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The song "New Math" is actually funny to listen to, I think most people don't pay attention to what's happening in it. In the first verse, he's mocking the idea

The song "New Math" is actually funny to listen to, I think most people don't pay attention to what's happening in it. In the first verse, he's mocking the idea of teaching kids the concept of borrowing, as if it's a bizarre and confusing concept that almost no one would understand. He was completely wrong, of course, and it's the way almost everyone - at least in the U.S. - does subtraction now. 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: > Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that…Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169. "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. Tom Lehrer - a math teacher, it should be noted - was mocking the idea of teaching people what was actually happening with subtraction. We see a similar thing today, where people mock the idea of teaching math concepts with Common Core because they think people should just use an algorithm to get the answer, even if they don't understand what's happening.

RheingoldRiver 2y ago
Wait is he actually criticizing it in this song? I always thought that he was mocking people who didn't understand borrowing by sounding disbelieving about it, when it's in fact obvious how & why it works
- 123894893 2y ago
  Yes, he's mocking the idea of teaching borrowing. He gives the way he thinks subtraction should be taught in the intro. It's a way that, as far as I can tell, hasn't been taught in America in over half a century. After the intro, he goes on to mock the concept of borrowing, which was (apparently) part of New Math. Then in the final verse he mocks the idea of teaching bases. Here's a good animated version of the song that shows the different methods. You'll notice that the "silly" New Math method is the way that makes sense to Americans today, and the "simple" method preferred by Lehrer is very confusing for anyone who's not used to it: https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA Here's more of the intro which makes it clear what's going on. My explanations in brackets: > Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the “New Math”. > So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we’re gonna cover subtraction. [ New Math is confusing, and he's going to go over how confusing it is in a humorous way.] > Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that… > Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169. [This is how it was done before the confusing New Math, the "right" way to do it. "Three from two is nine, carry the one."] > 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… [The New Math approach is silly because it's so focused on trying to teach kids concepts that it leaves them unable to do basic arithmetic.] After this comes the first verse, which is designed to make the idea of "borrowing" sound so overly complex that few people would be able to understand it.
  - 082349872349872 2y ago
    The "right" way to make borrowing complex is to teach it as a special case of cocycles in group cohomology: http://timothychow.net/mathstuff/jdolan.pdf http://timothychow.net/mathstuff/jdolan.pdf
andrepd 2y ago
Which is doubly stupid given that teaching algorithms for arithmetic is a useless skill and has been for 40 years. Just focus on the ideas, maths is beautiful, pen and paper accounting is not.
thaumasiotes 2y ago
> 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.
deleted 2y ago
[deleted]
t-3 2y ago
The only thing I found strange about it was that he started from the least significant digits rather than the most. It's basically the way I've always done subtraction, just backwards.
- chmod775 2y ago
  If you do it the other way around you may have to backtrack if subtracting a less significant digit from another worked out to <0. You will also have to be careful about alignment, since in one number you start from the first digit, but in the other you may start from the middle. It's slightly more complicated than just starting from the least significant digit.
chmod775 2y ago
As someone who learned the "right" way in German public school about 20 years ago, the statement "teaching people what was actually happening with subtraction" is completely indecipherable to me. Maybe I never learned what subtraction is. To me it's like advocating for not using a ruler because that would somehow forego teaching people what measuring actually means. Both "measure" and "subtract" can mean doing any number of things in mathematics, depending on what you're dealing with. Intuition for things in the physical world specifically shouldn't be a goal of teaching mathematics at school. It's a prerequisite. If a child already can't grasp the concept of taking away from something, throwing mathematics and numbers at them is not going to help one bit.

RheingoldRiver 2y ago

Wait is he actually criticizing it in this song? I always thought that he was mocking people who didn't understand borrowing by sounding disbelieving about it, when it's in fact obvious how & why it works

123894893 2y ago
Yes, he's mocking the idea of teaching borrowing. He gives the way he thinks subtraction should be taught in the intro. It's a way that, as far as I can tell, hasn't been taught in America in over half a century. After the intro, he goes on to mock the concept of borrowing, which was (apparently) part of New Math. Then in the final verse he mocks the idea of teaching bases. Here's a good animated version of the song that shows the different methods. You'll notice that the "silly" New Math method is the way that makes sense to Americans today, and the "simple" method preferred by Lehrer is very confusing for anyone who's not used to it: https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA Here's more of the intro which makes it clear what's going on. My explanations in brackets: > Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the “New Math”. > So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we’re gonna cover subtraction. [ New Math is confusing, and he's going to go over how confusing it is in a humorous way.] > Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that… > Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169. [This is how it was done before the confusing New Math, the "right" way to do it. "Three from two is nine, carry the one."] > 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… [The New Math approach is silly because it's so focused on trying to teach kids concepts that it leaves them unable to do basic arithmetic.] After this comes the first verse, which is designed to make the idea of "borrowing" sound so overly complex that few people would be able to understand it.
- 082349872349872 2y ago
  The "right" way to make borrowing complex is to teach it as a special case of cocycles in group cohomology: http://timothychow.net/mathstuff/jdolan.pdf http://timothychow.net/mathstuff/jdolan.pdf

andrepd 2y ago

Which is doubly stupid given that teaching algorithms for arithmetic is a useless skill and has been for 40 years. Just focus on the ideas, maths is beautiful, pen and paper accounting is not.

thaumasiotes 2y ago

> 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.

deleted 2y ago

[deleted]

t-3 2y ago

The only thing I found strange about it was that he started from the least significant digits rather than the most. It's basically the way I've always done subtraction, just backwards.

chmod775 2y ago
If you do it the other way around you may have to backtrack if subtracting a less significant digit from another worked out to <0. You will also have to be careful about alignment, since in one number you start from the first digit, but in the other you may start from the middle. It's slightly more complicated than just starting from the least significant digit.

chmod775 2y ago

As someone who learned the "right" way in German public school about 20 years ago, the statement "teaching people what was actually happening with subtraction" is completely indecipherable to me. Maybe I never learned what subtraction is. To me it's like advocating for not using a ruler because that would somehow forego teaching people what measuring actually means. Both "measure" and "subtract" can mean doing any number of things in mathematics, depending on what you're dealing with. Intuition for things in the physical world specifically shouldn't be a goal of teaching mathematics at school. It's a prerequisite. If a child already can't grasp the concept of taking away from something, throwing mathematics and numbers at them is not going to help one bit.