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.
Overthinking my teaching 
1. I love this for so many reasons: My parents had conversations like this with me growing up, and I now love having conversations of this type (thought not always of this caliber) with my own kids.
2. Solid classroom management move, using the Jedi Mind Trick with Griffin: THIS IS NOT THE CONVERSATION YOU’RE LOOKING FOR, followed by pushing him out of frame.This is a good reminder that guided inquiry works best when you’re pretty sure nobody in the room is going to take away another kid’s intellectual need.
3. Exceptionally well executed. Your little girl is going to love math for all the right reasons.
Hello,
Thank you for your posts, it has been very informative to watch your children grow in their understanding of mathematical concepts.
It popped into my head that my answer to your question to Tabitha “You think I have this whole thing memorized?” would be yes. Didn’t you memorize multiplication tables?
What do you think?
Dave
This is a key moment in the interaction, isn’t it Dave? Here is how I see it. You may disagree.
Remember that—at this moment in the discussion—Tabitha’s working hypothesis (which I have indulged) is that this is an addition machine that gives wrong answers. I ask the question in that spirit. You think I have this whole thing memorized? is not about the multiplication table, it’s about an idiosyncratic inaccurate device. I am hoping that the proposition that I would have memorized the wrong answers on this machine will seem preposterous to her. I am hoping that the fact that I know these wrong answers (which, don’t forget, she has drawn my attention to) will suggest to her that there is something else going on, and that she will end up curious about what that is.
So yes, I have memorized my multiplication tables (although there are certain facts I still use the distributive or associative properties for, such as all the 12’s and a fair number of the 9’s). But this wasn’t yet a conversation about multiplication; it was about wrong answers to addition problems.
Another subtle point (and forgive the venture into grad school epistemology here) is that I do not have this whole thing memorized. I have something else memorized, which the makers of this device have presumably faithfully recorded on it. I think that subtle distinction matters. It’s not the means by which we deliver the knowledge that matter; it’s the knowledge. Khan Academy energy points (to pick an easy example) are meaningless on their own; instead, they are presumed to motivate students to learn something important. Far too often, assessment focuses on the delivery method (did you do your homework? did you complete your worksheet? et cetera) rather than on what students know.
Thanks for asking!
Ah, I understand. You were stoking her curiosity about the “broken adding machine” so that she would be interested in learning what is going on with it. I’m curious about your thought process in concocting this evil scheme…did it happen when you realized she thought it calculated sums? (the beginning)…or some other time? How easy is it for you to diagnose like this in class (with more people and possibly less one on one)?
I would like to improve my diagnostic skills for students, I don’t think very quickly on my feet and sometimes my questioning ends up taking wrong turns. Do you have generic “question pathways” for some topics or common errors? Or are you looking for some key words and phrases?
I also agree with your sentiments about the distinction, many students don’t see this and focus on their “energy points” or “test points” rather than learning.
Thank you again, (sorry it took so long to respond, I didn’t get a notification that you had answered.)
Dave
Pingback: The sequence machine | Overthinking my teaching