Beyond the intriguing assumption that an adult man might purchase this book, a manual of basic arithmetic, for the purpose of self-improvement, it's pretty much indistinguishable from what we have today. This is the treatment of subtraction:
> If any figure [digit] in the subtrahend is a number greater than the one above it in the minuend, it cannot be subtracted directly and the following method is used. A single unit (1) is "borrowed" from the next figure to the left in the minuend and written (or imagined to be written) before the figure which is too small. The figure of the subtrahend is then subtracted from the number so formed and the remainder figure written down in the usual way.
> The minuend figure from which the 1 was borrowed is now considered as a new figure, 1 less than the original, and its corresponding subtrahend figure subtracted in the usual way. If the minuend figure is again too small, the process just described is repeated.
> As an illustration of the procedure just described, let it be required to subtract 26543 from 49825. The operation is written out as follows:
7₁
Minuend: 49/25 [the 8 is struck through; I don't know how to type this]
Subtrahend: 26543
-------
Remainder: 23282
> Here the subtrahend figure 4 is subtracted from 12 instead of the original 2, and the subtrahend figure 5 is then subtracted from 7 instead of the original 8.
(pp. 10-11)
What do you believe were the New Math revisions to this? There weren't any; what made it New Math was insisting that people be familiar with the theoretical background that the textbooks had always provided. The algorithm, and the explanation of it, were not changed in any way.
(Older textbooks do use the "8 from 4 is 6" model instead, where carries are done into the subtrahend instead of being taken from the minuend, and they have a different explanation. They still provide that explanation for those students who care to know, which is very few people.)
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. In the intro he gives the way it was taught ("Now, remember how we used to do that…Three from two is nine, carry the one"), then he says "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…", then he immediately shows the borrowing approach, which is followed by the chorus "Hooray for New Math!" After showing "how they do it now", he then goes on to show the same problem in base 8 for the second verse (followed by a repetition of the chorus, and then the song ends).
I get that you don't think the approaches are different, or that they're tied to New Math. Lehrer and his audience did, which is the entire point of the song.
> 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.
Beyond the intriguing assumption that an adult man might purchase this book, a manual of basic arithmetic, for the purpose of self-improvement, it's pretty much indistinguishable from what we have today. This is the treatment of subtraction:
> If any figure [digit] in the subtrahend is a number greater than the one above it in the minuend, it cannot be subtracted directly and the following method is used. A single unit (1) is "borrowed" from the next figure to the left in the minuend and written (or imagined to be written) before the figure which is too small. The figure of the subtrahend is then subtracted from the number so formed and the remainder figure written down in the usual way.
> The minuend figure from which the 1 was borrowed is now considered as a new figure, 1 less than the original, and its corresponding subtrahend figure subtracted in the usual way. If the minuend figure is again too small, the process just described is repeated.
> As an illustration of the procedure just described, let it be required to subtract 26543 from 49825. The operation is written out as follows:
> Here the subtrahend figure 4 is subtracted from 12 instead of the original 2, and the subtrahend figure 5 is then subtracted from 7 instead of the original 8.(pp. 10-11)
What do you believe were the New Math revisions to this? There weren't any; what made it New Math was insisting that people be familiar with the theoretical background that the textbooks had always provided. The algorithm, and the explanation of it, were not changed in any way.
(Older textbooks do use the "8 from 4 is 6" model instead, where carries are done into the subtrahend instead of being taken from the minuend, and they have a different explanation. They still provide that explanation for those students who care to know, which is very few people.)