How Do You Go About Learning Mathematics? I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a few people out there will tell me how they go about learning math so I can try out their systems.
I need to know basic things. Should I use one book at a time or should I be reading many books on the same topic at once? Do you stop reading when you hit on a fact that you don't understand or do you keep reading?
Do you read all in one go or do you do a little bit and for how long (1 hr, 2 hr or more?)
Do you read all the chapters or do you do all the exercises before moving on from a chapter?
Do you adjust your technique in Calculus(calculation heavy) vs. Analysis (proof heavy)? If so, how?
When you make notes, what do you make notes about? Do you make notes while you read or after?
Is there some note writing system (eg. Cornell system) that you find superior for taking mathematics?
If you think these decisions all depend, can you say what they depend on?
I am really lost here. I would appreciate any input.
Full Disclosure: I have asked this question on Math Stack Exchange.  
I am looking for a diversity of approaches. I hope this question is on-topic here.
 A: My suggestion is, first, don't look for the optimal way to learn mathematics -how not to quote here Menaechmus' famous reply to Alexander: "there is no royal road to geometry". 
Second, speak with other people -here it may be interesting to discuss with somebody else following a different book.
In any case, remember that maths books are very dense; no surprise if reading is slow! But on the other hand, each new single book that you read may enrich you greatly. So, just go at your speed, don't worry about the time  it takes, and enjoy what you are learning.     
A: "What is the most effective way to learn mathematics?"
I have been trying to answer this question for myself, and one measure I've taken towards this goal is to record all of my mathematical reading, work, and random thoughts in a journal.  I highly recommend the practice as it has been very illuminating to me since I started a few months ago.  Reviewing my previous readings allows me to ascertain how much math I actually end up retaining from my study sessions, and keeping all of my work in one place (as opposed to throwaway scrap paper) allows me to spot any particularly common mistakes.
So far, I've found that my memory is far more tenuous than I had previously assumed.  I'd look at last month's entries and realize that I'd only retained 20% of what I had learned; fine details being especially prone to slippage.  Yet from analyzing my mistakes, I've also found that those very details are much more crucial than I had thought.
The result of all of this is that I've started to shift my focus from "learning new math rapidly" (which has been my focus since I am still an undergraduate) to "winning the uphill battle against memory loss."  From this new perspective, the old adage: "the only way to learn mathematics is through doing" begins to make a lot more sense. While active learning is far from any cure to forgetfulness, given my own mnemonic capabilities I have come to see that it would probably be a better long-term investment to spend a month on fully working and understanding a chapter, than to spend the same time blazing through several chapters but skipping the exercises (having done both.)
I emphasize again that this is my own conclusion based on my own characteristics, and that is precisely why I recommend everyone to find their own answer to this question by keeping their own math notebook.
A: I've found that the best books are the ones that make me pause when I read a paragraph or point because I suddenly feel that I've understood something well or that I've suddenly slipped gears and am bogged down.  At that point, I tend to walk away from the book towards a stack of blank pages and work out the problem as well as I understand it at that point.  Either I conquer my mistake and return to the book, or I find another book or example which helps me.  This is particularly true of my Differential Equations book from my undergraduate course, and I remember stopping at the catenary problem and marveling at the simple and elegant way of looking at it provided by differential equations.
The other direction which helps me is in having (i) a problem to solve or (ii) a question to answer already in mind.  I ended up rereading a book on Computation theory and Finite Automata because I wanted to refine the state space of a particular algorithm.  Having a question in mind helped me re-state the examples in the book in terms of what I found myself interested in.
Summary:


*

*learn the tools, 

*handle and use the tools (you won't break anything, except a sweat),

*work out the math and proofs on your own, convince yourself of the validity of a proof,

*look for another source if the book you're using seems opaque and too difficult to understand.  Do you need more examples to understand a mathematical point?  Look for a book designed for applications, e.g. computer science or engineering oriented books about math.

*the answer depends on the book and on your mental state.  On some days, it is worth the trouble to keep on attacking a problem until you can solve it.  On other days, if you cannot concentrate, it makes sense to walk away from the problem and the book, read a different topic or read a different book on the same topic, and sleep on it.  Try again the next day.

A: I doubt there is a universal formula for this, but here's my view (expressed before on MathOverflow):


*

*As much as possible, learn things together with others. A working seminar is a wonderful way to learn things

*Read as little as possible and try to work out as much as possible on your own. Read only enough to get the idea of what's going on and then try to work out the details yourself. Consult the book only when you get stuck or lost in what you're doing. Avoid letting the book do any work that you are able to do yourself.
