> Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average. It is theorized that being able to work in both base-10 and base-20 might have comparable advantages to those bilingual students have from engaging in two ways of thinking about the world.
I had a few "ooh, this is not only useful, but intellectually fun" experiences with math in school. Positional number systems was one of them. Distinct aha moment. So much of what had not made sense before suddenly did, including why we had just spent weeks on logarithms. Without at least a few experiences like that, I am sure I would have been even more tuned out in class, destined to hate math for life. I found the way it was taught most of the time to be a form of low-grade torture.
- they learned base 20 counting verbally at home, possibly simple additions, multiplications, and such too
- they learned base 10 english counting as well, and eventually written arithmetic with Arabic numerals
- but there was a missing link in having to translate bases that made it harder for them to understand the underlying patterns of written arithmetic or even algebra
- having the new written numerals provided that missing bridge, allowing them to round out their understanding
To be clear that's just armchair speculation on my part, but that's how I interpreted it. They went from having a challenge of a missing mental connection, to having the advantage of two perspectives.
But certainly there's confounders, like perhaps the new numeral system motivated more interest and family support in math.
The Mayan numerals are also base 20 but like 2000 years older.
The real world is messy
Nobody suggest this to the Danes, please.
It's sort of funny how both Kaktovik and the D'ni numerals were invented around the same time (1994 for Kaktovik and 1997 for when Riven was released).
I am curious as to why Arabic numerals (or any number system, for that matter) are inadequate for a language.
Now consider a language which uses a different base, base 12 for example. The same number might be read as ‘three gross three dozen seven’. But when you look at the number, there is no symbol representing three, and no symbol representing seven in the position of units. The number is hard to parse and read out, and it is also hard to write when going by the words.
A more familiar sense of disorientation of language mismatch would be when metric users encounter imperial units and their arbitrary bases.
Trying to operate simultaneously in a base 20 system and a base 10 system would be much worse than that, since all the digits would be different between representations.
Instead, we represent hexadecimal (base-16) in programming as 0-9 AND a-f, for example, A is 10, B is 11, C is the same as saying 12.
If you have multiple digits, like A4C in hex, it gets more complicated to figure out what number you're talking about in base-10. A is 10, 4 is 4, C is 12. To convert base-10 number you need to do this equation: (10 * 16 * 16) + (4 * 16) + (12) = 2636
(Now, you could easily augment or modify them to do that—and the creators  of this system initially tried that but were unsatisfied—the common way of expressing base-16 using arabic numerals plus the first six letters of the alphabet as added numerals is an example, but if you aren't using a language whose existing writing system conventionally users Arabic numerals, why would you?)
 middle school students!
Having numerals that map well to words reduces friction for practical arithmetic.
Dozens (twelves) were also quite common, and are still in everyday use for certain commodities (eggs, donuts, etc.).
Counting by scores isn't enough to be base 20. The special quantities designated by the system are still 10, 100, and 1000, an obvious sign that the numbers are conceived of in base 10. If the system pivoted around 20, 400, and 8000 (as the dozen/gross system you mention does), then you could (and should) call it a base 20 system.
I actually gave an example of this. "Four score and seven" = 4 twenties, 7 units. That's base 20.
There is also, per Wikipedia, apparently the mixed-base “small gross” / “great hundred” = 120, which I have never encountered in the wild.
As it happens, we've learned to make two-state devices way more cheaply than three-state devices, so binary wins in the real world, but if we figured out how to make three-state devices for at most 1.5 times the cost of two-state devices, base three would win.
It's fun to prove that base e and base 3 are theoretically more efficient than base 2 (and not all that difficult...only basic calculus is necessary).
Only in one dimension.
Using your thumb, start counting on the first bone in each finger. That gets you to four. Continue with the first joint (8), then the second bone (12), then the second joint (16) and finally the tips (20).
You'd use the thumb of one hand to count to 12 on the phalanges of the other 4 fingers, and keep track of every 5 passes like we would do with the second hand
It can also be used to avoid computer vision solutions from detecting numbers in an image, avoiding OCR, etc.