[Blindmath] using Braille for math

[Bill Dengler]

Hello,
I'm currently in 11th grade, taking Calculus this year.
At the moment, I use a screen reader and a text editor to work out math problems. Before, I wrote my problems in an improvised "calculator notation", where each line was written in a similar format to how it would appear on a scientific calculator (+ for addition, / for division, ^ for exponentiation, sqrt for square root, etc). This notation worked, but had several issues: it was ambiguous at times and hard to read for my sighted teachers.
As of last semester, I've been writing all of my math in LaTeX. This solves the ambiguity issues with calculator notation, and can easily be compiled to PDF for viewing by the sighted.
However, things like
$\lim_{h \to 0} \frac{(-\frac{1}{2}+h)^3-(-\frac{1}{2})^3}{h}$
can be difficult to keep track of in speech; I often have to pull complex fractions apart, bringing certain parts onto their own lines, simplifying and combining everything back into the complex fraction at the end. It's horribly inefficient and error-prone.

My Braille reading speed is fairly slow (around 55WPM), and the only experience I've had with Braille math was in elementary school, using a device called the Math Window <http://mathwindow.com>. That was, in short, a disaster; I was consistently lagging far behind my classmates when doing simple two and three digit addition and multiplication problems because of the time it took to interpret the Braille and manipulate the Math Window's tiles. I haven't used Braille for math since, using only a computer with a plain-text editor to do Algebra, Geometry, and Pre-Calculus. I have, however, used raised-line diagrams and three-dimensional models for Geometry and trigonometry.

Questions:
Would the use of refreshable Braille, hardcopy Braille and/or a device like the Math Window while doing math help me to conceptualize problems more easily, particularly where advanced and/or heavy Algebra is involved?
If so, would it be most effective for me to use it in addition to, or as a replacement for, speech? The biggest problem with using Braille is that my teachers don't read it, so I'd have to frequently transcribe back-and-forth from Braille to LaTeX. This would be relatively painless for refreshable Braille, but less so for the hardcopy variety.
If I used Braille for math, which math code (UEB or Nemeth) should I learn and use? From what I've heard, Nemeth generally takes up less space to convey the same content (important for refreshable displays with limited real estate), but UEB's presentation of that content is clearer. Also, Nemeth could be replaced by the UEB math code in a few years since it's the international standard now.

Thanks,
Bill