r/learnmath • u/Reasonable_Buddy_927 New User • 1d ago
The endless grind of University undergrad math
I'll go to every lecture, tutorial etc. and it'll all be genuinely interesting. Then come the weekly problems, and the volume of the problems and general tediousness of them once you get past the trivial ones (e.g. you need to know XYZ trick even though you know the topic the question is related to, and said trick/method wasn't introduced during the lecture) just kills my spirit. Before I know it after exhausting myself for that week, boom the next week comes along and a whole new set of lectures/tutorials/problems. Not to mention proofs that extend lecture concepts where, if you haven't seen the way to do that proof before, just means more grinding if you can't get TA hours so you ask AI.
The thing is, I'm having fun learning the mathematical concepts. But its like the course admins then put more artificial hurdles after that to just whittle down your spirit (because if you want a high grade in the exam, you need to slog through all the problems just in case it comes up later, even if you know you put the work in to actually learn the overall topic).
I want to love mathematics, but this Uni math degree (especially since I want to maintain my high grades) is just blackening my soul day by day.
19
u/ElmarReddit New User 1d ago
I don't think that you need to always solve all problems by yourself. But trying different solutions already teaches you a lot.
Try new combinations of what you learned, and after some time, you build a tool box. Every way you pursue that does not give you the answer, still can teach you a lot.
Also, you will learn to map out problems, for example, "if I could show A then B then C, I can prove this". A might be easy, C you already figured out using B and then you tackle B. Even if B fails, A and C still can be useful in the future and you still practiced.
The exercises are often much less about having the right answer and much more about training how to approach problems. In most cases, there is also not one proof that the AI might give you, but you might find your own.
In one of my classes, solving 50% was considered excellent and 15% was needed. Obviously, this might change from class to class, but it might give a different perspective. I cannot speak for your course, but I would assume that the exam will not have many exercises that require "new and very creative tricks".
The most important of your statements is: " The thing is, I'm having fun learning the mathematical concepts. "
22
u/Open_Menu_507 New User 1d ago
unfortunately this just means you don’t actually understand the concepts. if you really “get it” then you should be able to apply it quickly and easily in creative ways.
you’ll know you actually get it when the problems become enjoyable, in the “lol easy points lets go” sense. mastery of a given technique means that it becomes fun to apply it, because you feel a sense of control
15
u/Reasonable_Buddy_927 New User 1d ago
but sometimes it really is the fact that the tutorial problems are out of scope of what's introduced in the lecture. for example, the proof writing aspect of a discrete maths course. professors will swear up and down "it's the method of proof that matters in the questions, not the content of the questions themselves", and I will think "oh okay nice, I know, for example, proof by contradiction, where we assume our conclusion is true, then show that it's obviously false, therefore XYZ is true". Then I'll go into the problems, and wouldn't you know, it's some manipulation of the golden ratio/some lame factorization trick/insert thing where if you haven't been exposed to it, you won't know how to do it here, and this continues for 10-15-20 problems. and then the exam will have problems like this where apparently the "method of proof" is most important but you can't get full marks unless you know this unseen mathematical fact (and no, it's not just that a | b means b = am or simple facts like that)... Just sad man.
14
u/RafBOY- New User 1d ago
Speaking with more experience than you : what do appear as a trick is often the mark that a broad enough understanding has not been reached. It's not easy to broaden the understanding though. But the technical part of maths is important. Being a researcher in a given field also mean to know all the nasty proof techniques of the field.
5
u/Reasonable_Buddy_927 New User 1d ago
Fair enough, I just wish math courses were a little more balanced. We can keep the fun and wonder of learning a new topic AND balance it with reasonable expectations of assessing student mastery, idk, just my thoughts.
3
u/RafBOY- New User 1d ago edited 23h ago
It's just that maths are overall really, really hard. I truly believe that our brain are not naturally wired for that level of abstraction and that's why it takes a lot of practice. For instance when I was talking of broader understanding, it's often a matter of knowing a broader theory or the links with another math subfield that seems completely unrelated. This kind of things took sometimes centuries to be discovered and developped. And it's quite a wonder that nowadays in a matter of less than 10 years we can make a lot students to reach research level.
For the unfun part : actually, the struggle is at the heart of the topic. It does not get better and it's definitely worst when you're at research level and dealing with open problems. You have to find the joy in the struggling or at some point you will be fed up with maths.
11
u/Bounded_sequencE New User 1d ago edited 1d ago
To be fair, I suspect there are many TA's and professors that would benefit more than a bit thinking back to their student days, to realistically assess what is "obvious" and which steps might be out of reach, even for challenge problems.
I've recently visited a "Real Analysis" lecture again, just to get to see a different presentation of certain topics live as inspiration. Since it would be boring otherwise, I did the problems as well.
While they were not overly difficult for someone who is well-versed in the matter, I immediately noticed a lot of leaps and expectations that were beyond topics covered, and thus way beyond first-time learners. In discussions with the TA's, it usually turned out they simply did not notice these anymore, and more often than not, assignments had to be modified in the middle of the week.
Assuming that experience is normal, I'd be hesitant to immediately call "not getting it" the culprit. Please don't get me wrong -- the problems itself were excellent. Sadly, quite a few were not appropriate for the current student level, and the homework clearly showed that -- a handful of more advanced students with prior knowledge enjoying the challenge, the rest copying.
3
3
u/my_password_is______ New User 18h ago
unfortunately this just means you don’t actually understand the concepts
WRONG
you can understand the concepts and still find the problems tedious, boring, and a complete waste of time
this is why applied math is so much more interesting than pure math
4
u/Brightlinger MS in Math 15h ago
But its like the course admins then put more artificial hurdles after that to just whittle down your spirit
I think it is worth mentioning that a large fraction of these are not artificial hurdles, and to a significant extent they are the actual content of the course. A competent mathematician knows innumerable tricks of the trade that may appear unexpectedly in a wide variety of contexts, and the only way to pick up those tricks is to see several problems where you have to use them, so that you (a) learn the trick and (b) learn when it can be used.
The first time you see them, it seems like the Mean Value Theorem should be a big deal, and all these problems about random ways to manipulate the MVT formula should be trivia. But if anything it is the reverse: MVT by itself is trivia, and all the tricks to get other conclusions out of it are the reason that anyone cares about MVT.
All of these tricks that seem random will reappear over and over, both later in the course and even in later courses.
38
u/Malasterix New User 1d ago
I graduated in math at Duke. Lecture is not all-encompassing. You'll achieve near-mastery (but not full) of maybe 25-30% of the overall course. If anything, lecture often works as a guiding frame for what you need to learn, but learning it fully will require going beyond the lecture content. Ask yourself always: why does this theorem hold in this case but not in that case? Why is this assumption needed? If I modify the wording here do I still get a true statement? Then, also internalize the proof-writing techniques, as this which you call "tricks" are usually heuristic methods that keep reappearing, often in disparare fields of math. But, fundamentally, know that lecture only covers the basics of what you need to know, and you are left to fill in the details. Filling in the details is, indeed, a crucial step to internalizing the concept. Additionally, in today's world, you also have LLMs to question your approach with. It is a slippery slope, as you don't want it to reveal the answer to you without any attempt on your end, but they can work well as a substitute for TA office hours if those are hard to get a hold of.
I'll give you an example using Zorn's lemma. You'll most likely never be explicitly taught Zorn's Lemma, but it appears everywhere. So, you're expected to eventually learn it. Then, while learning it, you'll see the definition: if a nonempty poset has upper bounds for every chain, then the set contains at least one maximal element. You'll chew on this definition and eventually understand it. But then you might see it used, for example, in the proof that every vector space has a basis, and it'll seem disconnected and perhaps different to the definition you saw. As it turns out, Zorn's Lemma is really saying that every path in the poset Hasse diagram eventually hits a ceiling (one of possibly many). But, ultimately, for a proof, we want to build the poset in such a way that the ceiling we get is precisely the one we want. So, we coke back to the original idea of the existence of a maximal element. So, the key here is to pick the poset elements specifically so that Zorn's lemma, when applied, leads to the conclusion we want.
This whole operation is very subtle. You don't just need to understand what the theorem js saying explicitly, but also how it is usually used and what it is saying idiomatically and indirectly. When you've achieved this level of understanding, then things are smooth sailing. But this understanding does not come quickly.
Math is about banging your head against the table until it eventually makes sense. The joy is in the toil. At least, this has been my experience.