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> But I already pointed out that the steps aren't different. They're the same style. Whether you use the term "carry" or "borrow" makes no difference to anyone.

by OskarS 2y ago

> But I already pointed out that the steps aren't different. They're the same style. Whether you use the term "carry" or "borrow" makes no difference to anyone. Obviously, it does. Like, that's what the song is about. You're not just disagreeing with the other commenters, you're disagreeing with the concept of the song itself. It's a different way of doing it, even if the underlying principle is the same. This stuff matters a lot in pedagogy, even if there's no difference in the underlying mathematics. I could say this: "you subtract two decimal numbers a_n a_n-1 ... a_1 and b_n b_n-1 ... b1 by successively calculating c_n = a_n - b_n - K_n-1 if a_n >= b_n, where K_n is the carry from the nth digit, or c_n = a_n - b_n + 10 - K_n-1 if a_n < b_n, and you set K_n to 1" or whatever. That's the same method, but it's a TERRIBLE way to teach a child how to do subtraction.

thaumasiotes 2y ago
It's not that there's no difference in the underlying mathematics. That would be true, by necessity, of any subtraction strategy that worked. There's no difference in the steps being performed. If you go through each approach, you'll notice that you do the same things in the same order, with the possible exception that carries might appear before or after the actual subtraction with which they are associated. All of your intermediate calculations are the same. Everything you write down is the same in both cases. Someone presented with your worked solution would have no way to determine whether you had "old math" or "new math" in mind as you worked it. Someone who watched you solve the problem would also have no way to determine that, because there is literally no difference in the method. The "new math" part of the problem isn't that you do the base-10 subtraction differently. It's that you're expected to be able to do the same subtraction in base 8 too.
- 123894893 2y ago
  > The "new math" part of the problem isn't that you do the base-10 subtraction differently. It's that you're expected to be able to do the same subtraction in base 8 too. That's the second verse. The entire first verse is mocking the "New Math" idea of borrowing, showing New Math subtraction in base 10.
  - thaumasiotes 2y ago
    Borrowing isn't even a New Math idea. Here's an American Old Math mathematics textbook from 1931: https://archive.org/details/in.ernet.dli.2015.509299/page/n25/mode/2up https://archive.org/details/in.ernet.dli.2015.509299/page/n2... 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.)
    - 123894893 2y ago
      > Borrowing isn't even a New Math idea. 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.

by OskarS 2y ago

thaumasiotes 2y ago

It's not that there's no difference in the underlying mathematics. That would be true, by necessity, of any subtraction strategy that worked. There's no difference in the steps being performed. If you go through each approach, you'll notice that you do the same things in the same order, with the possible exception that carries might appear before or after the actual subtraction with which they are associated. All of your intermediate calculations are the same. Everything you write down is the same in both cases. Someone presented with your worked solution would have no way to determine whether you had "old math" or "new math" in mind as you worked it. Someone who watched you solve the problem would also have no way to determine that, because there is literally no difference in the method. The "new math" part of the problem isn't that you do the base-10 subtraction differently. It's that you're expected to be able to do the same subtraction in base 8 too.

123894893 2y ago
> The "new math" part of the problem isn't that you do the base-10 subtraction differently. It's that you're expected to be able to do the same subtraction in base 8 too. That's the second verse. The entire first verse is mocking the "New Math" idea of borrowing, showing New Math subtraction in base 10.
- thaumasiotes 2y ago
  Borrowing isn't even a New Math idea. Here's an American Old Math mathematics textbook from 1931: https://archive.org/details/in.ernet.dli.2015.509299/page/n25/mode/2up https://archive.org/details/in.ernet.dli.2015.509299/page/n2... 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.)
  - 123894893 2y ago
    > Borrowing isn't even a New Math idea. 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.