Aside: I'm an old guy, it's been 40 years since I took physics and calculus--if this question has been asked and answered, maybe you could point me to the discussion, or give me some good search terms. I can imagine that people can have different levels of understanding of various math equations, and different approaches to manipulating / solving them. One might solve math equations in a very rote fashion, applying various rules that he has learned along the way (just to name one, integration by parts). Maybe someone else might have a, well, I was going to say very deep, but maybe it's not very deep, but a sort of intuitive / gut level understanding of the equation, and maybe even visualize the equation and the solution without applying various rules in some rote sequence. (I'm not sure I asked exactly what I wanted to ask--there are some things you get a gut level understanding of, maybe things like a length times a width gives an area, and times a height gives a volume, so if you think in those terms, the numbers you are manipulating have meaning beyond the numbers themselves in your head.) Maybe, somewhat similarly, in looking at some of Maxwell's equations and seeing a triple integral, you don't focus so much on the triple integral but think of that as representing a volume. My questions include the following: For physicists (or other scientists that use a lot of math) at the leading edge of their fields (doing original research), what kind of understanding and manipulation of math do they have and use--do they apply rote rules, or do they somehow have a gut level understanding of mathematical representations and transformations and do they do their thinking in that intuitive space? If they do their thinking in intuitive space, do you have any insights on how to help others develop such an intuitive / gut level understanding of mathematical representations and transformations? Thanks!