It is widely recognised that the Duodecimal number system is superior to the decimal system. However, it is plainly obvious that trying to introduce such a system would be difficult, especially in a classroom.
However, could some advantages of such a system be gained by using 12 (and its multiples and divisors) more in the classroom, while still using the decimal system for writing down numbers.
Examples I could think of are:
I High school: Graph paper with major lines at every 12th minor line, rather than the usual 10th line
In Primary school: "Coins" of denominations 1,2,3,4,6,12
Distraction from normal topics (which many kids need to work on, are not meeting state standards). Just in that it is "extra material".
Potentially confusing for kids struggling to master intricacies of the normal decimal system. (after the decimal point, scientific notation, repeating decimals, etc.)
Tees up the idea of alternate bases (this has been a GT "new math" topic for decades). However, I think it is better to use base 2 or 16.
I guess some forms of mental arithmetic might be easier (but this only if you do serious mastering of the new base so that it is automatic...which means a lot of time (Con A.1) as well as serious confusion unless they really master "two math languages" (and many kids struggle to master one), thus serious issues with Con A.2.
I think in effect it would just be a short special topic. So you would have a mild amount of Con A.1 but not Con A.2. Thus you don't get Pro B.2 either. You do cover Pro B.1 but at the expense of a more difficult to learn (versus 2) and less useful base than 2 or 16 (either).
(As with many questions on this site) the intro statement itself is in debate (that 12 is better). I have gotten by fine without 12. And I don't feel like debating/researching the topic (would not be appropriate for this board anyway since it is a math topic, not math ed). So bottom line, I don't "know" that the statement of it being better is correct (as I know 1 plus 1 equals 2 or even like smoking makes me at more risk for cancer).
The idea of trying to stealth change the world like this is sort of crankish. Worry about teaching the kids and getting them to learn fractions and stuff that they have a hard time on. Not stealth changing the world (since you never will anyway.)