Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?
In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.
This happened almost through all my formal education. The knowledge was just presented, as something that was obvious. There was very little discussion of motivation, or what problems was person/people trying to solve.
I understand that there is so much knowledge to cover during high school and college, but IMHO, this causes students to develop mindset, that you either see solution instantly, or you are just not smart enough to crack the problem.
This is a really good point I think! Up until the last few months, I've avoided doing any math with my brain that I'm not required to do (so, no brain math since college). Just recently I've been picking up some books about the history and philosophy of math like Fermat's Enigma and Everything and More (I'm a sucker for DFW, so this was a no-brainer when math was back on my radar).
Now every evening I find myself saying out loud, "why the hell didn't the math teachers explain it like this." For example, we all learn about the Pythagorean Theorem, but why didn't we learn about the Pythagorean Brotherhood[1] a little bit? And feel just a little bit of the excitement that Pythaogoras did when he found an exposed wire of the universe and figured out how to work it.
Fun fact: Pythagoras had to pay his first pupil to listen to him. No one gave a shit, probably not too different than how kids don't give a shit today. But the kids we're teaching don't even get paid, they're told their true reward is something intangible or some vague math-fu. All this while they have a black hole for their mind in their pocket where at least the fall is comfy.
> Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?
For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.
I had such a problem with this. At my college, the professor I had for my entire calc sequence once said, in response to a question about applications, "My degree is in pure math. I don't care how it's applied."
Thankfully(?) I was taking physics at the same time, so I did get to see how double and triple integrals and differential equations could be applied. But it frustrated me to no end that the person who was driving my math education cared so little about helping us understand it.
Do you know any books or websites that provide this information? I also feel more inspired and motivated when I know what problem they were trying to solve with that particular piece of mathematics.
Memorization is so underrated. Memorizing the fundamentals and having them available for instant recall is hugely valuable, especially when trying to grok a new concept.
I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.
I wouldn't personally find advice like this useful for me. I suspect it varies with "personal style". I grasped mathematical concept easily from an early age and I never made an effort to memorize things.
If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point.
Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content.
Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).
I think people tend to conflate "memorization" with "cramming". You're right that memorization is underrated. Are you familiar with the 'method of loci' (i.e. memory palace). It's an amazing technique.
I did that approach, too... up until my real analysis class. Then we'd have something like 50 theorems [edit: per test], and I couldn't remember them all. But I noticed that two or three theorems were used to prove the other 47, and so I memorized those. They gave me enough to survive the tests.
I found that grinding through the problems, over and over again, while using the textbook as a reference, resulted in the facts becoming committed to memory. I think it worked in two ways. First was the obvious repetition. Second, it created "slots" in my brain that were receptive to related material. There may have been an emotional factor as well: Feeling "on top" of the material made it easier to learn.
These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
In a similar vein, I've been enjoying PBS Infinite series. The original host recently stepped aside to finish her dissertation and the new hosts are doing great but still finding their footing a bit - I suggest looking at some of the older videos to get a good sense of the channel.
It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.
3Blue1Brown makes great videos and offers good explanations. However, particularly with his latest Fourier Transform video, he not only gives the incorrect explanation, he makes it unnecessarily and overwhelming complicated. It is still better than the linked circles approach, but the Fourier transform can be explained more accurately in around two minutes and in three if you were to include the the definition of frequency.
3Blue1Brown is better than most but far from best.
Some books on proof, theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence:
- Kevin Houston "How to Think Like a Mathematician"
- Keith Devlin "Intro Mathematical Thinking"
- "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)
I bailed on high school math, thinking I'm math dumb.
In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book:
It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me.
In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities).
Coincidentally, this book goes well with the technique described here:
What was the name of that 30-something woman who had never studied mathematics before, but then finished a degree in quantum physics in two years? Her name has been posted on HN at least two times before. I guess what she did is exactly how you should study mathematics if you want to become got at it.
I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, technique and mental models are left out of modern mathematics education and writing.
It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equat...
I studied a lot of math, pure and applied,
taught it, applied it, published research in it, etc.
so developed some ideas relevant to the
OP.
For
> To the mathematician this material,
together with examples showing why the
definitions chosen are the correct ones
and how the theorems can be put to
practical use, is the essence of
mathematics.
Is good, but more is needed.
(1) Plan to go over the material more than
once. The early passes are just to get a
general idea what is going on.
In such passes, for the proofs, they are
usually the near the end of what to study
and not the first.
(2) When get to the proofs, for each proof
and each of the hypotheses (givens,
assumptions), try to see where the proof
uses the hypothesis.
Next, try to see what are the more
important earlier theorems used in the
proof. So, sure, in this way might begin
to see some of how one result leads to or
depends on another and have something of a
web, acyclic directed graph, of results.
And try to see what are the core, clever
ideas used in the proof.
(3) For still more if you have time, and
likely you will not, can use the P. Halmos
advice, roughly,
"Consider changes in the hypotheses and
conclusions that make the theorem false or
still true."
(4) But, note that to solve exercises or
apply or extend the theory, need some
ideas. So, where do such ideas come from?
In my experience, heavily the ideas come
from intuitive views of the subject.
So, my best suggestion is to try to
develop some intuitive ideas about the
material. Definitely be willing to draw
pictures, maybe on paper, maybe only in
your head.
In the end, a solution or proof does not
depend on intuitive ideas, but finding a
solutions or proof can make use of a lot
in intuitive ideas.
For research, most of the above applies,
but IMHO there are more techniques needed.
Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.
Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards
This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discovering and rigorously proving your own theorems.
I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?
Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it.
A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d).
Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount.
An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :)
One thing they don't talk about is how you decide what to study. Before spending a lot of time on a particular topic in math, you have to decide whether it's worthwhile studying it at all.
It's apparently just assumed that you're taking a course so the decision is made for you.
One thing I figured out on my own that I wished I had realized sooner (which I'm trying - so far unsuccessfully - to impress on my 14-year-old son) is that, when reading math books, they follow a similar pattern. They describe a concept, show an example problem fully worked, and then discuss the ramifications of that concept, followed by another concept, followed by a fully worked problem, etc. I started making it a habit to try to work the example fully-worked problem by myself, based on the description that preceded it, before reading through the author's work. I was amazed how much better I was able to understand what he presented, and how much better I did working the exercises in the chapter afterwards.
This is good. I think two principal components should be emphasized when studying mathematics.
1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it.
Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them.
Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material).
Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them.
2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it.
Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter.
Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways.
Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.
Notation and syntax are areas that I have struggled with in higher level mathematics. Can anyone recommend a guide or resources useful to understanding things like converting set theory, sequence and series style problems into equations and solving them?
I always strugle with memorizing part. I am never able to reproduce word for word what I’ve learned or read, but I can put it into context and show my understanding pretty well. Here, with math theory, there is no alternative. I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge. I am having math exam on integral theory in 2 days and here I am sitting and lookin at screen. :)
You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again.
I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.
> I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge.
That sounds odd for college-level math courses (note that I am just a first year engineering undergrad and therefore possibly know nothing). Do you have oral part in your exams?
This text should be required reading for ANY university-level math course. It made me go from "ugh, not math again" to "yay, I'm going to learn something new!" in a matter of days.
[+] [-] dboreham|8 years ago|reply
Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?
In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.
[+] [-] mbag|8 years ago|reply
[+] [-] jackstraw14|8 years ago|reply
Now every evening I find myself saying out loud, "why the hell didn't the math teachers explain it like this." For example, we all learn about the Pythagorean Theorem, but why didn't we learn about the Pythagorean Brotherhood[1] a little bit? And feel just a little bit of the excitement that Pythaogoras did when he found an exposed wire of the universe and figured out how to work it.
Fun fact: Pythagoras had to pay his first pupil to listen to him. No one gave a shit, probably not too different than how kids don't give a shit today. But the kids we're teaching don't even get paid, they're told their true reward is something intangible or some vague math-fu. All this while they have a black hole for their mind in their pocket where at least the fall is comfy.
[1]: https://en.wikipedia.org/wiki/Pythagoreanism
[+] [-] jnbiche|8 years ago|reply
For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.
[+] [-] davehtaylor|8 years ago|reply
Thankfully(?) I was taking physics at the same time, so I did get to see how double and triple integrals and differential equations could be applied. But it frustrated me to no end that the person who was driving my math education cared so little about helping us understand it.
[+] [-] elbear|8 years ago|reply
[+] [-] cle|8 years ago|reply
I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.
[+] [-] joe_the_user|8 years ago|reply
If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point.
Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content.
Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).
[+] [-] newman8r|8 years ago|reply
[+] [-] AnimalMuppet|8 years ago|reply
[+] [-] Philipp__|8 years ago|reply
[+] [-] analog31|8 years ago|reply
[+] [-] fooker|8 years ago|reply
That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?
[+] [-] trentmb|8 years ago|reply
[+] [-] e0m|8 years ago|reply
These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
[+] [-] sophacles|8 years ago|reply
https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g
[+] [-] nothis|8 years ago|reply
[+] [-] ethn|8 years ago|reply
3Blue1Brown is better than most but far from best.
[+] [-] legionof7|8 years ago|reply
[+] [-] gtani|8 years ago|reply
- Kevin Houston "How to Think Like a Mathematician"
- Keith Devlin "Intro Mathematical Thinking"
- "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)
[+] [-] jacobolus|8 years ago|reply
Mason, Burton, & Stacey, Thinking Mathematically
Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols)
Schoenfeld, Mathematical Problem Solving
Larson, Problem-Solving Through Problems
[+] [-] dvfjsdhgfv|8 years ago|reply
[+] [-] BadMathBook3|8 years ago|reply
In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book:
https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/...
It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me.
In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities).
Coincidentally, this book goes well with the technique described here:
http://www.pathsensitive.com/2018/01/the-benjamin-franklin-m...
[+] [-] bjourne|8 years ago|reply
[+] [-] sebg|8 years ago|reply
If Susan Can Learn Physics, So Can You - https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...
and
So You Want To Learn Physics... - https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-l...
[+] [-] dqpb|8 years ago|reply
Maybe not who your thinking of, but similar story.
See: https://barbaraoakley.com/books/a-mind-for-numbers/
She also has a great Coursera class called learning how to learn
[+] [-] alexbecker|8 years ago|reply
It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equat...
[+] [-] graycat|8 years ago|reply
For
> To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics.
Is good, but more is needed.
(1) Plan to go over the material more than once. The early passes are just to get a general idea what is going on.
In such passes, for the proofs, they are usually the near the end of what to study and not the first.
(2) When get to the proofs, for each proof and each of the hypotheses (givens, assumptions), try to see where the proof uses the hypothesis.
Next, try to see what are the more important earlier theorems used in the proof. So, sure, in this way might begin to see some of how one result leads to or depends on another and have something of a web, acyclic directed graph, of results.
And try to see what are the core, clever ideas used in the proof.
(3) For still more if you have time, and likely you will not, can use the P. Halmos advice, roughly,
"Consider changes in the hypotheses and conclusions that make the theorem false or still true."
(4) But, note that to solve exercises or apply or extend the theory, need some ideas. So, where do such ideas come from? In my experience, heavily the ideas come from intuitive views of the subject.
So, my best suggestion is to try to develop some intuitive ideas about the material. Definitely be willing to draw pictures, maybe on paper, maybe only in your head.
In the end, a solution or proof does not depend on intuitive ideas, but finding a solutions or proof can make use of a lot in intuitive ideas.
For research, most of the above applies, but IMHO there are more techniques needed.
[+] [-] adamnemecek|8 years ago|reply
https://news.ycombinator.com/item?id=16390046
Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.
[+] [-] aje403|8 years ago|reply
Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards
[+] [-] sampo|8 years ago|reply
I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?
[+] [-] yiransheng|8 years ago|reply
A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d).
Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount.
An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :)
[1] http://longnow.org/essays/richard-feynman-connection-machine...
[+] [-] dboreham|8 years ago|reply
[+] [-] miobrien|8 years ago|reply
A related thread from two days ago for those who missed it:
https://news.ycombinator.com/item?id=16372454
[+] [-] skybrian|8 years ago|reply
It's apparently just assumed that you're taking a course so the decision is made for you.
[+] [-] sincerely|8 years ago|reply
[+] [-] commandlinefan|8 years ago|reply
[+] [-] backprojection|8 years ago|reply
I did my PhD in math, in large part because I enjoyed his class on advanced linear algebra so much, and later on real and functional analysis.
[+] [-] dsacco|8 years ago|reply
1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it.
Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them.
Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material).
Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them.
2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it.
Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter.
Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways.
Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.
[+] [-] FourSigma|8 years ago|reply
[+] [-] lowtec|8 years ago|reply
[+] [-] Philipp__|8 years ago|reply
[+] [-] jolmg|8 years ago|reply
You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again.
I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.
[+] [-] sannee|8 years ago|reply
That sounds odd for college-level math courses (note that I am just a first year engineering undergrad and therefore possibly know nothing). Do you have oral part in your exams?
[+] [-] bluetwo|8 years ago|reply
[+] [-] hollander|8 years ago|reply
[+] [-] Myrmornis|8 years ago|reply
[+] [-] Ldorigo|8 years ago|reply