> Borrowing and base 8 weren't created by New Math. Teaching them to students, at least according to the song, was part of New Math. Lehrer specifically says this.
No, he doesn't.
So first, we can observe with our own eyes that borrowing and carrying are the same thing, with only the label being changed.
But we can also observe that what was taught to students, as reflected in their textbooks, is the same thing that was taught under the label New Math and the same thing that is still taught today. Go ahead and look at the textbook.
The part that is specific to the New Math is the conversion of the problem to base 8. If you want to stick closely to the lyrics of the song, you might notice that they specify that the base-8 subtraction is the only problem posed by the New Math textbook; the base-10 version is something that Tom Lehrer provides to the audience to aid their understanding of the base-8 version.
This isn't just the clear message of the song, it's also what you'll learn if you read retrospective or contemporaneous coverage of New Math. You can see discussion in precisely these terms on the rather perfunctory Wikipedia page.¹ But most importantly, you might notice that working in alternative bases is actually new, in that - unlike the working of the base-10 problem in the first verse of the song - it doesn't appear in textbooks written a hundred years before the New Math was developed.²
The joke in the first verse is just that it's hard to follow a rapid patter. One specific joke in that verse is the set of lines "And you know why four plus minus one plus ten is fourteen minus one, 'cause addition is commutative. Right." Again, there's nothing new about this material, it's just that the explanation is superfluous to the process and paced in a manner that makes it hard to follow.
¹ Admittedly, the page's view of what was salient in New Math is pretty likely to have been influenced by Tom Lehrer's song, but that's still a radically different and more plausible interpretation of the song than what you're pushing for.
² That far back, it's all carrying into the subtrahend, but the approach of "here's an example showing each step of the process in detail, accompanied by a theoretical discussion of why it works" is already present. To get carrying out of the minuend, you need to go to just decades before the development of New Math, as the patter notes.
No, he doesn't.
So first, we can observe with our own eyes that borrowing and carrying are the same thing, with only the label being changed.
But we can also observe that what was taught to students, as reflected in their textbooks, is the same thing that was taught under the label New Math and the same thing that is still taught today. Go ahead and look at the textbook.
The part that is specific to the New Math is the conversion of the problem to base 8. If you want to stick closely to the lyrics of the song, you might notice that they specify that the base-8 subtraction is the only problem posed by the New Math textbook; the base-10 version is something that Tom Lehrer provides to the audience to aid their understanding of the base-8 version.
This isn't just the clear message of the song, it's also what you'll learn if you read retrospective or contemporaneous coverage of New Math. You can see discussion in precisely these terms on the rather perfunctory Wikipedia page.¹ But most importantly, you might notice that working in alternative bases is actually new, in that - unlike the working of the base-10 problem in the first verse of the song - it doesn't appear in textbooks written a hundred years before the New Math was developed.²
The joke in the first verse is just that it's hard to follow a rapid patter. One specific joke in that verse is the set of lines "And you know why four plus minus one plus ten is fourteen minus one, 'cause addition is commutative. Right." Again, there's nothing new about this material, it's just that the explanation is superfluous to the process and paced in a manner that makes it hard to follow.
¹ Admittedly, the page's view of what was salient in New Math is pretty likely to have been influenced by Tom Lehrer's song, but that's still a radically different and more plausible interpretation of the song than what you're pushing for.
² That far back, it's all carrying into the subtrahend, but the approach of "here's an example showing each step of the process in detail, accompanied by a theoretical discussion of why it works" is already present. To get carrying out of the minuend, you need to go to just decades before the development of New Math, as the patter notes.