The hardest thing I ever learned was how to translate geometry into algebra for one of my teachers.
Ok, most people don’t realize this, but once we figured out how to count with numbers beyond what fingers and toes allowed, the next big chunk of Math was geometry. Not algebra, calculus, groups theory… Geo. Any mathematical problem can be solved through geometry.
I think in geometry a lot of the time, rather than in any “human language”. To me, sometimes Spanish is as foreign as English. When I’m solving a complex problem, the data is geometrical blurbs: yellow-filled rectangles, red-edged clouds, joined by lines and arrows, intersecting planes and volumes. At those times, trying to explain the data in a way that doesn’t have to be drawn is painful.
My christallography teacher was one of those guys who don’t “see” geometry. To him, a symmetry axis was a matrix (sigh, no, an axis is by definition a line, the matrix is one of the possible ways to represent it); a symmetry plane was a different matrix (excuse me while I go get my tummy pills).
Once I managed to translate lines and planes into those matrices, I got 100% in the subject. The teacher was amazed. I wasn’t, I knew it was just a language problem…
Never was able to do the same with calculus, another one where I arrived at the solution through geo but the teacher wanted the equations. But at least with that one, I had the pleasure of hearing an actual mathematician tell the teached that my “geometrical solution” was correct (it was slightly different than the algebraic one, because in algebra infinity isn’t real and in geo it is).