> 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").
> 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".