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.
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?