The multiplication machine

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:

1. She thought it was supposed to give sums, and
2. 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.

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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.

Kindergarten addition strategies

A few months back, Tabitha (5) wanted me to ask her some math class questions. This led to some disagreement between the two of us on what constituted a math class question.

On our way back from our fall camping trip last weekend, Tabitha announced that she knows what 8+8 is.

Me: Ok. What is 8+8?

Tabitha: [longish thinking pause, eyes gazing up and to the right] 16!

Me: Wow. I could really see you thinking that through. How did you know that?

Tabitha: I counted.

Me: OK. Good. But how did you keep track of how much to count?

Tabitha: huh?

Me: How did you know when to stop counting?

Tabitha: When I got to 8. That was 16.

Me: Right. But how did you know when you had counted eight times?

Tabitha: I just counted to 8. That was 16.

Me: Oh! I know! Was it like this? Eight…nine (that’s one), ten (that’s two)…like that until you got to sixteen (that’s eight)?

Tabitha: Yes.

Me: How did you know to do that? Did you learn to do that in Kindergarten? Or was it your own idea?

Tabitha: [long pause] OK. I admit it; it was my idea.

I have written before about how sticking with this stuff pays dividends eventually. Tabitha announced that she knows what 8+8 is, but what she really meant was that she knows how to find 8+8. Tabitha knows that talking math is not about telling answers, but about actively thinking things through. That doesn’t come for free in classrooms, and it doesn’t come for free at home. It is a result of lots of prep work and lots of encouragement and lots of conversations.

Returning to the theme of the knowledge required for me to have this conversation with Tabitha, though, I needed to know something about possible strategies, and I needed to have thought some of those strategies through for myself-by trying them out.

If the sum in question had been 8+2, I wouldn’t have questioned her claim that “I counted”. You can subitize two, so you can know that you have counted twice without having a separate system for keeping track.

But counting 8 times without having a way to ensure that you have done so? This seems pretty much impossible. As an adult, I know to stop at 16, but that’s only because I already know the sum. As I count 9, 10, 11, 12, I am conscious that the only way I know I have counted four times is because I know that 8+4 is 12. I’m not able to keep track of the number of counts past three.

I had watched her fingers; no motion there.

So there had to be something else going on. I was only able to press because I knew what she knew (how to count) and what she didn’t know (addition facts) and some possible ways that this knowledge could be used to find the sum of 8 and 8.

The more I talk with these kids, the more impressed I am with the work of CGI. Much of what I know in this area is the result of studying their work.

UPDATE: The original CGI link in that last paragraph went to a tag search of CGI on my blog. That was confusing. It has been replaced by a particular post. Here’s the original link. And here’s a link to CGI on Wikipedia.

Compare and contrast

Digging deep in the comments to bring out some interesting observations.

Chris Hunter reports the two following interactions with his 7-year old daughter:

First

My daughter comes home from school and tells me she hates math. She completed 13 questions on “The Mad Minute” and compares herself to her best friend who completed all 30.

Second

Bedtime conversation:

Daughter: Dad, did you know that in some countries girls aren’t allowed to go to school? They won’t even get to know what 9 plus 9 is.

Me: Do you know 9 plus 9?

D: No, but I know 10 plus 10 is 20. We do that one a lot.

Me: Okay, so what’s 9 plus 9?

D: 18.

Me: How’d you get that?

D: I counted two down. They make JUST THE GIRLS stay home and do chores ALL DAY.

And Steve Prosser reports a friend’s progress:

third

[She is a] very bright girl, who because she was home schooled never did much math. Finger-counting still in fifth grade. Hated math. Believed she was terrible at it. After just over 1,000 flashes, she “GOT IT”. She was not bad at math, but had never done enough to internalize the basics. Now she loves math (her favorite subject), and is beginning to excel at it.

Follow up on that Tabitha post

Yesterday’s post recounted a conversation with Tabitha (5) in which she asked for a “math class” problem.

I focused the initial discussion on where she learned what constitutes a “math class” problem.

But there’s lots more in there that’s interesting.

It wasn’t just her affective response (rejecting the driving in a car context, asking for a naked number problem) that matters here. Notice the way she engaged with the two problem types.

When I relented and posed the question, What is two plus three? she guessed. I know that she guessed because she (a) took no time to process, (b) asked rather than told me her answer, and (c) it was wrong despite being within her grasp.

When I posed the exact same problem in a situation she could imagine (she and Griffy had been to the arcade just that day), she engaged quite differently. Her body position changed. She paused. Her fingers moved. Each of these is an indication that she was thinking. And when she had an answer, she stated it; she did not ask.

The central tenet of CGI and an important belief underlying the IMAP work is that children can use contexts to solve problems that they cannot solve abstractly. Here it is in action. 2+3 is meaningless to Tabitha right now. But her 2 tickets combined with Griffin’s 3 tickets? That’s got meaning.

The conclusion here is obvious, right? We start with contexts kids understand and can reason about (here, combining tickets). We move to the abstract mathematical representations (2+3). We don’t save arcade tickets for after the kid understands addition. We don’t wait for symbolic mastery before doing some applications.