I always associated the New Math (not the new new Math of today) with oddball things like matrices, different bases, and an emphasis on abstract relations like commutativity. But apparently another new-ish thing of the New Math was 'borrowing' in subtraction. Although the idea of borrowing is centuries old, apparently many older Americans were taught an algorism to follow that involved 'carrying the one' onto the lower number. Obviously the result is the same. And they were abjured from additional tick marks, or actually adding tens (i.e. borrowing) so that (rather illogically) they were taught "3 from 2 is 9" rather than "3 from 12 is 9".



I only learn this news by careful watching/listening to Tom Lehrer's "New Math", in which it looks like there were two different methods, not that it makes much difference.



The horrors of the New Math was that it wasn't just a mechanical process, but we were supposed to learn that one ten is the same as ten ones. And this is good. It adds some sense to the algorism, since the older version is somewhat more arcane (to me anyway).

The new new math, as I understand it, continues this process of making it clearer what the association is. You start from the lower number, and basically count up to the larger number. This establishes what the difference is between them.



Probably one could determine which is the most efficient, or which produces the fewest errors, or which is the most 'true' to the underlying mathematics. None of these mean much in the end, so stick to what you love.