How to read a book in mathematics?

58,605

Solution 1

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)

Solution 2

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.

Solution 3

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.

Solution 4

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.

Solution 5

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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pax
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pax

Updated on September 18, 2021

Comments

  • pax
    pax about 2 years

    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.

    • zrbecker
      zrbecker almost 11 years
      Usually I read sections in math books in multiple passes. On the first pass I just read until I am confused or finished. Then I go back over it slower, sometimes taking notes on important ideas. I think the first pass is really helpful because it will help you see the big ideas that your are leading up to.
    • gnometorule
      gnometorule almost 11 years
      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.
    • Tyler
      Tyler almost 11 years
      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."
    • Scott Kirila
      Scott Kirila almost 11 years
      I typically skim a chapter and then go through it (maybe the next day after things have had a chance to set in) with pencil in hand until everything makes sense. In my opinion, problem sets are a must. Many authors leave interesting results that aren't quite theorem-caliber as exercises. As for speed, if you find yourself constantly consulting previous chapters, then you're moving too quickly. That's my take on it, at least.
    • Gyu Eun Lee
      Gyu Eun Lee almost 11 years
      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.
    • Gyu Eun Lee
      Gyu Eun Lee almost 11 years
      Contd. This is also an excellent opportunity to polish your penmanship and note-taking skills.
    • orlandpm
      orlandpm almost 11 years
      @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.
    • Asaf Karagila
      Asaf Karagila over 10 years
      Yes. Not enough attention, except the fact that over 500 people visited the question, and over 20 votes the question and its accepted answer.
    • Red Banana
      Red Banana over 10 years
      @AsafKaragila Sometimes people do that to earn that badge. I confess I did that. :-P
    • Andreas Blass
      Andreas Blass over 10 years
      Concerning gnometorule's advice "Do read and understand all proofs": I agree with the "understand" part but if you can figure out the proof and understand it by yourself, without reading it in the book, that's even better. You're more likely to remember a proof (and the theorem it proves) if you work it out yourself. Also, the amount of work that you put into finding a proof can give you a useful idea of which parts of the subject are "routine" and which are the "meat".
    • Ragib Zaman
      Ragib Zaman over 10 years
      @gnometorule What you said about being able to do most exercises is usually true, but there are some texts where I think one could say they have learned a decent amount for a first course yet struggle with many exercises. Atiyah-Macdonald's commutative algebra and Big Rudin are examples IMO.
    • Guest
      Guest almost 7 years
      I would read proofs backwards at first. That way, you immediately see how the results are obtained, and the rest of the proof tells you how to get to this point. I think doing so gives a better idea how one would come up with such and such proof.
  • orlandpm
    orlandpm almost 11 years
    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.
  • Adam Rubinson
    Adam Rubinson almost 11 years
    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
    Adam Rubinson almost 11 years
    Now that I think about it, your method does sound ideal for Baby Rudin.
  • Martin
    Martin over 10 years
    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?
  • Chris Leary
    Chris Leary over 10 years
    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.
  • Admin
    Admin over 10 years
    Especially nice distinction in Read 3 regarding the readability of textbooks.
  • paul garrett
    paul garrett almost 10 years
    @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... :)
  • Manish Kumar Singh
    Manish Kumar Singh about 8 years
    this approach might work for a student but what about a person doing research, isn't he supposed to understand every paper published in his field, how do you make a transition!
  • R K Sinha
    R K Sinha almost 8 years
    If a textbook in math is written in the right manner, that is, keeping in the mind that the author is himself an uninitiated reader, then such a book will present no cause to worry.
  • galois
    galois over 7 years
    I feel like this is true for most good math texts. There are some university algebra and calculus books that seem long winded and poorly written (and somehow, they're always the books that the department has to use)... cough Stewart cough
  • Joao Noch
    Joao Noch about 6 years
    @galois are you talking about James Stewart, Calculus?
  • galois
    galois about 6 years
    @JoaoNoch yes, I'm not a fan for many reasons
  • Joao Noch
    Joao Noch about 6 years
    @galois what makes you say that? I am working through 6th edition now (borrowed from my high school library) and I like it. My only complaint is that it has too many exercises (eg 50+ for each part, very repetitive, I skip much of them). I'm only on ch5 but I would love to hear your criticism of the book.
  • Hikaru
    Hikaru almost 6 years
    @orlandpm at what step in this method do you work on the problems in the text?
  • orlandpm
    orlandpm almost 6 years
    @Hikaru, I try to work the one or two easiest problems on my first read -- the ones that are routine applications of the ideas in the chapter. Then I'll do more after subsequent reads. I've never finished all the problems in a textbook; I usually hit diminishing returns after doing 50% of them. But it depends on the book.
  • Bruno Reis
    Bruno Reis over 4 years
    That is awesome... Thanks!
  • kalashot
    kalashot about 2 years
    How do you have a mental capacity for reading a textbook for 8 hours per day? I manage to read a maths textbook for max. 4-5 hours and then I have absolutely no energy to do anything that day.
  • mhdadk
    mhdadk about 2 years
    (+1) very nice answer.