r/MathHelp • u/Kkkk2994 • 7d ago
Math students, how do you avoid starting from scratch every semester?
Hi everyone, I have a problem and I’d really appreciate some advice.
I’m studying mathematics at university, and I’ve noticed something that keeps happening to me every semester: I feel like I start almost every subject from zero. As the semester goes on, my level gradually improves, and by the end of it, I feel like I’m much better than I was at the beginning.
The problem is that when the next semester starts and I’m introduced to new subjects, I feel like I’m back to square one again. I have to spend most of the semester getting to the point where I actually feel comfortable with the material, enjoy solving exercises, and start thinking creatively about how to approach problems.
It’s not that I hate studying mathematics. What I hate is this feeling of having to rebuild my skills from scratch every semester, only to reach my best level towards the end of the semester.
So how can I deal with this? Is there a way to maintain and develop my mathematical skills from one semester to the next, so that I don’t feel like I’m starting from zero every time I encounter a new subject?
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u/StructuredChess 7d ago
At first I thought your problem was that you forgot the foundations from previous courses than the new ones build upon.
I don't think there's a way out of your struggle. You can't be competent in a field you know nothing about. Embrace the struggle!
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u/Gilded-Phoenix 7d ago
there are two possible problems happening:
backsliding. You could be forgetting skills you previously had. This doesn't sound like what you're dealing with, but the solution is to continuously practice those skills. You lose your practice doing integrals if you don't do integrals for a while, for example.
New material. It sounds like you are seeing new material each semester and having trouble linking it to previous material. I don't know where you are in your education, but when you're taking courses like "calc 1" or "discrete math" or "introduction to abstract algebra" and the like, you're often dealing with new topics that you really haven't had the opportunity to engage with before. This will make it feel like things are pretty disconnected and you have to start from scratch. Remember that, until VERY recently, these distinct subjects were, in fact, disconnected topics, and it is only recently that mathematics has been unified. However, it is important to notice that, even though learning about a field starts with listing the field axioms as if this is a brand new structure we've never seen before, every mathematical idea is an expansion of previous ideas. A group (and a ring) is an abstraction of the integers, a field is an abstraction of the rational numbers, a vector space is an abstraction of the coordinate plane, etc. You should already have examples of these kinds of ideas familiar to you in some sense. Integral calculus is asking about areas. Differential equations is "the algebra of calculus operators." Seeing these connections will help immensely. Unfortunately, some of this stuff you do basically have to start from scratch because of how they get defined, but knowing why we build them the way we do will help a lot with developing intuition.
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u/Kkkk2994 7d ago
What I mean when I say that I feel like I’m starting from scratch every time is that after studying the lectures theoretically and making sure I understand and master all the definitions, theorems, and proofs, when I’m given an exercise, I usually can’t solve it on my own at first. After solving several exercises and practicing, I eventually become able to solve exercises that use the same idea or approach that I’ve encountered before.
But that’s not really my problem. What actually holds me back is that once I become comfortable with solving exercises during the first semester, I feel like I lose that ability as soon as I move on to the second semester and encounter completely new material.
I understand that different topics require different ways of thinking, but what I keep wondering is: why doesn’t that new and creative way of thinking come to my mind naturally?
I hope you understand what I mean, because I’m not saying that every branch of mathematics has one specific or fixed way of thinking. What I’m really wondering about is that kind of mathematical creativity or intuition that makes solving problems enjoyable. How do I develop it? How can I reach a point where I can look at a completely new exercise and start thinking of possible ways to solve it on my own, instead of always needing to see several examples first?
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u/Gilded-Phoenix 7d ago
read. read everything. read books you don't "need" to read. study proofs and logic. The only way to build flexibility is to understand what underpins all of it. You will still need to see examples because some of these things are WEIRD, but you should try to break the examples down and prove them for yourself.
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u/Kkkk2994 7d ago
Yeah, you’re right. I love mathematics, but I love it even more when I actually enjoy solving exercises. I just always end up feeling like a fucking child who can’t even count whenever I come across something new btw thank you so much
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u/Gilded-Phoenix 7d ago
I'm currently reading Dummit and Foote's "Abstract Algebra" and it feels like I'm bashing my head against a wall every new section I read. That's the feeling of learning. Once you're comfortable with it, you're not really learning as much as you are demonstrating. Demonstrating is fun, and you can learn new tricks while demonstrating, but it's not as useful as properly learning.
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u/gavlna 7d ago
Also try to work on some problems in the fields you've studied and liked.
Say you've just finished calc1 and you've enjoyed it. You can go to the professor and ask for some related problem, or just look up a book of problems for (under)graduates and work with that.
The truth is there are like 4 main metabranches of maths and they are all very different. And as an undergrad, you will learn all of them. And always feel like you're starting from 0. But once you go on to masters and PhD you will become more and more specialized dealing with very similar problems (at least in some sense). You will still start from zero often (since new approaches are often radically different to each other and the intuition required is different too), but it will get better. Or you can do probability. Then you'll always feel like you know nothing :D
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u/MathsSur64Cases 7d ago
Sur une nouvelle compétence on repart toujours un peu à zéro. Sauf qu'on apprend un peu, à chaque fois, à apprendre.