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Comment by puffoflogic | Hacker News Reader

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.

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Imagine learning to code again. You're shown one for loop, a class header file, and a cartoon of a data structure. You write programs by hand on paper, submit them, and never get any feedback. And you're forbidden from looking at open source code. Some amount of struggle and self validation is fine, but unnecessarily wasting absolutely all your time on dead ends is a waste.

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.

I'm certain they grade homework and return it.

Grading the homework is not the same thing as providing feedback. A big red X tells you that something is wrong, but not what you can do to fix the problem or even why that thing is wrong.

Okay, so go to office hours or a math center (if available, apparently quite common).

I certainly don't want to be misquoted as claiming that mathematics must be learned in a vacuum. Just that access to a large collection of questions and answers is not helpful. Your own argument goes through just as well to show that access to the answer isn't helpful either: you know you got the wrong answer but you don't know why.

Sure, you can do that. But from a UX perspective, the system is clearly badly designed.

For proof based homework typically a grader will tell you which step failed with a short note saying why ("this mapping is not well defined", "here is a counterexample", etc)

It is cryptic at times bit ultimately developing rigor and intuition is something that in my experience will always involve a certain amount of struggle and confusion that no professor can or should alleviate

Right, but at the pace that many of these courses go, and with how the material continually builds throughout the semester, not getting answers until a week after the quiz, and hence two weeks after the lecture taught the material, is setting up a lot of people for failure.

By having access to multiple worked examples of problems, students at least have a fighting chance of learning how to solve problems _in parallel_ with the course material, rather than having to wait days or weeks to even know what they did wrong, while getting more and more lost.

Remember, these are Princeton students, some of the most brilliant and motivated minds of their generation, and the averages are _abysmal_. That indicates there's a greater systemic issue.

Unstated premise: the most brilliant and motivated minds of their generation ought to all be able to grasp mathematics. I for one don't believe this premise. University level mathematics requires a level of abstract thinking that most people are simply incapable of.

I respectfully disagree. Learning mathematics is just as much a craft as learning an instrument, or how to paint, or how to write prose, or how to layout a PCB; it requires a lot of hard work, but if you're consistent AND have the right tools, you can learn it just like any other subject.

Believe it or not, the number of practicing mathematicians and PhD students who have the innate grasp that a Terry Tao does are few and far between. The vast majority have to work hard to earn their skill.

Someone who is tone deaf will never be proficient with a violin, or a theremin.

No amount of hard work would have allowed Stephen Hawking to succeed at sports.

Some pursuits are simply unavailable to those without the requisite immutable attributes. This is not politically comfortable, but it is true. I think the only legitimate question is whether mathematics is such a pursuit, and I'd like to be proved wrong on that point, but trying to convince me that anyone can do anything they put their mind to is wasted effort as it is so plainly contradicted by the available evidence.

Note that the contradictions you mention are _physical_ in nature; it is true that there are some functions that some bodies can perform that others cannot. Mathematics is not one of those things.

Is the brain not a physical thing? Do you believe that all human brains are physically and functionally equivalent such that one brain should be able to learn to do anything that another can?

What about those people in this world with genetic abnormalities that affect brain function? Would you expect someone with Prader-Willi syndrome to be able to become competent at mathematics if they only worked hard enough at it? At what point does “hard enough” become “virtually impossible” or even “definitely impossible”?

While I feel your point is incredibly pedantic, perhaps I should have qualified with some form of "at least X% of people that do not have rare genetic disorders can learn math, X >> 50". However, I think such statements are usually implicit when talking in generalities.

Got it. Your previous statement didn't really seem like you were speaking in broad generalities and instead seemed more absolute, like math is something that could always be learned by any human brain. Clearly that's not the case, which was the point of gpp, I think.

> Is the brain not a physical thing?

Why, sure - from the physics standpoint all the human brains are essentially the same, they all consist of the same protons, neutrons, electrons... (Also, they are close enough to being spherical.)

People have wildly different personalities and creative abilities. why do people think the "math part of the brain" is exactly the same in everyone?

> University level mathematics requires a level of abstract thinking that most people are simply incapable of

You're begging the question. Please substantiate.

I had an organic chemistry professor who would not return anything until the end of the semester, homework, quizzes, test, nothing. I had the impression she was a procratsinator and just didn't do the work. It was beyond frustrating. She also got her ph.d from Princeton coincidentally!

One reason for not returning work immediately is it makes it harder to accept late work, because students will share the answers with each other.

The most lenient late policy I've experienced in my (ongoing) university experience has been that you can submit 1 day late for a 30% penalty. Most of my classes lock the dropbox immediately at midnight.

Simply returning work shortly after the late work window closes should suffice, no?

I did this at Loyola New Orleans (an intro physics lab class) and I'll tell you why. I didn't trust the kids to not lose their work. They could come to my office and collect it and if they challenged their grade, I had all the material to hand with me. Online reports were graded on Canvas at any rate.

You made your class educationally worse and blame the students because you didn't ask for a photo of their work?

Canvas had photos of it online - you misunderstand.

> I did this

Glad to hear you didn't actually do that. Returning digitally is still returning.

You might be surprised. My daughter took first year university linear algebra last year and graded assignments and tests were often returned weeks later, in some cases well into the exam preparation period. I told her this was unacceptable and she should complain, but you can imagine why a young, shy first year student might not be able to get over that potential barrier.

Another thing that one might not have considered is the perception of women and math. Of course your daughter might not want to protest against unfair mathematics teaching structures given that her struggle is societally used to represent that all women struggle at math. <relevant xkcd comic here>. I would also hesitate to protest unfair practices in situations where I was a minority because I know anything short of quietly excelling is a negative representation of my entire demographic.

this is an exceptionally bad take imo and seems extremely degrading towards women

> you can imagine why a young, shy first year student might not be able to get over that potential barrier.

If she continues on to functional analysis she can just tunnel through.

My university certainly didn't in most cases.

> 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".

> Instead I took the approach of trying to understand what the mathematical meaning of multiplication is.

There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there?

Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;)

Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio.

For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.

I think there is. For example, multiplication is not visually intuitive on a number line, but it is very intuitive shown as a grid of row x columns. Doing the math out helps build an intuition for things like commutative properties and what not.

Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.

> Doing the math out helps build an intuition for things like commutative properties and what not.

Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway.

I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.

> Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.

Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.

Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.

> Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.

That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa.

I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.

That is entirely the point. The number line is awkward for multiplication.

But mapping the relationship between multiplication and the area of rectangles is very visually intuitive. Geometric reasoning is powerful. Being able to use your understanding of multiplication rules to derive the formula for the volume of a cube, or a cylinder, or whatever is probably more insightful than realizing some arithmetic tricks.

> That is entirely the point. The number line is awkward for multiplication.

Strong disagree. The number line is one of the best tools for understanding multiplication of numbers (not just whole numbers, but any decimal). The problem is that it is no longer taught. Example: American rulers have both metric (cm) and Imperial (inch) units. You can use it to visually multiply and divide by 2.54.

To generalize multiplying positive numbers x and y, (1) Mark x on a number line, (2) create another number line with the number 1 placed where x would be on the original line, (3) find y on the new number line, and (4) the corresponding point on the original line is the product x*y. The point: Multiplication is scaling (stretching, shrinking, whatever you want to call it). The Common Core tries to address this.

As a kid, the row x column thing never made sense to me. I was horrible at understanding visual metaphors, I don't think I "got" the row x column thing until well into high school, and even then it was "I have to think about it".

I always had better luck with just manipulating the numbers mentally.

Metaphor seems like a poor term. Mapping concepts to geometric concepts is pretty important. Multiplication is simple enough that you can just do it.

But having that capability is pretty important for understanding other concepts. For example, many concepts in calculus.

Visual explanations of geometry, trig, and calculus made sense.

But as a kid I never got the multiplication thing. Even now as an adult I understand what they are trying to do, but it isn't "obvious" to me.

Then again I am one of those people who doesn't read comic books because all the pictures get in the way of the words. Heck as a kid I read illustrated books and didn't realize until I had finished the book that it even had pictures.

>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.

and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.

I don't know what the answer is, but we're currently going through this with our kids, but the opposite. There is now a push to not memorize the multiplication table, but treat multiplication as repeated addition. So to get 6 * 7 you do 6+6+6+6+6+6+6 or 7+7+7+7+7+7 (I forget which is 'correct' as it's been a few years). That's good because it gives understanding, but when you're trying to multiply 2 4 digit numbers, it's really useful to have your tables memorized! We're now going into a new school and their biggest advice to new students is to memorize their multiplication tables, as they'll have trouble otherwise. Maybe they should do understanding followed by memorization?*

> I forget which is 'correct' as it's been a few years

God I hope they're not teaching one of them is correct and one of them is incorrect.

Unfortunately it is according to some of the common core grading rubrics (in the US).

5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this.

I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.

I picked up this trick intuitively as a very young kid, perhaps 7-8

I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half.

I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head.

So say you want to find 86 * 32

86 * 10 = 860

860 * 3 = (2400 + 180 = 2580)

2580 + (86 * 2) = 2752

It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general

Multiplication tables memorized in school usually stop at 12x12. They're super useful for doing the exact thing you demonstrated, in your head, very fast, using some tricks to turn the problem into smaller problems (with memorized answers, so they're solved almost instantly).

Memorizing that multiplication table was a more valuable use of time than the vast majority of time I spent learning mathematics. It's useful daily. Can't say that for much else.

How do you know that 800 × 3 = 2400 if you haven't learned multiplication tables?

Addition

(8+8+8)*100

8 x 3 = 24

8 x 300 = 2400

Add two zeros

I meant, how do you know 8 × 3 without a multiplication table?

Ah I see what you mean In my head, I do this:

8 + 8 = 16

16 + 8 = 24

>This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math.

Also university-level math for everyone who doesn't get up to taking proof-based analysis.

I'm not sure that's a great example.

The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads.

This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.

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".)

> 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

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!

Learning by example is a thing in math too. Learning theorems and methods is not sufficient, they are tools and you need training to master them. Not everyone is able to generalize from one class example to other situations. For that you need to work with other examples. Not only the problems, the answers too.

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.

Richard Borcherds, a field medal winner, stated that he approaches new topics by doing a lot of examples before trying to understand the theorems and such. There is no "right" way to do math, contrary to what OP claims.

> Not everyone is able to generalize from one class example to other situations.

I found the same thing in my tutoring experience. I had several students in physics whom I couldn't "reach" no matter how hard I tried to present "theory" using the equations in full generality (symbols instead of numebrs). I taught them this way because they this is the way I learned and because it seems OBVIOUS to me that learning the abstract thing is more powerful/efficient way to get through it.

Imagine spending dozens of hours with them and they still didn't "get" anything to the point I was feeling discouraged and was like this person will never pass this exam that they have next week. I felt bad because these were friends and I really wanted them to succeed (more friends in STEM, more STEM conversations with friends).

Lo and behold, these same people whom I thought were "physics dumb," were able to pick up all the material in literally minutes, as soon as we started the exercises (one person) or having tried some exercises on their own (the other person). I don't mean "imitate the steps"-know, I mean completely solve 80% of the problems on the practice exams and adapt the use of equations as needed for each problem. I mean it's still PHYS101, so there are only 10 or so equations, so they didn't turn savants or anything, but the transformation was so sudden and drastic that I realized people simply have different learning approaches.

Some people like the abstract/general approach. Some people like the hands-on approach. The latter kind of people often think they are "not good at X" but really they would have no problem picking up X if they had learning resources suitable for them (often lacking in traditional textbooks and courses). I think that's why Khan Academy videos are so popular. Sal does a fair bid of hands-on approach, and many people need that!

My feeling is that what you say is true to a varying degree, dependent on the course in question. Maybe more particularly the level of the course. For proofs based math where the goal is to write a proof, I'm more inclined to agree with you. Especially since there can be more than one "correct answer" and seeing one may or may not help you at all in understanding why/how another one would be constructed.

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.

> 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.

I think there is a value in introducing the concept of verifying your answers early, as your ability to perform calculations is beyond useless if that isn't something you are thoroughly familiar with.

If you're just doing the calculation but not the verification, you're skipping half the exercise.

Sure but those two positions are not mutually exclusive. I don't think I ever took a Maths class that didn't introduce the idea of verifying your answers. But at the lower level, most of them used books that provided answers for some but not all of the exercises.

If nothing else, this is an advantage from a time efficiency viewpoint, as you can more quickly identify the problems you got wrong, and focus your review time on those. Those presumably being the ones most likely to highlight some misunderstanding of the material.

If time were infinitely available maybe the distinction wouldn't matter, but things being what they are...

Definitely agree with this. At least for me when taking undergraduate math, once you get to the proof based stuff, often times seeing the answer was in no way helpful. You either knew you proved it correctly or you weren't able to do the mental gymnastic it required to get there and seeing an answer was likely not going to help. I usually just had to re-read the book and try different things, take a break maybe, or something before I would finally get the intuition or whatever to really grasp it.

For calculation based math, which this course sounds like and based on a sample test I looked at, more or less is, having some subset of answers would definitely be helpful. I am not sure why they would make it more obtuse than it needs to be. I thought the course I took on multivariable calculus and linear algebra were quite easy but the number of people who dropped math entirely after those was pretty drastic. I don't think you really need to make these classes unnecessarily hard to weed people out. They'll do it themselves anyway.

Teaching yourself math by producing answers and then validating them seems like a great way to possibly completely fail at learning math. What if you are making some fundamental mistake, and now all that time you've spent studying reinforces that mistake? We see this in software all the time; people will latch on to bad patterns and stick with them for years.

I think that is the whole point of not giving answers.

You should be able to solve a problem in multiple ways and prove correctness.

It takes loads of time, but that is the only way to truly learn.

Most of students given answers will solve a problem in one way see that answer matches the correct solution and stop there. That is not the way to learn anything.

It is the same with software, it takes a lot of time to get good at.

In software it is easy to get correct solution with wrong approach, but often people just stop at "make it work" where they should step back and "make it work, make it right, make it fast".

> Attempting to learn mathematics as some sort of pattern matching,

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.

The course is "Linear Algebra with Applications" - and the OP explicitly wanted a course to learn linear algebra for non-math majors. It's not about "doing mathematics" in the sense of doing novel math, it's about learning basic linear algebra and being able to apply it to a non-math field.

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.

> Attempting to learn mathematics as some sort of pattern matching, [...] does not work; it isn't mathematics.

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.

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

Indeed, she says exactly that:

> MAT 202 is not a course that math majors typically take, but rather for underclassmen who are majoring in engineering or sciences.

from the coursepage

>Brief Course Description: More abstract than calculus, this course aims to develop basic algebraic tools for work with problems involving many variables. Starting from systems of linear equations and vectors in 2-space and 3-space, this course develops ideas about length, angles and resolving a general vector into useful components, identifying features of linear systems or processes in order to choose a basis that is well-adapted to studying a particular phenomenon and move between different points of view to reveal the essential underlying structure. Companion course to 201 (Multivariable Calculus). Discusses matrices and linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvalues and their applications to quadratic forms and dynamical systems.

Definitely NOT a math class

No. She was in a 202 class. "math for engineers" is usually locked behind a course of study or prerequisite classes so people who need those classes to graduate get them. Everything 200 and below is usually tagged as sophomore or below and is intended to be used cross major. 300 and up is when specialized topics start coming up.

You could argue it was "math for engineers" in that Liberal Arts / Social Studies don't require them and only the STEM programs do, I guess. You can graduate with a non STEM degree with just an Algebra course in most Universities in America. And those Algebra courses are usually just high school refreshers if you went to a good high school. Some of the non STEM require Statistics. But even there some schools do "Statistics for STEM majors" and "Statistics for non STEM majors"

> 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.

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.

From what I understood, the HW/exams were graded, but solutions were not provided. I've studied math at two universities, and every professor provides solutions after they've been graded. And if they didn't, they would give you the solution to a particular problem in their office.

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.

> 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.

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.

> Unfortunately it seems like no one has been willing to teach the author what mathematics is.

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.

No kidding. Doesn’t seem like they taught the author much of anything. Just threw problems at them and expected them to figure it out.

> 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 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 :)

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.

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".

Not to mention the article itself admits that she did pass the course just fine, she just had a miserable time doing so.

Very few of the maths textbooks I've read have had the answers to the exercises in them. If the author wants to put in a worked example, that will be in the chapter itself, rather than the exercises.

In maths, you know when you've got an exercise right, and so you don't need the answer written out for you. If you're looking up the answer before you get that feeling, you're cheating yourself. You won't learn the material.

It would seem weird if someone put the answers in the textbook, like having the answers to a newspaper crossword on the same page as the crossword itself. You'd have to exert willpower in order to not look at them.

The whole point of the exercises is to play with the ideas until you understand what they're for. If you don't at least partly invent the subject yourself, you never really 'get' it.

Textbooks are really guides to how to reinvent the field, what order to think about things in, what are the cleanest ways to break up the patterns, what are the best notations to use, so you can reinvent it all at a reasonable speed rather than having to spend a whole lifetime on each problem.

If you literally can't do the exercises despite having read the chapter, then it's a bad textbook. Throw it away and get a better one.

Very few of the maths textbooks I've read have had the answers to the exercises in them.

Well it's been a while - by my recollection is that (at fresh and sophomore level at least), the better textbooks always did have answer keys in the back (at least for a significant subset of the problems). But as we all know -- the better textbooks are few and far between. And college is filled with lousy textbooks that get thrown at students for who knows what reasons.

To the extent that they think lousy textbooks are the norm.

> In maths, you know when you've got an exercise right, and so you don't need the answer written out for you.

No, I might think I have the correct answer but I cannot know until I have seen the correct answer. I can justify any false answer that's not obviously false for myself which is why I need to compare.

Attempting to learn mathematics as some sort of pattern matching... does not work

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.

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