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How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs?

I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.

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Do read and understand all proofs; and do at least many of the exercises. Only when you manage to do the exercises as well, you get the book; and doing them will often make you read chapters again as you finally understand what they really mean. Personally, I do every last exercise in books I self-study (as you, say, Munkres chapters 1-5, 9, 11; and currently reading Artin); but that is a bit obsssive. None, though, you just cheat yourself: you read that book; but you know little. –  gnometorule Jan 15 '13 at 5:20
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Too bad you didn't mention you were reading Axler's linear algebra book. He says in the preface: "You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast." –  Tyler Jan 15 '13 at 5:21
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I prefer to take notes while reading the book line by line, and stop at any point where I don't follow an argument until I work it why it is true. I've found that this helps me slow down and really think about what I'm reading, and the act of writing down the mathematics helps me remember it (body memory?). I eventually end up with a notebook that's basically a condensed version of whatever textbook I was using, which is a highly useful and portable resource:) Try to do as many exercises as your patience allows, though it is tempting to hurry on to the new chapter with your newfound knowledge. –  kigen Jan 15 '13 at 7:17
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Slowly. Then I give up. –  Alexei Averchenko Jan 15 '13 at 7:59
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@TylerBailey on the other side of the coin, if you are spending an hour on a single page you have probably lost track of the big picture. There is a time to have such intense focus, but it shouldn't be on your first few reads. –  orlandpm Jan 15 '13 at 8:06

7 Answers 7

up vote 58 down vote accepted

This method has worked well for me (but what works well for one person won't necessarily work well for everyone). I take it in several passes:

Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book.

Read 1: Let your eyes jump from definition to lemma to theorem without reading the proofs in between unless something grabs your attention or bothers you. If the book has exercises, see if you can do the first one of each chapter or section as you go.

Read 2: Read the book but this time read the proofs. But don't worry if you don't get all the details. If some logical jump doesn't make complete sense, feel free to ignore it at your discretion as long as you understand the overall flow of reasoning.

Read 3: Read through the lens of a skeptic. Work through all of the proofs with a fine toothed comb, and ask yourself every question you think of. You should never have to ask yourself "why" you are proving what you are proving at this point, but you have a chance to get the details down.

This approach is well suited to many math textbooks, which seem to be written to read well to people who already understand the subject. Most of the "classic" textbooks are labeled as such because they are comprehensive or well organized, not because they present challenging abstract ideas well to the uninitiated.

(Steps 1-3 are based on a three step heuristic method for writing proofs: convince yourself, convince a friend, convince a skeptic)

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I am currently using this approach on Rotman's Algebraic Topology text. It has been extremely helpful because instead of explanations he gives explicit constructions, from which the reader is left to extract an explanation. An equation may be a concise answer to a question, but it is rarely a great explanation. –  orlandpm Jan 15 '13 at 7:32
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Ok. I am reading Baby Rudin atm. In the first chapter I went through all the proofs in great detail. Was this a mistake? I think Chapter 1 is different to the others because the proofs are very minimalist and axiomatic (see, for example, the proof on page 10). However, the proof is enlightening because there are axioms you would expect to need but the proof avoids. I think the rest of the book takes slightly more of an intuitive approach by the looks of it, but there are still concrete proofs everywhere. Would you recommend going through it page by page, or to use your method? –  Adam Rubinson Jan 15 '13 at 12:40
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Now that I think about it, your method does sound ideal for Baby Rudin. –  Adam Rubinson Jan 15 '13 at 12:42
    
Great answer. For those of us who work at small, liberal arts schools, but nonetheless try to maintain a research program, asking a friend might not be an option. Any alternatives you might suggest? Also, I'd be interested to know if you keep a notebook of details to proofs, problem solutions, and so on. –  Chris Leary Jun 28 '13 at 17:30
    
Especially nice distinction in Read 3 regarding the readability of textbooks. –  Andrew Jun 28 '13 at 17:42

From Saharon Shelah, "Classification Theory and the Number of Non-Isomorphic Models"; quoted in Just and Weese, "Discovering Modern Set Theory I":

So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights.

Sorry... I just love that quote.

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+1 for that is an adorably practical quote. –  skullpatrol Jun 23 '13 at 20:13

By accident I came to this question-discussion only today.

The theme of several answers and comments, that many readings in different styles is best, I'd second, at least up to a point.

I would disagree with all advice to refuse to move forward without "mastery of all details prior"... certainly for nearly all textbooks, and even many higher-level monographs. The reasons is that textbooks currently seem to have the style of belaboring every possible detail, in the name of "rigor", as well as being rather sub-verbal about it. That is, the relative significance of different details/lemmas/whatever is not at all delineated. Since at least 90 percent of details are not at all "dangerous", and not even terribly surprising or illuminating, this results in gross inefficiency. Textbooks are 10 times longer than they need to be, and the critical points are lost in a 10-times-larger mess of fussy details. Terrible.

The only serious approach to avoiding drowning in the faux-rigor fussy details is to make at least one pass through material to see the big points, the higher-level plot-arcs. This lends coherence to the lower-level details. "Hindsight" of a sort.

In particular, "exercises" are an extremely volatile issue. Contemporary textbooks "must" include lots-and-lots of exercises to please publishers and meet other expectations. Thus, one has scant idea of the nature of a given one! Also, one can observe the schism in many texts between the "theoretical" nature of the chapter, and "problem-solving" nature of the exercises, with dearth of prototypes in the chapter itself, to maintain a sort of misguided "purity".

So: distinguishing the relative significance of details, and seeing the larger story-arc, are the most important things to cultivate. Some acquaintance with lower-level details is obviously useful, but the purported "ultimate" significance of low-level details is mostly an artifact of the way mathematics is taught in school.

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Words of a professional mathematician! –  mStudent Jan 7 at 0:26
    
@mStudent Haha! But... maybe... yes. ?! ... Visibly, many of the questions asked here (as in all spheres of human activity) are more reasonably construed as questions about larger, often tacit/implicit hypotheses... but/and "the crowd" is (apparently) not interested in the obvious deconstruction, but ... in the fake game "as it must be played". Well, yes, we do live our lives out in human society... :) –  paul garrett Jan 7 at 0:58

Let me share with you the first paragraph of my math textbook's preface:

A math book requires a different type of reading than a novel or a short story. Every sentence in a math book is full of information and logically linked to the surrounding sentences. You should read the sentences carefully and think about their meaning. As you read, remember that math builds upon itself. Be sure to read with pencil and paper: Do calculations, draw sketches, and take notes.

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Apparently this is from Algebra and Trigonometry: Structure and Method, book 2 by Mary P. Dolciani, Robert H. Sorgenfrey, Robert B. Kane. Could you please confirm or else give the source? –  Martin Jun 21 '13 at 13:16
    
@Martin Yes, actually it's originally from Book 1. –  skullpatrol Jun 21 '13 at 16:30
    
Here is a link to more discussion on it. english-online.org.uk/askprof/… –  skullpatrol Jun 21 '13 at 20:43

It cannon be too strongly emphasized that a long mathematical argument can be fully understood on the first reading only when it is very elementary indeed, relative to the reader's mathematical knowledge. If one wants only the gist of it, he may read such material once only; but otherwise he must expect to read it at least once again. Serious reading of mathematics is best done sitting bolt upright on a hard chair at a desk. Pencil and paper are nearly indispensable; for there are always figures to be sketched and steps in the argument to be verified by calculation.

L. J. Savage

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lol when I read "bolt upright." –  skullpatrol Jun 23 '13 at 20:02

When you read a book,

(Lovely book!)

Read the first part and see how the layout looks.

If some sections are elective

Then don't read them, be selective

With your books books books books books books books

Be selective and you'll sail on through your books.


When you come to exercises

In your book

Keep a notepad and a pencil by your book.

Do each interesting problem,

All the easy, and some hard ones

In your books books books books books books books

Don't forgo the exercises in your book.

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It really depends on the book. There will be certain books that you won't like or wont be able to get on with. Other books you will sail through and enjoy immediately.

I have many books but tend to find that the majority of maths books are written to be concise rather than interesting (by this I mean the readers have to find the interesting bits themselves rather than the author transferring his/her interest). I am not a fan of this style but you should note that you can nearly always find some supplementary materials to fill in those gaps.

Sometimes it takes a mix of resources to develop sound knowledge in a specific area.

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