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I am currently relearning basic math - including; fractions, decimals and percentages. I am learning from various different sources, and what gets me is that they only teach math in forms of methods. This bugs me because they teach you the 'how' and not the 'why'. There maybe a good method to divide two fractions with uncommon denominators, but there is no explanation of why the method works. I want to understand the process, not just learn it.

Question - are there any reliable educational sources that explain the logic behind the methods? Or in other words, is there a way to understand the nature of numbers and how they are calculated? In this way I can understand math, rather than just learning the methods to calculate numbers.

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Well, there's always Russell and Whitehead, Principia Mathematica. Of course, the drawback to explaining the logic behind the methods is that it took them over 500 pages to get to the proof that $1+1=2$. –  Gerry Myerson Oct 22 '12 at 11:46
    
In general khanacademy.org gives very good understandable lessons (and explanations) of basic mathematics. Maybe you should try this. –  André Oct 22 '12 at 12:02
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Well -if- you are in America then I can understand why. Personally I hated the way math was taught in high school. Just like you said, method are taught, with no explanations. For example, the quadratic formula was given, and asked to be memorized. In the test, students scream when variations of problems of occur because they forbid direct application of memorized formulas. I believe this is caused by the fact that students want to spend very little time on the math, and learn quick ways to beat the test. For an educator to write books that care to explain the thoughts behind the methods, the writer must appreciate the beauty of mathematical structures. In addition, for the book to be sold well, students must be have enough curiosity about math. Unfortunately, what I saw was a trend of desire to stay away from math. People like you, are becoming less common. I blame the educators.

Try to take some elementary math class at a good universality. Most of the classes are big so you can just walk in and listen. You will find lots of crazy math professors and may be you will love it.

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Things are equally bleak where I live (Austria). –  fgp Oct 22 '12 at 12:19
    
To be fair: it does depend on the teacher. There are grade school teachers and high school math teachers who make an honest effort to explain the various methods that they’re teaching. But time pressure created by an externally imposed syllabus and the pressure to teach to mandatory statewide proficiency tests in many U.S. states undermine their efforts. –  Brian M. Scott Oct 22 '12 at 14:51
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