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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.

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closed as off topic by Loop Space, Martin Brandenburg, Ryan Budney, Daniel Moskovich, Felipe Voloch Sep 6 '10 at 13:26

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

    
As there is already one vote to close, I have started a discussion on meta at tea.mathoverflow.net/discussion/650 –  Theo Johnson-Freyd Sep 5 '10 at 18:15
    
I vote to close, the question is offtopic. Besides, everyone must find his individual technique. For example, I tend to read proofs sometimes in a non-specific order and let my intuition fill in the rest of the proof. But of course, not everyone should do this. –  Martin Brandenburg Sep 5 '10 at 19:03
    
And yes, all these questions depend on your mathematical experience, on your situation, how many time you want to spend, etc. pp. –  Martin Brandenburg Sep 5 '10 at 19:06
    
In Theo's meta thread I've posted links to some of the more relevant related threads. Browsing the "soft question" tag is a slightly less efficient alternative way to find these threads. –  Ryan Budney Sep 5 '10 at 19:32
    
@Martin: I expect the answers to be personal. However, my hope is if I get several answers, then among them, one might be useful for me. I can also mix and match variations of what other people are doing. For example, music tastes are highly personal but if a large enough number of people shared their favorite song then perhaps I would find one I really liked also. (Looking at other people's top ten music lists has been a way I've found new favorites in the past.) –  user9028 Sep 5 '10 at 20:13

4 Answers 4

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

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

  2. 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.

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+1 for the second point. However, usually it may take some time for beginners to realize that this is very useful. –  Somnath Basu Sep 5 '10 at 21:15
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@Deane I dunno if that really works.I fully agree that serious math students have to force themselves to produce as many proofs as they can without looking them up.But I don't know if this kind of brute,"info only need to know" minimalism produces the kind of deep insight working through several treatments of the same material does.I know many very talented math students at top programs who do this.The global result to me,is somewhat less then stellar.They usually have huge gaps in thier knowledge-usually in the most basic of concepts. –  Andrew L Sep 6 '10 at 6:37
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@Deane continued For example,I had a friend who was researching nonassociative algebras at Stanford by the age of 20.He had no clue what filters or nets were. He also had never heard of the classification of compact surfaces-which was particularly shocking given his area.I'm not saying you're wrong.I understand the need to force yourself to create math.I just don't know if you have to go to that extreme to accomplish this.I read everything actively,but I don't STUDY like that-when I'm studying,the only thing I don't produce from whole cloth are definitions and some examples. –  Andrew L Sep 6 '10 at 6:42
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Andrew, to clarify, I'm not suggesting that a student try to create his own math. I'm just saying that, given the logical rigor of math, you can often work out at least some of the more obvious rigorous details of a typical proof yourself from just knowing the intuitive idea of what's going on and that doing this is worth the effort. I certainly don't think it's realistic for a student to work out entire statements and proofs of theorems without consulting references carefully. And I don't believe I said anything that implies an overly narrow focus in what you study. –  Deane Yang Sep 6 '10 at 12:46

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.

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"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.

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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.
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