How to read a mathematics textbook (2016)(drmaciver.com)
drmaciver.com
How to read a mathematics textbook (2016)
https://www.drmaciver.com/2016/05/how-to-read-a-mathematics-textbook/
38 comments
> If a certain book isn't working for you, try something else on the same topic
This. Sometimes it'll take 10+ minutes to understand a single idea when you can spend <10 minutes to find another book, article, blog post, or mathexchange question that explains the same idea more clearly.
This. Sometimes it'll take 10+ minutes to understand a single idea when you can spend <10 minutes to find another book, article, blog post, or mathexchange question that explains the same idea more clearly.
> if a certain book isn't working for you, try something else on the same topic
I spent a year with Baby Rudin in college, what would you recommend as a replacement fit for self study?
I spent a year with Baby Rudin in college, what would you recommend as a replacement fit for self study?
I think the writer is understating the importance of the exercises. In graduate level math textbooks, the exercises frequently serve as a truncated exposition of related topics and concepts.
> I think the writer is understating the importance of the exercises.
I think it's not so much that I'm understating the importance of the exercises as that I'm coming from a position where by the time I want to read a textbook these days I effectively have my own exercises to work on.
That being said, it's true that I've always had a more-than-healthy aversion to actually doing the textbook exercises, and it's something I should work on.
I think it's not so much that I'm understating the importance of the exercises as that I'm coming from a position where by the time I want to read a textbook these days I effectively have my own exercises to work on.
That being said, it's true that I've always had a more-than-healthy aversion to actually doing the textbook exercises, and it's something I should work on.
I agree for undergraduate math. I love math textbooks, but it never occurred to me that one reads them without doing the exercises. And yes to the other post that points out many math texts contain cumulative knowledge making picking a random topic a method from someone not doing math. Intro geology text, sure. Some math textbooks have a few core chapters, too. Generally not reasonable to struggle through a random topic in chapter 5 when its underpinnings are clearly given in the first three chapters.
In school I read all of my math textbooks, cover to cover. But, I never sat down to read them, until after doing the exercises. I go through a chapter with pencil and paper, doing all of the examples, then attempting the easy set of exercises. Then I go through the chapter, reading the text and reworking the examples, then do all the exercises. For differential equations and linear algebra this was sufficient to do the homework and ace tests. For calc I and II, I needed class also, calc III a study group helped, crystallography I worked with one other person. It's highly dependent on the topic.
Other than crystallography, I haven't taken any graduate math courses. I did work through maybe half of my mom's advanced engineering math text, and it's not overboard or harder than the first two years. The exercises are where you learn math.
In school I read all of my math textbooks, cover to cover. But, I never sat down to read them, until after doing the exercises. I go through a chapter with pencil and paper, doing all of the examples, then attempting the easy set of exercises. Then I go through the chapter, reading the text and reworking the examples, then do all the exercises. For differential equations and linear algebra this was sufficient to do the homework and ace tests. For calc I and II, I needed class also, calc III a study group helped, crystallography I worked with one other person. It's highly dependent on the topic.
Other than crystallography, I haven't taken any graduate math courses. I did work through maybe half of my mom's advanced engineering math text, and it's not overboard or harder than the first two years. The exercises are where you learn math.
This is true. Complete understanding rarely occurs until one can understand how to wield this knowledge.
My calculus teacher used to say that you learn calculus through your wrists, when you read the theory you get a grasp of how to solve some problems, when you learn to solve the problems is when you really can start to understand the theory.
The same is valid for writing software, reading code is great, but you need to write a lot to learn how to program.
The same is valid for writing software, reading code is great, but you need to write a lot to learn how to program.
In my experience I never learned a single thing except from the exercises.
This is a very underrated comment: https://news.ycombinator.com/item?id=18055053
I've came to this conclusion on my own years ago.
I wholeheartedly agree that there are books whose authors don't understand the subject.
There is no other explanation for writing books so badly.
I keep thinking that if you know a subject well enough, you'll surely be able to write a comprehensible book on it, be it maths or anything else. They say that Feynman said something similar: "If you can't explain this to kids, you don't understand it well enough". Not sure if I agree completely with that cause advanced math. subjects typically require prior knowledge (just the way it is, no way around it), but the concepts.. perhaps you can explain them to a certain degree. Some times, at least in maths, older literature is written in such a better way, it's unbelievable. Modern literature is terse and dry. This is the skeleton: -definitions -some axioms that we take for granted -theorems based on the previous two -and (usually) the simplest examples the authors can find
And then a list of exercises. Some of them extremely simple, some of them postdoc. research (say, Engelking's topology exercises - if you can solve them all, you can probably publish double digit amount of papers)
There's usually no explanation, no historic importance of certain results, no motivation for what truly moved the mathematician to discover/invent that, and so on.
It's like we don't get to the bottom of things, only to the surface..
I've came to this conclusion on my own years ago.
I wholeheartedly agree that there are books whose authors don't understand the subject.
There is no other explanation for writing books so badly.
I keep thinking that if you know a subject well enough, you'll surely be able to write a comprehensible book on it, be it maths or anything else. They say that Feynman said something similar: "If you can't explain this to kids, you don't understand it well enough". Not sure if I agree completely with that cause advanced math. subjects typically require prior knowledge (just the way it is, no way around it), but the concepts.. perhaps you can explain them to a certain degree. Some times, at least in maths, older literature is written in such a better way, it's unbelievable. Modern literature is terse and dry. This is the skeleton: -definitions -some axioms that we take for granted -theorems based on the previous two -and (usually) the simplest examples the authors can find
And then a list of exercises. Some of them extremely simple, some of them postdoc. research (say, Engelking's topology exercises - if you can solve them all, you can probably publish double digit amount of papers)
There's usually no explanation, no historic importance of certain results, no motivation for what truly moved the mathematician to discover/invent that, and so on.
It's like we don't get to the bottom of things, only to the surface..
My thing is the symbols. Stopped studying math in college, and every time I've tried to get back into it, seems like my options are either starting over from 4th grade or make sense of a formula that appears to be written in some form of ancient hieroglyphics. I think I can grasp the concepts if I could understand the syntax, but the best source I've found is the Wikipedia page on math symbols, which is little more than a list (and quite a few have a good half a dozen meanings).
It'd be cool if there was a good book focused on just understanding the symbols in as plain english as the concepts allow, and teaching you what context you need to look for when the symbols have multiple potential meanings. Maybe there is a book like that (and if so, please let me know, as that would be super exciting).
It'd be cool if there was a good book focused on just understanding the symbols in as plain english as the concepts allow, and teaching you what context you need to look for when the symbols have multiple potential meanings. Maybe there is a book like that (and if so, please let me know, as that would be super exciting).
What's an example of a subject you struggled to learn? In a good introductory calculus book you shouldn't see anything more outlandish than an integral.
Not OP but you could write a whole book on "what exactly does dx (and its variants) mean and why can you sometimes manipulate it algebraically?"
I find that with anything where I feel I have to "start over" every time I try to get back into it, the solution is to consciously memorize the key points and the notation of anything I'm learning, using (Anki) flashcards. Michael Nielsen recently wrote a great overview of the process recently: http://augmentingcognition.com/ltm.html
Making sure the basics stick in long-term memory, you can then build on that anytime in the future without having to review everything.
Making sure the basics stick in long-term memory, you can then build on that anytime in the future without having to review everything.
I think I know what you mean about symbols but thats because I used to try to read math at normal speed. Actually theres a only small set of common symbols you could pick up in an afternoon and everything else is domain specific and therefore defined in place or needs to be looked up. The real issue is getting the math ideas in your head takes much longer than simply reading an equation, like you might see “e” or log and feel frustrated you don’t remember the symbol, but really the issue that everyone has to deal with is figuring out why that term is there which can take much more brainpower. Im not an expert but learned a lot through Coureras Introduction to Mathematical Thinking.
Step 1 to reading a mathematics textbook is finding 4 hours per day to work through it. Most people cannot do that.
I think even an hour a day is enough to concentrate on an author's discussion.
I nearly failed calculus until I took a couple weeks off and read the damn book. By read I meant: start from the beginning, skip nothing. Skip no exercise, skip no example, skip nothing and only continue when you've understood the chapter's material. I got a hundred on the final. Just school, but do wish I had that kind of time.
I'm curious if any of you full-time tech worker folks have ever read a significant portion of a math textbook and done the exercises just for fun/fulfillment. I've tried starting Concrete Mathematics myself a few times during multi-day breaks from work, and it's tough to keep it up when work starts back up again.
It's frustrating to me that it seems like this window during which you can get to some of this knowledge (college) closes and the to lack of a teacher, the structure provided by a course, and sufficient time seem to be unavailable.
It's frustrating to me that it seems like this window during which you can get to some of this knowledge (college) closes and the to lack of a teacher, the structure provided by a course, and sufficient time seem to be unavailable.
I took a Masters in Maths with the Open University for no other reason than fun/fulfillment. A textbook, some problem sheets, nine months and an exam. Repeat four more times and then write a dissertation.
About five and a half years from start to finish; when I totted up the hours I'd spent on it, it did come out as somewhere around nine to ten months of full-time study, spread over the five and a half years.
About five and a half years from start to finish; when I totted up the hours I'd spent on it, it did come out as somewhere around nine to ten months of full-time study, spread over the five and a half years.
Last year I read a book on combinatorics [1], and did a fair number of exercises each chapter. Normally, worked on it for an hour in the morning, and then 4 - 8 hours a week spread over nights / weekends.
Looking back, it was pretty taxing--I think if I'd been meeting with a tutor once a week it would have been a lot easier!
[1]: Basic Techniques of Combinatorial Theory, by Daniel Cohen
Looking back, it was pretty taxing--I think if I'd been meeting with a tutor once a week it would have been a lot easier!
[1]: Basic Techniques of Combinatorial Theory, by Daniel Cohen
How long did it take for you to finish the book?
aren't there literally hundreds of "rent-a-tutor" services available now? if you really think you want to pursue it seriously i'm sure you can negotiate something with a teacher on freelancer.com or something like that
Is it also reasonable to include "Read good textbooks"? Some books are just bad because the author is bad at explaining...
Or because the author does not actually know the subject: https://twitter.com/TheRealDrMcCoy/status/104379662417314201...
The key to reading a math textbook is learning to be generative. In other words, it is not enough to be able to verify a proof, but to be able to construct the proof yourself. One of the most common failure modes is reading a textbook and nodding along as the author presents proof after proof. Most theorems, proofs, and definitions will sound similar to each other or give a tautological "why is this even a theorem" vibe [1], and reading by verifying is how to fall into this trap. (Cue a metaphor about NP-complete algorithms being able to verify a proof but (probably) not able to construct a proof in polynomial time.)
This is why, like other comments have noted, doing exercises is so important. However, this is not the only way to train "generativity". I read textbooks very slowly by reading theorems and constructing the proof myself. (Hard mode: don't read the theorems and try to guess the next theorems and lemmas.) I like this approach because you can get feedback afterwards by reading the solution, while most textbooks don't have solutions for their exercises.
It sounds like OP's strategy is another approach for being generative that I've yet to try myself. I like his approach because it seems to be more effective at filtering out a lot of the noise from linear reading and, instead, focusing only on the results and definitions that end up being used later. But no matter the technique, it seems that the common theme is to spend more time staring at your scratch paper than at the book.
[1] https://xkcd.com/2042/
This is why, like other comments have noted, doing exercises is so important. However, this is not the only way to train "generativity". I read textbooks very slowly by reading theorems and constructing the proof myself. (Hard mode: don't read the theorems and try to guess the next theorems and lemmas.) I like this approach because you can get feedback afterwards by reading the solution, while most textbooks don't have solutions for their exercises.
It sounds like OP's strategy is another approach for being generative that I've yet to try myself. I like his approach because it seems to be more effective at filtering out a lot of the noise from linear reading and, instead, focusing only on the results and definitions that end up being used later. But no matter the technique, it seems that the common theme is to spend more time staring at your scratch paper than at the book.
[1] https://xkcd.com/2042/
Maybe its me being dumb, but I don't get the description of what the author actually does. I guess he picks some random theorem and then has two lists?
Here's the gist of it as I read it... the current list is a topic you want to learn, and as you're working on it, everything that you don't understand is added to the current list. As you check off the subtopics (and their subtopics), you'll have the requisite knowledge to understand the topic you wanted to learn in the first place. The pending list is stuff that will become the first item at the top of your next current list.
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A bit of shameless advertising: for those software engineers interested in learning mathematics, I am very close to publishing a book called "A Programmer's Introduction to Mathematics." I hope it will accommodate readers, specifically programmers, in a way that most mathematics books usually do not. After reading it, you should know how to read a mathematics textbook, and have a baseline of background knowledge to make the rest of mathematics more accessible.
Sign up here: https://jeremykun.com/2016/04/25/book-mailing-list/
Publication ETA is end of this year.
Sign up here: https://jeremykun.com/2016/04/25/book-mailing-list/
Publication ETA is end of this year.
For an even more demanding programme: www.topology.org/tex/conc/mathlearn.html
The suggestion to take material out of order is a bit of a mixed bag. A typical book will have a few core chapters, and some extra topics later on. Most chapters will follow a similar outline. You can skip the optional chapters and the optional sections within chapters, but if you skip chapter 2, you probably won't be able to follow much of chapter 3.
One last comment: there are huge variations in how well different books are suited for self-study. Rudin's "Principles of Mathematical Analysis" is a collection of true statements with zero motivation or exposition, and it's impossible to follow on your own. On the other end, something like Herstein's "Topics in Algebra" clearly spells out the reason for everything and is a joy to read. Most books are somewhere in between, so if a certain book isn't working for you, try something else on the same topic.