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I'm a new math student in college and so far I have done terrible in all my math classes besides one which is basically a much easier rendition of high school math competitions. I did decently well in math competitions in high school, and I genuinely do like math. However, I just don't seem to grasp college math that well. I have trouble coming up with proofs, even though a lot of them end up being quite straightforward. I make mistakes with really trivial observations, things like missing obvious cases where the mathematical object would behave differently than expected etc. Things that I really shouldn't be making mistakes on. Overall, my brain seems to just turn itself off when faced with college mathematics.

So how do I fix this? I am not sure what I am doing wrong. It seems like doing practice problems doesn't even help me that much. I also don't understand why the transition has been so bad for me even though I have done some college-level math in high school as well. Thank you in advance.

Hiroshi Kobayashi
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    You should expect to make mistakes and struggle against the material in a good math class. Grades are a poor measure of ability so try not to worry too much about the results. Just keep working problems until they become familiar as you move through the material. It's less about talent and more about frustration tolerance so your willingness to fail and continue to try will build the skills you need to do math. – CyclotomicField Jan 20 '24 at 17:01
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    I was in your situation as well. Read your notes, do your exercises, discuss with others. Youtube and books are also good sources to give additional points of view on the same problem. I started to be able to "do maths" at the end of my Bachelors. – Jfischer Jan 20 '24 at 17:02
  • Make sure you intuitively grasp the subject as good as possible (if you grasp a formula/theorem completely the formula/theorem is trivial to you (though sometimes intuition is hard to come by and it will come at a later stage)) (to achieve this think yourself, ask friends, ask here, ask in the exercise groups etc...) + do a lot of exercises (this will help you come up with proofs in the future, if you keep at it it will become very natural, it will also help you unwind confusions you didn't notice beforehand). –  Jan 20 '24 at 17:04
  • @eulersgroupie In basic courses, intuition is actually the easy part. Calculus is very intuitive, at least most of the time. However, turning the "obvious" intuitive argument into a formal $\epsilon-\delta$ proof is what students usually struggle with. – Mark Jan 20 '24 at 17:13
  • @Mark I was emphasizing intuition since I've seen people learn stuff by heart with no intuition (which I highly discourage, since it is not fun, not useful, does not help you to solve problems, etc...). I agree with what you are saying though. –  Jan 20 '24 at 17:51
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    welcome to math.SE, as long as you like math, nothing will diminish this appetency you have for math. Some results might be difficult, but just 'postpone' them and do math that match with you level. – niobium Jan 20 '24 at 18:05
  • @eulersgroupie : I think that you're both right. It's harder to understand what to do with $ \epsilon $-$ \delta $ if you don't have an intuitive grasp of Calculus; but even with that intuition, it's not easy. – Toby Bartels Jan 20 '24 at 18:21

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You are not the only one who faces difficulties at this transition point. The major step is from an algorithm-driven point of view, to a pure logical point of view. What was $$a\cdot b=0 \Longrightarrow a=0 \vee b=0$$ is suddenly no longer automatically true. It depends on where $a,b$ are taken from. Prime numbers are now called irreducible numbers and primality gets a new definition. What was formerly a continuous function if it can be drawn in one line becomes a complicated case of epsilontic. Logic substitutes algorithms. You have to doubt everything, nothing is automatically true anymore. 'Why' becomes the most important question, and you cannot go on without answering it. That's scaring if you think about it. So, take your time to get used to it. Here is an interesting interview by Prof. K.E.Smith that could be helpful considering time management.

Marius S.L.
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  • That's funny, I think of formal logic as "algorithmatizing" the reasoning about algorithms :P –  Jan 20 '24 at 17:17
  • I understand what you mean, but to be honest, I've never seen anyone use the term "irreducible number" instead of prime number. Most number theory books stay with the elementary school definition of a prime number, and the equivalent definition is given as a proposition. But yeah, in general integral domains, the terms are not equivalent, and in some sense prime numbers should probably have been called irreducible elements. – Mark Jan 20 '24 at 17:20
  • It is crucial in abstract algebra and ring theory to distinguish the two and not to confuse irreducibility with primality. Nevertheless, it only served as an example that many things in studying mathematics differ from how they are taught at school. There are no automatisms anymore. School mathematics is meant to provide a basis for every student to calculate things in life or begin e.g. an engineering study. Mathematics is more the art of conclusion, closer to philosophy - historically and in principle - than to computer science. – Marius S.L. Jan 20 '24 at 17:28
  • @MariusS.L. Strongly disagree with your last sentence. Mathematics yields EMPIRICAL facts, so I wouldn't say it's closer to philosophy than to computer science (not saying it ONLY yields...). –  Jan 20 '24 at 17:34
  • Mathematics existed long before computers were even invented. It was historically part of philosophy. During the 17th to 19th centuries, mathematics was closely related to physics. Mathematics as a language for natural sciences and computer science is a relatively modern perspective. As my example of irreducibility versus primality demonstrates, it is still much about logic, definitions, and rigor. These are very abstract terms for a description. – Marius S.L. Jan 20 '24 at 17:43
  • @Mark : I agree that people still say ‘prime number’. However, when we generalize from natural numbers to elements of arbitrary rings or semirings, then we start saying ‘irreducible element’, and ‘prime element’ means something different (which, for the ring of integers, includes $ 0 $). That's one example of the redefinitions that can confuse students. – Toby Bartels Jan 20 '24 at 18:19