course


May 2011 
If you've done hardcore math before, the previous two reviews will probably apply to you. Otherwise, they really won't. I hope that this one somewhat applies to the rest of us. Granted, even after going through this class, I'm still math majoring, so take my review with a grain of salt. I like to think I'm somewhere in between the (substantial number of) Honors Math A/B dropouts, and the ridiculous kids who didn't come to class yet never missed a point on anything. CLASS: In terms of content, the class covers linear algebra and calc III  IV from a proofbased perspective. It also gave students an idea of what's to come, should they start taking upperlevel mathematics courses... so definitely not the standard calcIIIIV curriculum. The two biggest additions are proofs (of course there are proofs in the calc sequence, but here they're covered more rigorously, and mostly everything is proven), and adding in ndimensions, which makes things abstract and nonvisualisable. Personally, I thought the smaller asides (usually irrelevant to the course's express topics) were very interesting, once I figured out what the hell they meant. The class is also supposed to help new freshmen think in the "language" of higher math. This includes the (sometimesabsurd) notation and thinking in terms of theory, proofs, and ndimensions  rather than just pictures and examples (not that these aren't helpful, because they really are). "Help" is a subjective term, though. I don't know if it' the course's intention or it was just Friedman but I felt like this class (especially first semester) was a "baptism by fire" into higher math. To give the analogy one of my classmates called out in lecture one day: we the students, were like helpless babies thrown into the woods, at the mercy of the wolves, and expected to figure out how to survive. I thought I was good at math. So did everyone else who signed up, I'm sure. Not everyone left feeling that way. The first semester was linear algebra. we went through the topic in a very proof/theory based. Very abstract. Friedman started with linear algebra. The second semester, calculus (with some topology, differential geometry, etc. sprinkled in for good measure) was more computational and less theoretical... again, I don't know if that was Friedman's doing, or just how the course works. First semester had probably 7080 kids... on the first day of second semester, though, it was as if half of the class had discovered how to shift into dimension n, n>3 (they disappeared) PROFESSOR: Friedman is, well, one of a kind. It's hard to separate the course from him, because he really defined it and made it harder than in years prior, at least from what I hear. He mumbles at times. He sometimes physically (and maybe metaphorically) stands in front of what he's writing/talking about. In class, he chugs through proofs nonstop like the little engine that could. These things are all negatives in a class where a good portion are freshmen who haven't done higher math before. I'm sure any mathematician, or even any math major in the Help Room, could easily transform Friedman's chickenscratch into concrete calculation. It wasn't that easy for me (and a lot of other kids in the class). Oh, and his handwriting/drawings suck. I thought that he really improved in the second semester, though. He started giving more examples, rather than just the proof/theory behind concrete concepts important for those of us with no previous exposure. I felt that he just generally explained the material more clearly. Class became actually helpful, not just an attempt to set the Guinness World Record for most proofs written in an hour (I use these absurd analogies because I thought the first semester was kind of absurd). Still, if you miss a beat (whether because you space out, or because you don't get how something logically follows in a proof, or anything), you're probably going to be lost for the rest of lecture. In office hours/review sessions, he was totally different. Very helpful and took his time in explaining the concepts. Unfortunately, his review sessions were like therapy sessions: he allotted exactly an hour to this session, and we got exactly an hour  no more, even if lots of people still had questions. Similarly, he himself was only available about four hours a week, and the TAs had 1 office hour between them (in the Math Help Room). Then again, pretty much anyone in the Math Help Room can help you with problems you're having, so it's not the worst thing in the world.
Apr 2011 
Prof. Friedman is a pretty cool guy. He occasionally writes too faintly on the blackboard or goes off on tangents, but those tangents are always fascinating and will give you a greater appreciation of the subject. If you snooze in his class, that's okay too, he doesn't judge. He posts class notes online, and they're super helpful when you're studying for exams or just going over material to complete the homework. (Only thing is that he writes really fast sometimes, so it might be a good idea to bring a copy of the current chapter in the notes to class and follow along.) Class isn't ever stuffy or boring. He makes a fair number of (slightly lame, but endearing) jokes to lighten it up. His lectures are always really wellprepared and at a good level of difficulty. He'll usually review the hardest part of the previous lecture at the beginning of class, so come to class on time. If you ask him a question or go to office hours, he's always really eager to help and answer your question as fully as possible. I found the course itself (Honors Math A/B) incredibly enlightening. In addition to basic linear algebra/multivariable calculus stuff, it introduces you to a whole host of theoretical math topics that give you a taste of "higher math." Some of it relates to physics, and I found that seeing the same material twice, from different points of view, helped me understand it better. This course is a definite musttake if you're considering majoring in mathematics.
May 2006 
Good professor overall, always has time for his students. Only two complaints: 1. Green Board + White Chalk + Flourescent lighting glare in Room 203 makes me one sad panda. (ie. bit hard to see). But that's not his fault. 2. Occasionally have to erase an example when something goes wrong along the way, but that's acceptable as it is not common. Recommend this class if you are a math major / need linear algebra too asap / want to challenge yourself. Engineers probably better off taking Calc 3/4, but this is still a very good class. Bit challenging at times, but grading is always fair. Overall, probably an A for Julius, though. He's very, very good.
May 2006 
i have taken 5 classes with thaddeus (too bad he went on sabbatical senior year, otherwise i might have taken more). these five semesters of "thaddeology" have helped me build a very rigorous and exciting foundation for studying higher math, and also introduced me to one particular way of thinking and talking about math. thaddeus is a very nice guy (i'm writing this because there seem to be many people who find him a little intimidating, but don't worry), very willing to answer your questions, and lectures very clearly. he encourages class participation, asking questions to the audience, and creating a questionfriendly atmosphere. he lectures with notes, but not from a book, and organizes his presentations very well so that they are not that hard to follow. he is very available during office hours: i remember him answering my silly settheory questions freshman year with the same patience as the (probably equally stupid) questions about serre duality or sheaf cohomology later on. honors 3+4: his lectures were vey engaging, and this definitely turned several students into math majors. thaddeus is a very eloquent speaker, and this actually helped my english as much as lithum did. although the lectures were fast (or so we thought), they were very funny at times: at the end of first semester, he read poetry to us, and at the end of second semester, he gave away fig newtons and leibniz cookies, to celebrate the inventors of calculus. i would say he is surely the best choice if you would like to learn calculus. topology: on we went after honors, to enjoy more thaddeomathematical theatre. the original plan was to take differential topology as well in the spring, but for some godawfully trivial reason that class was cancelled. the one semester of basic topology was a very nice undergraduate class, taught at a faster pace than honors. professor thaddeus seems to work very hard (just walk by the department at 2 am and look at which offices are still lit), which suggests that he wants his students to do the same. i remember spending a lot of time on the homework assignments, maybe too much. fewer problems might have given me more time to digest the material, but then worrying about finishing homeworks is a sign of undergraduate immaturity. although the material isn't that exciting (pointset topology), this class was probably the best organized one. i have a friend who is planning on typing up the lecture notes. thus, it is probably a very good idea to take an (advanced) undergraduate course with thaddeus. commutative algebra: the journey continued, and the few undergraduates left tackled their first graduatelevel class and (of course) had to struggle a lot. thaddeus' lectures were still very clear, but also very detailheavy (maybe because of all the undergraduates there), which made some of us lose focus on the conceptual understanding of the material. he covered a lot of topics, and expected very hard work from us. but commutative algebra is a subject that never ends  maybe realizing this is far more important than knowing the few selected topics that this course taught. (i have seen the final for the same class a year later with jacquet, and it's much, much easier than ours.) algebraic geometry: only very few of the undergraduates from honors continued all the way to this course. algebraic geometry made me work harder than for most of the undergraduate courses combined. following the lectures was not too hard as usual, but the homeworks were a huge workload. during the last month, thaddeus started (finally) to not focus too much on the details, and left out various proofs. personal conclusion about grad courses: i later took other graduate courses, which emphasized the global picture of the material, and didn't focus on the local details ("the proof can be found in any book on[subject name]"). they also didn't include any homeworks, and i learned a lot more in those courses (basically from studying all day in butler  on many days, and more importantly, talking with grad students). my guess is that now, i would prefer these grad courses over thaddeusstyle ones. but i also feel that taking two grad classes with thaddeus made me mathematically very mature. this maturity was absolutely necessary to get into a rhythm of properly working through, and truly beginning to grasp unknown mathematical theories and thoughts. if you feel you lack this maturity, taking a grad course with thaddeus wouldn't be a bad idea. he truly is a very amazing teacher, and columbia is lucky to have him. finally, i know that math professors read culpa a lot. thus, i would like to thank him for the very enjoyable fivesemester journey!
May 2006 
The first semester was markedly better than the second. He seemed more interested in the singlevariable material because it was more theoretical and abstract whereas the second semester got dampened by some pretty technical math and boring machineryas such, he made mistakes and was routinely unprepared. Julius made up for it by being a pretty lenient professor, making easy exams and being fairly generous about giving back lost points on homework. He explained things well in office hours for those who bothered to go. His exams were really easy and never mothing more than the homework problems.
Apr 2006 
Great teacher, at least for Honors. He is really smart, and it shows. The class itself is quite difficult, and I would not recommend taking it unless you are willing to spend alot of time working on the problem sets. The class is geared for those who want to be math majors/concentrators, and I would not recommend taking it and spending the necessary time unless you want to continue in math. The class teaches you how to think mathematically and how to work through/write/figure out proofs. Julius Ross is a great guy and is always willing to help on the homework. He will spend much/most of his office hours with his Honors students helping them and guiding through the tough problem sets. He is understanding about conflicts and helps you work out any problems. I was hesitant about taking this class in the beginning of the year but this class has convinced me to be a math major
Apr 2005 
Professor Wang is by far the best mathematics professor I have ever had. He can reduce the incredibly complex presentation of Apostol's Calculus to concrete, useful examples and provides much better, cleaner proofs than those in the book. That said, these classes are hard. When you think that they can't get harder, they do. Multiple problems from Apostol are misprints and cannot be solved but in failure, you will learn more about the process of solving these problems than otherwise. The bad part is his annoying habit of assigning problem sets on Thursday evening to be due on Monday. It really messes with your schedule. I will be taking as many classes with Professor Wang as I can.
May 2004 
Amazing Professor. Fantastic Professor. Best professor in the whole department, maybe even the whole school. Explains things really ,really well. A subject that will really turn you on to math, since it is everythign you already know from an advanced prooforiented viewpoint.
Nov 2003 
This is the first math course I've enjoyed in years. Say goodbye to computations, drills, and halfassed explanations. If you want to understand calculus, this class will make you go "ohhhhh". Be warned, however, you will be required to put in the necessary time and effort. Professor Thaddeus is organized and coherent as a lecturer. The entire class consists of definitions, theorems, and proofs, and he goes through many, many blackboards every lecture. His presentation is clear, rigorous when needed, and allaround excellent. Beware, however, that the homeworks are not easy, especially if you have never taken a proofbased course before. If you do not enjoy mathematics or prefer to ape manipulations and plugandchug, this is not the course for you; you need to think a lot here. I'm not sure if an engineer who has never taken MV calc before should take this class, because you don't really learn how to do the calculations. It's definitely a great math major/physics major course. Highly recommended.
May 2003 
As yet another Honors III dropout, I have to say that what both the other Honors reviewers said was true. The class started with about 22 people, when it ended I think there were 8. The textbook we used was full of errors. Beware of taking this class without a STRONG math background, or else be willing to work and study HARD  or else be a genius. Any of those (especially the first and last) will do. I am a math major (and I love math!) but this was too much for me. It's not like high school where you can always take the hardest course available.
Apr 2003 
I just wanted to add the set of glowing reviews of this professor. He was the best teacher I had this year and quite possibly the best teacher I have ever had. The lectures are just incredibly organized and clear and he gives them in such a natural way that you get the feeling that it is effortless on his part although he obviously spent a great effort on preparing the lectures for the class. He knows just when to be rigorous and just when to be handwavy while always explaining what he is doing so you can understand the distinction. He is very lively and says some funny things and he always breaks the chalk.
Mar 2003 
Without a doubt, the best math teacher I've ever had. He's always energetic, friendly, enthusiastic, understanding, and funny. He genuinely cares about both the students' understanding and the rigor of the logic upon which the course is based. Although I love Math, I was slightly intimidated going into an entirely proofbased class, but Thaddeus makes such an endeavor seem not only easy, but natural. I definitely struggle with the homework assignments, but I honestly don't care because I love this class so much. Every other Math class I've taken here has been completely subpar, but every day I go to this class I am reminded why I'm interested in the field and am given new insights as to how interconnected it all is. I wish he could teach every other Math class I take, because I couldn't imagine lacking the precision and comprehensiveness of Thaddeus' lectures. I only hope that I will be able to appreciate other Math professors in semesters to come. Honestly, if you don't think you're good at Math but are interested in its more theoretical aspects, definitely audit or pass/fail this class. You probably would be completely scared off by the problem sets, but honestly they're the kind of thing that are ten times easier when you don't freak out. Just interest yourself (if necessary) in the concepts and the proofs will follow naturally. This is easier said than done, of course, but my enthusiasm for Thaddeus and his course makes me unhesitant to make such idealistic statements. The only reason I dread the final is that it will be my last contact with this class. If he doesn't teach any advanced classes while I'm a student here, I might have to switch majors. Okay, I think I've made my point. Thaddeus rules.
Jan 2003 
This is the class that has made me decide to major in math. This is how much I loved his class. It's 5:30 in the morning, my friends are over and I'm writing this. This class is for those diligent, yet modest minds who believe in the beauty of knowledge and proof. You'll know who you are after the first few weeks... From his flowing golden locks, to his quirky clothes, to his jovial jokes and prose reading, to his smiles, his vivacious personality only added to his truly amazing lectures. Never have I seen a person so eager to teach, but one who can teach in such an effective and congenial manner. The ambiance in class was alway one of ease, relaxation. Go to office hours. Don't go to recitation. Although the TAs are smart, they're stupidly boring and useless, although they do grade well. I mean they grade hard, but reading over your graded homework does help the most when studying. Get ready to be put to the test. Take notes (copious unhealthy amounts!!), and ASK pertinent questions. I've never put so much effort into a class before, but I've also never had such a wonderful teacher before and never have I benefited so greatly. Take him if you ever see him teaching.
Jan 2003 
This class is like no other math class I've ever taken. Having done all the material it's supposed to cover was no help, except maybe having a conceptual understanding of linear algebra. There are no problems; there are only proofs. Doing the work at first was tough; high school courses really don't prepare for this completely different kind of problem. Many people drop after the first few weeks, and those who stay are antagonized by the harsh grading of nitpicking TA's. Those who have always done well in math may have trouble spending 6 hours on a problem and getting zero points for it. The excruciating detail of the grading does, however, cause students to recheck proofs, so that when the midterm does come around, they find out that they've really learned something. If you're thinking about taking this class, do it only if you're thinking about majoring in math, and only if you're willing to do the work. Copying from people in a study group all the time will kill you on the midterm. Thaddeus does a great job teaching the class; his lectures are organized so that he squeezes as much into the time as he can, so expect to write notes for the full class. Problems and proofs are often intimidating, and Thaddeus is always willing to help in office hours, and often just provides words of encouragement. He really seems to want students to do well, and to get into what they're working on. He ended the semester by reading a poem. He also seems to be one of the few in the department who speak clear English. It's probably easier for him to be the "good guy" in the class since the TA's grade everything but the exams. This class is really, really hard; I worked harder for it than any other class, but I also got the most out of it. If you decide to drop, it's not hard, and you're definitely in good company.
Aug 2002 
As an Honors Math dropout, I can say that most of what the other guy said is very true. I would probably add a couple things. Though, MuTao Wang's English is passable, our TA could not speak English for shit, which is a pretty big problem if you need him to explain stuff. The homework is not consistently difficult. Usually it will be a lot of easy (somewhat timeconsuming) problems, and then one or two problems that are a real kick in the balls. I remember one time MuTao assigned this impossible proof. At the recitation, the TA said, "This was solved by the famous mathematician BLAH BLAH BLAH ... so let us follow him." And I was thinking, "what if I hadn't shown up for the recitation? Does he REALLY expect us to singlehandedly accomplish a proof that only some famous Russian guy could do?" I didn't understand the TA's explanation either, because, as I said, his English sucked. That's when I left. So, if my experience is any indication, I would say this class is definitely not for the casual math enthusiast (like myself). Even though I got A's in BC Calculus and a 5 on the exam (booyah), I was definitely out of my depth. Even my exgirlfriend, who was brilliant in math and took the exact same class at Yale (the textbook was written by a Yale math professor) said the class turned her off to the subject. What the other guy says about "theoretical underpinnings" is true. If you take this class, you won't learn multivariable calculus or linear algebra. Really. You won't even see numbers that often. My exgirlfriend, who took the class both semesters, feels like she didn't learn anything. I would agree with this guy that you should only take this class if you are very serious about math, and definitely going to major in it. But even if that is the case you should probably take calc 3s and linear algebra at the same time as the honors class. I always got the impression that the class tacitly assumed a prior knowledge of these subjects. Though the math department says you should start with Calc 2s if you took BC in high school, that's bullshit. The first half of that class is stuff you already learned (like improper integrals and series) and the second part is incredibly easy. You could teach yourself everything in that class that you haven't yet learned in a few hours, a weekend at most. Again, I speak from experience. I took Calc 2s the following spring. It's cake. So the bottom line is, you should only take this class if you're definitely going to be a math major, and even if you do take it, you should supplement it with Calc 3s and Linear Algebraunless you are some math prodigy who already studied that stuff in high school, in which case you're good to go.
Aug 2002 
Before you even THINK about taking this class (with or without Prof. Wang) ask yourself this question: "Am I going to be a Math Major?" If you answered "no" or "maybe" then don't take Honors Math, take IIS. In addition to being very difficult, this course is abstract and proofbased to an extreme. I feel like I have an excellent understanding of the theoretical underpinnings of linear algebra and multivariable calculus. If I were required to say... build a bridge using my knowledge of the subjects, I would be absolutely hopeless. There are NO applications in this class, this class is all proofs and abstract concepts. Having said that, this was the best math class I've ever had in my life. I am a math major, I love proofs, and I don't really care for the practical applications of math. Prof. Wang gets a B+ on English (it's not at all an obstacle) and an A for everything else. He's approachable, willing to repeat himself, available after hours, and willing to reschedule a takehome midterm for you because you have tickets to see the Dismemberment Plan that night. The workload is hard, but fare, and he knows that the students in his class are working hard and shouldn't be making Cs. The bottom line is: if you're serious about math, love doing really difficult proofs, and willing to work, then by all means TAKE THIS CLASS! However, on the first day of class there were 40 students. Only 11 stuck around for the midterm. It's a hard class, that is not what most students are after, and if you think you fall into this category, IIS is a much better class for you.
May 2002 
If you are a (prospective) math major, or just a lover of math, you NEED to take a class with robert friedman. if not, run far, far away! as a fellow number geek, robert friedman rocks my mathematical world. his lectures are engaging and interesting...he goes at a very fast pace, but thoroughly covers the material. he is a fair grader and his exams are straightforward and almost predicatable. if you put a lot of effort into completing and really understanding the homework, you should easily do well on the exams/in the course, since generally the homework is much more difficult than the exams. highly knowledgeable and somewhat quirky, friedman is an excellent prof. although, i would advise any one who is not a math major/who has no interest in mathematics to avoid friedman at all costs.
Jan 2000 
He was HORRIBLE! He didn't order a book before the class began. We finally got a book a few weeks into the class, but it was uninteligable. The professor didn't teach in the order of the book, didn't use the same terminology as the book, and didn't even give us page numbers in the book for reference. There were no assigned readings. That would be ok, except that the lectures were equally useless. We spent the class watching the professor prove theorems, even though the homework never involved giving proofs. The lectures bore almost no relation to the homework. Many students realized that, so by midway through the course literally half the students regularly skipped class. We had a discussion section, but the TA often didn't know how to do the problems, and he had a thick accent. The professor showed very little care for the class. He handed back the midterms without even looking at the TA's grades or even the distribution of grades. (He said so himself!) The clincher is... we only got through half the material we were supposed to cover. The professor's reaction? "Well, that's ok, your next math teacher can catch you up" Outrageous! Prof. Hamilton left for a semester to teach at UCSan Diego. But he plans to return to Columbia. Mr. Hamilton, don't come back!
Jan 2000 
Professor Friedman is evil. He is a boring lecturer who often refuses to answer reasonable questions which he dismisses as "stupid." His tests are difficult, harshly graded, and there's not always a curve to help you out. His status as a native english speaker does little to make his lectures comprehensible. If you are taking one of his notebased classes, I'm sorry but he stands in front of whatever he just wrote on the board, and then erases it when he moves out of the way. Stay away. Stay far, far away.
Jan 2000 
Acquired by Columbia for research, not teaching. Very nice guy, but not a good math teacher unless you are extremely selfmotivated and willing to do the learning outside of class.