If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics?
My favorite accessibility-increasing tool is the computer. Doing math shouldn't involve so much circus math, i.e. doing things just for show, since a computer does so much immediately and accurately. We already use graphing calculators, and notation-wise graphs can be a useful tool in themselves in elaborating an idea (see also Feynman Diagrams, or electric circuits), but there's so much more calculators can do, let alone actual PCs, cell phones, and web apps. By chance in 9th grade "Intermediate Algebra/Algebra 2" I had a teacher not wholly opposed to modern technology and so he only had us do a small amount of those "solve this system of equations using a 3x4 matrix by hand, showing each matrix transformation to reach the row reduced form, taking up some pages of paper" problems before he brought in a classroom set of chonky TI-92 calculators and showed us the rref() function. That Christmas I asked my mom to upgrade me from my non-graphing scientific calculator that had served since elementary school to a TI-89 Titanium that served me even through college until I learned and got used to various PC programs. The lesson that there were powerful tools around stuck with me pretty fast though, and I wrote some simple programs on the calculator for that and other classes throughout HS (mainly just automating calculator input steps, no fundamentally new algorithms the calculator couldn't already do); in HS physics I also had learned more programming and did a little simulation with pygame and it was fun to enter numbers in the program, run it, see the mass trajectory animate and show some computed values, and then do the actual experiment and get the same results. I only wish I had been shown some PC programs earlier.
I met a friend many years later who sadly was still forced to do that rref()-by-hand for even larger systems of equations in university! That left no time to actually learn anything useful in linear algebra. Madness.
https://theodoregray.com/BrainRot/ has some nice ranting about this (though it does go a bit off the rails when it starts talking about video games).
> Why is it that every time any subject about mathematics comes up there is always a complaint about notation?
The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.
I really only think calculus notation is an issue. Calc 1 books absolutely demand that you view d/dx as an operator and not a ratio, likewise with the dummy variable and integration symbol. Then later the book teaches about unit normals and it’s implicitly acceptable and treat dx/dz and dy/dz as ratios.
i.e.
|u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2).
This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.
Sure, an introductory notation could be created to bridge the gap, and that could be more human friendly.
That said, as someone who uses notation for math regularly, I want to keep using notation. It’s a helpful tool, and it is an efficient and precise language.
See Brett Victor’s 2011 proposal: http://worrydream.com/KillMath/