> 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.
> 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").