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I am an undergraduate student in Mathematics. I am writing here because I think I have a problem in the way I study Math. My grades are pretty good, but I think that I am not able to effectively remember what I studied.

I will try to explain myself better: I passed Calculus and Analysis (I am from Italy, we usually learn those subjects together) with a very good grade, but I struggle to remember some basic stuff to series series or some techniques to solve differential equations. I am able to remember effectively things in short or medium periods of time, but I do not have a way to remember things in longer periods of time (years). I realized this today, during a lesson of Analytic Number Theory for undergraduates: we needed to use the classic series for $\log 2$, the harmonic alternating one, and I was not able to recall it. I have realized that this is a general problem for me in Math, but it is very clear in Calculus/Analysis.

Do you have any suggestions on how to overcome this problem? How can I study in order to remember better thing in Math (both theorems, their proofs and how to do exercises)? Thanks

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    $\begingroup$ Write down everything on a sheet of paper, try to figure out the proofs on your own, make lots of drawings.. $\endgroup$
    – Berci
    Nov 15, 2019 at 22:38
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    $\begingroup$ What works for me is to remember the methods by which these things are derived. For example, the only formula I can remember for $\cos 2 \theta$ is $\cos^2 \theta - \sin^2 \theta$. I always need to use that formula to derive the alternative formulas $2 \cos^2 \theta -1$ and $1- 2 \sin^2 \theta$. $\endgroup$
    – Robert Shore
    Nov 15, 2019 at 22:39
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    $\begingroup$ "I am able to remember effectively things in short or medium periods of time, but I do not have a way to remember things in longer periods of time (years)". I imagine most mathematicians/people in general are like this. People have better short-term than long term memory. That's not really a big deal. That's why University modules sometimes have a "refresher" section. But obviously the more examples you do of something and over a longer period of time, the longer you'll remember it for. It doesn't seem to me that your problem is that serious... $\endgroup$
    – Adam Rubinson
    Nov 15, 2019 at 23:15
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    $\begingroup$ we needed to use the classic series for $\log 2$, the harmonic alternating one, and I was not able to recall it --- For what it's worth, this seems like a bad example to me. I don't remember what the series for $\log 2$ is, and in fact it never occurred to me that this is something anyone would want to memorize unless doing a lot of work where it might come up, in which case after the 4th or 5th time you had to look it up (or quickly derive it from expanding $\log(1+x),$ convergence at $x=1$ from alternating series test, which by the way is worth remembering), you'd probably then know it. $\endgroup$
    – Dave L. Renfro
    Nov 16, 2019 at 8:06
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    $\begingroup$ (To continue) More important than remembering the series for $\log 2$ is knowing that there is a simple series expansion for $\log 2,$ even if you don't remember exactly what it is. Otherwise, if you're working on something where the expansion could be useful, then you might not think of using the expansion. Also useful to know is that, when you see the expansion, you know right away that it converges rather slowly, so for approximation purposes, rather than when using the entire series, it's not all that useful. $\endgroup$
    – Dave L. Renfro
    Nov 16, 2019 at 8:15

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This happens to me also and I'm sure that to many mathematicians too. You cannot remember everything, I generally remember theorems or tricks that I used a lot, or that I have some picture in my mind, or that I have some "conceptual deep understanding".[*]

I realized, when started to learn maths, that I forgot many things after some time. Then I started to do more exercises about the topics that I was studying, and also I started to write these exercises in some way that I could recover easily after some time has passed, so I started to write the solutions of the exercises that I was doing in digital text files.

This helped me a lot to remember some things (generally rare theorems or theorems that I did not used many times) when the time pass.

Also there are topics that you will understand very well and you will "absorb it", and other topics that you dont. In my experience this is usually related with the way the information is presented, there could be an abyss of difference in the way a topic is presented from one textbook to other, so take a look to some different textbooks or recommendations before to decide from what book to study.

I hope this helps. Anyway my background in mathematics is amateur: I study mathematics in my free time, as a hobby. Probably in university is different because there is an imposed timing.

[*] What I try to say by the words "conceptual deep understanding" is that you can explain something really abstract to anyone in simple words (non necessarily mathematicians, and no technical words). Some times this kind of understanding can be achieved, and it is reflected many times in the way some mathematicians present a topic in a textbook.

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