Why won’t anyone teach me 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/C...

Edit-

Now, the course says it's for students considering a major in STEM. I wonder if the business side of the college means to say: "Students who got into Princeton ostensibly to study some area outside of STEM but who are now thinking of going into STEM. They may take this course then add to our STEM student roster by switching majors and we don't want that for business reasons. So here's a course that will push back all but the students who have a good case to be in STEM."

Still, it seems there's bits of departmental strategy that are a bit kooky. Why provide practice problems (not for credit, it seems), and then never share solutions? What is the point in the student practicing without feedback about correctness? Wrong practice is just as likely as right practice.

On the other hand, you cannot conclude from one case that math teaching is broken.

This class seems to be a filter class. A class designed to sift out students that would not cut it in a engineering/science curriculum. They are not designed for students to learn, they are designed as a barrier. That in itself is fine if it happens early enough so students can regroup and reevaluate what to do next with their lives. What I don't like about this is that there is a mismatch between what colleges are saying they want to do and what they are actually doing. They should actually state that in the name and description of the course. I wish schools would be transparent so they don't waste the money and time of the students. This won't happen of course, we will just continue the kabuki dance.

I ask because I went to a "top" school with a small CS department, and I thought it was pretty easy - now I'm at FAANG and not having too hard of a time either. I'm wondering what kinds of opportunities I missed out on by not having a CS program with this level of rigor.

https://www.youtube.com/watch?v=7J-wCHDJYmo

If I recall the TLDR was if you want to graduate with a STEM degree go to a strong second tier school and be the absolute best there instead.

Yes, you can't expect to start from zero at this course. But it seems that they failed at teaching the course.

"Filter course" maybe (as another commenter suggests), but my opinion is that it's a BS concept created by the universities. Sure, lots of places have "sink or swim" evaluations. But it is more a product of the universities pride than anything of real value.

Part of the problem is that we rarely (even from elementary through secondary school, much less university) hold teachers accountable for teaching. And I don't mean unreasonably accountable. I'm not trying to fire a bunch of teachers, but I do want teachers to really want kids to learn the material above all else.

all this to say i think undergrad math education is poorly designed/ incentivized and run in my experience and leads to a huge loss of talent from the practice and art of mathematics.

This meant the university was comfortable with a low grade average, because school in Israel is subsidized - they're not competing for students, and the proportion of people with a BA/BSc in Israel is among the highest in the world. The only thing that mattered is more research, which means more grants, which means more staff, which indirectly means more professors to throw at teaching.

In all my time at TAU, I only encountered one Maths professor who wasn't faculty, so it seems the system was working.

Outside of biology and a small handful of other grant driven fields, notably not including math—-the customers of the side gig are paying for everything. The least the people charging such high prices could do is put in a bit of effort.

Quant background coming out of high school? Good grief.

For example, the 100 level math classes mostly did not touch proofs, just calculation. The 200 level math classes were almost exclusively proof based, but they didn't teach you how to write them. Either already knew how or you were going to have to teach yourself the mental framework on the fly. Contrast that with 100 vs. 200 level humanities courses in my college, where there really was a focus on teaching you how to argue points in writing.

Does it stand to reason that I have no business studying CS?

What are students who come from a less privileged background to do, self select out of good programs?

1. Most people would only take courses from their ‘major’ which they would apply specifically for rather than taking various courses from different departments. There was choice, e.g. science students were all grouped together and could choose from ‘normal math’ and ‘extra math’ with some people (e.g. those interested in physics) encouraged to do the harder course. Another counter example was a classist I vaguely knew who took one of the hardest final-year math courses (it began with ZFC and nonsense like proving that x -> x, and quickly ramped up from there)

2. Generally people were marked on exams and (at least in mathematics) everyone took the same exams and answered questions from their courses. Homework did not count towards final marks and wasn’t even graded.

Some thoughts regarding the article or courses at ‘top universities’

1. For some students it can be good to just get a lot of practice at doing a long chain of operations without making mistakes. This is a common problem from school where most questions don’t have many steps and have lots of checkpoints, e.g. (a) prove trivial step 1. (b) hence do trivial step 2, (c) hence do trivial step 3; rather than something more like ‘solve the problem by figuring out what the 5 steps are and following through without making serious blunders’. Maybe that was part of the point of the course. I don’t think my university would have wasted a course on this point though. We did have some early courses that were partly just building mathematical maturity though.

2. Not getting any answers to exercises is bullshit. Doing the exercise is meant to teach you something, and knowing the answer (or rather the thing you were missing to solve the exercise) feels pretty necessary. Towards the purer end, it’s quite typical that an exercise is basically ‘try to prove this thing that we give you the tools to easily prove in a few chapters’ and (after a time) you are generally expected to be able to work out for yourself if your solutions are correct. But having a good solution can be very helpful. It feels like it is usually good to have students suffer on an exercise for a bit and hopefully solve it before showing them the ‘nice way’ but it is still important to learn the better way to do the thing you figured out. For example, you could blunder around doing some horrid algebra and then be shown an easier way through, or some property you ought to have spotted. [1]

3. The professor (or better, someone who knew what they were doing in small groups) should have gone through the exercises and the answers. (And they would ideally be exercises where you would learn something rather than just doing calculations). The way my courses worked is that people would get given ‘homework’ exercises, attempt them (typically alone) and then some phd student/academic would read the submitted homework and go through the answers with students in small groups of 1 or 2. There would typically be problems that were too hard but whose solutions (or incorrect proof attempts) would be instructive[2].

4. Possibly they were expected to come to office hours for help but didn’t because they couldn’t work out how to interact with the professor or didn’t realise that that’s what they were meant to do because they were new/from the wrong class.

5. Knowing about matrices is useful and necessary for many other things but it feels like a poor introductory course. I think it’s better to focus on something that is new, mathematical (in the sense of doing proofs not calculations), and doesn’t have many dependencies. Like ‘introduction to some number theory and how to prove things’ or ‘elementary group theory up to the first isomorphism theorem’ or if they already know what integration and differentiation are, maybe ‘analysis with epsilons and deltas from sequences to Riemann integrals’. It’s also possible to have a good course with matrices (see my other comment on this thread) without having a bunch of pointless manipulation of grids of numbers.

[1] two examples come to mind: 1. Consider the problem of giving someone a gift such that they get $x after deducting a flat tax rate. You can imagine giving them $x and then topping that up by $(x - tax) and then topping up again summing the geometric series to get the answer, or you can just do x/(1+tax rate). Similarly there’s the problem of the fly going back and forth at constant speed between two trains moving towards each other. 2. The question was ‘find a space X and retractions from X to the annulus and to the Möbius band’ there is an easy visual answer: take a solid torus (S1 x D2), parameterize in R3, and carefully define your functions. But there was also some even easier answer, something like take the product space of the annulus and the Möbius band and the retractions are trivial. It was useful to know how much easier things could be. (Possibly the question was about deformation retracts).

[2] An example from an earlier course would be ‘construct a function R->R that takes every value on every interval’. I think I wrote some nonsense like the limit of tan(nx) as n->infty, which at least led to some discussion. The canonical answer is Conway’s base 13 function. A classmate of mine came up with a scheme based on something we’d proved earlier: any convergent but not absolutely convergent series (that is a sequence a_n such that sum_1^n a_i converges as n grows but sum_1^n a_i| does not) can have its terms reordered to make it converge to any value. Other examples could be logic/AOC puzzles about infinitely many prisoners suffering cruel punishments, hard proofs related to the topic, or ‘what is yellow and equivalent to the axiom of choice?’

At Caltech, the expectation was to spend 2-3 hours studying for every hour of lecture. I found it to be pretty accurate, spending maybe 50 hours a week total. I was never bored :-)

Princeton in particular prides itself on being focused on the undergraduate experience compared to other research universities. Research is (supposed) to be secondary.

Hopefully they make that clear to all the naive students before they sign up! :-)

I had a very similar impression of non-COS STEM classes while I was at Princeton, however... both course descriptions and other students pretty strongly discouraged me from exploring STEM classes.

As a lecturer, I've had several job interviews where half of the questions were some variety of "are you ok to put up with having way too many students and not enough time?". I had the luxury of not having to take these jobs, but obviously someone ended up taking them.

This wasn't even really intentional discouragement, but it was perceived as such. Hopefully, you learned the higher lesson.

What I mean to say is that this issue stems from your misunderstanding of the meaning of the word “mature” and nothing else.

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.

There's a problem with making a statement like that and not following it up with what you need to do gain the needed "maturity". Even adding "maybe you should take this course over here and come back to this one in a semester..." could make a world of difference.

Or if you had the maturity to accept criticism like an adult.

LOTS of non-math-major science/engineering students face that problem as soon as they start grad school.

It's a consequence, IMHO, of math curriculums failing to focus on the fundamentals rigorously enough. There's too much focus on what we used to call "plug-n-chug" mathematics where students learn just enough to apply formulae to get their problems solved. It might seem like "just enough" but it ends up pushing out higher mathematics and keeps students from being able to apply more advanced mathematical concepts that they would have gained from a much deeper dive into Real Analysis, abstract algebra, differential geometry, etc, and the grind of the theorem-proof cycles. Many of us felt this right away upon taking graduate level courses.

I wish that I had been counseled to take additional math courses before (or even at the beginning) of grad school. It would have reduced my suffering A LOT and I would have gotten more out of the degree.

I had a similar experience - and while it was a blow to my confidence I’m happy that he didn’t tiptoe around it with feel-goodisms. What I was doing as a student, and what had ben successful so far, was not working and doing more of it or doing it harder wasn’t going to help.

Honestly leaving that program was the right thing to do, I probably shouldn’t have been in it, though I’m confident a few years later I would have been successful, but I lacked the maturity and mathematical sophistication to get it then.

I, very briefly and quite honestly, attended a lower tier state school that was an absolute joke. Shortly after I started, there was an outreach program for local high school kids where if they showed a high school ID, they got admission for the upcoming year. The head of my dept dismissed it and quipped, "It's ok. Anyone that wasn't meant to be here will just drop out after a year anyway and we have their money."

I, thankfully, ended up at a much bigger P5, well-known university. One of the controversies was the engineering departments were trying to get as many students as possible to the point of exceeding infrastructure and admitting lower quality students into the program. Upperclassmen I talk to said, "that's why everyone hates <1st semester math for engineers> and <2nd semester math for engineers>. They immediately weed out those that don't qualify."

"Weeding out" is what college admissions should be doing, not the departments. When the departments weed someone out, they've wasted at least a year of their life and a year in tuition, and demoralized them.

This is what really the "college for everyone" mentality has led to. If the kid shouldn't be there, or should have started in junior college first, we need to be more upfront about that earlier.

What a clueless, absolute piece of shit..

Clearly he lacks the emotional maturity to deal with students (or just people in general) and unequivocally should be fired as a teacher and restricted to doing research.

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.

I don’t even buy that. Instead of trying to take away the examples and spatial content of a very concrete subject like basic linear algebra, much better would be to give people a lot of concrete (geometrical and otherwise) examples so that their algebra course is better grounded, before just dumping them into abstract definitions and expecting them to symbol-twiddle their way to solutions of unmotivated problems.

For anyone wanting to learn about e.g. group theory, let me recommend Nathan Carter’s book Visual Group Theory http://web.bentley.edu/empl/c/ncarter/vgt/

The second half of linear algebra was a bunch of stuff about eigenvectors and eigenvalues. We never were given an explanation why you'd care about them, and I still have no idea why you'd care about them.

I’d say this misses the point of the subject by a mile. Linear algebra is the study of finite dimensional linear maps. Saying it’s about “matrix adds and multiplies” is like saying the study of algorithms is about “filling arrays and comparing numbers”

Arguably even computer science suffers from the same problem. Keeping your domain as an example, it will go one level higher - and teach you why this math is useful for i don't know rotating a 3d object in space - but not actually teach you any of the libraries (OpenGL, Direct3D, Unity, Unreal) to actually get it done.

In either case you come out with a degree with a lot of theoretical understanding, but you are useless to practical applications without further training.

This is by design, but it's debatable if it's a good design. Humanities does a better job of welcoming laypersons and giving them SOME basic awareness of history/politics/philosophy/psychology to survive in the world. Math and Computer Science do no such thing, and so we have people who have no idea how to do their taxes, or automate a basic repeatable process on their computer. Maybe we should fix this.

- a refresher on matrix multiplication and eigenvectors/eigenvalues for those who hadn’t learned them in high school

- inverting arbitrary matricies (done with the horrific formalism and rigour that first year undergraduates sometimes like to see)

- basis transformations

- special matrices (orthogonal, unitary, hermitian, etc)

- suffix notation (by far the best bit)

- bilinear/quadratic/sesquilinear forms

- tensor products/contractions/basis changes (but without using tensor notation—it was all done with suffix notation thankfully)

And an exam question might be like:

- invert or diagonalise a 3x3 matrix of nice numbers. Or do Gaussian elimination to a 4x4. (These would be warm-up/small questions). If you get eigenvalues, maybe make a geometric interpretation.

- do some algebra with some variables known to be some kind of special matrix to prove something. (I.e. apply a property of the special kind of matrix to do the algebra)

- do some suffix notation algebra (in particular using some kronecker deltas or antisymmetric tensors and associated identities)

- something about bilinear/quadratic/sesquilinear forms/algebra

They mostly tried to avoid bullshit exam questions that were just computation. There were some other opportunities to do a bit of matrix computation like solving/linearising/sketching certain ODE systems, or computing Jacobians.

In second year we had a course that was actually called ‘Linear Algebra’ which touched on (numeric) vectors and matrices but mostly talked about linear maps, had lines of juxtaposed symbols with juxtapositions having different meanings between different symbols, and mostly involved statements like ‘let V be a vector space over F and let e_1,e_2,…,e_n be a basis for V’. I think the exams did not involve any computation and maybe involved producing some of the proofs from the course or applying them to something.

Problem 1 teaches about the rank of a linear transform giving different number of solutions.

Problem 2 from this single problem set is the only one you base a critique of an entire course upon. It is a simple 2D problem to teach that 2D problems often have nice matrix formulations.

Problem 3 is a 3D projection, also nice for geometric intuition.

Problem 4 is about rank, the image of a linear transform, and the kernel of a linear transform, extremely important concepts to go further in math, nearly zero use for games programming, and nearly as unused for engineering.

Problem 5 is about basis changes, also of great use in theory.

Problem 6 is about a least squares solution to a higher dimensional transform. Not used in games.

Problem 7 is about finding the determinant of a special arbitrary sized matrix. The determinant map is crazy important in higher math, and this is an entry to starting to think about that.

Problem 8 is about eigenvalues and diagonalization.

Problem 9 is about understanding the relationship between characteristic polynomials of linear transforms and how that relates to rank.

Problem 10 is about looking at quadratic forms as matrices, and being able to compute interesting things from this relation.

Problem 11 is about singular value decomposition, very useful in higher math, very useful in many engineering places, and I've never once seen it used for games.

Problem 12 is a matrix formulation of a dynamical system, and solving the resulting ODE.

The point of all 12 problems is to teach general linear algebra. The students are not being taught game programming. The book types you list teach pretty much none of this.

> Books on graphics programming cover this problem. But they usually do it much better, with graphical examples.

With all due respect to your impressive graphics programming experience, I think even you would have something to learn from this class in general and this problem specifically.

The columns of a matrix are where the basis vectors of the domain are mapped. As such, this problem can be done by keeping track of what these operations due to the vectors e_1 and e_2. In particular, it is easy to this problem in your head, never multiplying any matrices along the way.

> 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.

Did you... look at the actual transformations? They are incredibly simple; nothing about this problem is even minorly challenging to do with pencil and paper. The matrix for "scale by a factor of two" is [[2 0] [0 2]]. The matrix for "reflect about the x_2 axis" is [[1 0] [0 -1]]. Watch me compose them in my head: [[2 0] [0 -2]]

(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".)

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".

But for lower level "grind and chug" Maths classes (High-school algebra, intro Calculus, non-proofs-based Linear Algebra, etc) I think there is value in having access to "the answer". Certainly at least for a subset of the problems one works on, IMO.

Also if no one liked this course there is certainly a problem that can't be dismissed in a handwave like you did in your answer.

This type of bullshit is normal in crappy STEM classes. Universities are so concerned with keeping secrets that they won't give you the information in the first place. It's a sign of bad course design that's unfortunately common.

I took a couple classes that were like that, but I swapped the class for another subject those semesters and waited until it was offered again by a better professor. Same material, except I actually learned it instead of getting stuck at the starting gate.

It's possible the author was in a "math for engineers" class

Where a math student may see a technique as a swiss army knife that can be used for loads of things, an engineer might only care for the one blade that lets them check the stability of their boost converter and pass their power supply design class.

The math I did before university was doable with basically pattern matching ; I then still had a close to photographic memory (I could skim materials and recall them exactly) and got A’s in math because of mostly that. I could do exams by filling in methods I memorised. There was no understanding. Then in uni I got a good scare as none of that worked anymore; now I had to actually know what I was doing. That was obviously much better but I was not prepared for that at all.

You're making the student out to be some kind of simpleton. She's not.

She just wants what any decent textbook has: An answer key to the practice problems, so she knows whether her answers were correct or not. Being as they are, after all, practice problems. Meanwhile her instructor is telling her, in so many words, to buzz off.

That's what is at issue here. Not (as you presume) her being some kind of a lunkhead, who thinks it's all about "pattern matching".

As someone deeply familiar with/scarred by courses just like this one, in-classroom introductory mathematics pedagogy at elite institutions can vary wildly because mathematics graduate students and professors are incredibly specialized in their fields of study and accustomed to very specialized types of intuition - with perhaps more of a disconnect from the introductory student's learning experience than in any other field. Language barriers can exacerbate this. Proofs are thrown on a board, time management on the lecturer's part is nonexistent, and examples are inevitably skipped over as "an exercise for the student."

In contrast, many computer science programs recognize this tendency and optimize for pedagogy. I remember my university's CS department hiring and paying upperclass CS students to guide other students through their first times debugging code outside of the authority structures of official teaching staff. No similar program existed for the mathematics department, and arguably it should have - but who would volunteer? Once a mathematics major, you're pulled into a realm that's quite isolated from the broader student experience!

You're assuming there's only one answer to "what mathematics is" and that it coincides with your view. You appear to be a fan of doing math, but far more people use math.

I'm not a mathematician. To me, it's a tool that can be used to solve the problems I care about. I don't get joy from spending hours proving results just to say I did it - but I understand that others do. In spite of that, if I can understand the conditions in which an algorithm will converge to the solution, I can benefit from studying math.

They may not give it as a handout, but more likely would solve the problems in class for everyone to see. But one way or other, they would give it.

IMHO for everyone who wants or needs to "learn actual mathematics" there are at least ten people who need "applied mathematics" in the sense of being able to apply well-known theorems from a well developed field (like linear algebra) and with zero need to ever prove anything novel, even trivial things; they need to understand the concepts and know how and where to look up the appropriate formulas to do the linear algebra calculations they need for their non-math domain. This OP wanted a course for the latter need and this course failed at this goal.

For a random example I heard recently; there's applications of differential equations in pharmacokinetics, so people studying pharmacokinetics need a sufficient math background for that. However, the appropriate math background for these students involves the general concept of differential equations and being able to understand a couple specific equations which characterize their physical domain, and do calculations involving them. For that use case there is no need for a skill in "doing math" - all the mathematical properties of those particular equations are known and they can and should look them up instead of learning how to derive them from first principles, much less making any new assertions.

You're right, but you need to know the basic mechanics pretty well in order to start to understand the deeper philosophies and ways of thinking that math _is_.

It's like learning to ride a motorcycle without ever having ridden a bike; Riding a bike is not riding a motorcycle, but you're going to really struggle to make progress past a certain point of learning since you're mastering the absolute basics of _both_ things at the _same_ time.

It's the same thing with math. If I never see what well-formed answers to problems look like, never gain familiarity with the symbology or jargon, then I can never start to even learn proper math because the professor can't communicate with me.

China (and other countries who put American math education to shame) disagrees.

For you, math may have been a deduction from first principles, but to us mere mortals, math is a decade-long slow reveal through wax-on-wax-off repetition.

That is exactly how I got through university math so I think you are wrong. I worked on what felt like millions of problems the teachers gave out and then on some more I found in books. Then I looked on the answers and tried to figure out what my errors were. Then I worked on more problems. What didn't work for me was looking at lecturers writing out proofs for various theorems on the blackboard. I had almost zero use for that.

This is how mathematics works for most students. They get a set of problems, then a (poorly explained) set of tools and they try to apply those tools to solve the problems during tests. Then they leave school and never use those tools again - they often completely forget about them. In Eastern Europe they even call it the "memorize->get drunk->forget" cycle.

Arguably, this is the experience of most people, in most fields of study. This pattern matching does not only happen in maths. Most Bachelor's and Master's thesis are basically a compilation of quotes from various books and papers - without any new ideas, or original research. Maybe I am cynical, but in my opinion you are discouraged to make any original research even while getting a PHD as well. Your thesis is more about quoting the "right people" (= quoting papers of the people who will be promoting you), than doing anything new, or original.

But coming back to the question of actual education: the article clearly shows the experience of most people at Universities. If they want to 'learn for the sake of learning' you will get very discouraged fast, your experience will be very miserable - the university is not there to broaden your horizons, it is just a combination of a testing facility and a diploma mill. What is even sadder is that if you are laser focused to learn something that will be useful at work.. then you will get disappointed as well. The things taught at universities are often not even applicable at academia, not to mention "real" work. Writing a master's thesis teaches you how to write a master's thesis (assuming someone even properly teaches you how to do that, for me they just told us to write one, without even bothering to show few other ones as examples - I had to get those myself). It still teaches you something, but that something is not good enough for real academia or real work. In addition, since lots of lectures are graded on basis of tests, universities often feel like a glorified highschool. And you know what? Probably we cannot have it better: most people wont make original research - because it is hard, so universities are stuck with the tests and papers (that are a collection of unoriginal quotes). Also you need to learn to walk before you run, how are you supposed to write a good original thesis, if you didnt write a single non-original thesis, what is much easier?

The main reason why the blog's author was disappointed is the systematic problem of rewards for university professors. They arent really rewarded for doing more than the minimum. They have their own rat race, with their own reward structure, that rewards "research", understood mostly are publishing quotable papers, sometimes getting grants. Changing the reward structure would help a lot - those who focus on teaching, should be rewarded better. Universities supposedly are for students, but most of the time it feels like students are an unwelcome afterthought for the research professors. Also, professors seem to like to say that "if you want to learn a trade, go to college - university is there to broaden your horizons" - what is obvious bullshit, college seems to be the same as university, but just with a worse diploma at the end. [as a thought experiment: would the professors teach better, if their pay would be related to the salaries achieved by their students - ignoring GDPR, PII for a moment? if the professors could get a "share" of their students incomes, would they teach better? now in USA the students are obliged to pay their tuition with 0 impact on the quality of the received education, or in Europe receive free education]

Then we have those "good universities". USA has this corrupted system of admissions (probably so that students with best results dont take 100% of the available spots - universities want those sweet donations from rich people for taking in their children... so they use the admissions process to allow them in, what is completely not meritocratic). Yet in most good schools you are still supposed to know the subject before you even start to study it: if you go to study French, you are supposed to learn French in high school (reality is that: probably your rich parents sponsored you tutoring, there are very few kids who do it on their own), if you are supposed to learn programming - you are supposed to learn it before, on your own time... Which kind of proves the point that universities fail at teaching. Those students who knew how to speak French before joining the university, or those who knew how to program before coming to a CS course -> they were already good before they even joined the university. Are they now "great" because they later got an university education? Or did they get into a good university, because they were already destined to be "great" and the diploma is just a rubber stamp? One could say that perhaps they were unrefined stones and the universities polished them into diamonds, but I somehow doubt that. Were your fellow students unpolished diamonds, or more like morons?

Coming back to this "pattern matching". This is how education work (life?) goes for most people - do repeatable things again and again. The running joke among programmers (who are a creative job) is that code is routenely copied from stackoverflow and glued together - isnt that the definition of pattern matching? The hard stuff is done by others (either those few very smart, or those laser-focused on one thing), so most of the time you try to find the pattern that solves your problem. For every programmer doing complicated algorithms, there are probably 99 adding a button to some CRUD [source my ass], what can also be complicated, but for other reasons :)

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

Those are my favorite courses!

Then I can stay at home and read the text book myself

It is much better than professors who want everyone to engage in their lectures, so they do not give out lecture notes, and if you miss one class, you risk failing the exam

Man, what a lame excuse.

As if there were not hundreds of books with solutions.

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."

That being said, I seem to recall that when I took math classes in college, we had books that had even questions that the solutions were in the back, and the odd questions did not (they were only provided to the teachers). The questions were largely the same, so if a student had an issue with an odd question, they were able to just go to the cooresponding even question and work it out. I felt that was a reasonable compromise.

It's likely the point is to talk to the professor about the ones you're having trouble with, or where your study group disagrees.

However, I found out the hard way that also in respected universities, the actual assignments in "hard" courses were long-known, extremely challenging problems that were not really meant to be solved at this level, especially on a weekly basis.

Folks handing them in were often using solutions that were quite obviously inspired from online sources, if anything because of the weird methodology needed to solve some of them, without the course teams caring at all.

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.

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.

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.

https://tutorial.math.lamar.edu/

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you can see exactly where you were. Then you enter the next dark room."

I find this rings true--I didn't really understand the purpose of some of the material in differential calculus (Calc I) until I studied integral calculus (Calc II), and I didn't understand some of the material in integral calculus (II) until I took multivariable calculus (Calc III). Funnily enough, it was only after taking a combination of real analysis and discrete math (two years into undergraduate studies) that I finally had the A-ha moment when my real analysis class taught me how to actually read and write proofs, and my discrete math class finally made sets clear. THEN Calculus's proofs and its set-based conditions finally made sense.

I could go on, but there's really something to the frustration that beginners in math have. It's ironical, but the best way to learn calculus is reproducing calculus's underlying proofs; but at the same time, the best way to learn how to read and write proofs is to use calculus as an example of a system you already know intuitively, and just need to learn the formal language of proofs to explain it.

I guess my point is that there are genuine structural challenges/interactions in math's component knowledge systems which make the difficulty of learning math natural, if not to be expected. All of this to say, chin up lads, the only way out is through, and sometimes you're running blind. But that's okay, Andrew Wiles did too and he proved Fermat's Last Theorem.

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.

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.

The only thing special about math here is that this is especially transparent.

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?

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.

How is this at all different from the concept of a "section" that AFAIK exists in most every other university?

> Most of my friends at Princeton were considerably smarter than the professors who taught us,

Seems like a very confident, insufferable and non-falsifiable statement to me. Maybe this was more true 40 years ago than today.

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.

At the high end, humanities and 'softer sciences' courses can require a huge amount of reading and rote memorization. You end up skimming through entire tomes and trying to figure out what you actually need to read in detail, because it just isn't feasible to deal with the workload any other way. That's just as hard and painful as a course in number theory, just in a different sense.

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.)

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.

Learning math has been the only time that I have ever been truly humbled by the aesthetic beauty of an idea. I didn't intend on studying math, I wanted to be an english major, but the woman I was dating at the time was somewhat of a math prodigy and I wanted to understand her world more // impress her. This may seem silly to some of the more mathematically mature people here, but the moment I fully understood Cantor's theorem it felt like my mind was dunked into this sublime understanding that left me in a daze for a week. Russell sums this up pretty well,

""" Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. """

> 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.

This wont be much of a problem if you're interested in pure math as numbers bigger than 10 are rare in that domain. I would recommend picking up an intro to proofs type book like [1]. The reviews are mixed but I would take that with a grain of salt as I suspect many of them were written by bitter students.

[1] https://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Mat...

The hard part is guessing what would get you surprised enough.

I don't think I have enough information about you to answer this, but I'll make a suggestion anyway. Since you did 30 years in tech, try Donald Knuth's Art of Computer Programming. Despite the name, it really is a book for the math-oriented. He has thousands of exercises with complete solutions for each, and every exercise has a "rating" (estimated number of minutes to complete), and exercises that require more math are marked as such.

The first volume covers fundamental stuff, the second numerical analysis (with a nice discussion of random number generators), the third sorting algorithms, and the fourth combinatorics problems. You can jump into any volume without a strong background in math.

You can get them used and in great condition for a good price. Lots of people just buy them and let them sit on their shelf for years.

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?

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.

> 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.

Yep, that's something I heard from my kids while I was trying to help them with their math classes all throughout elementary school - "just teach me it!" If nothing else, learning math helps you get to the point where you can appreciate that there are things that you can't learn from just being told by somebody else who knows the "secret".

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.

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.

The teaching of different ways of making the journey is (often) the destination.

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.

Of course you could. Take a look at K-12 schooling some time, where the prevailing educational theory is that "students must learn the math by themselves", and are expected to devise "their own methods" to do so, including "guessing" and doing calculations in their heads, not on paper. Is it any wonder that even "getting the right answer" has been de-emphasized, never mind "show your work"? This is what passes for math education these days, courtesy of "educators" who have never gotten a proper education at the college level in the actual subject.

As a university professor, given the availability of all these resources, I'm not sure why you'd want to take advantage of them and take a university course if you're not interested in the credentialling. Universities are a place to learn the deepest knowledge of content experts, and I don't want to downplay that; but these content experts are usually not pedagogical experts, and they often tend to be less skilled at teaching less advanced material. That's not to say that there are no good introductory-level teachers out there—there are lots, and they do heroic work—but that a random university professor, even a very good professor, probably won't be as good at teaching introductory material. (With that being said, I don't regard Linear Algebra as introductory material—but, then, I'm not teaching at Princeton.)

Certainly, if I were advising a non-math major who wanted to take an introductory math class, I would encourage them to think very carefully about an auto-didactic approach to see if they like it well enough to continue. If you want to learn lots of mathematics, then a university math department is the place for you; but, if you just want to dip your toe into it, then it may well not be, because so many of those courses are set up as 'service courses' for people who don't want to be there but have to be, and that inevitably shapes the tenor of those classes.

- 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

https://www.forbes.com/sites/michaeltnietzel/2021/09/13/us-n...

(U.S. News & World Report's Best College rankings)

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.

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.

However, if your goal is to learn mathematics for the sake of art, I recommend Kolmogorov's Elements of the Theory of Functions and Functional Analysis.

I constantly assert that I'd have actually been successful with mathematics throughout school if I were able apply it in code (which was not a thing in my educational environment); not with my broken brain where I fuck up numbers on paper and use my fingers for arithmetic.

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

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

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

       /         \   / \     / \
      |  2  -1  5 | | x |   | 7 |
      |           | |   | = |   |
      |  1  +1 -2 | | y |   | 3 |
       \          / |   |    \ /
                    | z |
                     \ /

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)

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

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.

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.