3 ms·

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

by 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.
      - 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
  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 don't even look at the tens place, and you do - something. I'm still not sure what they were actually doing with "3 from 2 is 9, carry the one." You could mentally change the 2 to a 12 and subtract three (which would be closer to borrowing, though the steps are out of order), but the fact he doesn't say 12 and says 3 from 2 makes me wonder if this wasn't the case. Because 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. He mentions two ways people are taught to do the next step - the first is that after "carrying the one," you subtract it from the number in the 10's place. This basically creates a situation where subtraction is the same as addition - if you have extra with addition, you carry the one and add it. If you have an extra with subtraction, you carry the one and subtract it. The other way is carrying the one to the number you're taking away and adding it to it. So 4 - 7 in the tens place becomes 4 - 8 when you "carry the one." [I'm using tens and ones place for clarity, it could also be the hundreds and tens place, the thousands and hundreds place, etc.] So there are certainly differences. 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. You see the same thing when Common Core approaches come up - it might be fundamentally the same thing, but the changes in the steps that you take can throw people off.
  - thaumasiotes 2y ago
    > I'm still not sure what they were actually doing with "3 from 2 is 9, carry the one." You could mentally change the 2 to a 12 and subtract three > 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".
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.

by OskarS 2y ago

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.
    - 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
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 don't even look at the tens place, and you do - something. I'm still not sure what they were actually doing with "3 from 2 is 9, carry the one." You could mentally change the 2 to a 12 and subtract three (which would be closer to borrowing, though the steps are out of order), but the fact he doesn't say 12 and says 3 from 2 makes me wonder if this wasn't the case. Because 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. He mentions two ways people are taught to do the next step - the first is that after "carrying the one," you subtract it from the number in the 10's place. This basically creates a situation where subtraction is the same as addition - if you have extra with addition, you carry the one and add it. If you have an extra with subtraction, you carry the one and subtract it. The other way is carrying the one to the number you're taking away and adding it to it. So 4 - 7 in the tens place becomes 4 - 8 when you "carry the one." [I'm using tens and ones place for clarity, it could also be the hundreds and tens place, the thousands and hundreds place, etc.] So there are certainly differences. 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. You see the same thing when Common Core approaches come up - it might be fundamentally the same thing, but the changes in the steps that you take can throw people off.
- thaumasiotes 2y ago
  > I'm still not sure what they were actually doing with "3 from 2 is 9, carry the one." You could mentally change the 2 to a 12 and subtract three > 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".

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.