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> The "right" way that he presents at the beginning of the song is a way that I've never encountered anyone doing. It's actually fairly interesting: > "Three f

> The "right" way that he presents at the beginning of the song is a way that I've never encountered anyone doing. It's actually fairly interesting: > "Three from 2 is 9, carry the one"; it seems to be a completely algorithmic way of doing the calculation, where you end up with the right answer at the end, but it's completely detached from what's actually happening. I can't understand what you're trying to say. The only difference in the patter is whether you carry the one before or after you subtract 3 from 2. "Both" approaches have you do the same thing in the same way. What's the contrast?

OskarS 2y ago
Yeah, ultimately it's the same algorithm, it's just a question of what are the details of the procedure, which makes it confusing if you've learned one way and not the other. The whole point of the song is that Lehrer thinks this is a new, less good way of doing subtraction. Look at this video if you haven't already: https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA The first way he does it seems INCREDIBLY confusing to people who learned arithmetic in the last 30 or so years (the sentence "8 from 4 is 6" is nonsensical to most such people), but the "modern" way he's mocking is perfectly understandable. I think the song is very funny and charming, but I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote.
- thaumasiotes 2y ago
  > 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 to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote. 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.
  - 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.
- kloop 2y ago
  > I think the song is very funny and charming, but I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. How much of that is that you're familiar with this method (the same way Tom was familiar with the old method)?
  - OskarS 2y ago
    It's a good question, I'm not sure. I do think it's clearer what's going on, and the steps are more obvious. Like, there's a joke in the song about what to do with the carry, if you add it to subtrahend digit or remove it from the minuend digit ("if you're over 35 and went to public school...") which to me indicates that it's rather arbitrary and "learn algorithm by rote". Like, the "borrowing" thing just much better describes what is actually happening, rather than having to memorize a subtraction table and then have arbitrary rules about how to proceed with the carry. But who knows, I wasn't taught the other system, maybe it's equally obvious. I do think it's indicative that the "borrowing" system is nowadays much more common (that's how I learned it in Sweden in the 90s), which probably indicates that it does have some pedagogic value. I don't think for a second either way is "more efficient" than the other: once you get the hang of the borrowing system, you do it very fast.
    - thaumasiotes 2y ago
      The other system, as described in 19th-century textbooks, says this: ---- [What if the digit in the subtrahend is bigger than the one in the minuend?] Imagine adding 10 to both numbers. Obviously, the difference between them will not be changed. But adding 10 to the digit in the subtrahend is the same as adding 1 to the digit immediately to its left. So, add 10 to the [current] digit in the minuend, add 1 to the [next] digit in the subtrahend, and then perform the subtraction.

OskarS 2y ago

Yeah, ultimately it's the same algorithm, it's just a question of what are the details of the procedure, which makes it confusing if you've learned one way and not the other. The whole point of the song is that Lehrer thinks this is a new, less good way of doing subtraction. Look at this video if you haven't already: https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA The first way he does it seems INCREDIBLY confusing to people who learned arithmetic in the last 30 or so years (the sentence "8 from 4 is 6" is nonsensical to most such people), but the "modern" way he's mocking is perfectly understandable. I think the song is very funny and charming, but I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote.

thaumasiotes 2y ago
> 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 to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote. 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.
- 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.
kloop 2y ago
> I think the song is very funny and charming, but I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. How much of that is that you're familiar with this method (the same way Tom was familiar with the old method)?
- OskarS 2y ago
  It's a good question, I'm not sure. I do think it's clearer what's going on, and the steps are more obvious. Like, there's a joke in the song about what to do with the carry, if you add it to subtrahend digit or remove it from the minuend digit ("if you're over 35 and went to public school...") which to me indicates that it's rather arbitrary and "learn algorithm by rote". Like, the "borrowing" thing just much better describes what is actually happening, rather than having to memorize a subtraction table and then have arbitrary rules about how to proceed with the carry. But who knows, I wasn't taught the other system, maybe it's equally obvious. I do think it's indicative that the "borrowing" system is nowadays much more common (that's how I learned it in Sweden in the 90s), which probably indicates that it does have some pedagogic value. I don't think for a second either way is "more efficient" than the other: once you get the hang of the borrowing system, you do it very fast.
  - thaumasiotes 2y ago
    The other system, as described in 19th-century textbooks, says this: ---- [What if the digit in the subtrahend is bigger than the one in the minuend?] Imagine adding 10 to both numbers. Obviously, the difference between them will not be changed. But adding 10 to the digit in the subtrahend is the same as adding 1 to the digit immediately to its left. So, add 10 to the [current] digit in the minuend, add 1 to the [next] digit in the subtrahend, and then perform the subtraction.