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As Geoff Pointer commented:

[...] As a composer I've learnt a lot from studying famous composers why wouldn't that also apply to studying maths and mathematicians of note as well? [...]

Are there any (auto)biographies, studies, or references of mathematicians on

  • how they learn and study,
  • apprehend new maths,
  • or how they had done so while they were students?

In the Area 51 page, someone recommended this book, what are some other examples?

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    $\begingroup$ Perfect question. Nevertheless, I would recommend editing it, so that miscellaneous content isn't much longer than the actual post. $\endgroup$ – dtldarek Mar 15 '14 at 16:38
  • $\begingroup$ @dtldarek: Thank you for your support. I've streamlined it. Please advise if it can be recommended. $\endgroup$ – Greek - Area 51 Proposal Mar 18 '14 at 17:07
  • $\begingroup$ It was unable to put it into words, so I've edited your question as I would have written it. I hope I didn't changed the original meaning too much, please review/reedit/rollback it as you wish to make it even better. (I've also removed the remark about upvotes. I see such comments unprofessional if given without an explicit reason why the upvote; with students I'm always trying to be specific and I think it's a good strategy in many other cases too.) $\endgroup$ – dtldarek Mar 18 '14 at 17:21
  • $\begingroup$ I think structure of this question is not suit here. I think this type of questions must be at Meta section. $\endgroup$ – Huseyin Mar 18 '14 at 22:01
  • $\begingroup$ Huseyin: M;eta is for questions ABOUT THIS SITE, so this Q should definitely not be at meta! $\endgroup$ – kjetil b halvorsen Mar 20 '14 at 15:04
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How Does One Do Mathematical Research? (Or Maybe How Not To), by Lee Lady

Mathematics as a creative art, by Paul Halmos

I Want to Be a Mathematician: an Automathography, by Paul Halmos

Also, this MSE thread ("what is mathematical research like?") might be of interest.

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  • $\begingroup$ Halmos' book seems close to optimal here $\endgroup$ – Andres Mejia Dec 10 '17 at 20:27
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Here are two from famous mathematicians who have tried to explain how they approach mathematics:

George Pólya. This book is actually targeted at introductory students. It has great examples, and an explanation of a method to approach them.

Terence Tao is one of the best problem solvers of our time. He explains what I believe you're looking for in great detail.

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G. H. Hardy's A Mathematician's Apology is a nice read.

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    $\begingroup$ Yes, but not everything in there should be taken litteraly. $\endgroup$ – quid Mar 16 '14 at 1:00
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The Princeton Companion to Mathematics has a section Advice to a Young Mathematician, which seems available for free, where each of Atiyah, Bollobás, Connes, McDuff, and Sarnak give advice and most of it is of the form that it fits what is asked for in the question.

Terence Tao's blog contains various related information under Career Advice and On Writing; note that this is mostly different from the book 'Solving Mathematical Problems' mentioned in another answer, which he wrote more than two decades ago.

Cédric Villani has written a book Théorème Vivant that reads somewhat like a diary and generally feels very authentic, even including reproductions of some of his email correspondence. It might be of less practical value regarding finding advice or behavior to adopt, yet as far as conveying what it can be like "to live as a research mathematician" goes this is the best I know. The original is in French according to some rather recent information on his website it is also available in Italian, German, and Serbian while English, Romanian, Bulgarian, Japanesen and Korean versions are planned.

As a final note reading such things one shoud never forget that (as written by Connes in the text mentioned above) "each mathematician is a special case".

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Not to blow my own horn, but a significant portion of my doctoral dissertation was devoted to an analysis of memoirs and biographies of mathematicians at work. Chapter 2 is probably the most relevant.

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For paedagogical use, the book "Amongst mathematicians" by Elena Nardi might serve your intentions.

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