New Math was an educational movement in the 50's/early 60's (?) to increase the rigor of mathematics education in grade schools. Among other things it stressed doing arithmetic in different bases. This of course caused a great deal of consternation among parents as they brought home there homework and told there parents with great conviction that 3+2=11 (and of course they were right in base 4). Lehrer's song parodies parental exasperation.
It is widely assumed that we have a base 10 system because we have 10 fingers, our hands being the original calculators. I read in Number (Dantzig) that many have advocated giving up are base 10 system for base 12. The reason being there are twice as many base divisors (2,3,4,6) than 10 (2,5). Divisibility is extremely easy to test for base divisors. Think about how easy it is to tell whether a number is divisible by 2 or 5. In fact most of our temporal and customary measures are base twelve. There is also a competing logic. The great mathematician Lagrange argued for a prime base like 7. Can you guess why?