[roi]

"...study actively. Don't just read it: fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?...it is not a good idea to open a book on page 1 and read it, working all the problems in order, till you come to the last page. It's a bad idea. The material is arranged in the book so that its linear reading is logically defensible, to be sure, but we readers are human, all different from one another and from the author, and each of us is likely to find something difficult that is easy for someone else. My advice is to read till you come to a definition new to you, and then stop and try to think of examples and non-examples, or till you come to a theorem new to you, and then stop and try to understand it and prove it for yourself --- and, most important, when you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not."

(P.R. Halmos)

[dmos62]

There's a classical book titled "How to Read a Book" by Mortimer Adler. It's about reading texts for information, arguments, etc., but not pleasure. The heart of it is to not read the book linearly from cover to cover. Ideally you'd do multiple passes, often non-linearly, at different speeds, skipping different parts, concentrating on different parts. You'd start with parts that are known to have most information, like introductions and conclusions, the table of contents. Then introducing and concluding paragraphs, then introducing and concluding sentences. All the meanwhile, you'll be getting a top-down view of the material and you'll be naturally honing in on the parts that are pertaining to you.

Reading mathematics has a whole series of additional challenges. In my limited experience with reading those: nurturing comfort in incomprehension is important; it's often better to skip a part than get stuck on it; look for multiple accounts of the same concept.

[xtiansimon]

Holding my copy of this excellent book, I'm reminded of Mortimer & Van Doren's comment on 'How to Read Science and Mathematics'

249.2 "Until approximately the end of the nineteenth century, the major scientific books were written for a lay audience. Their authors--men like Galileo, and Newton, and Darwin--were not averse to being read by specialists in their fields; indeed, they wanted to reach such readers. But there was as yet no institutionalized specialization in those days, days which Albert Einstein called 'the happy childhood of science'. Intelligent and well-read persons were expected to read scientific books as well as history and philosophy; there were no hard and fast distinctions, no boundaries that could not be crossed. There was also none of the disregard for the general or lay reader that is manifest in contemporary scientific writing. Most modern scientists do not care what lay readers think, and so they do not even try to reach them.

Today, science tends to be written by experts for experts. A serious communication on a scientific subject assumes so much specialized knowledge on the part of the reader that it usually cannot be read at all by anyone not learned in the field. There are obvious advantages to this approach, not least that it serves to advance science more quickly. Experts talking to each other about their expertise can arrive very quickly at the frontiers of it--they can see the problems at once and begin to try to solve them. But the cost is equally obvious. You--the ordinary intelligent reader whom we are addressing in this book--are left quite out of the picture."

The authors apologize for not being able to offer specific advise to guide the reader of technical books, as those texts are on subjects of which they do not possess the necessary expertise. They suggest the reader fall back on 'scientific popularizations'.

I've found this to be a very useful strategy. Beginning with a readable book, well researched book. Then systematically read the book's bibliography, recursively.

The one problem I ran into, specifically in mathematics, was it seems some notations are not universal. Furthermore, some authors take no pains to define their symbols at any point in their texts. (I quite understand a paper would assume a specific audience, but for a published book I think it's just rude.)

For this I needed to rely on communities such as math.stackexchange.com.

Good luck. The journey is part of the reward.