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How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up.

EDIT: Could anyone suggest books for someone with a math education that just barely touches on high-school Algebra (got into parabolas, rationalizing, some graphing and functions). This is what I am currently doing: attending high school as a Junior.

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    $\begingroup$ Embrace failure. It is part of life. Don't give up. $\endgroup$ – ldog Apr 23 '14 at 23:40
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    $\begingroup$ There really is no other answer: practice. Take classes, read books, do examples, talk to other people about math.... This also depends on your age and background, of course. $\endgroup$ – mojambo Apr 23 '14 at 23:40
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    $\begingroup$ This question is clearly far too broad - Stack Exchange is about questions with hard answers. If there was a simple hard answer to the question "How can I become excellent at math?", I wouldn't be here right now. The book request is possible a bit better, but I think you should edit your question to focus on that, and be more specific about exactly what kind of book you need ("books for a high school junior" is still extremely broad). $\endgroup$ – Jack M Apr 23 '14 at 23:56
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    $\begingroup$ Maybe sometimes you just have to accept the facts you are just not that smart, I wish I could lean this when I was an teenager. No, I am not telling you to give up Math, I am suggesting you to take a more realistic approach to math. $\endgroup$ – Ave Maleficum Apr 25 '14 at 5:31
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    $\begingroup$ I've posted a reopen request. $\endgroup$ – Alexander Gruber♦ Apr 26 '14 at 4:04

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Be honest with yourself about what you do and don't understand. Don't fall victim to "proof by intimidation," where someone attempts to shame you into saying you understand something by implying that you're dumb if you don't. Always ask questions until you really get it. Similarly, don't let yourself move on before you understand something fully; pretty much everything will come back to bite you at some point down the line. If this seems like a pessimistic attitude, it's not - it is simply humility, and humility is the path to genuine knowledge.

EDIT: I just noticed your question about books. How To Prove It is a great transition to more advanced mathematics. After that, check out some of the Dover books, they are all very cheap and most are decent introductions to their respective subjects.

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    $\begingroup$ Good advice! Perhaps one small thing to add: at some times you probably will find you move on, thinking you understand something, and later you will realise you didn't understand it properly. In that case, don't be afraid to back up and do some more work on the topic you thought you understood. And don't let that discourage you - it happens to all of us. $\endgroup$ – David Apr 24 '14 at 12:15
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Buy a huge whiteboard and think of math like a puzzle. Hours can pass by quick with space to scribble and self-motivation. Mathematics has to become a hobby for you to actually come to understand it. Know your basics well, even if it takes a while longer than expected to have a solid standing.

And in terms of books, check Amazon. I surf the web if I've got a specific topic in mind though.

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    $\begingroup$ I imagine the OP was asking for recommendations of specific books. Telling them to look through the hundreds of thousands (millions?) available on Amazon isn't really helpful: most people already know that Amazon exists. $\endgroup$ – David Richerby Apr 24 '14 at 16:32
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    $\begingroup$ The "whiteboard" method really is an excellent way to expand your math abilities. Why, only the other day I figured out a way to impose a real vector space structure on smooth closed planar curves, along with an isomorphism from them to a particular subset of scalar fields! Math is fun! $\endgroup$ – AJMansfield Apr 24 '14 at 23:43
  • $\begingroup$ If you're someone with a sensitive sniffer, you may want to get a blackboard instead. Yes, chalk dust can be messy, but dry-erase fumes are nasty. $\endgroup$ – apnorton Apr 25 '14 at 12:44
  • $\begingroup$ I also recommend using a blackboard. Some specific paint might do the trick : houseoffaux.com/2010/04/easy-chalkboard-paint-finishes.html $\endgroup$ – pitchounet Apr 25 '14 at 15:38
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I would quote this excellent article by Peter Norvig.

It's about programming, but applies to all other domains as well.

Researchers have shown it takes about ten years to develop expertise in any of a wide variety of areas, including chess playing, music composition, telegraph operation, painting, piano playing, swimming, tennis, and research in neuropsychology and topology.

The key is deliberative practice: not just doing it again and again, but challenging yourself with a task that is just beyond your current ability, trying it, analyzing your performance while and after doing it, and correcting any mistakes. Then repeat. And repeat again.

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  • $\begingroup$ +1 ... deliberate practice is more effective than passion. I spend at least an hour every day exploring; or trying to derive or prove things by without peeking. (I try first for proofs that aren't "symbol manipulation" proofs so I get a better intuition.) $\endgroup$ – Michael Deardeuff Apr 25 '14 at 7:20
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There is a nice book by Polya "How to solve it". It does not teach particular mathematics but rather "a way of mathematics". It's also quite elementary.

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  • $\begingroup$ This is the answer I came here to give. Perfect for the level the questioner mentioned. $\endgroup$ – msouth Apr 24 '14 at 20:45
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Practice makes perfect!

Also ask for help from teachers, peers...even have a study group!

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If you're looking for a place to start, pick a video that covers something that you already know from this list of PatrickJMT videos, watch it, and then watch the videos after that in order. Try coming up with problems, and solutions, yourself. You could also try googling the name of the problem with "practice question" or "quiz" to get some premade questions.

This is also an excellent resource if you come across a problem and don't know how to proceed. Always try to solve the question after getting some insight before watching the whole video and 'copying' the answer.

Good luck!

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Just try to learn about proofs and logic. If you have difficulty with proofs this book is a good way to start: How to Prove It: A Structured Approach by Daniel J. Velleman

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You may find it useful to start with http://www.khanacademy.org/

Once completed, look for additional resources and side by side track how much you completed from http://mathworld.wolfram.com/

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Alexander Gruber's suggestions are excellent. Definitely take them on board. I'd also counsel patience and an appreciation for mathematical elegance. It's often the case that once you understand a concept, you're surprised at how simple and concise it really is. But don't rush to understand it. Enjoy the process of learning.

There's a fair amount of research into the connection between emotion and learning (e.g. http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.elsevier-9bf96239-1b71-30a0-a875-8878e78f28c4/c/main.pdf), so don't neglect that emotional connection. Most mathematicians have it to a greater or lesser degree.

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One thing that I cherish myself is I ask myself after learning a new topic or subject for a class is: Do I really understand everything that is going on?

If you don't understand everything going on you might struggle mightily with an application of the topic or further work in the course relying heavily on that specific topic.

If you don't think you understand it well then go ahead and write down what questions you have about the specific topic and seek answers. Whether it be MSE, your textbook, or your professors, you are likely to find the answers to those questions $somewhere$.

Not only will this help you in your current studies, it will help further your cementation of mathematics in your mind, which is always good because mathematics is a field that builds upon itself in many ways. (For example, If you struggle doing arithmetic, you are likely to struggle in courses that use it heavily.)

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