How do you learn math?

I learn math by understanding the fundamental intuition behind concepts. I agree with you, in that the American high school math curriculum is failing its students. For example, I remember in 6th grade, I questioned my teacher why something was the way it was - the answer I received was "It's just the way it is." Teachers introduce a mathematical concept, and yet they fail to comprehensively elaborate on it. No proofs are offerred, not even simple reasoning. When I try to learn mathematics, I do not accept what is being shown initially. I always question and ask, "why is it so?" I do not accept a concept until I can fully understand why. I believe this is a much more effective way of learning math instead of just mindlessly memorizing formulas. The American public has been taught mathematics wrong for such a long time that they characterize math as a subject that "you memorize a bunch of formulas for". Not so.

artofproblemsolving.com is a great website that has tons of resources for math and problem solving in general. Anyway to address your other questions, for me the best way to learn math is, like others have said, to try and prove anything I come across (The additional benefit is that a) you completely understand it and b) once you know the proof you can derive any formula quite easily instead of having to, as Alan put it, mindlessly memorize them.) as well as constantly challenge myself with difficult problems.

And I share your belief about most classrooms not being as well adept for handling gifted students. I hated mathematics until I was 15 because in class it was all about mindless memorization and repetitive application of formulas. At that point I decided that I couldn't put up with 3 more years of high school math and decided to teach myself Calculus just to be exempt. I used Spivak's textbook, and it was a challenging and beautiful work that made me love math, as for the first time in my life I had to think about the problems and was challenged by them. I feel that if other student had the exposure to an advanced and challenging collection of problems they would also love mathematics and problem-solving as well, but traditional schooling generally doesn't provide that.

I just attempt to proof (or at least learn the proof) of every formula i come across.

I originally fell into 'the calculus trap'; I wanted 'challenging' math so I went ahead and took calculus early thinking it would be difficult, and then realized that typical uni-curriculum calculus was not nearly as hard as some of the seemingly easy-to-understand questions here on Brilliant that require only basic algebra and trig (these require thinking instead of lots of computation). So I started reading into some of the discrete math topics, and I find that they really help you understand how math works. I think that if you have a firm grasp of the more 'basic' areas of math such as set theory and combinatorics, you might find math easier. As for sites/resources, I find books (and pdfs) more helpful than videos because you can go at your own pace and you don't have to wait for an teacher to speak. There are loads of really old math books that you can find online for free, and articles on the Brilliant Blog and betterexplained.com are helpful too. I found Dover math books extremely helpful in teaching me the basics and giving me interesting problems and proofs (I started with 'Sets, Sequences and Mappings' - Anderson and Hall). The only downside to these books is that you have to understand everything perfectly and do all the problems before you move on or you'll be completely lost a few pages later.

I use websites like khanacademy.org, brilliant.org, youtube.com/edu and a couple others. Till grade 10, I used to loathe math. In grade 10, (Current year) I met this totally awesome teacher who helped me learn maths in a much more fun way than the usual dull method. Though we have to stick with the sluggish curriculum of Pakistani educational system designed decades ago, every weekends, we choose a new topic we are not familiar with, start with developing a good sense of the basics and go all advanced. That's all. I find this sad that most of my classmates are ignorant and think that this is stupid and useless.

The Brilliant Blogs are a great way to learn new concepts in math! They have great material for all levels in math. They also have practice problems to ensure you understand the proofs.

I think the key thing to learning math, or really any subject, is having fast enough feedback loops so that you can quickly validate whether your understanding of the material is accurate.

So if a student is learning fractions and they come to believe that 1/2 is less than 1/3 because 2 is less than 3, then giving them the ability to quickly test this hypothesis against a number of challenges where they will be given feedback (probably a visual representation of 1/2 and 1/3 with the ability to play around with different numbers) can help them disprove their current hypothesis and develop a better one.

I'm trying to do this with some projects I'm working on. Here's one that can help students learn about graphing and equations:

http://puzzleschool.com/puzzles/xyflyer

All it's trying to do is provide appropriately challenging levels that have instant feedback loops. I haven't validated the theory completely yet, but students engage with it very easily and I think it can be part of a process that leads to learning that is both efficient and enjoyable.

It's always been a challenge for me in public school to be challenged. In recent years however, I have gone much on my own in my expanding of mathematical knowledge. There really is no technique, besides truely having your heart in math, or maybe it's math in your heart. It's what gives you the drive to continue. As for myself, I enjoy pure mathematics. And I keep myself entertained. I participate in all my school has to offer, including the amc 12. I even am starting up a math club. Outside of that, USAMTS, and things like brilliant or practicing for the amc, are what occupy my more competative math side. However, when i just wish to learn new math, I read any of the several Dover Publications books that I have (I am currently reading the Foundations of Modern Analysis). I attend math camps in the summer too. Always keep your mind wondering, and always keep your eyes open for new opportunities!

For me, I'm learning maths relying on myself, because in classroom, the teacher give us just the basics, so I do my researches, to understand the origin of each thing, and I'm trying sometimes to demonstrate formulas because using this way is very good to know how to solve some problems even they are so complicated, adding to this, i use intuition sometimes in geometry specially in Trigonometrical Calculus

if u take maths as just another subject to be learnt it would be difficult . Maths is about understanding concepts and applying them first in your mind and then in the environment.ask yourself what is going on . why this way and what is the importance of this .

There are different schools of thought (e.g. the Moore Method, from the late R.L. Moore) but they tend to overlap on a lot of key ideas, like the importance of understanding and connections.

I think it's a right problem. There should be challenges along with rewards.

If to talk about sites, I've used this one to solve my math homework and sometimes Reddit to get help from people, but it was a little bit risky. Tutor has explained me some theory, but if you really want to know math, you should consistantly educate yourself on your own. Also, when you get stuck with a topic and can't unterstand it, visit YouTube and try to find something valuable there. I have a list of useful viseos and chanels. You can find a math coursebook with answers, which fits you the best. I can suggest you to take a course Learning How to Learn, it's really helpful! By the way, train yourself with apps related to mastering your mental skills. It's necessary to train your brains every day! And, the last point is... philosophy and logic. If you can cope with them, you're a superhero and can understand everything.

i juz using my brain n wash up wit question on book. lol dont even to try any site. for me math wasnt difficult subject.. it easy actually as you know technique and skill as well. no special technique.