referred to or suggested it in class. I tried reading it for a few weeks but it was too advanced and of little help in solving the homeworks. Instead Friedman uploads his own typed notes but they are very confusing, the notation is bewildering and completely unintuitive to a student with little higher math background. As for the lectures, he goes really fast and writes really fast. Most of the time what he said went completely over my head and his handwriting and presentation on the board is abysmal. Basically you either try to take notes or just listen. I would recommend listening, you might absorb something. The notes will just be a somewhat extended version of what he posts online. Going to lecture is probably a good idea just so that you can have a vague idea of what topics/ideas were taught as there is no textbook to refer to. He did not show up to office hours the day before the final and did not send out an email or tell anyone that he wouldn't be there. Everyone who went just waited for 20 minutes and then went home. You would think that the professor who show up to office the day before the final exam but I guess not which just makes me think that he really didn't give a damn about teaching the class.
Jan 2011 
CULPA warned me not to take Friedman's class, but I did anyway, because, unfortunately, he was the only teacher for Honors Math A. I can't say that I'm sorry I took Friedman's class simply because Honors Math A was definitely the right choice for me since I had already completed the calculus sequence in high school, but IF YOU HAVE ANY CHOICE IN THE MATTER, AVOID FRIEDMAN LIKE THE PLAGUE! If you are on the fence about taking Honors Math A anyway, and Friedman is the only teacher, then I would say don't take it. The material in the course is extremely fascinating, but it is abstract, and Friedman is incapable of stringing together a coherent sentence, taking notes in an organized or legible fashion, and answering students' questions. On the first day, he told us that if we did not understand something, we should ask, but he never answered a single question for the entire semester. He usually just ignored raised hands, but when he did call on someone, he would either say, in irritation, I already covered that, I'm not explaining it again, or, We're not up to that yet, I'm not explaining it yet. I stopped going to class after Thanksgiving break, and it in no way interfered with my understanding of the material or my grades in the classthis should give you some idea of how useless the lectures were. The recitations were equally useless. I went to one and it was just the TA repeating what Friedman had saidonly it was slightly worse, because Friedman didn't bother to tell the TA what he was teaching, so a lot of the stuff was completely irrelevant. Furthermore, at least Friedman is brilliant and has a deep understanding of the material he is teaching, even if he doesn't know how to teach, and doesn't much care either; it was obvious that the TAs barely had a handle on the information they were presenting. The grading is something of a nightmare. Two TAs take turns grading the problem sets. They just do whatever they want and Friedman does not interfere. One TA refused to give above a 70/80 no matter what while the other always gave me about a 70/80, even though my problem sets were all about equally correct. Every student in the class had a similar experience. Furthermore, the "nice" TA then decided to grade out of 40 while the "evil" TA graded out of 80...Friedman failed to realize that when taking an average, this meant the "evil" TAs homeworks counted for twice the "nice" TAs homeworks. Comments on the homeworks were minimal and useless in terms of understanding how to improve. Only a few problems a week were graded so you ended up feeling like you wasted a lot of time AND it was just luck whether that one problem you didn't fully understand counted for a third of the homework. Friedman himself graded the midterm, which was surprisingly easy, yet no body did as well as they expected to/should have. Why? I never thought it was possible in math until this class, but Friedman's grading was extremely SUBJECTIVE. That said, I got an A, thanks to the curve, and I think most people got As or Bs. The lowdown? If you love math, have a strong background in math, are good at math and a creative thinker, and are willing to put in the time to teach yourself the material in addition to doing the absolutely endless (albeit interesting if you understand what's going onsee Workload) problem sets, and there is no other teacher teaching the course, then suffer through it for your love of math and the material, knowing that it's basically like getting credit for teaching the course to yourself. The one useful thing Friedman does is post a textbook of course notes he wrote himself online. While not ideal, they are infinitely better than his lecture notes, and I used them to teach myself, and I do think I learned a lot. He never uses or refers to the textbook you have to buy. I looked at it once and it just confused me because it uses different notation and terms than Friedman.
Jan 2010 
This semester is singlevariable calculus and some linear algebra from a rigorous standpoint. If you are interested in applications of math and not proofs, this is not the class for you. He freely admits that he doesn't care about the former. If you want to be a math major, take this class so you can tell early on whether you actually want to have 3.5 more semesters of abstraction. This course is based on Apostol. This is not the best textbook out there (maybe secondbest) if you want a deep and rigorous understanding of singlevariable calculus, although it works. I'd prefer a different choice, but it's what Columbia has at the moment, and it works fine. Thaddeus is an awesome lecturer. A number of people dropped the class but came to classes anyway just to hear him speak. He is engaging, funny, thorough, and excellent at explaining things. He's also quite friendly and receptive to questions, as long as they are not stupid ones. I know that a large number of people will disagree with me on this one, but the class is not as hard as anyone else says it is, at least if you've ever seen a proof before. The psets, the midterms, and the final are all pretty easy, although somehow the median on the final was a 57/100. Given that this is the course recommended to math majors and supposedly the hardest math course available to freshmen here, I hoped that this would be really hardcore, possibly on the level of Harvard's Math 55. It is not. There was around one difficult problem set (out of 13), and aside from that, there were no challengingtoimpossible problems that you had to turn in. It's by no means an easy course, but I wanted more difficulty.
Dec 2009 
Thaddeus' enthusiasm for mathematics and his patience with students make for a perfect professor. In addition to the scheduled weekly office hours, he encourages students to knock on his door whenever confused and tries his very best to help and enlighten you by offering explanations in a different way. His quirky, hot outfits and obsession with wearing a glove while writing with chalk on the blackboard add to the entertainment. On top of that, he weaves together proofs of theorems so elegantly that will absolutely dazzle you. On the last day of classes for the fall semester, he read poetry on Euclid by Wordsworth towards the end of the period and received a huge round of applause by the entire class, which shows how much everybody loves him. But, a huge warning  this class is especially difficult for people who have not been exposed extensively to the theoretical aspects of calculus and not meant for slackers. You've got to work and think smarter and harder than you have done before because the basic foundations of proofs can take some time to sink in. The homework are graded somewhat harshly by the TA, who is picky with little details here and there, but this is understandable given the care you need in constructing legitimate proofs. Studying from the corrections to the mistakes can help better your understanding of the material. The exams are difficult and one suspects that the average must be pretty low. But the problem is that Thaddeus does not announce averages due to the unhealthy competition it would bring about and offers no concrete indication of how you are doing in class until after the first midterm (the last date for dropping a class in CC), so Columbia College students are taking a big risk. For SEAS students, many dropped out later in the semester given the longer deadline for dropping a class. So, in conclusion, this course is for people who simply enjoy the spirit of learning and not obsessed with getting the highest grades possible to boost their GPA. This is not like high school when you can always take the hardest classes and skate by with an A, but the new stuff you learn is definitely much cooler and interesting than anything you've been exposed to previously.
Dec 2009 
The course is, in a nutshell, must for math or physics major but not very much so to most other students who 'love mathematics' and want to take hard course. Especially, if you primarily want do something like econ and try to concentrate on math or applied math JUST because they are used a lot in econ, I'll say it's useless. Or if you think you are good at math and you have to take math but you do NOT want to major in math, just take calc 4 and feel good about yourselves; you will either think this is not real mathematics because it doesn't feature sharp drawings or intuitions, or give up your love of mathematics. However, this is the real math, which features rigorous proofs of everything we use. If you are going to be math major but doesn't like proofs, just bear with it, since every math class after this would be prooforiented. The course is basically elementary analysis and algebra. It is rather challenging because it basically sets up your mathematical foundation in two semesters in a rigorous way. It's definitely not easy; another warning for those who doesn't really need those things. Grades are not that generous either, and hard work often doesn't pay off right away, but it does pay off eventually, at least I believe. If you seek only A, it could be risky. About the professor He is a nice guy. He has some quirky sides, but they are not harmful to students and rather funny. He is not desperate to convey those intuitions and graphs like high school instructors, which could come off as being 'not good' or 'not passionate', but he does explain physical nature of definitions and theorems when needed. The course is hard, but might have been little harder if it weren't for him.