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So here's my question, in math, there are several places in which understanding why something works helps for a deeper understanding in mathematics. However, there are also other ways in which knowing the why is pointless unless you are specializing in that specific field. So in general, how do we know what we should look into in-depth to enrich our understanding in math and what do we simply pass over? Is there any way to differentiate between the 2 kinds of learning?

I know my question is very philosophical, however, I think it's still something useful to know when studying mathematics.

Thanks!

Clarification: Say you are just studying math as a subject in grade school, and to extend it, you do research at home, where would you go in-depth and crazy to the basics and where will you stay on the surface and know that it works?

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The answer varies depending on what your goals are - if you want to be an engineer, applied mathematician, pure mathematician, etc, and then it depends on what subfields you want to study. E.g. if you know you'll want to study algebraic geometry, you'll probably be fine glossing over the precise construction of Lebesgue measure. Students are generally bad at predicting what will and will not be important later, so it is best to have some sort of mentor to guide you. –  Ragib Zaman Aug 17 '13 at 15:33
    
Having better ways to spend your time is not the same as it being pointless to spend it on something. –  nullUser Aug 17 '13 at 15:34
    
@ragibzaman Say you are just studying math as a subject in grade school, and to extend it, you do research at home, where would you go in-depth and crazy to the basics and where will you stay on the surface and know that it works? –  m.smakg Aug 17 '13 at 15:35
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You make weighted judgment calls based on what you know you're interested in or might in the future be interested in and on educated guesses about the varying levels of depths various facts or techniques have in their respective areas. –  anon Aug 17 '13 at 15:35
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If I'm understanding correctly, "grade school" refers to the schooling most people receive between the ages of 5 and 11? If so, you should understand everything deeply at that level. If your goal is to be anyone who knows math well (be it an engineer, mathematician, actuary or whatever) you should definitely have a deep understanding of everything up to and including high school math. It is only perhaps around the upper undergraduate mathematics level where you start being more selective in what you learn deeply and what you just gloss over. –  Ragib Zaman Aug 17 '13 at 15:40

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