How to read a mathematics textbook (2016)
drmaciver.com
drmaciver.com
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.
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.
I spent a year with Baby Rudin in college, what would you recommend as a replacement fit for self study?
The same is valid for writing software, reading code is great, but you need to write a lot to learn how to program.
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.
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.
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..
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).
Making sure the basics stick in long-term memory, you can then build on that anytime in the future without having to review everything.
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.
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.
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
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.
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