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I am a first year undergraduate student, currently in second semester.

So basically I learnt most of the first year stuff in high school, so I have a lot of free time in this year (currently in second semester), and thus I'm self studying math in the free time. In the first sem I tried reading some measure theory and lebesgue integration, some galois theory and some algebraic topology (I think I had the prequisites -- I did half of baby Rudin, most of Artin and Topology, Willard in high school).

So here are my problems:

  1. I tend to pick up very little "big picture" intuition while self studying. I was learning measure theory from Stein Shakarchi, and my experience - it's hard to describe in words - was like I can follow the theorems as a standalone basis, I don't have much difficulty solving the exercises but I was heavily bogged down in the details so I could neither see the big picture nor any non-trivial/deep connection between what's done in a page and what was done say three pages ago.

  2. I have an extremely terrible memory. My memory is so bad is that while I recall working on an abstract algebra book till Fundamental theorem of Galois Theory around in October, now I forgot what's a simple field extension (or even worse, I forgot the exact statement of Fundamental Theorem of Galois theory ! Only thing I remember is that under some conditions on $K \subset L$ it gave a one to one correpsondence between intermediate fields between $K$ and $L$ and the subgroups of $Gal(L/K)$, and one direction is not hard but the other requires linear independence of characters)! This is extremely frustrating, when you forget what you learn one/two months ago completely.

  3. I was a relatively good problem solver in high school (did quite well in national/regional math olympiads) but my problems solving skills don't seem to be that good in college level math. For example, I tried proving Hilbert's Weak Nullstellensatz (when I was studying it from Artin) couldn't prove it even though the proof is "short". If you say it's short but tricky -- well I couldn't even prove the fact that the characters are linearly independent (I recall trying for like 30 minutes and then being frustrated since it seemed a pretty easy proof and then seeing the proof). Due to #2 and lack of mathematical maturity, I tend to give a shot at the proof of some theorem myself before seeing the solution, but this way consumes a huge amount of time and also I fail to prove the theorems when it's nontrivial in most of the cases.

So basically I still have plethora of time in this semester (just started; four months long) and I plan to self study some complex analysis, manifolds and algebraic number theory but the points above are heavily discouraging me to self study math or pursue math in general. Whenever I start to read a book I feel depressed as the pointlessness of me studying a book since (a) I will not have any big-picture understanding of it and (b) I will forget what I did after a while anyway (if I don't use it).

Also I am currently the topper of my batch (which consists of one IMO medalist and many national olympians) -- but that's solely because I knew most of the course materials beforehand so I didn't need to put in too much effort outside the class to understand the stuff. As I mentioned in the points above, it's hard for me to get used to and not forget new and hard math so sometimes I feel insecure about performing extremely terribly in the upper division courses which I didn't know a priori (basically I wouldn't have any edge in those courses), so this is discouraging me more to study maths.

Am I making any obvious mistake while self studying ? Is there any global change which I should make to end up with much better understanding/retention of the material ? Would be a wise choice for me to not stay in academia and move on to CS/other applied math courses ?

Note: People can suggest actually sitting in the lectures of the courses I want to self study, but that's not feasible for me: While most profs are friendly towards second year people or higher auditing any course (basically sitting in the lecture w/o crediting it), they're mostly cold towards first year people sitting in the lectures.

[Crossposted from Math SE]

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    $\begingroup$ Closed @ MSE as "off-topic." $\endgroup$ – Joseph O'Rourke Jan 7 at 20:20
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    $\begingroup$ Sometimes, depending on your interest and the nature of the topic, you may find that writing careful explanations/summaries (with various illustrative examples you come up with or find in various books) for "your future self" will help. I've written many, many such mini-manuscripts over the years, some of which I've lifted large parts from for use in Stack Exchange answers! (example) $\endgroup$ – Dave L Renfro Jan 7 at 20:27
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    $\begingroup$ Those are quite tough subjects that are often not explained very clearly. In particular, textbooks on such subjects often don't try hard to provide intuition or to explain how one might have thought of this stuff. This is a case where you could find some comfort in Von Neumann's quote: "In mathematics you don't understand things. You just get used to them." Hang in there and keep it up and a year from now you will have gotten used to this material, and it will seem much more clear (and you will understand it). Also try other books like A Radical Approach to Lebesgue's theory of integration. $\endgroup$ – littleO Jan 8 at 2:32
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    $\begingroup$ It seems that you read a lot. But do you also practice problems? In our group, the main difference between the people who "got it" and who just couldn't do much was almost entirely in the amount of problems they have solved. This is especially true for things like integration - you'll go through thousands of increasingly difficult problems before it really sticks in your head. The trap is that the simple problems are easy to do with a bit of brain, so you might be tempted to skip the drill - but the drill allows you to spot patterns in teh more complex problems. $\endgroup$ – Luaan Jan 8 at 13:52
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    $\begingroup$ (I do NOT think that working endless exercises from ends-of-chapters is useful to understand mathematics. Yes, it is exactly useful to understand commercialized textbook versions of things, much as any other commercialized thing. For that matter, failing to understand a commercialized version is not a serious failing, any more than not understanding why you should buy a thing advertised on TV that you don't need. Rather, one should practice/rehearse thinking how genuine mathematical things are real, and be able to respond usefully to practical questions, not artificial.) $\endgroup$ – paul garrett Jan 10 at 1:35
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Perhaps you should seek texts that emphasize the high-level viewpoint that you are missing in the details of the more advanced texts. Three examples:

(1) Bressoud, David M. A radical approach to Lebesgue's theory of integration. Cambridge University Press, 2008. MAA Review.

(2) Hajime Sato. Algebraic Topology: An Intuitive Approach. Transl: Kiki Hudson. Transl. of Math. Mono., V. 183. AMS, 1999. MAA Review.

(3) Ghrist, Robert W. Elementary applied topology. Vol. 1. Seattle: Createspace, 2014. AMS Review

If you grasp "the big picture," then perhaps your retention will improve, as then the concepts are pinned to a mental map rather than floating free.

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    $\begingroup$ I agree that knowing the big picture can help you to connect all the tiny details into a unified schema. Perhaps sitting down during office hours with a professor who appreciates your motivation can also help to make more connections. $\endgroup$ – Matthew Daly Jan 7 at 17:51
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    $\begingroup$ I might add Hatcher's text on algebraic topology to that list. $\endgroup$ – Xander Henderson Jan 8 at 1:22
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    $\begingroup$ By any chance, is there a list of intuitive/understanding books for other branches of mathematics? I'm particularly interested in probability and statistics, and representation theory/group theory. I have acquired good enough intuitions in calculus and linear algebra after lots of trials and errors but the other branches still avoid me. $\endgroup$ – THN Jan 9 at 8:31
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    $\begingroup$ @THN: Good question. Try Are there other nice math books close to the style of Tristan Needham?. $\endgroup$ – Joseph O'Rourke Jan 9 at 12:03
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    $\begingroup$ @THN: I've heard praise for: Taboga, Marco. Lectures on probability theory and mathematical statistics. CreateSpace Independent Publishing Platform, 2017. $\endgroup$ – Joseph O'Rourke Jan 9 at 13:14
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  1. You need to pick pedagogically appropriate texts. Not the Rudin ballbusters. Pick ones that have explanations and were written for students with occasional imperfections in their previous knowledge. Don't go too much off of what people say on the net is teh ultimate book to use. SINCE IT ISN'T WORKING FOR YOU. Find other texts that you can handle. Obviously the tougher texts will be made more accessible if you know some of the material already and then only have to deal with the shitty pedagogy.

  2. Also, you need to have a source of problems that includes more progression (some easy problems, not just the skullcrackers, at least a section of easy, section of medium, section of hard). As a self studier, it is important to have the answers to the problems as well. (Ideally, worked solutions, but at a minimum answers...so you have a feedback loop.)

  3. I also question moving straight into real analysis versus doing other topics first (e.g. differential equations). At a minimum, I think you'd find it easier and thus a better choice to self study. Could then do real analysis with the benefit of instruction.

  4. You don't read math, you work it.

  5. As far as memory and concepts and the like, this is related to (1) and (4). You are doing insufficient drill problems, particularly insufficient basic ones. Saying "I got it" is not sufficient, if you don't "got it" a few days later. Grind the groove in. You need to be Umbridge's pen and your mind needs to be Harry Potter's hand.

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    $\begingroup$ (I have to disagree with "drill"... I do seriously think that "developing facility/intuition" is far too often confused with whatever "drill" means... exactly because the majority of exercises in standard texts are artificial, not constructive, not purposeful, not useful. E.g., denial that one's physical intuition is reliable is a terrible, perverse thing to try to teach people.) $\endgroup$ – paul garrett Jan 10 at 1:40
  • $\begingroup$ @paulgarrett I think it depends on the person. Some people have a better ability to remember vague concepts and then use that to attach concrete examples to, some people have better ability to remember concrete examples and use that to attach the theory to. Even changes depending on the subject matter for each person--for example, with discrete math I can usually remember the definitions, and then use that as a memory hint for how stuff works. For physics, I can more easily remember how stuff works and use that to hint to myself what the definitions are. $\endgroup$ – user3067860 Jan 10 at 14:41
  • $\begingroup$ I'm referring to issue (2). If he does drill using the basic concepts, in calculations, applications and simple Q&A or small proofs, than the concept/definition will become second nature. He can then rely on it as a building block in more difficult constructions. This is a pedagogical and psychological issue, not a math logic point. Applies in math/science/sports/music/shop. In this case, it is brutally clear that the trainee needs more drill on the basics since he is HAVING ISSUES using them in more advanced work. That makes it not even a question of if drill is excessive or insufficient. $\endgroup$ – guest Jan 10 at 15:36
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Your question sounds to me like:

  • I have a terrible memory
  • I can't intuitively understand some of the new things i come across

I've seen this plenty of times, but not in maths. I've seen this in linguistics. Languages as a whole, aren't built from logic. They just kind of happened, and we attributed some logic to them. Due to this, any adult language learner, trying to learn a second language for the first time will have to Memorize thousands of root words before reaching a good deal of fluency.

How can someone memorize so much? The solution : Spaced Repetition.
https://www.fluentin3months.com/spaced-repetition/

When it comes to the question of how often, I prefer short doses, with long breaks between them, on a daily basis. A soft rule is an hour a day : 15 minutes in the morning, 30 minutes in the afternoon and 15 minutes in the evening.

Personally, i believe that one can only be fluent in a language when they understand it intuitively. That, is less a question of intelligence and good memory, and more a question of practicing the above, for a length of time.

How much time : About 2-3 months.
Personally, from studying Japanese over the past 2 years, i found words and characters become intuitive after 3(ish) months. Between you and me, i believe that intuition and habits stem from the same thing, as both are reflexive behaviors. Here's a solid article on habit formation:
https://jamesclear.com/new-habit

I like to think of a new piece of knowledge like a single piece of grain:
Plant it in the spring, tend to it during the summer, harvest it in the autumn. Never harvest too early and never expect to harvest without tending to the crops.

Finally, how do language learners study new words and phrases : Flashcards.
https://apps.ankiweb.net/

You need to make flashcards that are 1 to 1. Questions need to have a single output. For example, in english, the word 'set' can have 464 meanings. Sticking all of those into the back of one card would be terrible. However, creating 464 cards, with 464 situations where the word set is used to convey it's meaning, that'd be better. After learning all 464, you may not know the exact linguistic description of the word, but you would end up with an intuitive understanding.

Now, do take this all with a pinch of salt, as this is completely from my own experience.

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This is probably not a real answer, but its too long for a comment.

Last year I spent some time working on Stein and Shakarchi's Volume 2 with several advanced math majors. I actually think that text actually belongs in the list Joseph O'Rourke sets forth in his answer. In fact, they do make a large effort to explain some general themes and intuition before burying you under some mountain of inequalities etc. Now, I did not spend as much time as I really should have to deeply understand it, but I did find I could at least hack through proofs in real time with the help of the students who often see things more clearly. Certainly there are particular portions of the text where the intuition is just much more lucid. That said, it seems to me there is much more attention to intuition and motivation in Stein and Shakarchi than many other older texts.

(I do think you should read something a bit more pedestrian before hacking through their complex analysis)

Anyway, I think the comment by Dave Renfro is on point. In my experience, I really only understand something when I go through the trouble of rephrasing it in my own words. The process of writing it down for yourself is a good way to help remember a lot more. This is actually quite connected to "guest's" admonition to "work" the book, not just to read it.

I would add, I think you need to pick an example from each topic and make it your own. I've noticed, some of the greatest teachers when asked about a particular circle of topics will always come back to the same example. If you can find that one example which collects all the ideas and their interconnection then it can be very helpful in remembering the rest of the story (theorems, definitions etc.)

Larger point, don't be too discouraged. It sounds like you're on the right path. There is always and ebb and flow to the excitement of learning. I doubt you'll find the same satisfaction outside math.

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Regarding retention, doing lots of problems can help with this. Also, asking lots of questions about what you’re reading in the book, and trying to figure them out. You mentioned field extensions, so one question might be:

“It’s a bit counterintuitive that this book spends more time talking about field extensions than about fields themselves. What’s so special about them?”

An answer might include things like:

  • A field, by itself, isn’t a vector space. In order to call something a vector space, you need to be able to say what field it’s a vector space over, and we need the smaller field in the field extension for that to be defined. For example, even a basic statement like "the complex numbers are two dimensional" isn't really a statement about the complex numbers, it's a statement about the field extension $[\mathbb{C}: \mathbb{R}]$; when we look at the complex numbers in isolation we have no notion of dimension.
  • We often want to look at fields created by adding the roots of a polynomial to some other field, in which case it’s useful to distinguish between the “new” field and the “old” one.

Another question might be: “This book defines a field extension as a pair of fields and an inclusion map from one to the other. But isn’t that pretty much the same thing as taking a subfield of the bigger field? What’s the point of explicitly having an inclusion map?”

And an answer to that might be something like “Well, sometimes we want to construct a bigger field from a smaller field, in a way that the smaller field embeds canonically into the larger field, but technically isn’t set-theoretically the same as its image in the big field (e.g. taking the splitting field of a polynomial). In this case, the definition with the inclusion map makes things a bit more convenient if we want to be super rigorous.”

Other questions might be something like “this proof seems unnecessarily complicated, is it possible to do it this simpler way?” (usually the answer is no, but you’ll still get a better understanding of why the complexity is necessary), or “why is this condition necessary in this definition? What breaks without it?”

I partially agree with Jhal's answer; the part I agree with is the last sentence about taking it with a pinch of salt. What you want, presumably, is deep understanding, not rote memorization of many superficial facts.

Regarding the weak Nullstellensatz, I think trying to prove it yourself was a good idea, but I wouldn’t get discouraged about not being able to do it. I think the root of the problem there may simply be that math is hard (there is a reason Hilbert is famous you know). And you still learn a lot when you try to prove something and don’t succeed.

Regarding big picture: you seem to be trying a bottom-up approach, where you expect the big picture to emerge as you read the book. You might want to try the top-down approach, a complementary strategy: instead of expecting the big picture to emerge as you read, you start off with some vague understanding of the big picture before reading, and reading the book is how it becomes more detailed. Here are a few concrete suggestions to help you do this:

  • Pay attention to what you are curious about. Ask lots of questions and let those questions guide your learning. That way, you'll never feel like what you're reading is pointless, it's always relevant to some question you are curious about. I consider it a major advantage of self-teaching that it gives you more freedom to learn this way; by all means you should take advantage of it.

  • Try to find a high-level explanation of the topic. Wikipedia is sometimes good for this. If you know someone who already understand the topic well, you can also try asking them questions about it.

  • Skim over a part of the book before reading it, just looking at the statements of the theorems and definitions without looking at any of the proofs. This can give you a sense of the general trajectory of that section. Sometimes, skipping ahead in the book can also help, e.g. if there is some big important theorem you are aiming towards, you might be able to get some vague idea about how some of the concepts introduced earlier relate to it.

  • Continue to try proving theorems from the book yourself sometimes. Even if you fail, you will end up with a better appreciation of the proof when you finally read it. You will better see why it has to be done that way. Also, this will be great for your problem-solving skills. Another thing to try is briefly glancing over a proof in the book to get hints / a general outline of the proof, before you try to prove the theorem yourself, using the hints you glanced from the book. This is like an in-between strategy that’s easier than proving it all on your own but still a better exercise than memorizing the proof.

Finally, while I hope that something I’ve said here will be useful to you, it important to remember that math is difficult, and requires lots of patience and persistence, and there is no advice anyone can give you that will change that.

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For (3), it happens a lot for me. If the theorem is nontrivial, maybe you should try to spot the nontrivial part and read only that part and try to prove the rest. Don't feel too guilty for not solving the problem, just view this as picking up some new tricks to your bag.

For (2), I agreed with most the answers here. The method I used is taking notes. When the terminology comes up again, try to see if you remember it before looking back to the notes. The last thing is that do exercises so that you can repeat previous thing a lot of time. If you forget, you go and look back. Again, don't feel bad because you don't remember (as I did before), because it will happen a lot. The main thing is that when you reread it, you need to check for yourself that your learning speed has to be faster.

For (1), I don't have much experience in this. The method I want to use in the future is that: try to spot the main theorems in the section you are reading, look up the internet ...

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Start meditating. Your memory will improve, along with your energy, mood, concentration, etc. I use the Headspace app. A growing body of research suggests this is due to the fact that meditation reduces inflammation and furthermore that inflammation ay well be a main cause of numerous mental illness. Other things that reduce inflammation are hydration and exercise to name a couple.

Might not be the answer you were looking for, but for me and for many people I know who are now converts, meditation has been nothing short of life changing.

  • I have two science degrees and a healthy scepticism of pseudo-scientific remedies, but this works.
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