Wait is he actually criticizing it in this song? I always thought that he was mocking people who didn't understand borrowing by sounding disbelieving about it, when it's in fact obvious how & why it works
Yes, he's mocking the idea of teaching borrowing. He gives the way he thinks subtraction should be taught in the intro. It's a way that, as far as I can tell, hasn't been taught in America in over half a century. After the intro, he goes on to mock the concept of borrowing, which was (apparently) part of New Math. Then in the final verse he mocks the idea of teaching bases.
Here's a good animated version of the song that shows the different methods. You'll notice that the "silly" New Math method is the way that makes sense to Americans today, and the "simple" method preferred by Lehrer is very confusing for anyone who's not used to it:
Here's more of the intro which makes it clear what's going on. My explanations in brackets:
> Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the “New Math”.
> So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we’re gonna cover subtraction.
[ New Math is confusing, and he's going to go over how confusing it is in a humorous way.]
> Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that…
> Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169.
[This is how it was done before the confusing New Math, the "right" way to do it. "Three from two is nine, carry the one."]
> But in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer. Here’s how they do it now…
[The New Math approach is silly because it's so focused on trying to teach kids concepts that it leaves them unable to do basic arithmetic.]
After this comes the first verse, which is designed to make the idea of "borrowing" sound so overly complex that few people would be able to understand it.
