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As I wrote in my autobiography: -I'm unable to do mathematics, but I want do it-.
I'd like to have any hints about how to improve my understanding of mathematics.
Till now, I've received hints like:
-Do as many exercises as you can.-
-Read a lot of mathematics.-
-Don't give up.-

But, how do I stratify the efficacy of the above suggestions? What are other useful hints? Equivalent: if you've had difficulties understanding mathematics, as I have, how did you improve?

P.s.: -For example: as I read a math exercise, often I understand what it says and how to do it. But, while writing, I can't convert my thoughts into math sentences.-

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closed as primarily opinion-based by Mark Fantini, Adam Hughes, Kirill, ncmathsadist, apnorton Oct 5 '14 at 2:00

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ @rustyn Thank you for the edites. $\endgroup$ – MadMathFourierRoad Oct 3 '14 at 17:29
  • $\begingroup$ you're welcome. I hope it's ok $\endgroup$ – Rustyn Oct 3 '14 at 17:40
  • $\begingroup$ @Rustyn have you any hints? $\endgroup$ – MadMathFourierRoad Oct 3 '14 at 17:42
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    $\begingroup$ Dispel these "I am unable" words from your mouth and your consciousness forever. Nothing worthwhile comes easily. If you are willing to put in the hard work, you can do math. It takes time, and comes easily to no one I know (I know some powerful smarties). You enrolling in courses is likely the best follow through on any advice anyone could have given you. Now prepare to work. There are 24 hours in a day. Use them wisely. Do not be a little crybaby when you feel you "can't" (<-get rid of that word) figure something out. Just do the hard work. If you fall on your face, give up or do it again. $\endgroup$ – J. W. Perry Oct 4 '14 at 5:10
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    $\begingroup$ Use math.stackexchange! you mentioned you want to do linear algebra and analysis, well ask a question about whatever problem you are having when you learn linear algebra and analysis. See math.stackexchange.com/questions/160056/… and math.stackexchange.com/questions/474780/a-good-book-on-analysis . There are no shortage of "where can I find a good book on [insert math subject here]?" questions on Math.SE. $\endgroup$ – Joao Oct 4 '14 at 7:11
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An old teacher of mine used to say,

"There are no shortcuts to learning Mathematics"

So put simply, if you are like me - ungifted and only equipped with average intelligence - strenuous hard work is the only answer. It's not easy. But I'll promise you this, you will get there.

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Mathematics is the physical science of the amazing aspects and implications of quantity. Don't listen to the nonsense about abstraction and reasoning and rigor and cognition and problem solving and skill and philosophy.

Mathematics begins with quantity meaning how many, including none. Then strictly for purposes of measurement it develops fractions as formal–ratio and decimal forms. (Formal-ratio form is ingenious as are many of the so-called "definitions" in mathematics. Understanding them means seeing what's ingenious about them.) Then for situations where it is meaningful to represent a location in one dimension as the distance on either side of a relative zero, we begin the development of vectors which are quantities with directions. In one dimension we denote these by signed numbers, an ingenious improvement over some kind of "above" and "below" or "right" or "left" notation. Then we expand that idea to two dimensions and come up with the ingenious but misleadingly named "imaginary" and "complex" numbers. Then we look at the idea that two quantities – though unspecified – can be related. We call such a relationship a "function" and the question then is how we can represent it. It turns out that there are a number of different ways. All of these things have properties. Differential calculus is the study of one property of functions. And so on.

Look for what each topic is really about. Look for what alternative notations are possible to see what is ingenious about the ones that we use. Look for the distinction between objects and their properties. Look for clarity in what exactly the objects are and what exactly their properties are. Most of all, keep in mind that mathematics is completely sensible even when the sense is not explicitly shown to you (as it usually isn't but should be). And most of all, question everything and think for yourself.

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