If I have to study difficult material for the first time (the kind of 100 page books that take you days and days of studying), I am often inclined to just keep on reading whenever I get stuck for on something (like a proof, derivation or idea) for too long.

My question: is it efficient to do this? That is, is it useful to skim through the material first before 'diving in deeper'? Or should one try to go very slow from the beginning and make sure to understand everything before reading on?

I realize that this question might be somewhat subjective and vague, but I am sure that a lot of you recognize what I'm saying, and that there ought to be at least a somewhat general educational scientific answer to this question.


closed as primarily opinion-based by anomaly, Qwerty, Henrik, Daniel W. Farlow, Joey Zou Aug 16 '16 at 23:31

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  • $\begingroup$ Yes it is too vague. Could you tell what is your level of study ? What are you targetting at present time ? A diploma ? A qualification ? What are the kind of mathematical books you are reading ? Unless we know more about you, the advices we could give are general : yes, it is good to skim, but stay for example half an hour on a difficult point, etc. $\endgroup$ – Jean Marie Aug 14 '16 at 17:10
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    $\begingroup$ Relevant : "How to read a book in mathematics?". $\endgroup$ – Raymond Manzoni Aug 14 '16 at 17:12
  • $\begingroup$ @JeanMarie I'm at the end of an undergraduate program. I would say that abstract algebra or differential geometry are good examples of the sort of courses I'm talking about. $\endgroup$ – Michael Angelo Aug 14 '16 at 17:13
  • $\begingroup$ I suppose you are autonomous enough. Abstract algebra, till a certain level, isn't as difficult as differential geometry (DG). For the latter, it is easy to be lost in the sands. Looking at a good book before beginning your lectures is not a bad idea. But you need to be advised/coached on such huge subjects as DG : I strongly advise you to manipulate on a good computer algebra system, with the aid of very good books such as the book by Alfred Gray "Modern Differential Geometry" with Mathematica. $\endgroup$ – Jean Marie Aug 14 '16 at 17:25
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    $\begingroup$ A 100 page math book often takes me far longer than "days and days" to learn. $\endgroup$ – littleO Aug 16 '16 at 19:07

Yes, I think it's good to learn math and read math textbooks in a "big picture first", coarse-to-fine manner. Before you learn your way around a city, you first look at a map of the earth to decide which city you want to visit.

I think usually reading the entire textbook thoroughly may not even be the right goal (unless the book is fundamental to your research area and you really need a deep mastery of it). The ocean of knowledge is infinite. You can never understand all the drops of water in the ocean, but you can soar over the water like a seagull, occasionally diving down to catch some prey.

Here's a description of how the mathematician Peter Scholze (who is said to be revolutionizing arithmetic geometry) learns math:

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat's last theorem, which says that the equation $x^n + y^n = z^n$ has no nonzero whole-number solutions if $n$ is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”

Elon Musk, who has created a grade school called Ad Astra, makes some interesting related comments in this video.

Let's say you're trying to teach people about how engines work. A more traditional approach would be to say, we're going to teach all about screw drivers, and wrenches, and you're going to have a course on screw drivers, a course on wrenches, and all these things, and that is a very difficult way to do it. A much better way would be like, here's the engine, now let's take it apart, how are we going to take it apart? Ah, we need a screw driver, that's what the screw driver's for. We need a wrench, that's what the wrench is for. And then a very important thing happens, which is that the relevance of the tools becomes apparent.

Richard Feynman, in Surely You're Joking, Mr. Feynman, mentioned that he quickly skims the whole book to get the big picture and see how the ideas fit together, before digging in to the detailed arguments.


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