The textbook was _An Introduction to Analysis_ by William R. Wade, the same book that had been used for "Real I" in Spring 2013. It felt in bad taste for reasons that are hard to precisely articulate. I want to say the tone is patronizing, but don't feel like I could defend that judgement in debate against someone who doesn't share it. What I love about Schröder is how it tries to simultaneously be friendly to the novice (the early chapters sprinkling analysis tips and tricks as numbered "Standard Proof Techniques" among the numbered theorems and definitions) while also showcasing the fearsome technicality of the topic in excruciatingly detailed estimates (proofs involving chains of inequalities, typically ending on "< ε"). In contrast, Wade often feels like it's hiding something from children who are now in fact teenagers.
-The assignments were a lot of work, but that was good. It was what I was there for—to prove that I could do the work. I could do most of the proofs with some effort. In 2012–2013, I remembered submitting paper homework for a lot of classes at SFSU, but now, everything was to uploaded uploaded to Canvas. I did all my writeups in LyX, a GUI editor (I know) for LaTeX.
+The assignments were a lot of work, but that was good. It was what I was there for—to prove that I could do the work. I could do most of the proofs with some effort. At SFSU in 2012–2013, I remembered submitting paper homework, but now, everything was uploaded to Canvas. I did all my writeups in LyX, a GUI editor (I know) for LaTeX.
-[TODO: pull out a graf about LLMs being new, and being useful enough but still confused about some things
-When I got stuck, I sought help at office hours or from a large language model (citing the fact that I got help in my writeup).
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+One thing that had changed very recently, not about SFSU, but about the world, was the availability of large language models, which had in the GPT-4 era become good enough to be useful tutors on standard undergrad material. They definitely weren't reliable, but it was so convenient.
+
+I adopted the policy that I was allowed to consult LLMs for a hint when I got stuck, citing the fact that I had gotten help in my writeup. Prof. Schuster didn't object when I inquired about the propriety of this at office hours. (I also cited office-hours hints in my writeups.)
Prof. Schuster held his office hours in the math department conference room rather than his office, which created a nice environment for multiple people to work or socialize, in addition to asking Prof. Schuster questions. I came almost every time, whether or not I had an analysis question for Prof. Schuster. Often there were other students from "Real II" or Prof. Schuster's "Real I" class there, or a lecturer who also enjoyed the environment, but sometimes it was just me.
In contrast to what I considered serious math, the course was very much school-math about applying particular techniques to solve particular problem classes, taken to the parodic extent of quizzes and tests re-using worksheet problems verbatim. (You'd expect a statistics professor to know not to test on the training set!)
-It was still a lot of work, which I knew needed to be taken seriously in order to do well in the course. The task of the quiz #2 was to derive the moment-generating function of the exponential distribution. I had done that successfully from the recitation worksheet earlier, but apparently that and the homework hadn't been enough practice, because I botched it on the quiz day. After the quiz, Prof. Mujamdar wrote the correct derivation on the board. She had also said that we could re-submit a correction to our quiz for half-credit, but I found this policy confusing: it felt morally questionable that it should be possible to just copy down the solution from the board and submit that, even for partial credit. (I guess the policy made sense from the perspective of schoolstudents needing to be nudged and manipulated with credit in order to do even essential things like trying to learn from one's mistakes.) For my resubmission, I did the correct derivation at home in LyX, got it printed, and bought it to office hours the next class day. I resolved to be better prepared for future quizzes (to at least not botch them, minor errors aside) in order to avoid the indignity of having an incentive to resubmit.
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-I continued my practice of using LLMs for hints when I got stuck on assignments, and citing the help in my writeup; Prof. Mujamdar seemed OK with it when I mentioned it at office hours.
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-I was apparently more conscientious than most students. Outside of class, the teaching assistant who graded our assignments recommended that I make use of the text's solutions manual (which was circulating in various places online) to check my work. Apparently, he had reason to suspect that some other students in the class were just copying from the solution manual, but was not given the authority to prosecute the matter when he raised the issue to the professor. He said that felt bad marking me down for my mistakes when it was clear that I was trying to do the work.
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-The student quality seemed worse than "Real II", at least along the dimensions that I was sensitive to. There was a memorable moment when Prof. Mujamdar asked which students were in undergrad. I raised my hand. "Really?" she said.
+It was still a lot of work, which I knew needed to be taken seriously in order to do well in the course. The task of quiz #2 was to derive the moment-generating function of the exponential distribution. I had done that successfully from the recitation worksheet earlier, but apparently that and the homework hadn't been enough practice, because I botched it on the quiz day. After the quiz, Prof. Mujamdar wrote the correct derivation on the board. She had also said that we could re-submit a correction to our quiz for half-credit, but I found this policy confusing: it felt morally questionable that it should be possible to just copy down the solution from the board and hand that in, even for partial credit. (I guess the policy made sense from the perspective of schoolstudents needing to be nudged and manipulated with credit in order to do even essential things like trying to learn from one's mistakes.) For my resubmission, I did the correct derivation at home in LyX, got it printed, and bought it to office hours the next class day. I resolved to be better prepared for future quizzes (to at least not botch them, minor errors aside) in order to avoid the indignity of having an incentive to resubmit, and mostly succeeded.
+I would end up doing a resubmission for quiz #8, which was about how to sample from an exponential distribution (with λ=1) given the ability to sample from the uniform distribution on [0,1] by inverting the exponential's cumulative distribution function. (It had been covered in class, and I had gotten plenty of practice on that week's assignments with importance sampling using exponential proposal distributions, but I did it Rust and used the rand_distr library rather than what was apparently the intended method of implementing exponential sampling from a uniform RNG "from scratch".) I blunted the indignity of my resubmission recapitulating the answer written on the board after the quiz by also inverting by myself the CDF of a different distribution, the Pareto.
-[TODO: A procedure that we performed
+I continued my practice of using LLMs for hints when I got stuck on assignments, and citing the help in my writeup; Prof. Mujamdar seemed OK with it when I mentioned it at office hours. (I went to office hours occasionally, when I had a question for Prof. Mujamdar, who was kind and friendly to me, but it wasn't a social occasion like Prof. Schuster's conference-room office hours.)
-Late in the semester, I learned from non-course reading that the
+I was apparently more conscientious than most students. Outside of class, the grad student who graded our assignments recommended that I make use of the text's solutions manual (which was circulating in various places online) to check my work. Apparently, he had reason to suspect that some other students in the class were just copying from the solution manual, but was not given the authority to prosecute the matter when he raised the issue to the professor. He said that felt bad marking me down for my mistakes when it was clear that I was trying to do the work.
-gap between the eigenvalues
+The student quality seemed noticeably worse than "Real II", at least along the dimensions that I was sensitive to. There was a memorable moment when Prof. Mujamdar asked which students were in undergrad. I raised my hand. "Really?" she said.
-I tried pointing this out in class; I don't think anyone got it. I don't think the professor got it, either (not because she couldn't have, but because it was unepxected); it just wasn't that kind of place, whereas Prof. Schuster's class was, to some extent ]
+I was only late in the semester that I was alerted by non-course reading (specifically a footnote in the probabilistic graphical models book by Daphne Koller and the other guy) that the stationary distribution of a Markov chain is an eigenvector of the transition matrix with eigenvalue 1. (And taking such a linear-algebraic view has interesting applications: for example, the mixing time of the chain is determined by the second-largest eigenvalue, which is less than one, because any starting distribution can be expressed in terms of an eigenbasis, and the coefficients of all but the stationary vector decay as you keep iterating.)
+The feeling of enlightenment was outweighed by embarrassment that I hadn't independently noticed that the stationary distribution was an eigenvector (we had been subtracting one off the main diagonal and solving the system for weeks; the operation should have _felt familiar_), and, more than either of those, annoyance that neither the textbook nor the professor had deigned to mention this relevant fact _in a course that had linear algebra as a prerequisite_. When I tried to point it out during the final review session, it didn't seem like Prof. Mujamdar had understood what I said—not for the lack of linear algebra knowledge, obviously—let alone any of the other students.
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-[TODO—
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- * didn't quite have the same rapport in office hours; lanauge and culture might have made a difference
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+I can only speculate that the occurrence of a student pointing out something about mathematical reality that wasn't on the test or syllabus was so unexpected, so beyond what everyone had conditioned to think school was about, that no one had any context to make sense of it. A graduate statistics class at San Francisco State University just wasn't that kind of space. I did get an A.
#### The Putnam Exam
projects / An_Algorithmic_Lucidity.git / commitdiff