"Don't just read it; fight it"
jsomers.net
jsomers.net
My advice is quite the opposite: do not try to understand the article left-to-right, top-to-bottom. Skim (omitting the hard parts) as many times as necessary to get the general ideas and only then attack the hard parts.
"I wish I had read the first 10 pages of many more books -- a splendid mathematical education can be acquired that way."
"Another way I keep active as I read is by changing the notation; if there is nothing else I can do, I can at least change (improve?) the choice of letters. Some of my friends think that's silly, but it works for me... Finally, a small point, but one with some psychological validity: as I keep changing the notation to my own, I get a feeling of being creative, tiny but non-zero..."
And the part that reminded me of your advice:
"Learning a language is different from learning a mathematical subject; in one the problem is to acquire a habit, in the other to understand a structure. The difference has some important implications. In learning a language from a textbook, you might as well go through the book as it stands and work all the exercises in it; what matters is to keep practising the use of the language. If, however, you want to learn group theory, 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."
"To know and not to do is not yet to know."
There is something very real in the action taken from knowledge that looks so minor in foresight, and is so huge in hindsight. Like Goethe's famous quote about the power of action.
There is also the book "The Knowing-Doing Gap", which takes this idea into the organizational context.
*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?
http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...
Thinking about a concept creatively to see a different view is great but only understanding something as something else can prevent one from deeply understanding the concept. Even if one begins to comprehend a concept through analogy I think it's important to eventually think beyond it, and see "through" the analogy to the deeper meaning. Think of leaky abstractions.
I know that I've been on an information skimming trip, and although I feel that I have enlarged my impression of the world considerably, my actual knowledge of the world has only grown marginally - along the lines of confirming intuitions, and occasionally challenging preconceptions I've had.