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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.

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

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

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.

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

> 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?

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

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.

> 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.

> 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.

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.

> 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.

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/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.)

> 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.

> 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.

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.

> 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".

Maybe they used "borrow" in the "new" method to avoid having both 2-3=9 and 2-3=-1, compared to explicit radix+2-3. But if you actually wanted to memorize subtraction table then "old" way is maybe easier, because your table is nice square grid instead of wider triangle (and if you actually need negative result you can do second lookup for 10-x).

Also try doing something like 2000-1111 in "new" method and you go on huge side quest to propagate the borrows and go back to the beginning. Compared to "old" method where you progress one digit at a time without backtracking.

> 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".

The ten's compliment of 3 is 7.

> 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".

12 - 3 = 9 is quite different from 2 - 3 = 9 carry the one. The latter requires a separate explanation for what's actually happening.

> But they got better marks than the new math students, because they weren't graded on whether they understood.

People seem to do subtraction just fine with borrowing, and I've never heard anyone claim that the old method is superior outside of Lehrers song.

> As I just said, Lehrer knew that people couldn't understand it either way.

This is clearly false, though. Most people today understand borrowing just fine, while (at least according to Lehrer's song) people who studied the old approach had so little understanding of what was happening that they couldn't even grasp the concept of borrowing. If you look at what's actually being said, all of the stuff in the first verse that Lehrer is presenting as mindlessly complex for adults is completely intuitive to anyone with a decent grasp of modern elementary school math:

"You can't take three from two Two is less than three So you look at the four in the tens place Now that's really four tens So you make it three tens Regroup, and you change a ten to ten ones And you add 'em to the two and get twelve And you take away three, that's nine Is that clear?"

The sarcastic "is that clear?" is there to show how confusing this is. But it's actually quite clear for people with a modern education. The problem is 342 - 173. You don't do 2 - 3 ("You can't take three from two, Two is less than three"), so you borrow a ten from the 40, changing it to a 30 and the 2 to a 12 ("So you look at the four in the tens place, Now that's really four tens, So you make it three tens, Regroup, and you change a ten to ten ones, And you add 'em to the two and get twelve").

> The ten's compliment of 3 is 7.

Not a good look for someone extolling the benefits of understanding the theory behind an algorithm. This is only true if you're working modulo 10.

I'm suddenly very curious what you think the ten's complement of 12 is.

> 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)?

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.

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.

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.

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.

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.