Among all those symbols only ">" and "<" are somewhat intuitive, all the others you have to learn what they mean.
Even "=" is derivative of "<" and ">" because by reasoning you can understand that you get to it by rotating the 2 lines about 30 degrees after realizing that you are dealing with 2 numbers which are in fact the same and not one being bigger than the other
That said, all notations --including the written alphabets of many spoken languages-- are impenetrable until you learn them. As a personal example, for me, learning French and German was a million times easier than learning Chinese and Japanese. In the first two cases I could read and write the languages right away. In the case of the latter two the notation imposed both a significant time drain and a cognitive load that got in the way of learning. I did a lot better with Japanese than Chinese. And BTW, I would not dare say I know these two languages. I can rattle off a bunch of phrases in Japanese and understand them if spoken slowly. My brain has yet to synchronize to Chinese.
My point is that specialized notations have been a part of the human experience forever. From cuneiform to modern written languages. Our brains are pretty good at learning notation. I would not fault mathematics for anything other than, perhaps, practitioners assuming everyone reading a math-heavy text understands the notation as they do.
Personal example: One of my kids is going though an MIT CS class on edX. He got scared when he was presented a formula with a huge sigma "Σ" sign in front of it and numbers below and above it.
It took less than a minute to explain that this just means a sequence of sums, maybe ten seconds. I just wrote down something like: "(a0 * b0) + (a1 * b1) + ... + (an * bn)" and said: "This is what it means. Summation". Done.
The point is, notation doesn't have to be hard.
I think the real world feedback is quite different, given that math can be explained textually with words , why should we not do it?
The burden of the proof is always on the institution trying to do something. In this case the US government trying to make the US population better at math.
The population is quite okay with the present day situation, it's the government's job to make stuff happen and change things around to obtain the desired result, that is an improvement compared to what we have today.
Math proficiency is in line with new notation foreign languages proficiency from your examples (Chinese, Japanese, Austrian and German to a certain extent), that's because as you said both math and those languages have a different notation.
Given that (unlike foreing languages) math can be explained WITHOUT having to teach a new notation, then why don't we do it?
New notations are necessary for Chinese, but not for math, so why don't we remove this barrier to entry?
New notation is part of the human civilization but it has to be acquired early on to become like a second skin, which is what Latin letters are for us.
One has to be realistic . Mathematical notation will always take the backseat vis-a-vis literal notation. Kids just don't learn (and aren't taught) mathematical notation the same way they learn (and are taught) latin letters.
Instead of fighting against windmills we should take that as a given and try to influence what can be influenced.
As I said the institution trying to make a change in end results, must consider changes in the process...otherwise nothing happens.
A kid can learn the notation for whole, 1/4, 1/8, etc. musical notes and their positions on the staff very easily. An immediate relationship is created to the key on the piano or the fret on the guitar. I have been to math classes where the professor simply vomits formulas on the blackboard for one hour and you are left to figure out what they hell happened. That is a problem. Not the notation. The way math is taught.
Can you explain, say, orbital mechanics, without math notation? In a way where someone can determine where a satellite will be at a particular time given its position and velocity at a prior time taking into account disturbances to the ideal orbit caused by the Moon and Sun (we'll stick with just those 2 and pretend the Earth itself is a perfect sphere).
I don't mean explain in a pop-sci sense. That's actually feasible with very little math (though you will probably want some diagrams), I mean explain in a way that the audience can then apply this math-but-not-in-math-notation to solve real world problems.
