Two views of addition, two weeks apart

Now that I'm watching my kids learn skills like addition and reading which go above and beyond the obvious natural language acquisition, it's still amazing to watch the process of acquisition, and to see how suddenly things click so that what seemed impossible becomes totally natural in a matter of just weeks.

Since a colleague of mine who used to be an elementary school teacher mentioned to me that "counting on" was something that took kids a while to get, I've been paying attention. Sure enough, though Grace can happily count numbers and add them, she's spent quite a long time doing addition by putting two groups together and then counting all the items, rather than simply starting counting from one set and then up to the other. It's kind of fascinating how stubborn she is in this, and how much math she can seem to know when she still can't get the basic idea that if you have five things and you add four you only have to count up four from five.

Anyway, here's two facebook posts (the new journal? ack?) of mine documenting the shift -- because they have dates, I can actually see how little time it took to make the transition from absolutely not counting on to doing it like it's nothing.

From March 7th:

Watched Grace add 8 and 5 today to get 13. I noticed she did it all on the fingers of one hand and seemed to be counting up to thirteen. I asked her how she did it, and she explained she did 4 + 4 +5, because she somehow already knew 8 was 4 + 4.

Doesn’t it seem kind of weird that she can break that problem down that way in her head all to avoid using two hands to count (I think maybe she was holding something with the other hand?), but still can’t simply count on (i.e. start at 8 and then count 5 more up to 13)?

From March 22nd:

Grace magically started counting on today!

(in the context of a card game where this matters...) Me: Grace, what's 4 + 7
Grace: Do you mean 7 + 4?
Me: Sure -- that's the same thing.
Grace: Okay... [counting to four on one hand], 8,9,10,11... 11!
Me: Grace, who taught you to do that?
Grace: No one, I just do it.
Me: Huh.


So I can tell you with reasonable certainty that 2 weeks ago Grace couldn't count on when I tried to coach her to do it and looked at me in total confusion when I suggested it as an approach to handling numbers that didn't fit on her fingers. Now she's doing it like it's the most obvious thing in the world, with no awareness of having learned it.

The question, as ever: did she "learn" this from the inadvertent teaching I've been doing as I've been trying to understand how she does math in her head, or did something in her development just click today so she was ready to do it now?

I realize the answer is almost certainly a little of both, but I certainly lean toward the development side of this question. It amazes me how many skills seem absolutely unlearnable until some magical threshold is passed and they're learned seemingly effortlessly...