I grokked math as a kid, but it was precisely because I was able to make the leap that the language of math was arbitrary and substitutable while other kids were stuck not understanding the meaning.
Then we need to teach them that, not a new set of symbols. Again, I think the crucial insight here which you uncovered is that people are distracted/confused by the symbology, perhaps trying to take everything too literally.
By the way, the way to "eyeball" a third is to use your two hands (rotate them so palms facing each other) to divide into sections A B and C; since we can very accurately eyeball a 50/50 split, you simply compare A to B and B to C, then adjust your hands until A=B and B=C. Bam, you have thirds. Once you get good at this you just mentally visualize invisible dividers instead of actually using your hands.
When it comes to a circle, if you are staring at pies all day then maybe you are better than average, but many studies have shown that humans are horrible at discerning angles other than 180 and 90 degrees.
Then we need to teach them that, not a new set of symbols. Again, I think the crucial insight here which you uncovered is that people are distracted/confused by the symbology, perhaps trying to take everything too literally.
By the way, the way to "eyeball" a third is to use your two hands (rotate them so palms facing each other) to divide into sections A B and C; since we can very accurately eyeball a 50/50 split, you simply compare A to B and B to C, then adjust your hands until A=B and B=C. Bam, you have thirds. Once you get good at this you just mentally visualize invisible dividers instead of actually using your hands.
When it comes to a circle, if you are staring at pies all day then maybe you are better than average, but many studies have shown that humans are horrible at discerning angles other than 180 and 90 degrees.