What took me a long time -- and is still a skill I'm developing -- is to both verify and "read" the math at the same time, to see the proof and the story at the same time.

At one level, you're observing a technical construction and trying to ensure that it's (mostly) sound; but at another level, you're trying to understand the broader picture of how it fits in, what the builder was trying to accomplish or what perspective of the world they're trying to share.

Mathematics is -- like any language -- just the articulation of an experience, of an insight, of an understanding. As you get further into mathematics (and possess more technical skills of your own), it becomes more important to see "Oh, he's trying to apply the machinery of homotopy to type theories as a means of discussing equivalence" than it is to get bogged down in the technical details. Often, the details are wrong in the first draft, but in a fixable way. (This is extremely common in major proofs.)

> There is a point where one starts to see "behind" the symbols. It's a strange sensation, as if one could understand the ideas in a non-verbal way

I think at some point, you have to compile mathematics to non-verbal ideas for computational reasons -- your verbal processing skills are simply too slow and too simple compared to other systems. Your visual and motor systems are way more powerful and (in the case of motor systems) operate in high dimensions. Much like GPUs in computers, if you can find a representation of a problem that works on a specialized system, you can often get a big computational boost; in mathematics, we have to push our understanding of self and experience to the limits to find more efficient representations of ideas, so we can operate on more interesting or complex ones.

I think most mathematicians work in extremely personal, non-portable internal representations, and then use the symbols as a way to create an external representation that the other mathematicians can compile into their own internal representations.

If you see mathematics as extremely high level code meant to be compiled to equivalent internal representations on thousands of slightly different compilers, I think the language starts to make more sense -- it's meant to be a reverse compilation target for machine code that's been under revision for ~3000 years, so of course it looks a little funky.

Ed:

I will say this --

One thing I've noticed as I've gotten older is that we do a really poor job of teaching students the story of mathematics -- the human motivations, the community, the long standing projects (some have gone on for hundreds of years; some are still ongoing).

I sincerely believe that for young kids (less than, say 10), it would be better for their development to teach skills 4 days a week and simply tell them part of the story on the 5th. It would make mathematics much more relatable and understandable.

> Ed: ...

A few people have thought about this very idea. You may take a look at:

  "http://www.vpri.org/html/words_links/links_ifnct.htm"

I liked but didn't love mathematics in high school and as such I just did what I had to do and moved on. A decade later I worked through a CS degree and gravitated towards books about mathematicians and now I have a deep fascination with mathematics and I wish I read these books when I was in high school!

A survey of how mathematicians think about mathematics [citation needed] found 80% visually, 15% kinesthetically, and 5% symbolically (i.e. in terms of notation).