MATH 335
This is my course review for MATH 335, Advanced Calculus. This is the second quarter of the MATH 33X sequence, which started with MATH 334. I took the course in Winter 2016 with James Morrow and teaching-assistant Will Dana.
Notes
Here I’ll keep some observations of the course as I go along.
- The first two weeks have been just a lot of computations. I’m not sure I’m gaining a lot of understanding so far. Also apparently our class is currently a few weeks behind of last year’s class.
- I’m getting more and more frustrated that I’m having to spend so much time on math, when I consider some of the other activities I do, like Wikipedia editing to be of higher value.
- When it comes to the divergence theorem and Stokes’s theorem, and other multi-dimensional ideas, I really feel like Folland’s book doesn’t contain enough illustrations.
From 7th week:
I’m getting frustrated once again with the time it takes to type up problem sets. I think that almost all the learning comes from just doing the problems, so typing up solutions just so it’s legible and coherent to the grader isn’t actually helping me learn anything. This is one of the big differences I see from high school math. An annoying bug that comes with college-level proof-based math courses is the emphasis on having homework that is clearly laid out. In high school, math problems were mostly computational, which made them boring, but it was also acceptable to present just scratch work instead of coherent thought processes (indeed, homework was always completion-based and not actually graded for correctness1).
Besides the point-set topology in 334 and some of the basics of analysis (which I had learned on my own in high school), there hasn’t really been any math in 334/335 that I could call “beautiful”. In fact, it’s a bit tiring to do problems week after week that are, in effect, factory-produced; they are problems written not for intrinsic value, but rather simply to be solved. In other words, these problems are not the sort of “beautiful” or “interesting” problems that one would naturally gravitate toward (the sort of problems that are merely and divinely present because of their inherent value, and meant to be solved by elites), but rather something mass-produced, to be consumed by the “mathematical masses” (i.e. those who are competent enough to do undergraduate math, but who haven’t developed a good taste for problems), or written due to the practical desire to evaluate students.2 These problems are in fact no different from whiteboard interview questions in programming, which have been criticized well elsewhere3.
I will even say that because of this emphasis on cardboard-like problems, I’m not getting any practice in refining my taste in finding problems to work on. Indeed, each week I am thrown a new set of time-consuming but straightforward and unenlightening problems to work on, which certainly gives me practice with some of the standard tools I am expected to know (in a similar way to, but on a more advanced than, doing many long division problems), but also completely drains me of the curiosity to find interesting problems of my own (this might change to some extent in 336, where a term paper on a topic of one’s choice is assigned).
I should also note that my interest in math as a whole has been going down steadily, which could contribute to part of why I find the problems uninteresting. In fact I am often amazed at how much the other students seem to be working on the problems, and often wonder if it is idiosyncratic to me that I am hopelessly bored by the problems.4
I’ve been realizing more and more that Morrow likes to discuss topics differently than in the book (in 334 he gave a different definition of continuity), which makes it difficult to follow sometimes (especially since I tend to skip lecture a lot), although he does post PDFs for (most of?) these (but with little organization on the course website).
This quarter in general has been rather stressful for me, and looking at my performance so far and in previous quarters, I can’t help but to recall Raman Shah’s quote “It is entirely possible for a school to break you and award you decent grades at the same time.” (Although UW is nowhere as intense as Caltech is.)
This is something I’ve been feeling since I came to UW, but more and more I feel it is simply impossible to learn everything that is part of a course such as this one (and even in my other two courses, CSE 332 and CSE 351) if one hasn’t seen the material previously. This of course assumes that one is spending a reasonable amount of time on the courses (though perhaps less than what if officially expected—which is I think 3 hours per credit?).
The trend continues in which I become even more disillusioned with UW and academics. It feels incredible to me how everyone else can wake up, attend class each day, dutifully complete homework each week, etc. I suppose it’s easier for people who believe more in the human capital narrative of education, but even then, it seems so incredibly exhausting to do this each year. It’s difficult for me to explain this precisely, and sometimes I feel that one either intuitively understands the mundaneness, repetitiveness, and stressfulness of being at UW or else one is completely incapable of understanding it5. Indeed, I think UW is no better than being in high school6. I am reminded of days in high school when I would search online for reasons others had come up with against school, because it seemed so obvious and intuitive to me that I shouldn’t be at school, yet difficult to verbalize why exactly that was.
I suppose one part of my complaint is to ask the question, Isn’t it possible to imagine some other life than to be stuck at school doing this? There is an eerie lack of creativity across most of the students here, where they cannot—or do not allow themselves to?—imagine a life significantly better than their current life.
This is something I see in other subjects as well. High school tests and projects were graded, but daily/weekly assignments were often graded solely on completion (possibly because there was usually only one teacher who couldn’t possibly have graded everything each day), whereas in college even weekly problem sets are graded (and still account for a nontrivial portion of the final grade).↩︎
See also this quote from “Why are modern scientists so dull? How science selects for perseverance and sociability at the expense of intelligence and creativity”:
↩︎Also, in the past educational ability was more often measured using relatively-infrequent, timed and supervised, previously-unseen formal examinations during which the examinee would need to work fast to organize their knowledge. Such formal examinations are likely to be more ‘g-loaded’ (i.e., correlate more strongly with IQ) than the greater emphasis on frequent ‘course work’ which has characterized educational systems over recent decades – course work tends to reward Conscientiousness over IQ compared with formal exams and be preferred by more Conscientious and less intelligent students.
See in particular Raman Shah’s answer to “I’m doing a computational physics PhD. Is working as a software engineer really a viable option after I graduate?”, which contains:
↩︎People obsess a lot over the data structures and algorithms stuff because Silicon Valley firms have made a cargo cult over being able to whip out solutions to crazy algorithmic problems on a whiteboard. I feel this is unfortunate - it discriminates against candidates like us while driving up the prices for some contributors who are not that great at their jobs and are nasty human beings but who are really good at the whiteboard stuff. The limiting factor in real-world software engineering in my experience seems to be that eye for simplicity - maybe it requires a lot of intelligence and working memory, but most of the meaningful lift in software engineering to me seems like organizing your sock drawer, just with really good taste. Anyway, you’ll probably have to join the cargo cult to get the job even though the material is almost irrelevant to the day to day of the software engineering I’ve seen. What’s nice is that the subject is pretty delightful, so it won’t feel too onerous. So maybe take a Coursera course and back it up with selected readings from Cormen, Leiserson, Rivest, and Stein. I have almost a religious antipathy to interview prep books, but I guess they can be useful. I certainly have avoided touching them to this date, and I’ve yet to be unemployed. I’m not certain if my quirks translate to good general advice; your mileage may vary.
I am reminded here of another quote by Raman Shah, from Raman Shah’s answer to Why do so many Caltech students procrastinate?:
It’s kind of amazing, really: Caltech manages to severely damage the mental health of many bright young people every year, and many people assume it must be brutally competitive or have abusive instructors or something. But none of those things are true at all – all it does is just hand them a few dozen pdfs per quarter with interesting-looking math and science problems and due dates. That’s all it takes.
However, my experience differs from Raman’s in various ways. (1) I am not attending Caltech, so my coursework is (presumably) not even close to the intensity of a typical Caltech course load. (2) The problems in MATH 334/335 (and even CSE 332) do require some amount of creativity, but they are mostly pretty straightforward. There are few problems that are “really easy lay-ups in the homework to buoy your spirits” (to quote Raman in his answer), which is different from high school. However, most of the problems in e.g. 335 are in the class of “more complex problems to make sure you grasped the way things fit together at a more sophisticated level, but still a level that was explained to you”. So I don’t experience the sort of pressure to be creative under a deadline that he mentions. However, I still experience the continual series of deadlines that slowly numbs me each quarter from everything interesting in the world. (3) With the noted caveats, I still relate strongly to the quote, because my intense frustration at UW is really just the result of apathy and lack of awareness of a system that hands me deadline after deadline after deadline, not something maleficent.—Indeed UW is a very lonely place in the same way that doing archival work is lonely, because nobody else seems to care about the things I intensely care about.↩︎
For if one feels the need to interrogate me with questions like “What’s so bad about it?”, “Isn’t college the best time of your life?”, “Aren’t you studying what you’ve wanted to study?”, and so forth, then I cannot take the time to answer them. The only correct answer for those capable of understanding—and therefore those who intuitively grasp the suffering at UW—is simply “I understand”.↩︎
See for instance this thread on /r/udub/ for others who agree.↩︎