Thanks for posting that Robyn!It's interesting how rhymes travel!!

I learned "Who Stole The Cookie From The Cookie Jar" when I was growing up {I don't remember how]. But I'm almost positive we didn't use the introductory phrase "One two three and a zing zing zing". However, that phrase changed around to "Zing Zing Zing And Ah One Two Three" is very widely used for a number of handclap rhymes.

I've only experienced "Who Stole The Cookie..." played as a sit down hand game, though I've read that some people have played it standing up. In my childhood I remember being seated in a circle, and clapping our own hands to one beat, then hitting our knees, and then clapping our hands again. This clap-slap-clap-slap beat is maintained throughout the entire chant. The tune to this song isn't very uptempo.

In my childhood the children said numbers in sequential order {so maybe this was a school thing}. But when I taught it to the Pittsburgh kids a few years ago, I used the version a Black Pittsburgh woman had shared with me. In that version, before the chant begins, every child is given a specific number. The game always starts with a leader voicing the line for "number 1". After number 1 recites that game's standard line, he or she picks another number at random. That person must say that standard line etc. and then must pick another number at random. Because they never know when their number is going to be called, each girl and boy must be alert to the possibility of her or his number being called, and must respond accordingly. * Technically, the game is supposed to end after every number is called.

* It's my opinion that some recreational games have [or had] a purpose beyond mere fun. I think that "Who Stole the Cookie Jar etc" may have been a means of reinforcing among children the need to remain alert & and aware, to think quickly, and respond appropriately. In similar fashion, I think that "let me see your motion" ring [circle] games where one child is selected at random to be in the center of the circle] also taught {teaches} children to be alert & aware, and think fast and act appropriately. In those circle games, a child is suppose to come up with a motion {usually nowadays a social dance move}. If someone before you performs the move you had thought to do, you're suppose to think fast and come up with another one {or better yet, in anticipation of the fact that you may be called to go in the 'middle' of the circle, and someone might have 'taken' the dance step you were going to do, you should have another dance step in mind as a 'back up plan'. Training and exercising their minds in such a way was [is] important for children anywhere, but perhaps even more important for children living in a dangerous, oppressive society {such as the mid 19th century American South, not to mention 21th century American South, East, North, and West.

In my neck of the woods, it seems to me that "Who Stole The Cookie.." is not a child initiated game anymore {if it ever was}. It appears to me that this game is most often taught to children by adults, and that adults lead children in playing this game. And even then, that's doesn't happen very often. Around 2003 or so I introduced "Who Stole The Cookie..." to a group of African American girls and boys-ages 5-12 years old. It was clear that they had never heard of the game before. Maybe it was because the group was too large {20 children or so}, but truth be told, the group quickly got bored with that repetitious chant, so long before all the numbers were called I switched up and moved to another game.

Here's the Pittsburgh, PA version of "Who Stole The Cookie..."

WHO STOLE THE COOKIE FROM THE COOKIE JAR
Group-    Who stole the cookie from the cook-ie jar
[leader]   Number 1 stole the cookie from the cook-ie jar
#1         Who me? Couldn't be
Others    Then who stole the cookie from the
          cook ie jar
#1         [Picks a number at random]
          Number 8 stole the cookie from the cook-ie jar
#8         Who me? Couldn't be
Others    Then who stole the cookie from the
          cook ie jar
#8         [Picks a number at random]
          Number 3 stole the cookie from the cook-ie jar
{etc etc etc}   

Btw-I should also confess that I changed the words to "who took the cookie from the cookie jar" as I didn't want to promote stealing...

Maybe the word 'stealing' was an American substitution. I have a vague memory of reading a Caribbean version of this rhyme that said something like "Someone took my cookie, was it you?" and a similar pattern continued from there...