The Map is Not the Territory

My kid LOVES the vegetable pakora appetizer at a local Indian restaurant. Yesterday she had some serious dental work done and I sweetened the stressful day with an offer of pakoras for dinner.

Our plan was to get one serving of pakoras to eat there and one to take home to share with the papa. We talked on the way to the restaurant that we needed to find a way to split the two orders evenly between three people.

Our in-house plate of pakoras came first; there were eight of them.

"So," I said, "How many should we start with?"

9yo: Two for me and two for you.

[Eating commences. Yum!]



Me: So we have four left. What should we do now? If we take one more each, there'll be two left on the plate and...Oh wait, I just realized we don't know how many pakoras there will be in the take out box!

9yo: Probably eight.

Me: So how could we split 16 Pakoras evenly between three people?

9yo: [Turning slightly and looking to her left for a couple seconds] Five and one third.

Me: I wonder how you got that?

9yo: Well, 5 times 3 is 15 and then you split the last one into three pieces.

Me: But what if there are only 6 pakoras in the take out box? How will we split 14 pakoras so it will be fair for all three of us?

9yo: Well, 4 times 3 is 12...

Me: But that leaves two. How would you share those last two pakoras between three people?

9yo: Well you could cut them in half and each person gets a half.

Me: What would you do with the last half?

9yo: Give it to me? <sly grin>

Me: Well...but what could you do to share that last half fairly?

9yo: Hmmm. Cut it into thirds?

Me: Cut the last half into thirds? That would make what, exactly?

9yo: Um....really tiny pieces?

Me: It would make sixths!

[Take out order arrives!]

Me: Let's see how many are in this box! There are EIGHT!  Great, what do we do now? We've each had three, let's take two more each ... Okay, now Papa has his five in the box, and there is one left. Wanna cut it into thirds?

9yo: Sure! ... well they're not really equal ... I'll take the biggest piece!
............................

I love how we ended up talking about two ways to share between three people. First, many pieces into three shares. Second, only two pieces into much smaller shares. Both fractions, but of slightly different natures I think. In my mind "fractions" refer to really small pieces. But that's obviously not the case with my share of five whole pakoras!

During the entire conversation I felt really proud of myself! Helped by projects such as Talking Math with Your Kids and also a recent video showing two boys figuring out how to share sausages, the whole pakora conversation just flowed. Good modeling is definitely the key to learning how to 'talk' math.

And, as we were waiting for our check we also got a bonus #dswyk (Doing Science with Your Kids)!

Me: What do you think is happening there? 
9yo: There's a rainbow on my hand! Lol.

You know that alphabet game you play on long car trips? You can play it a bunch of ways -- look for a word on a sign that starts with A, then B, etc. or just look out the window and look for things whose names start with A, B, C...

Today my kid and I were driving to the Indianapolis Zoo. It takes over an hour to get there. Near the end she suggested playing the sign version of the alphabet game and I said, "What if we did a math version?!"

She was all for it. Right away she said, "I see angles!"  I don't remember everything but here are some highlights:

[C] Congruent shapes: I found this one and a nice way to introduce that language.

[C] Curves: This was hers. She saw curves in the telephone wires.

[D] Degrees: I saw a 90 degree angle in a roof

[E] Equal: She used this word in relation to the shapes she saw. I mentioned the word congruent again in relation to this idea.

[H] Height: Some trees were higher than others.

[I] Interval: The telephone pole were spaced in equal intervals. In this game I don't think it matters if the adult ends up identifying more things than the child.  It's engaging for everyone to be looking out the window and a great opportunity for talking math with your kids in a low-pressure, (hopefully) highly enjoyable context.

[P] Parallel lines

You get the picture, right? Math talk can be about numbers but also about relationships (higher, smaller, faster) and other ways we describe our physical world.  Next time it would be fun to look for [A] Arrays, [M] Multiples, [N] Numbers or even the math version of "Are we there yet?"

[Z] Zeno's Paradox...you can't get there from here.
"

A riddle, by my newly 9 year old daughter, presented in 20 Questions style, at 7:00am in the morning.

Her: Mama! I have a great riddle!

Me: I'm trying to read the paper, I'll be there in a minute...

Her: But this is SO great.

Me: Okay...

Her: I'm thinking of something that is whole, but not whole.

Here is what my guessing revealed:

1. Not alive
2. Not an object, but can be (her emphasis)
3. Made of many parts
4. Parts look similar to each other
5. Not a reflection
6. It's in our house and other houses too.
7. We use it every day.

Her: Here's a hint, it relates to math.

8. It's an idea.
9. We use this idea.
10. The idea helps us do stuff around the house
11. Idea helps us do housework.
12. A specific kind of housework? Sometimes yes and sometimes no. Maybe.
13. Helps us clean.
14. Helps us clean inside and outside.
15. Helps mama think about our day

Me: Is it the calendar? [And, because I thought we had gotten to 20 questions I continued] Okay, just tell me!!

Her: IT'S A WHOLE FRACTION!!!  Like when you cut carrots into sixteenths [we have never ever counted carrot pieces]. Look...[she takes my pen and writes something like this]:

1 = 1/2

Me: But one isn't equal to one half, is it?

Her: No, but it's a whole half. It's a whole fraction but still a fraction of something.

Me: So you mean the 1 is whole, but when you divide in half, the 1/2 is a new whole?

Her: YES!!!  Can I watch Frozen?

I think this calls for a celebratory viewing of this wonderful video:

Some logic and equivalence #tmwyk* this morning:

8yo (trying to get me to do something for her): Please plus please!
Me: What does that mean?
8yo: It means yes.
Me: So, what does a thank you + thank you mean?
8yo: That means yes, too.
Me: How about a no thank you plus a thank you?
8yo: No.
Me: And a no thank you plus a no thank you?
8yo: Maybe.
Me: How does that work?
8yo: Because there are two no's and two thank you's.
Me: You mean there's an equal number of both?
8yo: Yeah.

A little later, about our cats Max and Lucy:

8yo: Max + Lucy = Trouble
Me: That's for sure.
8yo: Actually, cat + cat = Trouble.
Me: So, Max + Lucy = Cat + Cat = Trouble!
8yo: NO!
Me: Why not? If cat + cat is trouble, and so are Max + Lucy, then don't Max + Lucy = cat + cat? They both equal trouble...

She had reached her limit with my questions by then, but I found the whole series of ideas rather fascinating.  First because this was the first time I've had this particular kind of conversation with her. It seems Christopher Danielson's Talking Math with Your Kids is really having a positive influence on my conversational abilities! Second because the whole idea of equivalence in equation form seems to flummox kids. The equal sign often seems to mean "an answer" and not a relationship. It was fun to test/push her a little on this idea and see how it would play out in conversation.

*#tmywk is the hashtag for Talking Math with Your Kids on Twitter. You can also go to Talking Math with Your Kids.  You'll be glad you did!!

Math related conversations with my 8 year old tend to pop up unexpectedly. These are often around something she's making, and are often a gorgeous little gem of a surprise. Today was no different.

I found her creating a grid of tape on top of a piece of origami paper. She was coloring it in when she said, "You know, mama, patterns don't necessarily have to be colors in order."

Oh my gosh!

Me: What do you mean by that?

As you can see from the picture, below, she was coloring the whole page somewhat randomly, sometimes following the columns down, sometimes not. 


Her: Well, the colors don't have to be regular, they just need to be in the windows."

Me: You mean the windows are the places where the tape is not covering the paper, that's what you're looking at? It doesn't matter what color windows are, just that they're colored in?

Her: Yeah.

So here I pause to do a happy dance. My biggest discontent around patterns is that many kids grow into adults who think that "patterns" are only linear repetitions of colors. It is clear she has not internalized that particular reality.  The other reason I'm happy is that Prof. Triangleman once said to me:

"Math is when you say exactly what it is you want to pay attention to, focus only on that attribute and ignore everything else."

She was doing this!  Her pattern has 'windows' that are colored in, but the pattern is not defined by the colors themselves. She was consciously creating a pattern of windows (spaces between!) and consciously excluding the colors. I am thrilled to have caught a glimpse of this multi-layered attention in action.

This really goes to show you that it's worthwhile to keep your ears open while kid is focused on making or building something.  Even if it's after the fact, ask your kids to tell you about what they did, even if it doesn't look like much.  I mean, just look at the taped/colored piece again. It's pretty much a bunch of scribbles and it'd be super easy to pass it over, to think it was nothing special. In reality, though, there was so much thinking going on while she worked.

Here's what the piece looked like when she took the tape off:
Her: Oh.

Me: You seem surprised. Did that not turn out the way you expected?

Her: Why do you sound like a journalist?

I had a very interesting conversation with my eight year old over math homework this morning revolving around the commutative property.  Interesting because of her thoughts and also interesting that, for the very first time, I backed away from mathematical correctness, and truly listened to what she had to say. It was fascinating.

The homework asked for factors of various two digit numbers. For 24, my kid put 2x12, 1x24 & 24x1.

I said, "Those last two are the same thing, what other factors can you figure out?"

The response was immediate and somewhat intense. She was convinced that 1x24 and 24x1 were different because that is what the teachers said.

I mentioned we had read about the commutative property in Beast Academy 3B but, sweetly, nothing could sway her loyalty to her teachers and her opinion about what she thought they had taught her.

It was at this point I thought back to all the things Christopher Danielson has written about Cognitively Guided Instruction and the wonderful modeling of his Talking Math with Your Kids project. These approaches show the worth of conversation around math with an emphasis on the adult really listening to what the child is thinking.

She continued. "See! One times twenty four is [pointedly counting] one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve...twenty four.  Twenty four times one is...[pausing, then saying emphatically] twenty four."

I nodded. "Oh, I see what you're thinking. The first way means you have to count by ones 24 times. The second way you just have to say 24 once."

In her mind it's the process of getting to the final answer that makes the two facts different. Never mind that she gets the same answer both ways. Never mind that she knows all about the "twin facts" on multiplication chart. Never mind that we're having fun finding different ways to memorize multiplication facts including sneaky guerrilla tactics. Nope. This is her reality and it's not going to budge by quoting official definitions.

All I said was,"You can put those two facts on the paper, but your teachers may want you to put some others as well."  In the end she found all the factors of 24, but wrote each combination twice (e.g. 6x4 and 4x6).

In the process of writing my new book, tentatively titled Meaning in the Making: The Body Learning Math, I've been doing a lot of reading and thinking about how the processes of doing and learning math are just as important as the product.  In this case, she can easily figure out factors of two digit numbers, but it's by watching her process closely and engaging in conversation about her thinking where we really get a glimpse into what she knows and how she knows it. Specifically, we can see how she is literally making and reasoning out her own meaning of how multiplication facts are combined. 

We only get half of the picture if we look at the final product/answer (double facts). I know how to watch for and identify understanding through the processes of making math and dance at the same time, but now I'm really learning about how it works with numbers, too! Fun stuff.

"Mama, is a curve a line?"

"What do you think?"

"I don't knowwwww...."

"Oh, come on, you asked the question, I bet you have some thoughts about it."

Showing me her Etch-a-Sketch and turning the knobs: "Okay, this is a line [drawing a straight line horizontally to the left] and this is a line [drawing a line vertically upward] but this is not a line [squiggling the line back and forth]."

"Why isn't that back and forth drawing a line?"

"Because it has to go in one direction. This curve is a line, but not when it goes back and forth..."

I left it there because the kid was sick, it was time to start resting and I know answers often rely on more questions about and interactions with the idea at hand. I'm sure this'll come up again at some point in the future. 

Approximation of original by the mother.  As users of Etch-A-Sketches will understand, the original got destroyed with an inadvertent shake.