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> I'd be curious to hear takes on what would make you think Astra is yet to be AGI

Give someone 10 remote employees for a few months, 5 of them human, 5 of them AI. After a few months, check to see if the humans (manager, other coworkers) can figure out who is AI and who isn't.

Would that be sufficient? I'd have to think about it. But AGI is supposed have human level capabilities, so this would be a necessary prerequisite.

None of the models are anywhere close to this.

> 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.

> 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.

> 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 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.

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 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.