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Short version of my question: What are good and motivating working habits for a mathematician?

N.B. There are similar questions to mine: see this (how to read maths) or this (efficient study) or this (good habits). But none of these covers my question. I have also heard the standard answer: develop your own working habits. But I feel that I am neither efficient enough nor fast enough at studying mathematics, so I want to improve my working habits.

Long version of my question: For me, when reading a highly theoretical book, it is quite possible to get lost and become demotivated. This is because sometimes there is not enough motivation to read complicated definitions and study full proofs again and again (sometimes full proofs are not available, definitions are fuzzy or 'advanced' etc.).

I would like to ask if there are some 'gamification' tricks that convert boring-looking (since I know they are very rarely boring) things into the puzzles I enjoy. Advice about reading would also be more than welcome, because the problem could be my reading strategy: maybe there is a more efficient reading strategy that keeps things alive and attractive.

To sum up, I would like to ask for your strategies for serious mathematical study and how to turn it into enjoyable puzzles when it does not seem very attractive in itself.


Note also that there is plenty of advice about working for exams, but that is not even close to a potential answer to this question. That is because working for exams is, by definition, equivalent to working on or memorizing boring things to pass the course and get the grade. In a course setup, there is usually nothing about motivation for real learning. I am not asking about motivation to study mathematics for a course or exam, but about how to motivate oneself to think through complicated things which at first sight make little sense when one has not acquired the background.

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closed as primarily opinion-based by m_t_, Bookend, RecklessReckoner, Jack M, Slade Dec 27 '15 at 0:25

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

Those books more often than not have exercises. When delving into new theory I need to start working on them after at most a chapter or two. Otherwise I get lost. Some need to do it sooner, some later. I also always try to "ground" the theory I'm learning to as many other theories I have learned earlier as possible. This helps in building the bigger picture. Sometimes this doesn't help (say the first time I tried to read Hartshorne :-). This is a sign that you aimed too high and need to strengthen your background in some area(s). Do whatever works for you! – Jyrki Lahtonen Apr 28 '13 at 13:20
Jyrki Lahtonen, thanks, you are right. I discovered that, reading a book in a linear fashion, chapter by chapter, theorem by theorem, is rarely working for me. When I skip something, I generally find the real motivation for working on it in later sections. I suppose, this is itself a good strategy (and may be it is a well-known fact I do not know). I am just try to construct a general strategy for myself. :-) Thanks again. – oeda Apr 28 '13 at 13:27
I have edited this question, but I left this bit unchanged as I was not sure what oeda meant: But I feel that I am not so efficient and fast enough, so I will try to improve these habits themselves. I thought it might be meant as But I feel that I am neither efficient enough nor fast enough at improving my working habits (and so am asking the question), which makes sense in this context. But perhaps it should be But I feel that I am neither efficient enough nor fast enough at studying mathematics, so I want to improve my working habits. – PJTraill Dec 8 '15 at 2:26
@PJTraill you're edit was absolutely fine. I approved it, but couldn't understand why someone else had rejected it. – Shailesh Dec 10 '15 at 1:34
Community users, not moderators, closed the question. I personally could see it being closed as too broad as well. Just because this question isn't entirely within the scope of the website (which has evolved over time) doesn't mean it isn't a good question or that it was closed to upset you. – pjs36 Dec 26 '15 at 20:44

Henri Poincaré had a good way of working. He'd work for a couple of hours in the morning, and a couple of hours in the evening, and the rest of the time he'd let his subconscious work on it. I find that works for me too :-).

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Ok, so this is not about "working habits of a mathematician", but about "how to read mathematics".

Some of the observations in the question are absolutely "right-on": school-math, as in study to get a good grade on exams, is a very highly stylized version of a tiny subset of mathematics. In particular, it is not at all aimed to persuade, or even appeal to, the students, but to be fairly hostile to them. In particular, quite oddly, "criticism", in the legitimate intellectual sense, is seemingly completely disallowed. This is crazy and anti-intellectual.

Given that, the question of how to maintain enthusiasm enough to "get through school" is not easy to answer...

On another hand, the question of how to view scholarly treatises on mathematics is more answerable. (Note: "textbook" is not at all "scholarly treatise". One should not necessarily trust textbooks' picture of mathematics any more than one should trust paid advertisements...) One critical point is that for many decades now there has been a very strong stylistic skewing, so that one doesn't "let on" what the real trick is at the outset. Define stuff, have some lemmas, make up some terminology, and if anyone's still paying attention at the end ... :)

That's not a complete exaggeration. A sort of opposite is some part of the contemporary physics mathematical literature, where it's often very difficult to tell what's meant as "vague heuristic" as opposed to "proof-ish".

My default advice, and my own default behavior, is (as suggested in questioner's comments) to skip forward over boring parts (that, in particular, seem to not be explaining anything to me... it's ok to go back later, after all).

Edit: to add more to the actual "prescription": indeed, one should try to not think in terms of some external authority's commands... nor in terms of compliance to an instructor's insinuations about "what will be on the final exam", ... but in terms of one's own interests, curiosity, satisfaction-with-explanation, and so on. Don't be cowed... Consider, as an antipode to the usual, that it is the author's obligation to convince you ... and that an inadequate author may fail. The complication is that this inadequacy does not mean that the subject is deficient, but only that the writer was not up to the task. And this challenge becomes all the more severe with regard to more substantive mathematics (or anything else).

So, in some sense, I'd recommend just being more realistic, rather than being fixated on the idealizing mythology of mathematics (or of other subjects, presumably, in their bailiwicks). If it quacks like a duck, it probably is a duck, etc.

Do not too much distrust your own sense... just a little.

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This may help. "HOW TO Solve IT" by G. Poyla. I don't know if it's still in print. Princeton University Press was or is the publisher. Good Luck.

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This does not answer OP's Question. – user45099 Apr 28 '13 at 14:08

protected by paul garrett Dec 26 '15 at 23:41

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