Learning math is hard. Those that have braved some of its depths, what did you discover that allowed you to go deeper?
This is a place to share insights, methods, and tips for learning mathematics effectively, as well as resources that contain this information.
Self-teaching math is a skill in itself. The hardest thing is to recognize what it feels like to be confused, and to attack the source of your confusion (it's way too easy to think "meh, this makes sense" when it doesn't.)
Read with a notebook, like a monk, copy things down as you go. When you finish a book you should have a (somewhat paraphrased/shortened) copy of your own. Do the exercises if there are any (yeah, this will make you feel stupid. The more you can face this feeling, the more math you'll know.)
This is good advice, to which I'd add: once you're done studying some particular area, be sure to have a clear and systematic "bird's eye view" of the basic definitions, lemmas, and theorems, how they depend on each other, and what the salient point of each one is. Because if you don't use this knowledge for a few years, it's surprising how thoroughly you can forget almost everything -- and in case you ever need it again, you'll be in a much better position if your knowledge decays into a still-coherent outline of this "bird's eye view" than a heap of disorganized fragments.
I find it scary how thoroughly I've forgotten some large chunks of math that at some point I knew so well that I would have be able to reconstruct them, with proofs and everything, given just paper and pencil. Those I still remember very well after 10-15 years are either those that I drilled so intensely that it developed into an irreversible skill like bike riding, or those where I organized my knowledge into a very systematic outline (even if I never had a truly in-depth understanding of all the logic involved).
I also find that scary/frustrating. But don't you find you can relearn those forgotten chunks much more rapidly than the first time, if you need to?
Oh, yes, definitely. But the amount of effort necessary to relearn them is much smaller if you remember something resembling a coherent outline than if your knowledge decays into incoherent fragments.
My own experience is that it is fairly easy to identify points of confusion, and the hard part is finding a book or whatever, at the right level, to address that specific point. This is a tough problem to solve with self-teaching.
Interestingly, I never found this to be a problem with mathematics, although I did find it a problem many times I tried to teach myself physics. In my experience, textbook-level mathematics is almost always a perfect self-contained edifice of logic, whereas textbook-level physics often leaves unclear points that you can clarify only by asking an expert or finding another book that addresses that specific point. (I suppose things might be different if you're reading bleeding-edge math research papers.)
Out of the large number of mathematical texts I've read, I can recall only one occasion when I felt genuinely confused after making the effort to understand the text in-depth. In this case, it turned out that this was indeed a fundamental conceptual error in the text. (I can write down the details if anyone is interested -- it provides for a nice case study of what seems "obvious" even in rigorous math.) Otherwise, I've always found mathematics to be perfectly clear and understandable with reasonable effort, as long as you can locate all the literature that's referenced.
I agree the problem is even more pronounced in physics.
Also, I am interested in and would appreciate the details of the case study to which you refer.
The case to which I referred was when I first studied calculus as a teenager. The book I was reading took what I think is the standard approach to handling trigonometric functions, namely first prove that the limit of sin(x)/x is 1 when x->0, and then use this result to derive all kinds of interesting things. However, the proof of this limit, as set forth in the book, used the formula for the length of an arc. But how is this length defined? Clearly, you have to define the Riemann (or some other) integral before it makes sense to talk about lengths of curves, and then an integral must be used to calculate the formula for arc length based on the coordinate equations for a circle -- even though that formula is obvious intuitively. But I could not think of a way to integrate the arc length without, somewhere along the way, using some result that depends indirectly on the mentioned limit of sin(x)/x!
All this confused me greatly. Wasn't it illegitimate to even speak about arc lengths before integrals, and even if this must be done for reasons of convenience -- you can't wait all until integrals are introduced before you let people use derivatives of sine and cosine -- shouldn't it be accompanied by a caveat to this effect? Even worse, it seemed like there was a chicken-and-egg problem between the proofs of lim(sin(x)/x)=1 for x->0 and the formula for arc length.
This was before you could look for answers to questions online, and it was unguided self-study so I had no one to ask, and it took a while before I stumbled onto another book that specifically mentioned this problem and addressed it by showing how arc lengths can be integrated without trigonometric functions. So it turned out that I had identified the problem correctly after all. But considering that I was a complete novice and thus couldn't trust my own judgment, I had an awfully disturbing feeling that I might be missing some important point spectacularly.
Thanks for this. I guess this goes to show how hard it can be to communicate math well. When I learned the sin(x)/x limit I accepted the "proof" by geometric intuition with no protest and was not alert to any deeper source of confusion here.
Come to think of it, the rigorous treatments of sine that I've seen probably all use power series definitions. To see that it's the same function as the one defined using triangles I expect you have to appeal to derivative properties, so that approach would not skirt the issue.
(yeah, this will make you feel stupid. The more you can face this feeling, the more math you'll know.)
This is probably true not only for math. It suggests a general principle of rationality: seek out situations in which you might be made to feel stupid.
This is much easier said that done. Unpleasant feelings are unpleasant, it burns will power.
I think positive conditioning on this is vital. However if this conditioning can be generalized or if it remains field specific is another question.
Do the exercises if there are any (yeah, this will make you feel stupid. The more you can face this feeling, the more math you'll know.)
This.
I would also recommend studying the proofs and making sure one is capable of proving a few key concepts in different ways.
Some relevant posts on MathOverflow:
How do you go about learning mathematics (this version in math.stackexchange also seems useful)
An excerpt from advice given by Ravi Vakil:
Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)
His description provides a way to make sense of John Von Neumann's famous quote:
Young man, in mathematics you don't understand things. You just get used to them.
Advice on the basis of my personal experience:
Learn to pinpoint your confusion.
Train yourself to follow your confusion to its source. Often, doing this is most of the work of clearing it up.
Develop intuitions through simple, canonical examples.
While you need to be able to express concepts formally, these expressions should flow from your mathematical intuitions. You can build these by using simple, representative examples, and by connecting the new concepts to intuitions you already have.
Connect the definitions and theorems of an area into a conceptual map detailing how they relate to each other (Vladimir_M elaborated on this here).
Try to do this across mathematical areas as well, to get a coarse map of all of mathematics.
Work problems (either given by others or of your own creation).
It's tempting to think you understand the material by just reading. While you can avoid feeling stupid this way (as SarahC mentions), you won't get to test your understanding. Problems also help to build and sculpt your intuitions.
One frustration I find with mathematics is that it is rarely presented like other ideas. For example, few books seem to explain why something is being explained prior to the explanation. They don't start with a problem, outline its solution provide the solution and then summarise this process at the end. They present one 'interesting' proof after another requiring a lot of faith and patience from the reader. Likewise they rarely include grounded examples within the proofs so that the underlying meaning of the terms can be maintained. It is as if the field is constructed so that it is in the form of puzzles rather than providing a sincere attempt to communicate idea as clearly as possible. Another analogy would be programming without the comments.
A book like Numerical Recipies, or possibly Jaynes book on probability, is the closest I've found so far. Has anyone encountered similar books?
I agree with your remarks here and share your frustration. While books of the type that you're looking for are relatively uncommon; over the years I've amassed a list of ones that I've found very good. What subject(s) are you interested in learning? (N.B. There are large parts of math that I'm ignorant of - in particular I don't know almost anything about applied math and so may not be able to say anything useful - I just thought I'd ask in case I can help.)
Thank you, my main goal at the moment is to get a handle on statistical learning approaches and probability. I hope to read Jaynes's book and the nature of statistical learning theory once I have some time to devote to them. however I would love to find an overview of mathematics. Particularly one which focuses on practical applications or problems. One of the other posts mentioned the Princeton companion to Mathematics and that sounds like a good start. I think what I would like is to read something that could explain why different fields of mathematics were important, and how I would concretely benefit from understanding them.
At the moment I have a general unease about my partial mathematical blindness, I understand the main mathematical ideas underlying the work in my own field (computer vision) and I'm pretty happy with the subjects in numerical recipes and some optimisation theory. I'm fairly sure that I don't need to know more, but it bothers me that I don't. At the same time I don't want to spend a lot of time wading through proofs that are unlikely to ever be relevant to me. I have also yet to find a concrete example in AI where an engineering approach with some relatively simple applied maths has been substantially weaker than an approach that requires advanced mathematical techniques, making me suspect that mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem. Although I'd love to be proved wrong on that point.
Upvoted for a thoughtful comment.
I don't know anything about statistical learning theory.
I don't know what kinds of probability you're interested in learning, but would recommend Concrete Mathematics: A Foundation for Computer Science by Graham, Knuth and Patashnik and William Feller's two volume set An Introduction to Probability Theory and Its Applications.
I would second the recommendation of the Princeton Companion to Mathematics but would also warn it does not go into enough depth for one to get an accurate understanding of what many of the subjects discussed therein are about. This is understandable given space constraints.
The edifice of pure mathematics is vast and the number of people alive who could give a good overview of existing mathematics as a whole is tiny and possibly zero.
As a matter of practice, much of the information about how mathematicians learn and think about a given subject is never recorded. See this comment by SarahC and Bill Thurston's MathOverflow question Thinking and Explaining.
On average I've found reading math books that adopt a historical approach to the material therein to be considerably more useful than reading math books that adopt an axiomatic approach to the material therein.
Based on my (limited) impression of applied math, it's not uncommon for people to use advanced mathematical techniques to solve a practical problem because doing so makes for a good marketable story rather than because the advanced mathematical techniques are genuinely useful to analyzing the practical problem at hand.
There is an issue of a high noise-to-signal ratio in mathematics textbooks corresponding to the fact that many authors of textbooks don't have the depth of understanding of the creators of the theories that they're writing about and correspondingly do not emphasize the key points.
Concerning your suspicion that "mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem" - there's great variability among mathematicians here. Two essays which discuss dichotomies which are not identical to the one that you draw but which I think you'll find peripherally relevant are Timothy Gowers' The Two Cultures of Mathematics and Freeman Dyson's Birds and Frogs.
Those mathematicians who seek profound insight into problems often seek profound insight into problems within pure math rather than problems that arise in engineering.
Looking at your website, you might find it useful to check out the Brown University Pattern Theory Group. I don't have any subject matter knowledge of what they do, but the group includes David Mumford who is of extremely high caliber, having earned a Fields Medal in the 1970's for his work on algebraic geometry.
While I don't know enough to point you in the right direction to help you with your research, if you're interested in learning about pure math out of general intellectual curiosity then there are many books that I can recommend.
The edifice of pure mathematics is vast and the number of people alive who could give a good overview of existing mathematics as a whole is tiny and possibly zero.
In the 3,000 categories of mathematical writing, new mathematics is being created at a constantly increasing rate. The ocean is expanding, both in depth and in breadth.
By multiplying the number of papers per issue and the average number of theorems per paper, their estimate came to nearly two hundred thousand theorems a year. If the number of theorems is larger than one can possibly survey, who can be trusted to judge what is 'important'? One cannot have survival of the fittest if there is no interaction. It is actually impossible to keep abreast of even the more outstanding and exciting results. How can one reconcile this with the view that mathematics will survive as a single science? In mathematics one becomes married to one's own little field. [...] The variety of objects worked on by young scientists is growing exponentially. [...] Only within the narrow perspective of a particular speciality can one see a coherent pattern of development.
Thank you very much for your great reply. I'll look into all of the links. Your comments have really inspired me in my exploration of mathematics. They remind me of the aspect of academia I find most surprising. How it can so often be ideological, defensive and secretive whilst also supporting those who sincerely, openly and fearlessly pursue the truth.
My capacity to read proofs went through the roof once I went through a few chapters in Velleman's How to Prove It: A Structured Approach. If you feel shaky with less intuitive proof techniques like proof by contrapositive or , for example, how to prove a logical disjunction of propositions, you should at least skim parts of this (or a similar) book.
Echoing what other posters' have said: always read with a pencil and notebook in front of you and write down all the definitions you read. Draw pictures. In my experience, a large part of being able to comprehend more advanced mathematics is being able to take complicated definitions and chunk them into a single concepts without losing the fine details of the definition. I've found that asking yourself dumb questions about definitions is useful for trying to do this.
Terence Tao's career advice page seems to contain a lot of useful advice, both on learning mathematics and doing new mathematical research.
First you need to learn about mathematics. Read general books and surveys; browse a good encyclopedia of math; get a feel for how the different topics fit together. Two good general books are Davis and Hersh's The Mathematical Experience and Eric T Bell's Mathematics: Queen and Servant of Science (this has the problem that it is dated and doesn't even mention computers and has little on probability and statistics, but is still worth reading for algebra and analysis). Also check a college catalog for required courses, and read the detailed description of the courses, what they cover and how.
Second, unless you just want to learn about math, you need to practice. Mathematics is a skill, and you must practice it to be able to use it. Work examples, work problems, create your own problems to illustrate questions you have.
Third, for self-education, get multiple textbooks. If you run into problems understanding a specific technique with your primary text, read what other books have to say. Different authors approach problems from different angles, maybe a different approach will help you. Old textbooks can be really cheap, there is no good reason not to have several different versions for subjects that are important to you. This is less important if you have someone you can ask about problems, but can still be important, not many people are very good at explaining what they know.
[edited - I had misremembered the authors of The Mathematical Experience]
Read general books and surveys; browse a good encyclopedia of math; get a feel for how the different topics fit together.
I disagree, but not very strongly.
If you find yourself interested in a topic, focus on that topic. Your interest and motivation is more important than being well-rounded or comprehensive.
Most people who are interested in math for itself already know enough that they aren't going to be too interested in this sort of post. For the majority who learn math for other ends, engineering, computer science, physics, etc, browsing will help them orient themselves, and to find out what interrelationships there are between different fields and techniques. And what they are missing when they discover they need background that they don't have for a book they are working through.
I really appreciate this thread. I'm getting back into the learning math (topology and tensor calc) and finding it more difficult than I have in the past.
I also find that scary/frustrating. But don't you find you can relearn those forgotten chunks much more rapidly than the first time, if you need to?
Oh, yes, definitely. But the amount of effort necessary to relearn them is much smaller if you remember something resembling a coherent outline than if your knowledge decays into incoherent fragments.