Math Doodles is an app for iPad and iPhone. It is a ton of fun.

I was initially nonplussed but playing with Tabitha turned me around in a hurry.

Tabitha was playing the *Connect Sums* game by herself and guessing haphazardly. In this game, the player is shown a 4-by-4 grid of numbers, and is offered a target number. The goal is to click on numbers in the grid until you reach the target number. Then the numbers you used disappear and a new target is offered. Et cetera.

Haphazard guessing didn’t seem to be a particularly valuable activity for Tabitha, so I sat down with her. Our first target number was 12.

Me: What do you know about twelve?

Tabitha(five): That it has a 1 and a 2.

Me: Yes. Why does it have a 1?

T: Because when you count to twelve and write it, there’s a 1.

Me: I see. Actually it’s because twelve is made of 10 and 2. So if you can make 10, you can make 12.

T: Ten is two 5s. So 5-5-2.

Our next target number was 8.

Me: What do you know about 8?

T: That it’s two circles; one on top of the other.

Me: Yes. But what do you know about how much eight is?

T: I don’t know.

Me: Do you know a number that eight is more than, or less than?

T: It’s more than 6, but less than 10.

Me: Yes. How much less than 10?

T: Two less.

Me: Nice. So a minute ago, you did ten as 5 and 5. Now we need two less than that. Do you know what’s 2 less than 5?

T: Three. So 5 and 3.

Our next target number was 9.

Me: What do you know about 9?

T: One more than 8. So 5-3-1.

Our final target number in this round was 7.

T: That’s one less than 8.

Me: Yeah. But we don’t have a 5 this time.

T: Uh huh! Two and three!

Me: Oh nice. Two and three to get five. Then two more.

There was a whole lot of learning going on there. Of particular interest to me was her shift from numeration (how we write a number) to quantity (how much a number represents). In our first two rounds, she told me what the numbers looked like; she was telling me how to write numbers. In the third round, she told me about quantity when I asked. And in the last round, she knew to start with quantity.

Playing the game a few days later, when she was a bit less patient, we got stuck on the target number 11 on a board that had only 4s and 3s remaining. I couldn’t see how to use what she already knew to build towards 11 as 4-4-3.

I can, however, see that various decompositions of 10 are really useful, and that halves would be as well.

We’ll have to talk about those things.

I’ll let you know how it goes.