My goal with most *Talking Math with Your Kids* posts is to demonstrate how easy it is to do this at home.

That is not the point of this post.

I do not pretend that the average parent is ready for the conversation I document here. This is graduate level parental math talk. I’m just showing off here. Sorry about that.

But I do hope what comes across is the pure joy of a child exercising her mind. The video runs five minutes. I promise it’s worth your time. Transcript with commentary follow. [Link to video]

Here’s the set up.

We have this multiplication machine. It’s a multiplication table with spring-loaded buttons. You push the button and it pops up; you read the product off the side of the button.

Tabitha was playing around with it one day and declared that it had a bunch of wrong answers. She showed me what she was talking about and I quickly diagnosed two things:

- She thought it was supposed to give sums, and
- We were gonna have a good old time with this.

I set up my iPad to record the conversation and away we went.

Tabitha(5 years old): If Griffy used this, he must have gotten his homework a little bit wrong.Don’t you think so?

Cause it’s lying.

Isn’t it?

I set the stage for our conversation by recapping its beginning. I chose my language carefully. I didn’t want to lie to the kid, so I spoke of “6 and 1” not “6 plus 1”.

Me: OK. So we’ve established that 6 and 1 gives us 6.And we expected 7 and 1 to give us 8 but it didn’t; it gave us 7.

What do you think 8 and 1 is gonna give us?

T: Eight.

Me: And what should it give us?

T: Eight. Because that’s the right answer.

Wait. Has she picked up on what’s going on so quickly?

Me: Wait.

T: There’s seven and then one would be eight.

Got it. She’s still on 7×1. Let’s reset.

Me: Right. But what about when you do 8 and 1? What should that be?

T: Nine.

Me: And uh…What do you think it’s…What do you think this is gonna tell us, based on what we saw from the other ones?

Do you think it’s gonna say 9? It got all these other ones wrong. You think it’s gonna say 9 here?

T: No.

Me: What do you think it is gonna say?

T: Eight.

Now we’re on the same page. She thinks it’s supposed to give sums, so she expects 9, but with some support she is able to apply the pattern that the machine is giving us one less than the right answer.

Me: Why is that? Why do you think that?

T: Because one less than than the one over here. This [7×1] was seven, so I think it’s gonna say 8.Aw god! This is so…

Me: What’s this one [9×1] gonna say?

T: Nine.

Me: What should it say?

T: Ten.

Sure enough.

T:Awww! This thing is a liar!

Now it’s time to change things up. We have worked our way down the left-hand column. So she has done 7+1, 8+1, 9+1. She has *one more than nine* nailed down. Can she as easily do *nine more than one?* Does she know that addition is commutative?

Me: All right. What about this one [1×2]?

T: It’s 2.Oh wait. Oh yeah, it should be…

Me: What about this one [1×5]? What should it be?

T: Six.

She is banging these out as quickly as she did the others. She seems to be using the commutative property of addition. Dig it.

Me: But?

T: It’s five.This thing’s a liar!

Me: All right…what about…

T: This thing is a total liar.

Me: What about this one [2×2]?

T: It has to be four.

I have no idea what she means by this. She clicks before I can ask.

T: Some of these are right.

Her brother, Griffin (8 years old) wanders into the room. The jig is up, I fear, so I chase him off.

Griffy check out this one.

Check out this one. It should be 6 but look; it’s 5.

This thing is a total liar.

Me: All right, except this one it got right.

T: Yeah, but what about the other ones?

Me: What about this one [2×3]?

T: Five!It’s six. But it’s supposed to be five.

I don’t think she noticed that this wrong answer is one *more *than the one she expected, while the previous ones had been one *less.*

Me: Interesting.What about this one [2×4]?

T: Six!

Me: What do you think it’s gonna say? Wait wait. It should be 6. What do you think it’s gonna be instead?

T: Five.

Of course it’s not five. It’s eight.

T: What? That’s totally not right!

Me: OK. Umm…Let’s try a hard one.What should…

What should this one be [2×6]?

T: Eight.HEY!

Is this [6×9] [unintelligible]?

Thirteen!

She clicks. It’s not thirteen; not even close. But she’s not really sure about 6+9 anyway.

T: What was it supposed to be?

Me: It was supposed to be 15.

T: WAAAA!

Me: Fifty-four?

I have to think quickly. How can I ramp this up?

It occurs to me that she may think these are arbitrary wrong answers; that this machine is some sort of random bad answer generator. I want to pique her curiosity by demonstrating that there is *something else going on*.

So I propose working backwards.

Me:Ooo. OK. Let’s play…are you ready for this?You choose one. Choose a hard one from down in here.

You press it.

Tell me what it says; not what the problem is, but what the answer is and I’ll see if I can tell you what the problem was.

She chooses a problem. Out of the corner of my eye, I can see that she is focused somewhere near the bottom of the machine.

T: Thirty-six.

Me: Thirty-six. So I think you did 9 and 4.

T: Yes.Is that right?

Me: Nine plus four is not 36.

By this time, Griffin has wandered back into the room.

Griffin:Nine plus four is 13.

Me: So how…so how did I know? How did I know what it was?If 9 plus 4 is not 36, how could I have gotten that right?

T: Because you’ve seen it before.

A reasonable explanation.

Me: You think I have this whole thing memorized?

T: Yeah.

Me: Here, do one more.Do a really hard one.

What did you get?

T: Uh…I don’t know what…

G: [whisper] Sixty-four

T: Sixty-four.

Me: Sixty-four. I think you did 8 and 8.

T: Yes. How did you know?Is that right?

Me: It’s not what 8 plus 8 is.But here I’ll tell you…

The iPad ran out of memory at this point.

But that’s OK. I told Tabitha about the difference between a multiplication symbol and the addition symbol. I told her that multiplication is about groups. So 2×4 means *two groups of four*. We talked about how many things would be in two groups of four.

And then the moment passed and we were on to other things.