Earlier quoted context omitted.
> the sentence "8 from 4 is 6" is nonsensical to most such people Anyone who understands "7 from 3 is 6" must necessarily also understand "8 from 4 is 6". Nobody learns how to subtract from numbers that have digits of 3 without simultaneously learning how to subtract from numbers that have digits of 4. > I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way…
The whole point of the song is that Lehrer thinks that teaching it as borrowing is so different that it makes it incomprehensible to people. > But I already pointed out that the steps aren't different. They are, with borrowing you make the change to the tens place first, "getting" the extra ten ones, then explicitly add it to the ones place, then do the subtraction. With the old way (Lehrer's preferred method), you d…
> you could also take the tens complement of the number being subtracted and add it to the number you're subtracting from
> Or simply memorize a subtraction table, the same way people memorize a multiplication table
Well, you can't take the ten's complement of the number being subtracted, because it's infinitely large. One obvious difference between subtracting 3 and adding ...99999999999999999999999999999997 is that it's possible to write "3".
You definitely can memorize a subtraction table, and that's the approach being taken in all cases you've mentioned so far. Including the new math approach; indexing your table entry under "12" and "3" is not a different approach from indexing the same entry under "2" and "3". As with "borrowing" versus "carrying", it is a purely cosmetic difference, where you have the same literal object with a slightly different name.
That's the reason the textbook wants you to do the same problem in a different numerical base; the author is making an attempt to force the student to solve the problem from first principles instead of relying on a memorized algorithm. This doesn't work unless the student cares about the material. But note that the author recognizes, as you seem not to, that regardless of how much theoretical background you provide for why the subtraction algorithm works, the student won't pay any attention to it unless they have to. And the algorithm itself hasn't changed - what's changed is the inclusion of the followup problem "same numbers, base 8".
Tom Lehrer implies that this approach to pedagogy is misguided; under the old system, students learned to produce correct solutions to subtraction problems and didn't know why their approach worked, whereas under the new system, we asked tricky questions that successfully revealed that the students didn't know why the approach they were being taught worked, and therefore couldn't apply it to problems of the kind that never come up. He is correct that this is pointless; we already knew that the students didn't know why the math worked.
> These might not seem like big differences to you, but they're big enough that Lehrer, and apparently others, felt that people couldn't understand it when one was used rather than the other.
As I just said, Lehrer knew that people couldn't understand it either way. The contrast is between "getting the right answer" and "understanding what you're doing"; there is no implication that people who learned the old approach understood what they were doing. But they got better marks than the new math students, because they weren't graded on whether they understood.
I am aware of one other contemporary record of societal struggles with "new math"; it came up a fair amount in Peanuts. The only example given was the problem "write the 'new math' sentence for 'three is less than five'", and the correct answer was "3 < 5".
