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Why won’t anyone teach me math?

• graycat 5y ago
  Part I So, Princeton did a really bad job teaching a first course in linear algebra for non-STEM majors. I'll try to make some sense out of that and outline how a student might defend themselves. First, as a ugrad, I looked at college education as career preparation, essentially trade school and had no understanding of the issues of status, prestige, new research results, etc. Point: A lot of high end US research universities concentrate on the status, prestige, bright students, and financial support they can get from new research results, etc. They can regard teaching as a bit silly: The research results are in the library and available to anyone who wants them. Second I had been influenced by the US NSF propaganda that STEM would make a good career so wanted to major in physics. Uh, in simple terms, the NSF was trying to create a labor force for US national security. Soon I saw some really sloppy math in physics classes, e.g., a just awful attempt, basically 100% wrong, to prove Stokes theorem, guessed that the physics profs were so bad at math that if I stayed a physics major I would be so bad at the relevant math -- for Maxwell's equations, quantum mechanics, general relativity -- that my STEM career direction would be in trouble. So, I majored in math intending to return to physics. I went to grad school in math to get the rest of the math I needed for physics so I could switch to physics. That math department was not much interested in teaching me the math for physics. I got recruited to work around DC, mostly on US national security, and there studied math on my own. How: Get a highly regarded text, one section at a time, study the material, study any example problems, work the exercises. Lesson: That worked pretty well, and I recommend that students consider it. Soon, on both my job and my independent study, I ran into quite a lot of linear algebra and learned a lot of it. Then I returned to grad school. I got accepted to Princeton, Cornell, and Brown but sensed the contempt issue and went elsewhere. Presto, bingo, the department Chair taught an advanced course in linear algebra; yup, it was a flunk out course. I told the faculty that I didn't think I needed more linear algebra and wanted to get on with material I didn't already know. The faculty just gave me a patronizing smile. So, I took the course. I didn't intend to embarrass the faculty and/or the department Chair but, in the end, I did: The course had a lot of graded homework and tests. Early on the homework grader made a mistake on one of my solutions; I corrected him and he made no more mistakes. Unintentionally, I was totally blowing away all the other students on homework, tests, and the midterm. When the prof got to the polar decomposition result, I blurted out "That's my favorite theorem!". The prof was so flustered he didn't complete the proof. Lessons: (A) There is a strong propensity among US math profs to look for essentially superstar students, to have contempt for everyone else, and, in particular, to look for any flaws and, seeing one, to have contempt for the student. A remark from WWII was that the German military had a big weakness, that the officers had to prove themselves everyday. Well, in US college and grad school math, there is a strong propensity to do that to the students. There is a grand, Kryptonite-strong, better than anything even Spiderman has, way around that -- will mention that below. (B) The Princeton math department, partly due to its neighbor the Institute for Advanced Study where at one time were Einstein and von Neumann, has a reputation as the best pure math department in the world. That Princeton prof A. Wiles solved Fermat's last theorem likely plays a big role. Then it is easy to guess that there is high propensity to look for superstar performance and to have contempt for any students who hint about anything else. (C) There really are flunk out courses. (D) It's possible defend yourself from the flunk out courses and to blow away the other students and fluster the prof -- just learn the material before taking the course. (E) The really difficult math courses are theorem proving courses, and there the exercises and the tests are to prove theorems. There a student doesn't really need answers or "101 Solved Problems" because soon it can be clear when do have a proof: The difficulty is finding the ideas for a proof; checking the correctness of a proof tends to be relatively easy. Before the linear algebra course, I'd studied linear algebra and closely related topics from several texts, some that likely remain standard and good and some that were more advanced and specialized. So, here I'll outline what worked for me: As a ugrad, I had taken a course in abstract algebra. So, that was about sets, groups, rings, fields, Galois theory, vector spaces, quaternions, basic number theory, the fundamental theorem of arithmetic (each positive whole number can be written in exactly one way as a product of prime numbers) and the fundamental theorem of algebra (the complex numbers are algebraically closed, that is, each polynomial of degree n has n roots and, thus, can be factored into a product of n linear terms). Quite a lot of this material now gets used in cryptography and error correcting codes. For a while, there were some influential, maybe popular, texts on advanced calculus that did quite a lot of linear algebra. One of these was Nickerson, Spencer, Steenrod and from Princeton, and there in the early chapters can learn about vector spaces and subspaces, linear independence, dimension, linear transformations, inner products, orthogonality, the Gram-Schmidt process, etc. Another was from W. Fleming from Brown and there can also learn about convexity. Both of these texts then continue on to the exterior algebra of differential forms and, in particular, careful proofs of Stokes theorem. So, I got both of these texts and a few others and dug in. Actually that list of topics -- vector spaces and subspaces, linear independence, dimension, linear transformations, inner products, orthogonality, the Gram-Schmidt process -- has good intuitive explanations where can draw nice, helpful pictures, can be covered in not many pages, and, really, are, say, over 60% of what should get from a first course in linear algebra. I heard the linear algebra book by E. Nering was good, got a copy, and worked quite carefully through it. It was good; likely still should be considered good. Nering was a student of E. Artin, right, at Princeton. The main examples of fields are the rationals, the reals, and the complex numbers. But there are some more fields, e.g., integers modulo a given prime number. Linear algebra and its vector spaces need a field, and Nering does nearly the whole book assuming any field and not just the rationals, reals, or complex. So, Nering does a little extra generality -- that approach makes some arguments a little more delicate but otherwise causes no trouble. Error correcting codes has some applications of linear algebra using finite fields. Then I heard that the P. Halmos, Finite Dimensional Vector Spaces was a good book on linear algebra and a also a good introduction to the math of quantum mechanics. Right, that book was written by Halmos when he was an assistant to J. von Neumann at the Institute for Advanced Study at Princeton (the town, not really the university). The book can be regarded as a baby step into more general Hilbert space theory. So, I dug in and studied carefully. It's a good book! Halmos is one of my favorite authors. When I noticed that his proof of the Hamilton-Cayley theorem didn't work for finite fields, I wrote him a letter. Got back a really nice, enthusiastic response! Apparently at least then writing a really good book on linear algebra does not make one a rock star -- don't get a lot of mail! Then did more in linear algebra -- numerical linear algebra, applications to statistics, the fast Fourier transform, optimization, computational geometry, error correcting codes, and more. Lesson: Those efforts in learning linear algebra are why effortlessly and unintentionally I blew away the other students in that flunk out linear algebra course.
  • graycat 5y ago
    Part II So, for the OP and their "Why won’t anyone teach me math?", if you wish, can learn linear algebra like I did and outlined above. Then can effortlessly blow away the other students and even intimidate the prof even in an advanced flunk out course in linear algebra. Sure, can learn from Nering and Halmos if you want. Also looks good to me is Hoffman and Kunze, Linear Algebra, Second Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. apparently now available at https://www.math.pku.edu.cn/teachers/anjp/textbook.pdf There is also a text from G. Strang that is recommended for a first text. I will try to help get the frustrated Princeton student started: First, it is fair to say that a good start on linear algebra is just solving a system of linear equations. E.g., given numbers a and b, find the set of all numbers x so that ax = b Exercise: Show that depending on the values of a and b, the set of all solutions consists of none, one, or infinitely many values. The ax = b is one linear equation in one unknown, x. Well for positive integers m and n, we can have m linear equations in n unknowns. So, here are 2 linear equations in 3 unknowns, x, y, and z: 2x -y + 5z = 7 x + y - 2z = 3 Why are they called linear? Good, crucial question -- profound issue; will come to that! For reasons analogous to what saw with ax = b again the set of solutions of m linear equations in n unknowns has none, one, or infinitely many solutions. Exercise: We have already treated the case of m = n = 1. Treat the case of m = 1 when n > 1. Assume the number of equations m > 1: For finding the set of all solutions, the standard approach is Gauss elimination. Here is the key idea: If take a number a and multiply one of the m equations by the number a and add it to one of the other of the m equations, then the set of solutions does not change. That is called an elementary row operation (ERO). Exercise: Argue this point -- it's easy. So, with Gauss elimination just apply EROs to yield the equations with lots of coefficients 0 that permit reading off the set of all solutions easily. How to do this? Do an ERO to have a 1 as the coefficient in row 1, column 1 and 0s in the rest of column 1. Now do an ERO to put a 1 in the 2, 2 position and 0s below that. Continue in this way and end up with a triangle of 0s. Now can just read off the set of all solutions. Here start to see that get a lot more for your time and effort than you expected. Back to 2x -y + 5z = 7 x + y - 2z = 3 We rip out the variables x, y, z and write all this as / \ / \ / \ | 2 -1 5 | | x | | 7 | | | | | = | | | 1 +1 -2 | | y | | 3 | \ / | | \ / | z | \ / So we have three matrices, the one on the left has 2 rows and 3 columns so is said to be 2 x 3. The next one has the x, y, and z and is 3 x 1. The one on the right is 2 x 1. For the first two matrices, we define the matrix product to yield essentially the same thing we had with the 2 equations in 3 unknowns. So we have just rewritten the 2 equations in 3 unknowns. It turns out, however, that working with the matrices is a huge improvement, step up. Vector spaces? A matrix with one row and/or one column is called a vector. Sometimes we can be more specific and call the vectors with 1 row dual vectors. Then if we generalize a little we can prove the Riesz representation theorem that gets used in quantum mechanics. Point: Linear algebra is an introduction, simple elementary special case, of a lot more in pure and applied math. Quite generally we say that a function F is linear if F(ax + by) = aF(x) + bF(y) Here I have deliberately not given a careful definition of the symbols F, a, b, x, and y because (i) linearity is one of the largest pillars of math and (ii) there are lots of cases with different definitions for the F, a, b, x, and y. E.g., in calculus, both differentiation and integration are linear. For more, in electronic engineering and signal processing, every time invariant linear system just modifies the amplitude and phase of sine waves. The real world and applications of math to it are just awash in linearity. Well, matrix multiplication is linear! And that's why the equations 2x -y + 5z = 7 x + y - 2z = 3 are linear. Okay, suppose m = n. Then a matrix m x n is square. Given a matrix U, suppose for any x the length of Ux is the same as that of x. Then all U can do is do rigid rotations and reflections. Or suppose given a matrix H where all it does convert a sphere into an ellipsoid with mutually perpendicular axes. Now given a square matrix A, there exist U and H so that A = HU. The U is called unitary and the H, Hermitian. In quantum mechanics, the evolution of the wave functions is unitary, and the measurements are Hermitian. Principle components in statistics is based on Hermitian. The A = HU is the polar decomposition. So, linearity is simple: All the linear A can do is rotate and/or reflect and then stretch and/or contract on mutually perpendicular axes. Fill in all the details, and that should be 80+% of a first course in linear algebra. For more, take Gauss elimination and specialize it slightly and get the simplex algorithm of linear programming optimization. At one time, Princeton, along with Berkeley, played a leading role in linear programming. Then linear programming became the core of at least one Nobel prize in economics. Can use some of the basic theory of linear programming to show the saddle-point theorem of game theory, e.g., was done by different methods by von Neumann. To defend yourself from the propensity of pure math profs to dump you into the contempt bucket: Move to some advanced, specialized material, find a loose end, state and prove a theorem to tie off the loose end, and publish the result. Can work better than Kryptonite.
• cosheaf 5y ago
  The complaint here is about not having access to solutions for problem sets. On the one hand I can see why this would be annoying, on the other hand part of learning math is about coming up with your own questions and answers, a skill that is hard to develop if all the answers are just handed to the students.
  • lthornberry 5y ago
    There's a pretty solid literature on math pedagogy, and it's clear from that research that students need some way to tell when their answers are right or wrong, so that they can learn from the mistakes. For homework, that could mean getting problem sets back with the wrong answers marked. But it sounds like these were just practice problem sets, so the students literally had not way to tell if they were doing them right or not.
    • cosheaf 5y ago
      Then maybe Harvard is not the best place to learn mathematics.
• skywardavocado 5y ago
  > For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control. Wild. EDIT: A response (and +1) to other comments: of course, there are many solutions to different problems on the internet. But IMO, a big advantage of taking a class is to have shared context with a group of people, and thus interactive help (from other students, TAs, etc) working through specific problems together. Being able to know if you were "right" seems to be an important part of this process. So having solutions to the shared problems you're all looking at seems important. (Withholding solutions seems like a method for instructors to reuse questions instead of having to write more every semester.)
  • pfortuny 5y ago
    Departmental policy? Man, what a lame excuse. As if there were not hundreds of books with solutions.
  • Spivak 5y ago
    It's 100% because the problems come from a textbook and some other professors assign them as homework, as opposed to study, rather than writing their own problem sets. If it's from the professor's own problem sets then actually WTF. Textbooks written "for teachers" are the bane of every college student's existence if you dare to actually learn from them yourself. "Only solutions to odd problems -- cool guess I have to look on the internet or never be able to validate whether I got them right."
    • strikelaserclaw 5y ago
      you would think Princeton professors would think that their students wouldn't cheat. Must be those kids who paid to get in.
      • HWR_14 5y ago
        Cheaters, if they are good/hard to detect, would be more likely to be at Princeton than at a community college. By the time you get to Princeton, you're either good at studying or cheating (or being born to the right parents.)
• mindcrime 5y ago
  I left with the certainty that I would never take a math class again, and a lack of desire to explore other scientific fields for fear that I would have a similar experience. That is one of the most tragic things I can imagine reading. A student who really, genuinely wants to study Maths, just for its own sake... and is dissuaded from doing so by myopic, short-sighted, inane policies. It makes you wonder "could you fuck up education any worse, if you were intentionally trying?" On the other hand though... I believe a student with enough self motivation can overcome those inane department policies and learn the Maths by pulling in outside resources. Yes, it's more work, and yes maybe it's not "fair" that you have to do that in addition to attending (and paying for!) a class. But it's possible. Consider the availability of free resources like Khan Academy, Paul's Online Math Notes, hours and hours of Youtube videos from people like Professor Leonard, Gibert Strang, etc. One can also often find problem sets with solutions by just Googling around and finding previous years course websites for various courses that have been taught at universities all around the world. There are also online forums where you can go to get your answers validated, or get additional explanations about Maths problems. They vary in the extent to which they accept "do my homework for me" style questions, but almost any site will acknowledge somebody who has put in some work, come up with an (possibly incorrect) answer and says "Can somebody help me understand what's going on here?" Examples: * https://www.reddit.com/r/learnmath/ https://www.reddit.com/r/learnmath/ * https://www.reddit.com/r/cheatatmathhomework/ https://www.reddit.com/r/cheatatmathhomework/ * https://www.reddit.com/r/MathHelp/ https://www.reddit.com/r/MathHelp/ * https://www.physicsforums.com/ https://www.physicsforums.com/ * https://math.stackexchange.com https://math.stackexchange.com (note: but not so much Mathoverflow, which is more for research level Maths and is not a good place for straight up "do my homework" style questions at all) And if one is willing to pay a bit, the Schaum's Outlines books and similar books provide a huge catalog of worked problems, along with problems and solutions. There's also Brilliant.org and other paid educational resources for Maths that can supplement ones university course(s). One can also look into hiring a personal tutor as an option. Many (most? all?) universities also have something like a "math lab" or "learning center" or something where students can go for individual tutoring and additional support. My experience (albeit dated now) leads me to believe that these are probably drastically under-utilized. Should universities do a better job with introductory Maths courses? Almost certainly. But I'd encourage anybody dealing with this to dig in and try to overcomes such shortcomings by using other available resources as well. EDIT: inspired by @skywardavocado's answer, I also wanted to add that old-fashioned "study groups" are another valuable arrow in the ole quiver. If you join up with 2, 3, 5, whatever, of your peers to study together, chances are that somebody in the group will understand the thing that the others are struggling with, and can explain it. I didn't do a lot of this in college myself, probably to my detriment. But to the extent that I did occasionally join a study group, I'd say they can be wildly helpful.
• rossdavidh 5y ago
  Ugh. Something deeply wrong about our university system, that this article didn't really surprise me at all.
  • bell-cot 5y ago
    Princeton has picked a metric, and has gotten so good at optimizing for it that they're now #1 for eleven years and counting: https://www.forbes.com/sites/michaeltnietzel/2021/09/13/us-news--world-report-releases-its-2022-college-rankings/ https://www.forbes.com/sites/michaeltnietzel/2021/09/13/us-n... (U.S. News & World Report's Best College rankings)
• brunellus 5y ago
  I’m glad I’m not in college anymore
  • pirate787 5y ago
    It was the same way 30 years ago.
• AtreidesTyrant 5y ago
  Does anyone have good book recommendations on Math? I've always deeply enjoyed the subject, and I think I'd like to study on my own but I'm unaware of where to even delve into the subject.
  • Spivak 5y ago
    My username comes to mind. Spivak's Calculus is hands down the best intro Calc book if you want to learn math for its own sake. His book on manifolds is also amazing but definitely not intro. If you want stats I highly recommend Statistical Inference by Casella & Berger -- it's extremely dry but so many stats books out there try to "make it easy" but the simplification means that you can't actually grok what's really going on. HOWEVER if you want to actually apply anything in this book you'll need to grab something more practical as well. Going through an applied stats book after having done SI is like having superpowers.
    • l33t2328 5y ago
      Spivak’s Calculus is hands down an excellent book for a first course in real analysis, and I’ll die on that hill(it’s only real competition is Analysis 1 by Tao, in my opinion). Anyone that says “it’s just a calculus book; it’s not good for introductory real analysis” is invited to go solve every problem in, for example, the chapter which defines integration, and compare the difficulty with problems in “traditional” analysis books.
    • mjh2539 5y ago
      If they know how to do (rigorous) proofs, then Spivak would be good. Otherwise it would be far too advanced. It's essentially an introduction to analysis.
      • Spivak 5y ago
        That's fair. I don't really know what counts as advanced anymore. There just aren't that many books on theoretical math that aren't "graduate level" (which is way scarier than it sounds) -- Spivak was my Real Analysis 101 book as a freshman math major. Spivak at least has the nice property that it doesn't assume you know much.
• orangepurple 5y ago
  Is this an outlier? Or a sign of something more endemic in the University system?
  • colechristensen 5y ago
    Why not both? A particularly bad class demonstrating typical problems in an unusual intensity. Also an individual who had a particularly intense mismatch between her learning style and the course design.
  • lthornberry 5y ago
    I don't know if it's true of all schools, but I had a very similar experience at another Ivy League school. Took the first semester of linear algebra, got a very respectable grade, but really didn't learn much because the class wasn't structured to do that. It was there to make STEM majors prove they were good enough. I wasn't a STEM major, so didn't need to keep going with that kind of hazing.
  • peckrob 5y ago
    In my undergrad years every major seemed to have what students referred to a "weed-out course." These courses were intentionally made difficult and often has inane non-standard grading policies (for example, in one, anything under 85 was considered an F). They were often taught on odd schedules (like one semester a year or something) and at unusual times. And they were often taught by difficult professors who did little, if anything, to help students succeed. If you were lucky, there might be a TA that might be able to help you, but students were often expected to form independent study groups and assist one another. Basically if you couldn't pass this course - and in some cases department policy limited you to a single try - your chances of success in that major were slim to nonexistent, and you were encouraged by your advisor to find a different major. If you could pass the weed-out course, you could be reasonably assured you were ready to face the rest of your coursework, because nothing ahead would be as difficult as that course was. And, that you were reasonably certain this was what you wanted to do. They weren't necessarily directly related to the major. For some of the engineering and science programs, a math course was often the weed-out course. I can't say for sure that this was the author's experience, but her description of it sounds very much like a weed-out course.
• puffoflogic 5y ago
  Unfortunately it seems like no one has been willing to teach the author what mathematics is. Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work; it isn't mathematics. Now it's possible I'm misunderstanding the problem here: I am assuming that during lessons students have been given answers to problems worked in class, so that they can see what an answer looks like, which is probably one of the most important parts in learning introductory mathematics. Indeed, learning to validate one's own answers is not only the most important lesson in this stage of learning mathematics; it's also the only way to actually learn mathematics as it's the only way actual math gets done. (ETA:) So I guess what I'm saying is that the author is right that no one has taught her mathematics, but as a symptom of this she doesn't even understand what form actually learning mathematics would take.
  • cptcobalt 5y ago
    I think there are shades of correctness to your answer (at the upper echelons of math study, as it were, there will be no answer book for you to check), but the act of confirming your answers should be transitioned in and taught; which from this author's perspective hasn't happened. Further, there should be education, refinement, and strengthening of learning to validate one's own answers before leaving them to flounder in a 200 level STEM-focused course. (Honestly, I'd probably have a better relationship with math if it were taught to me in the method you seem to require as "actual math".)
  • sidewndr46 5y ago
    > Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. I remember the agony I experienced as a 3rd grader for not being able to memorize "multiplication tables". Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. This was to my extreme academic & mental detriment as we were not graded on understanding, but instead graded on how fast we could answer math questions. I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles. Obviously, my concern over this was totally pointless as I've never been in a career environment where I was told "just memorize the answer, don't bother trying to understand anything".
• CoastalCoder 5y ago
  I wonder if some universities should pay more attention to the impact of discouragement on some students. I remember attempting a grad class called "Hard problems in combinatorial optimization". I was struggling so I asked the professor for advice on getting my footing in the class. His answer was that I may just not have the "mathematical maturity" for the material. I was so discouraged that I dropped out of the program. What I didn't realize at the time was the particular meaning of that term. His advice was well intended and accurate. But what I heard was that I was hopeless in this topic area because I wasn't smart enough. So much pain could have been avoided if we just extended the conversation by a few minutes, or he invited me for a discussion over coffee.
  • treeman79 5y ago
    First day of college algebra. Teacher shouted out “75% of you will drop out. Please leave now” 4 hours a night 6 days of week and I got a C. Less the 25% made it to end. Don’t think many of remaining passed. Got a lot out of it though.
    • eigenket 5y ago
      That sounds like either an absolutely terrible teacher, or a terrible program. Why were so many people enrolled in a class that less than 25% would pass? I'm not American so I don't really understand the course names but I guess a class just called "Algebra" shouldn't be too hard? Stuff like basic groups/rings/fields or is it a more advanced class?
      • JadeNB 5y ago
        The title "College algebra" in the US indicates material that is well pre-groups/rings/etc.; it is about the basic skills of computations involving manipulation of variables, often involving such topics as solving linear and quadratic equations, and perhaps inequalities. At least at my university, the reason that such classes have a high failure rate is that the university is incentivized to have high enrollment numbers by admitting students who do not have the background to succeed at college mathematics at the level of calculus or above, and so need some pre-calculus courses; but someone who does not have a solid grounding in pre-calculus mathematics will often struggle to learn it when it is taught at a rapid pace at the college level—especially by professors most of whose teaching experience is with college-level math.
• cjbgkagh 5y ago
  Sounds like she would have enjoyed a stats class more. Applied math without as much emphasis on first principles and formal proofs. I didn’t like university level math (first year lvl) because I preferred programming and doing math without computer assistance seemed to involve a lot of wasted effort. Maybe I’ll revisit it with Lean.
• bsedlm 5y ago
  What I've found, is that often teachers (really anybody who's understood the subject and is willing to teach) will try to teach both the subject AND how they have learned; i.e. they do not only tell you about the math but also the way in which they have arrived at understanding it (how they've related to it). to use an analogy, when describing the math they tell one about a place where they are (a 'place' where the math is there, already understood) and how they got there (the analogue of route they took to get there e.g. "took public transport line 4, get off at station Example and walk until you're there"). This leaves little room for those who may prefer to ride a bike all the way there (the analogy has been stretched to the breaking point), cuz the teacher will knock you off the bike and insist that you ride the line 4 then walk, same as they did.
  • eigenket 5y ago
    For most maths, at least the stuff you'd learn before or during an undergrad degree the interesting maths is the journey. The teacher is attempting to teach you to ride a bike, and the student is like "this is stupid, I already walked to the end of the road". The teaching of different ways of making the journey is (often) the destination.
• temp8964 5y ago
  According to the author, the course is "for underclassmen who are majoring in engineering or sciences", but she is a humanities student who wants to "explore STEM fields." Isn't it obvious that the course is not designed for her learning purpose?
  • eropple 5y ago
    "A humanities student"? She's a humanities student who had taken up through linear algebra! I have a CS degree and I never learned linear algebra in high school or college--I learned about the topic later! The idea that a 200-level course should be unapproachable for an interested person with a stronger mathematics background than I have, with an ostensibly mathematics-based degree, is absolutely silly. I took 300 and 400-level classes in economics, in English, in history and philosophy. I never felt unwelcome or incapable.
    • Jtsummers 5y ago
      > I have a CS degree and I never learned linear algebra in high school or college Really? I mean, you probably did have linear algebra in high school, it was just "disguised" (you may not have seen matrices and vectors, but probably did solve linear equations of multiple variables). I'm surprised about the college thing, though. I thought every CS degree (in the US at least) required at least through linear algebra (linear algebra with applications, probably not linear algebra with proofs and theory).
      • eropple 5y ago
        I knew what a matrix was. I don't remember having to actually ever use them for anything. I learned what a (mathematical) vector was offhandedly in college, but I think it was from talking to someone, not in a class.
  • lthornberry 5y ago
    I'm assuming the issue is that there isn't a math pathway for humanities students who want to take math electives. This is the closest thing she could find. This was the case at the Ivy I attended. There were classes designed to fill the one-course math requirement, but they were all below the level of the math I'd done in high school. There wasn't anything for someone who was decently good at math and genuinely interested in it, but not planning on a math-related major.
• mathattack 5y ago
  I’d have more sympathy if this were someone from the inner city or rural America. But they’re in Princeton crying about not getting answers to problem sets. As if the first time they’re not spoon fed something, it’s time to trash the department. If you can afford Princeton, buy a linear algebra book with solutions. It’s as if a Math major said, “Oh I would love to read Hemingway but there is nobody in the English department to tell me what questions are on the test. How can a Math major learn to read? Where in the world might I find a clue about Hemingway?” Yes - math education can be awful, but this snowflake needs to lose the entitlement.
  • dboreham 5y ago
    If you read the article they say that they did achieve success in the class, but the experience could have been much better/easier. Only if you believe that an ivy league institution is there to provide a kind of S&M experience for its students does your comment ring true.
  • PaulHoule 5y ago
    I'm shocked with how inarticulate the author is. I'm not getting any idea of what math class they are taking or what they expected to get about it. You'd expect a humanities student to be able to explain themselves better. The one well-developed complaint is the one about not getting the answers. On one hand that sounds almost calculated to provoke the reaction that you had, but on the other hand for every math class I've taken I turned in the problem sets and had them get graded with corrections that were useful.
    • mathattack 5y ago
      This sounds like problem sets above and beyond the homework. As in “Prior tests have problems like these…”
    • Jtsummers 5y ago
      It's pretty clear from the first paragraph that it's a linear algebra class.
    • 5y ago
• DoneWithAllThat 5y ago
  So the author took one class that was unnecessarily difficult. That’s unfortunate and worth following up with the Math department on. But expanding that into a full-length article with the thesis of “Why won’t anyone teach me math?” seems like a stretch.
  • 100721 5y ago
    I don't think she had an issue with the difficulty of the course, but rather the pedagogical approach. It seems she walked away feeling like she didn't actually learn very much, despite passing the course.
  • tester756 5y ago
    I think being open about poor sides of higher edu institutions may result in them improving
  • newaccount74 5y ago
    The article is from the Princeton university student newspaper, so I assume the author meant "Why won't anyone teach me math at Princeton?". It's specifically about the experience of liberal arts majors at Princeton, not some general thing.
  • dematz 5y ago
    It seems like the issue was more that the class was not only bad for her but poorly structured in general - it would be a good thing for non math majors to learn topics like linear algebra but the department could set up classes differently to better teach them.
• tester756 5y ago
  I've been terrible at math in hs and then I started doing programming and realized that math ain't hard and actually liked doing it I managed to improve and then I had math at higher edu institution and oh boi, I instantly had no interest in doing it 4fun, just learn, pass and move on. idk, I feel like there's too much stuff to be done (a lot of courses + you gotta learn other stuff like "real world" computer's related stuff) and there's lack of time for 4lulz messing with fancy topics like maths, but on the other hand I've been working + studying, so maybe that's why
• dboreham 5y ago
  There isn't any supporting material in the article (the links to course review comments are behind an authentication wall) so it's hard to be sure, but it sounds like the answer is "because you had abusive teachers". By that I mean psychologically abusive people who derive pleasure from seeing students struggle in pain. It makes them feel better about themselves.
  • eli_gottlieb 5y ago
    No, those aren't abusive teachers. That's the standard approach to teaching STEM subjects in most of the worldwide. I'd say the abusive part is the often-unspoken expectation that students are going to sneak around department policies to find ways to check their work other than handing it in.
  • irrational 5y ago
    That isn’t what the article says. It says the teacher’s hands were tied because of department policy. Why does the department have that policy? What set of circumstances led to that policy being set and what would be a better policy to both address the set of circumstances and to not hinder teachers?
    • colinmhayes 5y ago
      A class doesn't become bad enough to write an op-ed about for that alone. There must've been multiple other issues for this class to have a 2.7 rating.
    • notreallyserio 5y ago
      Seems that is exactly what the article says. A professor that can get a job teaching undergrad math at Princeton can likely get the same job at any other university. So this professor is actively choosing to enforce this policy (it's their revealed preference).
      • irrational 5y ago
        This is a lower level math class. It isn't being taught by a professor. At best it is being taught by a grad student.
• NoNameHaveI 5y ago
  Sadly, in my experience, these was the case in introductory STEM courses. They could care less whether you understood or not. There were plenty of people in line behind you, so your personal experience didn't matter. This is a big reason why I encourage people to take as much STEM as possible at community college instead of public university. Faculty are focused on teaching, not research and publishing. You have a better chance of actually learning something.
• pseudolus 5y ago
  Reading the article am I the only one who had difficulty understanding what exactly the subject matter of MAT 202 was? She's indicated that she's taken courses up to linear algebra and then states that "she wanted to learn linear algebra". Is this a linear algebra course?
  • Jtsummers 5y ago
    I mean, she said she had taken math classes "up to linear algebra" and she took the course to learn linear algebra so I would guess that it was linear algebra. Google confirms: https://www.math.princeton.edu/undergraduate/placement/MAT202 https://www.math.princeton.edu/undergraduate/placement/MAT20... - Linear Algebra with Applications.
    • pseudolus 5y ago
      Thanks. Maybe I'm being pedantic but leading off with "so I selected MAT 202 - Linear Algebra with Applications" would have provided the article with some clarity.
      • bell-cot 5y ago
        (It's the Princeton student newspaper, so the target audience pretty much knows that.)
• wanderr 5y ago
  I believe there is a major misalignment of goals/incentives in undergraduate academia. As students we think the model is, we pay [quite a lot of] money to be taught useful knowledge in a rigorous way that means we will leave with real knowledge. Universities, on the other hand, seem to think the model is to churn out well qualified individuals in their field. There are two ways to accomplish that goal: doing the hard work of providing a real education to everyone; or doing the much easier work of filtering out those who don’t already "get it". This is why Universities don’t care that a class is reviled by all the students.
  • colinmhayes 5y ago
    In my experience undergrads are almost universally seen as a nuisance by professors, especially at the most prestigious schools where professors see themselves as godly researchers. It also sounds like this class isn't even for students in the department, which usually means no one wants to teach it. Basically becoming a professor has almost nothing to do with ability to teach, so I'm not surprised when the least desirable classes to teach are taught by people who have next to 0 ability to do so.
  • zozbot234 5y ago
    > doing the hard work of providing a real education to everyone; or doing the much easier work of filtering out those who don’t already "get it" I wonder how much of a problem this is in intro CS. Everyone knows that the camel has two humps, but are we inadvertently gatekeeping by not even trying to teach the low-scoring "hump" about the fundamentals of the discipline?
    • eli_gottlieb 5y ago
      I thought it was common knowledge that this is the problem with intro CS. There are places that get much better results by making sure that either freshmen test out of the intro class or they all know equal amounts of nothing when they take it.
• jbluepolarbear 5y ago
  In college they did the same thing, so I pirated the book the teacher was using for the class so I could get the answers to the questions we were being assigned. It helped me learn the material with multiple examples and solutions. I think it should be required to give the solutions to homework; especially, with classes like differential equations and vector calculus.
• ineedasername 5y ago
  >For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. Strange. But it also may no longer be true, or not to the extent that it was when this student was enrolled in the course. From the course website: "Try some old quiz problems for this course, but don’t just read the questions and solutions. Instead see if you can produce correct solutions to most of the problems in the allotted time." https://www.math.princeton.edu/undergraduate/placement/MAT202 https://www.math.princeton.edu/undergraduate/placement/MAT20... So clearly, now, there are archives of problems w/ solutions that can be reviewed after attempting a solution. Apart from lack of solutions, I don't see anything "off" here. I don't see other criticisms of the teaching approach. I am also not surprised that one of the best math & physics universities in the world does not cater even its lower level courses towards students for whom they are optional and not part of their intended academic career. In my school (which was not a top-tier program) courses for majors in those areas were also similarly difficult. The difference was that my Uni also offered a few "Math for non STEM majors" courses to choose from because they still did have requirements for all students in those areas. Basically a survey course of mathematical concepts and how they show themselves in the world & everyday life, along with practical applications of how they may need to use math in any way throughout their lives, things like how basic level probability & statistics will be useful no matter what they end up doing in life. And things like fractals, or the golden ratio, and other concepts along those lines that show up everywhere in life. The course was designed to answer the question, with concrete examples, "Why should I care about math if I'm barely going to have to use it in life & have a calculator?" I think Princeton simply expects their students to understand the answer to that question without putting their students through the trouble of an entire semester on the topic.* *Though clearly should address the "no solutions" issue if it has not been fully resolved already.
• mattgreenrocks 5y ago
  My entire undergrad schedule was arranged according to getting the best-ranking math teacher on ratemyprofessor.com. I had enough painful experience and had to early drop a math class or two to learn this fact. Bad math teachers essentially forced me to self-teach, and I did not have the ability to do that with math, as I wasn't a natural in math beforehand. It got so bad that I happily took 8am classes despite being a night owl. What it looks like now as an adult is just a failure of pedagogy and institutional incentives. I suspect the people teaching it had little reason to take the teaching part seriously, it was just something that got in the way of their research/grad studies. I went to a well-known state school. There were other departments like this.
• ngngngng 5y ago
  > This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s. Is there some prior knowledge about university exams I should have in order to understand what these figures relate to? Standardized set of 3 exams maybe?
  • eli_gottlieb 5y ago
    The grades are out of 100%, so these averages mean that the professor has to z-score the grades in order for any significant portion of their students to pass at all (the pass line in the USA is usually 65%). I'd consider this a strong sign of bad teaching.
    • andrewprock 5y ago
      Or the course may have been taught on a curve.
• jcranberry 5y ago
  This is a very vague article. The only fact I see is not providing solution to practice problems. That's not ideal but shouldn't be a dealbreaker. More details might be helpful. I do find it surprising that the whole class could have such a low grade in linear algebra, which is one of the easier introductory math classes. But I've heard of grade deflation in Princeton is pretty severe, so I don't know what standard grades are there. IIRC Princeton also has one of the premier math departments in the entire world, so I would expect the program to be very difficult. On the other hand, I think the interest of tenured professors in math at universities is almost entirely on research. This doesn't mean they can't be good teachers but the ones who are I think in general are thinking more of cultivating potential phd students and the like. A practical class for non math students I can easily believe would be completely blown off.
  • seanc 5y ago
    Well as a physics grad who struggled with first year LA and a parent of a current university student I can fill in some blanks. Linear Algebra specifically is a superpower these days and I was very excited to do the best I could to help by daughter get past the barriers I experienced and learn it well. I failed. I found her course to be not only sink or swim, but actively hostile to student learning. As mentioned in another post, assignments and quizzes were returned weeks late with little helpful coaching. Assignments very often worked with higher levels of abstraction than presented in the lectures, for example the first time she saw a matrix of functions was on a homework assignment. Furthermore there seemed to be no additional help for Linear Algebra. To my great surprise the math department had no facility for matching students with tutors. The university math help center was almost exclusively oriented towards calculus and had no resources for Linear Algebra. After weeks of searching online we finally got a call back from a third party tutor, only to be informed that this person (a grad student in math) couldn't help us because the assignments were "too specific to the particular course." The _only_ place I've ever seen someone show any creativity and ingenuity in teaching linear algebra is 3Blue1Brown, but obviously even though he's got a whole linear algebra series it's not a complete university course, nor is it meant to be. So yeah, it's a damn shame, and IMO inexcusable. Linear Algebra is the foundation of modern data science and machine learning. Math departments should consider it a sacred duty to bring as many students as possible to at least some level of comfort and familiarity, but instead they seem to treat it as an annoying distraction at best, or a weeder class at worst.
• trinovantes 5y ago
  Don't most universities have TAs around, especially for lower year courses, to help students falling behind? I remember in my undergrad when we had difficult courses, half the class basically camped outside our TAs' office before each assignment deadline to ask for help or clarifications.
  • gus_massa 5y ago
    I agree. I'm a profesor in the first year of the University of Buenos Aires. We have no official list of answer for the exercises (but there are a few unofficial ones). The students can ask me or the TA abut the exercises. We prefer to see what the student has attempted, and find the exact spot where the student made the mistake. It takes more time, but it's more helpful than a nice ideal solution. We also may recommend to make another exercise from the official list and come back in 10 minutes with a solution attempt, or make some custom exercise on the spot that is about the same subject. (Exercises about derivatives are easy to invent, integrals and linear algebra is harder.)
    • joey_bob 5y ago
      What portion of student's would you estimate ask you or the TA about exercises?
      • trinovantes 5y ago
        When I was in undergrad, I would say it was about 10-50% of the class depending on the difficulty of the course
      • gus_massa 5y ago
        It's difficult to know, because sometime the course have 100 students, and each day different students make questions to each TA. My guess is 10%-50%, but it depends a lot on the major the students are. One of the useful trick is to write an optional homework exercise in the blackboard. I get a lot more of answers than just selecting an exercise from the official list. Sometimes it's just an exercise from the list with different number and sometime it has an intermediate step to make it easier. But writing it in the blackboard increase the chance the students will write it down. Another trick is to write an old midterm and then go to each desk to talk to the students about what they are doing. We usually write the solution at the end of the class, and have some discussion about it, but the useful part is the discussion with each student (or small group of 2-3 students).
• wintermutestwin 5y ago
  I'll use this thread to ask a semi-related question: Over my years (probably more than most here), I have found no need for math beyond some simple algebra and statistics. This includes 30 years in a tech career. I'm now semi-retired and working in an even less mathy field. That said, I have always felt like I might have missed out on something important in my auto-didactic intellectual development by skipping math. Given my situation, how would you try to convince me that I should learn math and how would I teach myself the actually interesting parts without getting bogged down by doing lengthy calculations that a computer could do. The standard curriculum is so heavy on the latter and mostly absent of the former.
  • akomtu 5y ago
    A common trick to get someone interested is to present something surprising. That surprise gets you attention that can be turned into understanding by explaining the mystery. The hard part is guessing what would get you surprised enough.
  • pdonis 5y ago
    > Given my situation, how would you try to convince me that I should learn math I wouldn't. I would say, if you don't need math to accomplish anything you want to accomplish, and you're not interested in it anyway, why bother? If you are interested in it anyway, then you should not need anyone to convince you that you should learn it. You should just go learn it. > how would I teach myself the actually interesting parts without getting bogged down by doing lengthy calculations that a computer could do First, if you don't understand the "lengthy calculations", you will have no way of knowing whether the answer the computer spits out to you is correct. Second, doing the "lengthy calculations" is sometimes the only way to learn the "actually interesting parts". If you haven't done the grunt work of solving some problems from start to finish in a subject area, you don't really understand it. You might have a sort of vague conceptual overview of it, but you don't really understand it. So you need to decide whether a vague conceptual overview is enough for you (for many people it is, and if you don't need to know the subject for anything essential, it might be), or whether you want to really understand it.
    • deleted 5y ago
• grumple 5y ago
  My experiences as a math major at a prominent university were even worse. In upper-level courses, professors would work through one problem on the board for the entire class. Turns out, they'd do it wrong about half the time. The next class (usually a few days later), they'd spend 30 seconds saying how they made a mistake here or there. So your entire instruction set was incorrect, solidified in memory for days at best, then we're supposed to divine which part you did incorrectly in 30 seconds! And then many of the professors had what was, at best, a tenuous grasp of the English language - which matters a lot when teaching complex subjects in English to students. And we were supposed to learn in these classes? Jeez it was awful. And these are the people at the top of their field, paid to do research? Engineering classes were hit or miss - some were great, taught core concepts really well. Others turned into awful mathematical formula applications / regurgitation with extremely dubious accuracy. Needless to say I don't have a great respect for math as an academic discipline, despite really liking math.
  • Someone 5y ago
    > In upper-level courses, professors would work through one problem on the board for the entire class. Turns out, they'd do it wrong about half the time. In my experience, such errors always were in the mechanical parts of the proofs, not in their main structure, and easy to spot and correct for the students, in the sense that at least one spotted them, and told the teacher.
• glitchc 5y ago
  It's really very simple. Math is a self-study subject. The axioms and relationships are described on paper in very clear language. How to reason and internalize those axioms is entirely up to the reader. Each mathematician or practitioner builds their own models and tools to absorb the content. Princeton publishes a Mathematics Companion. It's a good book that covers a wide range of topics. The treatment per topic is rather brief and acts as a gateway into deeper study.
• vondur 5y ago
  This type of experience was typical for me as an undergrad Chem major many years ago. Basically sink or swim, figure it out on your own, professors didn't care that O-Chem had a 50% fail rate. My P-Chem teacher told the class we could bring whatever materials we wanted during an exam, it didn't matter, he was going to "destroy" us.
• moralestapia 5y ago
  IMO it has to do with the people around said math course, professors, students , etc... Some environments make academic life great, some others make it "miserable" as the author describes. The advice I always give to young students (and a topic I'm writing about in an upcoming website that deals with abuse in academia) is find better people. There's (almost) always an alternative, find it and move to a better place; it's not always easy but it's easier than dealing with pricks for 2-4 years.
• jonnycomputer 5y ago
  Well certain university courses in STEM are "weeder" courses, designed to be much difficult than they should be to push out the people who aren't serious or talented enough. I think this happens in particularly popular majors, where there are far many more people interested than spaces available. That's actually too bad--policy wise, artificially limiting the number of people with high-value skills isn't smart.
  • kwertyoowiyop 5y ago
    This is just bizarre and backwards, now that a university could put the overflow into a MOOC.
    • tzs 5y ago
      I think he got the purpose of weeder courses wrong or at least partially wrong. It's often about getting people out of majors that they won't be sticking with before they have put in a lot of time. In a lot of STEM fields there will be one or more required courses in the 3rd or 4th year of a 4 year bachelor's degree program that are significantly harder than anything in the first two years. You can go through 2 or maybe even 3 years in a major thinking that this is the field you want to make a career of, then you hit those harder course and discover that the field is probably not a good career choice for you after all and you should change majors. Changing majors at that point might mean you will need a 5th or even 6th year to get you degree in your new major. If your STEM program includes a required hard course in the first year or two, something to give a good taste of what you'll need to get through 3rd and 4th year, you can find out early that this is not the major for you and switch to something you are better at and still get your degree in 4 years.
      • kwertyoowiyop 5y ago
        That makes sense. They need to find the right balance between coddling and abuse! I remember my low level CS courses being difficult, but not unfairly so. Except for when the second assignment in my CS 101 essentially asked for a recursive-descent parser, haha. Wooops. Anyway…couldn’t a MOOC be just as good as an on-site course at filtering?
• mmaunder 5y ago
  Would be interesting to do the math on how much that particular class cost.
• 0xbadc0de5 5y ago
  The author appears to have a framing problem. Her education is not somebody else's responsibility - it's hers. If she legitimately wants to learn, she already has access to the sum of human knowledge at her fingertips - for free in many cases. It seems to me that her quarrel is rather with the structure and delivery of one particular course she chose to take. This may not be obvious to a lot of young people whose only life experience has been tied to the education system, but you don't need a formal course to learn a subject.
  • pdonis 5y ago
    > Her education is not somebody else's responsibility - it's hers. If she legitimately wants to learn, she already has access to the sum of human knowledge at her fingertips - for free in many cases. Exactly. My first reaction to the article was, if you are interested in linear algebra, why don't you learn it over the summer in your free time, using one of the many sources available for free online? That's how I learned ordinary algebra. Why is paying expensive tuition to Princeton the only option?
• tacoluv 5y ago
  My 2 cents: The author presumably goes to Princeton - the ivy league is in general a tough place to "start learning" things, especially STEM. Few of the staff would teach you the basics of anything, mostly because you are attending a research college, where teaching is the professors' side gig. I went to an ivy league school, and a large portion of the people in the CS program did competitive programming/knew number theory and discrete math from high school etc. All the problems we got as homework were really intense - I'd consistently do more than 60-70 hours of studying outside of classes to keep up. Mind you, for me CS was/is like crack - I feel like I'd have put in even more time if I didn't need to sleep or want to hang out with my friends. There are some intro classes, of course, but the quality of those varies a lot. Edit: I don't mean to discourage people with this post. I was actually one of the few people who didn't have much of a CS/quanty background in my CS classes. My advisor told me to have a backup major in case I fail the tougher required classes, but I made it through.
  • Waterluvian 5y ago
    I went to the University of Waterloo and took a "Intro to CS for non-math students" course. The thing was designed and taught by CS faculty, so naturally they taught you the basics of programming using math. I spent more time re-learning high school math than actually learning programming. And not because it was directly related, but because some lessons were like, "write a function in Java that factors a quadratic." So 90% of that tutorial was me re-learning what the heck a quadratic is and how to factor it. The experience really sucked and I gave up on university programming courses and just started learning it all practically and on my own terms. Edit: Enjoy this wonderfully styled course page: https://student.cs.uwaterloo.ca/~cs125/S08/Resources/Admin/CourseOutline.shtml https://student.cs.uwaterloo.ca/~cs125/S08/Resources/Admin/C...
    • artificial 5y ago
      I like learning files and sockets once the syntax is a familiar and then branching out to error handling followed by more advanced topics and then circling back around to testing. Project based learning resonates best with me, hands on cements knowledge rather than pure theory.
• yobbo 5y ago
  In some universities (my experience is with European state funded universities) the effective goal might not even be to teach at all. Rather, the effective purpose is to function as a filter so that individuals with beneficial circumstances (whether it be talent, time, or being well prepared) can get good grades/credentials and others simply fail. Generally, there is little unstructured interaction between students and instructors. One point in defence of this is, the skill/knowledge that is effectively trained is "how to learn", rather than specific courses and subjects.
• jonnycomputer 5y ago
  Slightly off topic, but when I noticed that there was sort of a market for prior exams for the CS courses I was teaching, I made it my policy to provide students with all prior exams. I'd have preferred that no-one had them (easier on me), but given how much an advantage having a prior exam is, I can't find it justifiable to favor those with the social connections that gave them access to such exams, over those who didn't-people with friends, people who belong to fraternities and sororities, etc. etc. etc.
• cjbgkagh 5y ago
  First the phrasing "teach me math" like it's a thing that is done to the student, or in this case, failed to be done, as opposed to something the student failed to do. An example alternate phrasing may be "help me learn math". I think this is a modern attitude that removes the student agency from the process. If the university failed to give worked solutions and the student feels they need it then there are other sources for worked solutions. I could understand lack of desire to spend additional money on a different textbook with worked examples when so much money was spent on the course in the first place. At the same time I could see reasons for a math course to push students to figuring much more out for themselves to ensure the early principles are deeply embedded into their brain.
• kazinator 5y ago
  > Unfortunately, it is difficult for students pursuing humanities and social science degrees to explore classes within STEM departments due to the inaccessibility of introductory courses. People in humanities and social science tend to avoid these topics in high school, and would have a tough time with something like Algebra 12.
• omginternets 5y ago
  Oh man, I can totally relate. I'm a self-taught programmer, and have been trying to learn math in the same way for the past couple of years. I've certainly improved a little bit, but learning math is very different from learning programming, and I'm frustrated. Every time I buy a book, I think I understand what I'm reading until the time comes to do the problems ... then I get hopelessly stuck and discouraged. I've never encountered something I couldn't learn with a bit of consistent effort before. To make matters worse, I can already appreciate the beauty of math from a distance, yet it somehow remains totally inaccessible to me, and this despite my honest-to-god best efforts. I don't know what I'm doing wrong. It's a terrible feeling. =( I harbor a secret fantasy of meeting a maths enthusiast who would take me under his or her wing, and help me struggle through problems for one hour each week.
  • akomtu 5y ago
    The secret trick is proving theorems. Prove the Pythagoras theorem or prove that quick sort is indeed a NlogN algorithm (the latter needs some real math skills). A proof counts only when you can explain even the most boring detail in the proof. Books on math only tell what proofs and in what order you need to learn. I'd compare math proofs to functions, and understanding of math is a lot like understanding how complex functions work down to assembly level.
  • tarentel 5y ago
    Maybe try different maths. I majored in math in college and even though I have a degree I would say I am pretty terrible at it. Some areas I understood almost immediately at least at the lower levels I took them such as algebra and combinatorics, but real analysis after like basic limits completely eludes me.
  • buescher 5y ago
    I am not a mathematician or math teacher and I would not call myself a maths enthusiast, but what were you trying to learn? For the usual undergraduate courses, there are Schaum's outlines, which are cheap and usually OK for self-study. There are also explanatory books that you might find helpful, that might not form a whole substitute for a textbook - an example would be Div, Grad, Curl, and All That for vector calculus. It's a bit of a trick to find them on Amazon and through other searches, but you can get practiced at it.
• kwertyoowiyop 5y ago
  Better headline: “How could MAT 202 at Princeton be improved?”
• chadash 5y ago
  > This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s. There's a big disconnect between STEM subjects and liberal arts in my experience. As an engineer, I was used to classes where exam averages were in the 60s (30's does seem kind of low), but I learned to only care about the curve and where I stood on the distribution. Contrast this with liberal arts courses where generally an 85 on an exam meant a B and a 93 or 94+ meant an A. It's fair enough to say "this paper about social dynamics in medieval Italy is a B quality paper". It's harder to say "this is a 55/100" and here's how it stacks up against the curve in the way that many engineering and math courses allow. At the end of the day though, this translates to inflated grades in liberal arts as compared to math. The other issue is that (pure) math at the university level is just harder. I suspect that many people who are good at number theory can probably wing it in a literature course. I doubt that vice versa is true. This isn't to downplay liberal arts degrees... the fact that you didn't have the toughest major doesn't make you less intelligent nor does it mean that the field is less important. In fact, in some cases it's the opposite. I know many people who did "tough" majors only to regret it when liberal arts folks with higher GPAs had easier times getting accepted to grad schools.
  • puffoflogic 5y ago
    > I suspect that many people who are good at number theory can probably wing it in a literature course. You may be right, but the worst grade I got in school (as a Mathematics + CS major) was in some absolute throwaway, freshman-level liberal arts class (Intro to Japanese Popular Culture) I took in my third year! Don't underestimate the value of staying within one's wheelhouse.
    • dc-programmer 5y ago
      I got crushed by a class in environmental humanities. Not due to a lack of comprehension of the material but because I didn’t understand the technical aspects of writing literary analysis (extreme focus on close reading of the language used where I tended to focus on the abstract conceits). It felt equivalent to taking a class in graph theory without first taking a proofs/discrete course and everyone else in the class had. Great class though - I think about the texts all the time
• password4321 5y ago
  If you or someone you know needs help, Jason Roberts teaches math remotely to middle school and high school students in Pasadena, CA: https://www.mathacademy.us/beta-test-information https://www.mathacademy.us/beta-test-information For more info, listen at https://techzinglive.com/page/1764/346-tz-discussion-better-late-than-never https://techzinglive.com/page/1764/346-tz-discussion-better-..., offset 36:40. > 36:40: Jason: You go to some .us site … that's about some school program, and then there's some beta link; and then you go to this big long stream of screenschots and information; and then you have to send an email to get a demo. I mean, talk about … the worst conversion funnel possible. Justin: Even the fact that … you've had those hundred people is … great. […] Jason: Given that we're almost making it impossible — you have to … run a spartan race to sign up for this course — … it's not bad. […] > 39:15: Jason: What the system does is it says, … "You're in this course calculus or algebra or whatever it is, … here are five tasks that are unlocked for you to work on." Some combination overview tasks, and some are new lesson taught lessons on new topics. … The problem is is that you don't really have a context for … "Why am I doing these tasks? … Where is this in the context of everything else on the graph?" And it doesn't give you a lot of freedom … to define your path. So … what we're doing now (because there was a little bit of discussion … from some of our high school students about this) is … you'll go in and you will be able to pick from the graph what you want to do. … Here are all these different major categories, … I want to do something on differential equations or we do something on number theory and you just click on it and then it shows the graph -- Thanks to audiotype.org for 1 minute transcribed for free, and revoldiv.com (from a reddit recommendation) that took a bit more to clean up so I just chopped a lot more out of it.
  • lordleft 5y ago
    Mathematics is such a creative discipline, and so applicable to so many scenarios, but the way math is taught really obfuscates that. If I have motivation or context for why I'm learning something, I'm dramatically more likely to learn it.
• steve76 5y ago
• mindslight 5y ago
  The elephant in the room is Covid. Was this a normal in-person course with OP living on campus or otherwise able to easily connect with other students? Heck, even just masks would probably make it harder to introduce yourself to people in an out-of-major course and look to study together. [0] I say this because I wouldn't think missing answers to sample questions would be a big deal if you have a peer group to bounce things off of. Someone else would either know they got a problem right, or you'd work together to figure out who had the right approach, or some combination. And if you sill couldn't figure it out as a group, some of you would be inclined to go visit the professor some time. Universities aren't so much a place for getting spoonfed learning but rather learning to collaborate with your peers. I can imagine this major aspect has been left by the wayside due to Covid, and been replaced with many more opportunities for students to fall between cracks. [0] #include <std-pro-pandemic-mitigation-disclaimer.h>
• analog31 5y ago
  My advice is to take the same course at a community college. In my locale the CCs are solely focused on teaching. The teachers are unionized, so it's a real career for them, and they're really dedicated.
• debbiedowner 5y ago
  You can get a tutor? TAs are pretty money hungry so almost any of them will do it for 25 an hour. I had a Prof say "no matter how I clown around up here, you're going to have to teach yourself these things". My perspective is that classes provide the motivation, then each student does what they do with it. I don't think departments are financially incentivized to create happy customers, and I do still think this is a good thing after my own multi-year "sufferings."
• 2bitlobster 5y ago
  At least you had math. At my school, they only taught "History of Math," and didn't have any affiliations with other programs/schools to fill in the gaps. Pretty awful!
• andrew_eit 5y ago
  I really wonder what percentage of difficulties that stem from math are due to its representation in the symbolic language used. By that I mean, so often, I am staring at a long, condensed equation. Factored to perfection by whomever was working on developing it. I stare at it for hours and just can't figure out what's going on. Then after much tinkering with the parameters, exploring the limits, plotting graphs, slowly I look at it and recognize what is going on and realise "gosh that is actually really F**ing simple" I wonder how much of "learning" math is really spent on decoding the representations we are provided, rather than understanding the concept they're meant to represent.
  • lysecret 5y ago
    That is a really good point. I spent some time putting ml equations into code and the code representation always seemed so much simpler to me. Many math equations seem a bit like too clever one liners in code haha
    • andrew_eit 5y ago
      "clever one liners" That's exactly it! It's basic clean coding style where it's better to use meaningful names and sometimes write something in 2 or 3 lines that is more interpret able than to write a perfect one-liner that no one will be able to unpack. In this regard I almost wish we had a de-facto standard in math of presenting the 'condensed' perfect form of the equation and a more chunkier version. What I like is that recently I saw some researchers annotating the symbols in their equations [1]. [1] https://twitter.com/sibinmohan/status/1480583840858996743/photo/2 https://twitter.com/sibinmohan/status/1480583840858996743/ph...
  • johnaspden 5y ago
    This is how all of maths is. Before you learn it, it's incomprehensible frightening gibberish. After you learn it, you wonder how anyone can not understand ideas so simple and obvious. How to get from one state to the other is the whole problem. The terse symbolic language is someone's best attempt to communicate the beautiful simple idea in their head. More than any other discipline, I think, mathematicians write to be understood, to be clear.
• zxcvbn4038 5y ago
  When I was in school the math teachers were just out of control, didn't really understand the topic, thought screaming at the students was the best way to make up for that. We observed that they basically copied everything out of the teacher editions and couldn't really answer questions beyond that, so we started swiping all of the teacher editions and chucking them in a dumpster or throwing them on the roof of the building. They couldn't re-order them easily during the year because the textbook budget had already been spent, so they had to work from the student books instead - really improved their attitude. Outside of the school systems there are some really well written books on basic and advanced maths and that is more the direction I went. I forget the title but there was one book in particular I bought that taught an algorithmic approach to math problems with an emphasis on how to solve problems with really large numbers so you can amaze your friends. I'm not sure I'd want to hang out with anyone that wanted to sit around doing math problems with me :), but it was a very effective book and more then made up for the public school system. So I guess bottom line is that there are resources available if you want to learn something new or need a refresher. You don't need an instructor led class to do it.
  • civilized 5y ago
    Did you really do this? I think this was the plot of a Simpsons episode.
    • zxcvbn4038 5y ago
      Prior art, I predate the Simpsons by a few decades. ;)
• dc-programmer 5y ago
  One unfortunate thing about undergrad is that students are punished for taking classes that push their capabilities or even classes in other disciplines (past the basic requirement version of these classes). Many ambitious students are striving to make a career in the humanities (probably like this author), to go into investment banking, or to get into an elite law or medical school. These fields are so competitive that it’s not worth risking taking classes you might not get an A in. This results in students filling their schedule with easy classes and professors and maybe not taking the classes they are interested in. Or worse, punishing students who don’t do this. Some of the smartest people I know were 3.2-3.5 students because they did not play the game and took challenging classes across disciplines. Conversely I know some absolute morons who got I-banking jobs at elite firms because they only took fluff classes
  • sh4rks 5y ago
    Are they really morons if they're working their dream jobs? I don't understand why people think that an undergrad class is the only way to learn about something. There are far better resources nowadays on the Internet where you can learn something from scratch at your own pace, without the stress of deadlines and grades. That's the unfortunate truth.
    • dc-programmer 5y ago
      Well, the few people I had in mind aren’t exactly tearing up the world of financial services to put it nicely. Maybe moron was too mean. A related belief I have is that college students over-index on trying to get their foot in the door of whatever industry. For example, students who skip stats and CS classes because they might hurt their GPA might find themselves behind the students who did 5 years down the line. Even in the humanities and social sciences. Another example is all the junior engineers who are leetcode experts but don’t know what DNS is. I actually agree with what you are saying. But why can’t college be the place to do that? You have a bunch of people together in one place who want to learn (and have the time), in theory guided by experts. Most people are highly social learners and need some accountability, and Khan academy videos aren’t going to cut it
• ccleve 5y ago
  I went to Princeton 40 years ago and had to put up with the same crap. The problem is the spectacular arrogance of the professors and their contempt for students. Anyone who gets to the level of ivy-league professorhood has reached the top of their profession, and has acquired an ego to match. I'm generalizing here, of course, not all of my professors were full of themselves, but large numbers of them were, and more to the point there was a culture of disregard for the opinions of students. Over the years the University has sought to control, limit or suppress student ratings of courses, and professors were never penalized for getting a bad rating. They didn't care about the opinions of students because they didn't have to. It's a lot like government. A person in this kind of organization does not advance by serving customers well. In fact, they often bristle at the idea that students, or constituents, are customers at all. Rather, they are sheep to be led or punished as appropriate. Most of my friends at Princeton were considerably smarter than the professors who taught us, but we were young, lacked knowledge, depended on the good graces of professors to get the grades we needed for our future, and so lacked the tools to fight back against the substandard product we were being fed. This, plus public conversations about courses or professors tended to be dominated by the few but prominent students who fawned, obsequiously like the courtiers, over the brilliant professors who filled them with inspiration and hope. We all threw up a little bit in our mouths when we heard this sort of embarrassment, but what could you say? Again and again, I encountered professors who would assign readings that were unreadable, would promote their own work over better work by others, drone on about drivel, advance idiot theories, and treat the slightest challenge as an affront. Nominally, Princeton isn't like that. Unique among universities, it has something called a "precept", which is a small group meeting once a week, led by a grad student or sometimes a professor, to debate and discuss the issues. It's a nice theory, but most students say little in precept, aside from a few suck-ups. We were so overwhelmed by the firehose of un-curated content gushed at us that virtually no one had done all the reading, and risked being exposed as the one who didn't know what he was talking about. Things won't change until professors start losing their jobs over bad ratings. All that having been said, my physics courses were pretty ok.
  • busyant 5y ago
    Your mileage may vary. You had a miserable experience. I did not. I'm not saying the courses that I took (or TAed) were all superb. It was a mixed bag. The prevailing philosophy was summed up by one of my profs in my 2nd semester. He said, "You're expected to aggressively eradicate any gaps in your knowledge by whatever means you feel are appropriate." Translation: You're going to have to teach yourself. But that doesn't mean teach yourself alone. If you have to corner me, or the teaching assistant, or your classmates, to understand the problem set, then do that. If you need to hunt down a book, then do that. The buck stops with you.
• r_hoods_ghost 5y ago
  Three things. 1) Gatekeeping. There are lots of professors in lots of disciplines in lots of universities, be they STEM, social sciences or arts and humanities, who are determined to keep out any student who hasn't followed the exact progression that they believe is necessary in order to take a course. Often there is no good reason for this. They're just petty. 2) Most academics are awful teachers, even worse course designers and have no formal training in pedagogy. This is why you get stupidities like the author describes where answers aren't given to problem sets, or courses based around lectures that are literally someone standing at the front of a lecture theatre reading from a text book in a monotone for an hour. Teaching and designing courses is not a priority for most academics as it is not what they are in academia to do, and being good at this usually won't help them in their career. They are often given little or no training in either course design or teaching by the university and so tend to fall back on trying to replicate how they themselves were taught. They then panic or are dismissive when faced with a student who asks questions or requires some guidance. Remember, the job of a teacher is to teach, not just to grade. 3) Some courses, even at a fairly low level, are so specialised that it only really makes sense for a major to take them, or they are not "stand alone" as they are intended to be completed at the same time as another related course with concepts from one helping illuminate concepts from the other.
  • someguyiguess 5y ago
    Your first point is definitely something I experienced in university a few times. Mostly in STEM classes. Your second point is very interesting because in general, the highest-tier schools tend to hire professors that are mostly academic researchers where teaching is more of a “side gig”. It seems this would directly translate to a worse learning experience. I definitely had experiences where a CS professor at the local community college was a better teacher than the equivalent professor at my university. The community college teacher worked as a software engineer in the real world. Had to onboard people, train junior devs, explain concepts to stakeholders, etc… From what I understand from reading this thread, most elite schools provide less in terms of instruction. The cheaper schools provide more “hand holding”. Just an interesting thought. Edit: clarification
    • 5y ago
• bernulli 5y ago
  The article perfectly reflects my experience teaching STEM in the last semester. Instructional material that led to rave reviews and amazing student outcomes the years before now results in similar reviews like “lost a lot of respect for the Math department after taking this course.” The main complaint is "Why won’t anyone teach me XXX": "it seems like the instructor is expecting us to learn the material ourselves". Sorry to say, but for many STEM topics that's the only way. You need to struggle, do, and solve the problem. And not only there: you won't learn figure skating / cycling / playing an instrument by watching a lecture (or video) on it. You have to actively do it. The instructor can lead you to the water but can't force you to drink. Just staring at the water and complaining does not help. I wonder what these students will then feel like in their first job. You hire engineers because you have a problem that needs solving. This kind of graduate on the other hand expects that someone solves the problem for them and feeds the solution in small packets. Also, of course, it's interesting to wonder why this is happening now and if it will go away. The connection to the pandemic seems clear, but how exactly?
  • whimsicalism 5y ago
    Sure, but none of that is an excuse to not provide students with worked problems or solutions for at least some of the problems they are given, after they have worked them. > The connection to the pandemic seems clear, but how exactly? Does it? If by clear you mean "relating to the anecdote you shared at the top of your comment", then sure - but that is far from my standard for "clear," at least.
    • bernulli 5y ago
      But the material is not the problem. You have the whole world knowledge at your fingers, in books, the internet, etc. Schaum's outline will give you hundreds of worked problems. The problem is that the student sees no possible solution to this if she is not handed the solution by her instructor. >Does it? If by clear you mean "relating to the anecdote you shared at the top of your comment", then sure - but that is far from my standard for "clear," at least. It's a temporal correlation that all my colleagues observe as well. Average grades drop by 50% using the identical material that was always successful pre-pandemic. So you find it more likely that lockdown, restricted gathering in learning groups, Zoom lectures, potential financial hardship had no impact on learning whatsoever? That's not quite up to my standard.
• AvAn12 5y ago
  I get it. After working in tech for 10 years, I went back to grad school for statistics recently. I came to the realization with every math class that I took - after the first couple of lectures I had to say to myself, it looks like I'm just going to have to teach myself this class. I think that's the deal with math. With some other subjects, you can get quite far by knowing the gist of the material. Since math classes are generally evaluated on one's ability to grind out problems, you have to just grind out a lot of problems - no shortcuts. Like learning to play the piano - there's only one way work your way to learn to play a mozart sonata or whatever - lots of grinding away at it. Definitely think it's open to debate if this fact us USEFUL to anyone. For example, making Calculus II students learn to grind out loads of integrals by hand which can be solved by Wolfram Alpha in a fraction of a second may be of debatable value - - let the debates carry on... But I think under the current idea of what it means to "learn" math, to my eye it appears it really is the case that one has to teach it to onesself...
  • williamstein 5y ago
    “There is no royal road to geometry.” -Euclid
  • rossdavidh 5y ago
    Some truth here, but some of what she describes (e.g. no example problems with worked solutions to learn from) are not sensible if you want people to learn. Imagine if programmers had only the definition of what a function does, but example code using it was banned from all documentation (and no Stack Overflow). Especially in a math class intended for non-majors, it's an odd policy.
  • arghnoname 5y ago
    There's an additional problem that you kind of glancing hit at, which in the US at least, there are at least two maths. Math for non-math major is grinding out some class of problems until you internalize the mechanism for solving that class of problems. You continue, adding a new class of problems on top of the old ones. This eventually culminates in something like calculus, where you do algebraic transforms until it's in a form you know how to mechanistically derive or integrate. Math for math majors is usually about finding proofs. This also just requires a lot of work, but it has a very different feel that the math classes for engineering students (source: was math and CS major). I found that it wasn't very difficult to learn the same mechanisms someone programmed into Wolfram Alpha (though it was tedious), but being good at proofs requires 'mathematical maturity,' and this is something else altogether. There seems to be something innate here. I was decent at it, but it took a lot of work for me to get there. Others were better than I'll ever be and they were 15 years old while I was a college junior.
• j7ake 5y ago
  Why are they treating students like children? The easiest solution would be to provide solutions to problems. The lecturer obviously has the solutions. It makes life easier for everyone. If you provide solutions you don’t even have to grade homework because students can grade it themselves. People generating extra work for everybody sounds pointless.
  • xyzzyz 5y ago
    > The lecturer obviously has the solutions. No, that’s not obvious at all. The lecturer can obviously sit down and solve these problems, but it doesn’t mean that she already had done so. Sitting down and writing clear solution that the students can follow is substantial amount of work. I worked as TA at university, and one of my tasks was to grade homeworks and produce example solutions. The lecturer did only provide the problems, and writing down solutions was my job. It took nearly as much time as grading homework from 15 people in my group.
    • adamnemecek 5y ago
      Right so providing the solutions and not grading would cut the work in half.
      • xyzzyz 5y ago
        Yes, but it would cut learning by 90%.
        • adamnemecek 5y ago
          I would like to see something supporting that claim. From my own experience, I don't think so.
        • j7ake 5y ago
          This is really narrow thinking. Provide exercises and release solutions one or two weeks afterwards. The lecturer can write out questions. TAs write out solutions. Everybody is happy. Nobody needs to grade assignments, students get a chance to work on problems without solutions, lecturer doesn’t have to deal with students coming to office hours asking about exercises. Anybody who needs to grade assignments in technical courses is doing bullshit work.
• jimmyvalmer 5y ago
  The only value of a university math class are the problems and their solutions. Lecture is worthless. MIT does this right. Students can boost their grade by submitting revised solutions after the initial marking.
• mbbutler 5y ago
  This same class did the same thing to me at Princeton except I was an engineering major at the time. I had done well on the Calc AP exams in high school which the engineering department said placed me in MAT 202. On the first day of class the professor started summarizing "what we already knew" from high school about linear algebra and I had seen basically none of it with the exception of basic matrix multiplication. The rest of the week was ego-destroying as I attempted to get help from the professor and was repeatedly told (almost berated) that I should already know the answers to my questions from my high school courses. I eventually dropped the course, left engineering entirely, and majored in a biological science. But the jokes on them because I self-studied a shit ton of math after I graduated and eventually went back to grad school doing ML+Physics. It turns out that I'm actually pretty good at Linear Algebra after all.
• someguyiguess 5y ago
  this reminds me of Charley Bucket’s Teacher from Willy Wonka. “Our test we normally have on Friday will now take place on Monday before we’ve learned it”
• deleted 5y ago
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• paxys 5y ago
  The first part is so relatable, especially for engineering classes. Professors would take pride in making the material complex and exams as difficult as possible so the average score was in the 50s or even less, and then the university would make them grade on a curve and hand out the required number of As and Bs. What even is the point?
• dleslie 5y ago
  > We are often told of engineering or STEM students exploring the humanities to their heart's content, but I feel that we rarely hear of students in the humanities being encouraged to take scientific or quantitative classes. Hey, that's me! I graduated with a minor in English, major in Computing. My two cents: I attended those courses because the discussions were fun, the topics were thought-provoking, and I wanted something of an academic vacation. The experience was altogether _social_; there was less an atmosphere of education and more an atmosphere of dialog. I didn't have to try in any of the humanities courses; not English, not philosophy, not Women's Studies, not a one. I didn't even bother to read the books on the syllabus for several of the courses, and I still managed honors-level grades because the necessary information flowed freely through discussion. Whereas Computing, Math and Physics courses were less about experiencing the thoughts of others, if at all, and more about building a mental tool set to solve hard problems. I like to think that the practicing, the quizzing, was akin to having a budding musician play an instrumental piece in repetition: struggling through and repeatedly applying the tools of mathematics was a purposeful attempt to guide students toward mastery. As with instruments, many students found the practice to be miserable and found their way on to other things.
  • zozbot234 5y ago
    > I didn't even bother to read the books on the syllabus You do realize by implication that you have not engaged at all with the non-trivial part of doing scholarly-level work in the "softer" disciplines, right? Any course can come with relatively "low" standards of assessment.
    • dleslie 5y ago
      > Any course can come with relatively "low" standards of assessment. Absolutely not. There's no way I could have used the same practice-free methods employed in excelling in humanities classes to similarly excel in the sciences. In the hard sciences, it was imperative that I develop the skills to apply the fundamental knowledge tools to solve the problems posed. I couldn't bullshit my way through with wisps of knowledge gathered from fellow students.
• ameminator 5y ago
  I'm sorry that this course (and seeming department policy) gave such a bad experience. I've had similar experiences at top schools - most classes were about learning the material on your own, even if it wasn't explicitly stated. On the other hand, there are lots of places to do math outside of formal schooling, and I thoroughly enjoyed those, even if it meant going through textbooks myself.
• symlinkk 5y ago
  College isn’t designed to teach you anything - that’s what the internet is for. College is designed to weed people out and give degrees to the survivors.
• pjungwir 5y ago
  I was an English major who took 300-level courses in math (and CS, physics, history, philosophy, and Chinese). For 25 years I've regretted not applying to places like Princeton and MIT. I'm sure I could have gotten into some of them. My own college was pretty second-tier. But articles like this make me grateful for choosing a teaching-focused place, where professors are promoted for teaching, and there are no grad students outside a few professional programs. I had tons of personal attention everywhere I went. It didn't help my career, but for someone who wished he could study eight years instead of four and major in everything, it was a pretty nice situation. Years later I was doing a Classics Ph.D. at Penn, and I could tell the difference. Actually I think in our department the professors were universally excellent teachers and very accessible, but it's true if you were in first-year Latin, your instructor was a grad student, and if you were in classical literature, you mostly saw the professor in a 300-person lecture hall, with a grad student leading your discussion group. But a 300-level course would have been led by the professor---and I've met Ivy League professors where that would be even worse. :-) EDIT: The article seemed pretty weak to me: I sort of agree with people saying she sounds whiny and entitled. I don't think she made much of a case that there's a systemic problem as hinted by the title. It's just a story about one bad math class. But at the same time I'm very sympathetic to curious students wanting to learn more outside their specialty. Trying to read generously, I think it's mostly an article about how universities that reward research can neglect teaching. I don't know if it's worse in math (or STEM), but certainly those subjects require you to know the fundamentals before you can go on, so being an outsider is tough. And I have never understood why some professors give tests expecting a top score of 50%.
• The_rationalist 5y ago
• dml2135 5y ago
  I went to Columbia and hands down the worst class I took there was Calc I. The professor (might have been a grad student) was just absolutely incapable of explaining concepts in plain English rather than mathematical proofs. Luckily I had already taken Calc in high school -- I was taking this class in order to get an easy A to fulfill my science requirement, which as a humanities major I was pretty cynical about. Last time I ever made that mistake. But my heart went out to the students that were learning this material for the first time, because all of them struggled due to the poor instruction. It was absolutely inexcusable given the high tuition costs at the school.
• dav_Oz 5y ago
  > I don’t see any inspiration for those of us who really want to learn math, or physics, or chemistry, but just don’t want to focus on it for 4 years. Truth be told (in physics at least with no prior extensive knowledge beyond school) it takes about minium 1.5 years (3 semesters; 40hrs/week) of practice in order to get really started. The most diffucult part for me was getting used to the pace and regularly attending to the exercises. The first two semesters were the most humiliating ones, I really struggled to translate simple concepts into mathematical sound equations, I've constantly missed a lot of nuance and lacked elegance in solving problems, a lot of my calculations appeared to me just brute-forced. Only in my 2nd year in I realized that I had actually gained some "competence" in mathematical modeling. So, yeah, in order to be able to play along with a band or orchestra one has to invest some time into learning the particularities of a given instrument. You can get some decent results after 3 months of practicing but really only after 1-2 years (without prior experience) it starts to really open up.
• throw0821374 5y ago
  Does the Princeton math department not schedule office hours undergraduate courses ? The percentage of professors I have met who complain that no one is attending their course office hours is close to 100%. Engineering schools in the UC system explicitly state during orientation that office hours are the central to the educational experience.
• Animats 5y ago
  Here's the class description for MAT 202, "Linear Algebra with Applications" at Princeton.[1] Here's a problem set with answers.[2] It's painful to read. Problem 2: "Find the matrix A describing the following linear transformation from R2 to R'2 : First rotate clockwise by π/6, then scale by a factor of 2, then reflect about the x2 axis." Anyone who programs video games recognizes that as a stack of transformations. You set up the matrix for the rotate, the scale, and the reflection (which is a scale of -1 on one axis). You multiply the matrices to get the result. This is a pain to do with pencil and paper. GPUs have hardware for it. Books on graphics programming cover this problem. But they usually do it much better, with graphical examples. The presentation here is done very abstractly, without any motivation. Most of linear algebra has graphical representations. It takes a useful and relatively easy area of math and makes it harder. This is apparently on purpose. The class description says: The calculations are relatively simple once you understand what you need to compute, but it can take time to master the abstract concepts well enough to understand what that might be. The challenge in 201 (a calculus class) is usually how to finish a problem as the technical complications mount, whereas in 202 the challenge is often in seeing how to start the problem. Right. In calculus, problems usually involve trying to integrate something, where you get stuck and need to transform the problem in some non-obvious way to make forward progress. It's puzzle-solving. In linear algebra, the actual operations are mostly matrix adds and multiplies. It's setting up the problem that's hard. It's annoying to see this for a useful area of applied math. If you're headed for abstract algebra, where intuition breaks down, this approach might be useful. But if you want to use ordinary linear systems to get work done, which is common in engineering, it's not. [1] https://www.math.princeton.edu/undergraduate/placement/MAT202 https://www.math.princeton.edu/undergraduate/placement/MAT20... [2] https://www.math.princeton.edu/sites/default/files/2018-04/MAT202Sample%20Problems.pdf https://www.math.princeton.edu/sites/default/files/2018-04/M...
  • ChicagoBoy11 5y ago
    As a high-schooler I did a summer course at Columbia about "graphics programming". In the morning, the instructor would cover the very basics of the theory of what we were trying to model, from the physics of it to the math required to represent the physics. The point of it was that, starting from scratch, we would build our own little ray-tracer in C++. The course lasted 4 weeks (and was intensive, we were in class 5hrs a day if memory serves me right). It may have been the most unique and interesting educational experience of my life. There was so much motivation to LEARN the math and physics, and I vividly remember going home and looking at objects, and thinking about the angles that the light would bounce off of objects, etc. Plus, there was something truly empowering with starting from a completely blank sheet of code and building something like this, fully understanding what each part of the code was doing. It was this course that introduced me to vectors and vector operations, as well as matrices, before I fully dove into them in school. It was absolutely perfect. Ever since I've had a soft spot for linear algebra, and am often disappointed at how frequently I've encountered it completely in the abstract, when it may be one of the topics in math that you can most readily connect to interesting, non-trivial real world problems.
• Hackbraten 5y ago
  > This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s. As a non-American, I’d like to know what this even means. Percentage? With 100 meaning aced, and 0 for “handed in a drawing of a spider?”
• ARandomerDude 5y ago
  I feel bad for the author. One day she will buy a well written, well illustrated, accessible math book for $15 on Amazon and realize the debt she accumulated from Princeton was not worth it.
  • chadash 5y ago
    People who have this POV often miss the point: - A degree from Princeton (or other elite schools) is worth a lot of money regardless of what you actually learned there. For the rest of your life it follows you around and helps your career. - You are likely to make connections there who will help your career even further - You get access to many of the brightest minds in academia to learn from - You get world class competition... for many people (myself included) I work harder when I'm in a class with smart people than i would if I were studying a textbook on my own - You get a 4 year experience that many people would consider some of the greatest years of their life
• aaron695 5y ago
• deleted 5y ago
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• lordnacho 5y ago
  I do sympathise with her, she wants to learn the stuff but it seems like nobody wants to get her up to speed. It would make a lot of sense to give her the answers to the questions, tough having done such sheets it's often of little use. This is also where I point my finger as a warning. This will sound awfully arrogant but I think it's true. You can always pick up the essay subjects on your own if you studied STEM. You can't pick up math subjects if you focused on essay subjects. I'm no expert on history or philosophy but I did read books on those subjects on my own, cover to cover. Biology and physics, superficially I can read them, but it ends where the interesting, current topics begin, or where the math gets important. I have an engineering degree and even with that I find it hard to pick up any old math book and just read it. Math books seem to just not work that way, the density of information is on another level. You're often asked to immediately apply a newly learned concept in the next chapter. With a history or politics text there's a lot of words but it's not building up as much as across. If you don't get something the first time, it will come again.
  • deanCommie 5y ago
    I sort of agree with you, and had the same experiences, but I think the difference is of scale and expectations. You can bullshit your way through a 1st, 2nd, maybe even 3rd year humanities essay with layman theories, and get a C+ or even a B-. But they're humoring you - you won't get into grad school with those marks. And you won't be able to actually work in the field without ideas backed by the same academic rigor we expect from the scientists - understanding deeply the work of all your predecessors and contemporaries, and adding your own interpretations not gleamed off twitter. Whereas if you try to join a math or sciences class you with the same attitude, you will just fail. You can barely pass, eek out a computer science degree, or even just drop out without one, then teach yourself all the practical skills, and build a career. I don't know how long that'll last, but it's definitely the case right now.
    • lordnacho 5y ago
      But grad school requires expertise regardless of what the subject is, I don't doubt that. Question is will you sound educated outside of academia when talking the basics of the subject. It's hard to do with math, compared to humanities.
• joelbondurant1 5y ago
• flint 5y ago
  I was trying to learn some math in my 50's. I hired a NYU math PHD to come to my office once a week for $100/hour (an NYU course was $6000/semester). I think it worked out well for both of us.
• mcguire 5y ago
  "The way the course was run did not at all set up students to succeed — or even learn math. For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control." The calculus courses when I was in college (not at Princeton, in 1986-1990) were taught by assigned reading and problem sets in the text book, which were graded by TAs, and a run through the proof that a given technique worked in the lectures. It is what it is.
• da39a3ee 5y ago
  It’s really not that difficult to find answers to math questions online. If not the exact same one, a sufficiently similar variant. Sounds like these kids deserve bad grades for lacking enough initiative / lateral thinking to use google.
• ozim 5y ago
  Why won't anyone teach me anything? Because it is expensive and if you are not putting loads of effort yourself, don't expect someone to do that for you. I was imagining that I will get a mentor, I am capable, I am smart... but that is not enough and mentoring is really expensive thing. I know we live in world where everyone feels like special snowflake but you really have to put "loads of effort" to become worthy to study under a master. Yes I feel like I am special snowflake as well. But I did not get FBI roping down from helicopter to my apartment so I can help them save the world... If you want to learn complex stuff you are on your own and really on your own until you can prove that you can work out hard stuff on your own and people super smart people will be willing to collaborate with you when you really contribute something... It is as easy as thinking about torrenting - everyone can be a leech, to provide new stuff you real havee to have stuff....
• cjohnson318 5y ago
  > For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. This made my jaw drop. Just find similar problems from the textbook that do have solutions, or check out other math books from the library, or use the internet. I feel like a Princeton student should be able to aggregate information from different sources, even if it's not their field.
• protomyth 5y ago
  I know we have seen students coming out of high school without a grasp on the basic math, never mind algebra. They just aren't fast enough at the basics to be comfortable going on. I would assume Princeton wouldn't have that problem, but I guess I could be very wrong. I've often wondered if a course called "Just Numbers and some logic" would be helpful. I've been collecting a lot of different math and logic material and wonder how far someone could get before algebra? I just think one of the big problems is students don't really have basic math skills. Given some of the techniques and problems my school age relatives do, I think there are problems at the beginning.
• DerArzt 5y ago
  >For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. This gave me flashbacks to my time in University. Several of my professors wouldn't give out the answer key for study problems in both math and physics because "if you are using the correct process you know you have the right answer" which was a load of poo.
• wisty 5y ago
  There seems to be 2 seperate issues: 1. The author seems to be arguing that the class was not well taught, given its goals. 2. The author disagrees with the goals of the class. The goal of a math for non-majors class isn't to give them a chance to explore math. It's to force them to learn as much applied math as they can handle, and to figure stuff out independently. I'm guessing it's presumed that students are doing one of the last math courses they will ever do, before going into a field where they need to be able to use mathematics. I bet that when there's a meeting between the science or engineering professors and the math professors, the the math department doesn't hear "give them more support so they learn what's on the syllabus properly", but "give them hell so they learn to figure stuff out for themselves, because after this they're on their own". From the article: > Princeton promises students a “liberal arts education,” and defines that as an education offering “expansive intellectual grounding in all kinds of humanistic inquiry.” Yes, this probably isn't the goal of most of the math (and even most of the science) courses. They are more about grinding out highly proficient professionals. IMO there should be more courses where there's scope for students to just chill and do some intellectually engaging stuff, but when should that happen? I don't think it's possible to run a fun course when the students are stressing out over assessment in another course and expect the students will get much value out of it. There's things like Terrance Tao's Masterclass, or Youtubers like Veritasium if you want to learn about science and don't care about a piece of paper. I get that it's an issue that there's no piece of paper to earn to say you got a "expansive intellectual grounding" in STEM in a humanistic way, but if it's a credential it needs to be assessed somehow (or you get a Python Paradox - employers will take it as a signal for passion, then it will be swamped with people who just want to signal that they're passionate). They could also make a third stream (2nd year math for non-STEM majors, as well as the math for STEM and math for math majors streams), but it may not be viable in terms of student numbers.
• Seattle3503 5y ago
  > For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control. My experience 10 years ago was that you need to go in to the "offices hours", or whatever they are called at your university and walk through problems with the staff. Even if they can't share the solution to that particular problem, they can walk you through a similar one.
• julienchastang 5y ago
  It does not seem like she is getting her money's worth for about $300,000 (tuition, room, board @ Princeton for four years) assuming she did not qualify for financial aide.
• zelphirkalt 5y ago
  At my university it was sort of 2/3 there. They did start with basics, alright, but I had never done any proof before. Then there were other students, who somehow had some experience proving stuff. Some of them were from specifically MINTy boarding schools etc.. I think there was only once at my school, when a teacher tried to make us prove something and then never again. That was already the advanced math classes at school. Given that, I was woefully unprepared for what awaited me at university (also a quite prestigious one). The lectures quickly ramped up and I was still unable to perform the simplest proof. Every time there was a proof in the homework, I sat in front of it for hours, not knowing what to do. I knew some facts about the problem, but just could not express my thoughts or see any way forward using the knowledge I have about the problems. Like sitting there, nd thinking: "Yes well ... so what now?" Then next week it would turn out, that I should have looked at the book and used some phrase (lemma? theorem? idk. whatever.) and I would think: "How the f am I supposed to know, that I could have used that?!". Basically I would have needed to consider things not taught or only taught in future lectures and things, that were not in the homework itself. Sort of "out of the box thinking". This was a big contrast to how I had gone threw my previous education at schools. I basically never had to do much to get good grades. Very low effort. I did not manage to flip the switch at university that well. Add to that, that the handwriting of the lecturer was unreadable and he refused to have the lecture video recorded with a lie as an excuse ("The video would not be good enough to read anything."), so that I could not watch again later and take it a bit slower to maybe understand it. The lecturer made many feel disrespected during lectures and was generally disliked. I did not go see him or his helpers for asking about stuff that I did not understand. Only once or twice I did, together with another student, but came back with more questions than answers. There were some extra hours, where a person tried to help us students, but that also did not help me much, because of how they explained things. It was not put in a way, that my brain would accept. My brain wants step by step, making real sure I understand each step along the way, not glossing over things, while the lecture felt like it jumped ahead way too quickly. Once I would get into the "I am confused." mode, I could basically forget understanding the rest of what was being presented. If I raised my hand, they would just give me an explanation, that presumes some other knowledge or fact I was not aware of and it would not help me. So I pretty much felt like a loser, when some people only needed to see things once and already could do any homework problem. I barely made it through and never needed any of it again. I did ace some other more practical lecture's homeworks though, sometimes with full points, like making models of software or for solving problems, UML and stuff, while people with no problem in mathematical lectures had issues there. Probably also didn't do so badly at coding homework, when others struggled to implement something correctly. Years later I know, that I am actually excellent at solving problems, when given all required input and when I know about the basics. I just don't have the mathematical education for solving math problems, that involve proving things, requiring stuff that is not given with the problem. I also understand mathematical problems better, when I write code to solve it. Perhaps that would have been the way to go for me, if it had been taught like that. It would have given me something to grasp, play around with, giving me motivation to make it work. No one ever taught me mathematics in a way, that worked well for me. I think it is up to me. Years later I read about a little book "Introduction to Mathematical Thinking", which I bought. I don't have much motivation to deal with that stuff, but I did read part of it, trying to understand every little detail, to perhaps catch up on what I missed, finally understanding things, that I did not understand back in mathematics lecture and all that. Turns out, that they did not manage to teach me the actual meaning of even the mathematical implication arrow correctly. I know now, that the teaching wasn't optimal for me, because so much of the basics were either missing, or not properly understood and not given much time in the lectures. There was never a lecture, that took apart what mathematical implication arrow actually means and how it differs from an implication in philosophy or just any natural setting, when talking about what implies what in the real world. That is only one example, that probably confused me countless times during the lecture and threw me off. When/If I find more time to continue reading the book, I am quite sure, that I will discover more basics, that were missing. In hindsight, I am pretty sure, that those lectures were also meant to weed out students. Good that I stuck and finished my degree. Looking back, sometimes mathematical things are very interesting, but I never wanted to become a mathematician anyway. Would it be great to know more math? Sure! It would be great to understand more and be able to express oneself more mathematically correct. Would it be applicable to my actual job? Rarely. I might never become a good mathematician, but I am a good engineer. Software development is different from mathematics in many ways. Similar in some others. I have seen code written by math and physics professors. Lets just say that it was quite underwhelming and wouldn't fly in any code review with me on the review team. There are simply so many more things to look out for than merely the algorithm, which mathematicians writing code often have no eye for, that I would recommend any mathematician to better work with an actual experienced software developer, to get the actual code done. Unless the mathematician has taken extensive time to study up on how things are done in software development, the result of writing the code themselves likely will not result in great code.
• lmilcin 5y ago
  > For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control. > This begs the question: what interest does a department have in making it impossible to study? Well... math is not about knowing solutions, it is about knowing how to find them. You can't learn math by memorising all math books in existence because math is dynamic activity -- it is about training your ability to find solutions to problems. Your math skill is trained by solving problems you don't know solution for, not by looking up the solution. Your teachers did not make it more difficult for you to learn math -- they made it easier. They were forcing you to do the right thing. The solution in itself is worthless. If you are really after problems with solutions -- go find any book that already has problems with solutions listed. There is a lot of these, you can look up solutions to problems to your hearts desire.
• harry8 5y ago
  The elephant in the room is there really isn't a huge amount you can do well and effectively in humanities if you don't know some reasonably serious math. The replication crisis is a thing and it's a huge embarrassment for modern universities. Are you convinced by this argument using statistical analysis as evidence or not? You need an answer. You can't shuffle that off to someone else to think for you and take your learning on the matter seriously. We've seen that pretty conclusively now. The math department needs to take its share of the blame for turning so many off this utterly essential pillar of education.
  • 6gvONxR4sf7o 5y ago
    I think it’s a symptom of a larger problem. Teachers can’t not pass students, and so people make it to (and through) college without knowing the material from the classes they passed. The only thing special about math here is that this is especially transparent.
    • harry8 5y ago
      This is the "go to" excuse for academics who are as lazy as all get out and refuse to make more than the bare minimum of effort in their thinking and preparation of teaching. "The students are too stupid and or lazy and we're not allowed to just fail them even though we have taught them next to nothing." Note that the replication crisis involves researchers, most of whom did very, very well academically having worked hard and been clever, before taking that success on to becoming academics. That's kind of how it works. So no "We can't fail people" is a non-starter here. It's totally bogus and needs to be treated with withering contempt along with any academic pushing such errant nonsense.
      • deleted 5y ago
        [deleted]
• mhh__ 5y ago
  The trick is to learn by yourself and then babble your way in by speaking the shibboleths
• onos 5y ago
  She seemed to make just one criticism in the full piece: no solutions were provided for practice problems. If that’s the only issue, pick up the relevant Schaums outline.
• sniperjoe 5y ago
  I went to Princeton and took linear algebra - at that time taught by none other than John Conway - of game of life fame. It was by far the best math class I've ever taken. This author's experience is likely valid, but I suspect the professor teaching the course had a large responsibility for its low rating.
• proposicion47 5y ago
  Re: Humanities student complains that math class are not run as humanities classes
• golemiprague 5y ago
• Glyptodon 5y ago
  I think math departments are for some reason especially miserable. One of the calc profs at my college wasted a whole class period on "RTFTB" - "read the fucking textbook" after somebody asked a relatively obvious question because of confusion. Have experienced intentionally discouraging practices in other departments of course, and there is something to be said for not telling the hopelessly ill-prepared "you can do it" while failing, but that's not the right way, and seemed particularly egregious in math courses.