More patterns on the multiplication machine

When we left off last week, I had challenged Tabitha to find a pattern on the multiplication machine so that there would be the same number of buttons up as down. This challenge followed up on her sophisticated argument that her down-up-down-up pattern yielded more downs than ups.

ups.and.downs

There are 81 buttons, so the task of evening out the ups and downs is not possible.

But Tabitha is 6 years old. She knows little about even and odd numbers. Searching for a way to share 81 things equally (between up and down in this case) is a good way to get her thinking about the idea.

You may recall that I had shooed Tabitha off to her bath on giving her this challenge. This is where our story picks up.

At the end of the bath, she puts on her jammies and announces…

Tabitha (6 years old): I know how!

She runs into the room to get the machine.

T: Now Daddy, I don’t know if this is going to work, so just keep your ideas to yourself.

This line is awesome, is it not?

I do as I am told.

She produces this:

Photo Nov 20, 9 41 52 PMT: Oh no.

Me: What?

T: These [she points to top and bottom rows] are both up.

She tries again, producing this:

Photo Nov 20, 9 42 20 PMT: Oh no. Still too many up.

At this point she gives me a look which I take to mean that I can have a try. So I go back to her first pattern.

Photo Nov 20, 9 41 52 PMAnd I start to share out the bottom row—half up, half down.

patterns.6Tabitha: But Daddy! That’s not a pattern!

So what do we learn?

The raw beauty of Tabitha’s line, “I don’t know if this is going to work, so just keep your ideas to yourself!” strikes every time I think about this conversation.

Children enjoy investigating their ideas. I have to work very hard to get many of my college students back to a mental place where they trust that they have mathematical ideas worth investigating.

The best thing a parent or teacher can do in this situation is be quiet and let the kid work it out.

Starting the conversation

As all interesting conversations do, this one had a trajectory. We started in one place (making fun patterns), focused our attention on one part of what we were doing (comparing the number of ups and downs) and finished off with a “what if” question (what if ups and downs were equal in number, what would that pattern look like?)

You can practice that with your child. It doesn’t matter whether any particular conversation goes anywhere (many of ours do not), eventually you’ll hit on something interesting to both of you and pretty soon you’ll notice that 10 or 15 minutes have gone by.

And then the next time will be easier. Soon it will be a habit.

 

Patterns on the multiplication machine

Tabitha (6 years old) has been playing with the multiplication machine off and on for a few months now. We have never once used it for learning multiplication facts, but we have had a ton of math-learning fun with it.

Recently, she carefully did an up-down-up-down pattern that she continued along all of the columns.

ups.and.downsWe discussed her patterning a bit before I sent her upstairs for her bath.

I followed behind and, when I knew the machine was out of reach, asked her a question.

Me: You know how some buttons are up and some are down right now? Are there more ups or more downs?

Tabitha (6 years old): There are the same amount.

This was the answer I expected. As we will see, her reason for the answer surprised me.

Me: How do you know?

T: I did two patterns. Up down up down… and Down up down up. Let me show you.

We head downstairs together.

Me: OK. In this one column [I point to the first column on the left], are there more ups or downs?

T: [She counts each type] Downs. But in this one [the next column to the right] there’s more ups.

Me: So how about the whole thing?

T: Same!

Wait.

I’m gonna figure this out.

1,1,2,2,3,3,4,4,5

There’s more downs.

Me: How do you know?

T: Watch. [She points to columns as she counts, one column for each number word] 1, 1, 2, 2, 3, 3, 4, 4, 5.

Me: Right. But what does that mean?

T: One. There’s more downs. Then one. There’s more ups. Then two, and two, like that. But five is more downs.

Me: Wait. One has more downs, then the other one has more ups. So together the 1s have the same number of ups as downs?

T: Yeah.

Me: So then what?

T: Same for the 2s and 3s and 4s. But 5 has more downs.

Me: So there are more downs total.

T: Yeah.

Me: I have a challenge for you. If you can, try to make a pattern that has the same number of ups as downs. But not now. Now you have to get in the bath.

(to be continued)

So what do we learn?

We have to keep an open mind when our kids are telling us what they think.

I knew there were more downs than ups because we started with a down in the upper left. Moving across the row, there is an up for each down. The last down at the end of the row has an up at the beginning of the next row.

Every down has an up.

Except for the bottom right corner. That down has no up to pair with. So there are more downs than ups.

This is not at all how Tabitha saw it. She made groups—each column has an extra up or an extra down. Then she grouped these groups—each pair of columns has equal downs and ups.

By keeping an open mind, I was able to listen to her thinking. This let me ask follow up questions, which helped Tabitha make her own thinking better and more clear. I try as much as possible to have explain your answer be about convincing each other of something. You think they’re the same? Convince me. You think there are more downs? Convince me.

Starting the conversation

Patterns. Notice them. Play with them. Ask about them.

Any repeating pattern will do.

Even if the pattern had been Down down up, down down up (i.e. two downs for every up), we would have had a lovely conversation about which there were more of. The keys to the thinking in this pattern were:

  1. There were too many things for counting to be a convenient solution, and
  2. We started thinking about it when we didn’t have the machine in front of us.

I would have been content if Tabitha had counted 41 downs and 40 ups. But I was very much hoping to push her to use the pattern she had created to reason rather than to count.

Big Cheez-Its

There are now BIG Cheez-Its (U.S. only, it appears). The package claims that they are “Twice the size!” of regular Cheez-Its.

image

Naturally, I bought some a few months back.

I asked Tabitha (6 years old) and Griffin (8 years old at the time) what they thought. I started with Tabitha when Griffin wasn’t around so I could get her pure thoughts.

She put one cracker on top of the other and proclaimed, “No”.

image

I wanted to know why she thought that. I thought she might be mistaking side length for area. That is, maybe she was paying attention to the lengths of the sides of the two crackers rather than to the amount of cracker. So I asked about it.

She pointed to the uncovered part of the BIG Cheez-It and argued that this wasn’t enough to make another full regular Cheez-It. So she was paying attention to the amount of cracker.

A few minutes later, it was Griffin’s turn. He ran like a chipmunk with his two crackers into the dining room.

I imagined that this chipmunk would be nibbling the crackers next door and that our conversation would be at an end.

I was wrong.

He was in search of paper and a pen. He carefully traced each cracker, cut out the uncovered part of the BIG one and attempted to partition and reassemble this remainder on top of a tracing of the regular cracker, which it did not completely cover.

Sadly the cut outs are lost forever.

His conclusion: BIG Cheez-Its are almost but not quitetwice the size of the regular Cheez-Its.

So what do we learn? 

Notice the differences between the children’s strategies. Tabitha, the six-year old, worked with the crackers. She put one cracker on top of the other and tried to picture whether the leftover space made up a whole cracker. She was very concrete in her thinking.

Griffin, the eight-year old, worked with representations of the crackers. He traced and cut out squares of paper which he could manipulate with more precision than the actual crackers.

The two children reached similar conclusions.

Neither child used tools to calculate areas.

Knowing whether one cracker is twice as big as the other does not require measuring how big either cracker is.

All of this is very typical for young children. Younger children tend to work with the actual things they are comparing. They are what we call concrete thinkers. Older children begin to work with representations of the things (e.g. Griffin’s cut outs). They are more likely to be abstract thinkers.

Starting the conversation

Investigate advertising claims. Have a healthy, skeptical attitude towards these claims, and encourage your children to wonder about them, too.

Be forewarned, though! You may create critical thinkers who question your authority, too.

And you may end up spending a LOT of time trying to figure out whether Double Stuf Oreos are really doubly stuffed.

How young children learn about numbers

“As in other areas of language development, it appears children infer the meanings of [multi-digit] numbers using whatever experiences they can access.”

This is one of several conclusions a group of researchers at Michigan State University and Indiana University drew from their study of 3 \frac{1}{2} through 7 year olds (pdf). (Read the Washington Post’s report on the research here.) In particular, these researchers were studying the place value knowledge of young children, trying to understand whether they learn multi-digit numbers logically through direct study or culturally through everyday experience.

Examples of Tabitha’s recent experiences with multi-digit numbers.

Their study made clear that children absorb a lot of information about multi-digit numbers through their everyday experiences.

These researchers provide compelling evidence that young children (as young as 3 \frac{1}{2} years old) connect number words (fifty-seven) to numerals (57). Children can use their ideas about these numbers to identify and to compare numbers.

Talking Math with Your Kids is a project based on this premise. Children don’t need iPad apps to teach about numbers, they need conversations about the numbers in their worlds.

If we are aware of the importance of these experiences, parents can provide more opportunities for children to think about these numbers. Some examples from this blog include Days to Christmas, The Biggest Number, Uncle Wiggily, and Counting by Fives.

Days to Christmas: Place value follow up

In yesterday’s post, I told of the Christmas Countdown cube calendar, and of how Tabitha (6 years old) changed my 06 days to 6 days by removing the leading zero. I challenged readers to consider her reasons for this.

Of course I asked her. I showed her the pictures I took and asked why she had taken off the zero. Here are the results of our conversation.

Me: I’m curious about why you took off the zero.

Tabitha (6 years old): Because there aren’t sixty days until Christmas.

A new piece of research has been making the rounds in the media recently. I will write about it at length soon, but for now you just need to know that the common headline is something like, “Young children know more about place value than most people assume they do.” In particular, the research looked carefully at the partial place value knowledge children have, rather than just calling wrong answers wrong.

Helping children develop partial knowledge into better knowledge through exploring and talking is the heart of the work here at Talking Math with Your Kids. Naturally I want to explore Tabitha’s knowledge here.

I get out a piece of paper, a pen and write some numbers, asking her to tell me what they are. I write down her responses verbatim so that I can share them with you.

25

Twenty-five

52

Fifty-two

A reasonable hypothesis for Tabitha saying that “06” meant “sixty” would be that she doesn’t pay attention to the order of the digits in a number. We now know this isn’t true.

More numbers follow.

60

Sixty

06

Sixty

Now things are getting interesting. There seems to be something special about that zero out front. We do some more.

600

Six hundred

006

Six

What? I was sure she was going to say six hundred for this one.

060

Six

Hmmm….

0

Zero

00

Zero

No hesitation on this one, which surprises me. I thought she might object to two zeroes.

3

Three

30

Thirty

030

Three

At least this one is consistent with 060.

I am beginning to wonder how she will work with numbers that have zeroes in the middle of them, such as 1002. And I am curious how she will work with numbers that use the same words, but in a different order. I think of 1002 and 2001 as a good example: one thousand-two and two thousand-one. So I build up to those.

2000

Two thousand

1000

A thousand

Uh oh. I didn’t expect this. I expected one thousand, not a thousand. I have to change my examples.

3000

Three thousand

3002

Three hundred two…er…three thousand two

2003

Two hundred three

Beautiful! This is what the researchers were saying—children have partial place value knowledge. What is going on here is this: 3002 looks like 300 with a 2 at the end. This is a very common error, and it represents trying to make sense of a complicated idea.

We do only one more; I can tell she is getting tired.

4030

Seven

I cannot tell if she is being silly or serious, trying hard or being clever.

Photo Dec 22, 12 35 43 PM

So what do we learn?

Just as with learning to speak a native (or foreign) language, learning about numbers is not an all-or-nothing proposition. Children have partial knowledge that is sometimes inconsistent. Tabitha self-corrected on 3002, but not on 2003. This conversation supports her thinking about the structure of numbers. She will think about it in the future and be more prepared to pay attention to it because we talked about it. When she wants to know more, she will ask.

Starting the conversation

Children learn about numbers and language in similar ways—through exposure in their everyday environments. We read to our children to enrich their language environment, and we can expose our children to numbers to enrich their math environment. Waffles, toys and calendars are all examples of everyday objects with mathematical structure that children can play with.

Have these things available. Ask about how your children play with them. Listen to their answers. Then ask follow up questions.

 

Days until Christmas

Each day, Griffin (9 years old) has taken great joy in setting the Countdown to Christmas cube calendar the kids recently received from my father and stepmother.

On Thursday, I was doing my end-of-semester grading from home and noticed after he had left that he had neglected to set it on his way to catch the bus to school. So I did.

christmas.06

My wife got Tabitha out the door later in the morning while I was at work in my basement office.

Later, I noticed that Tabitha (6 years old) had modified it.

christmas.6

Here I will pause to give readers a day or so to consider why. What might a 6-year-old have in mind that would cause her to remove the zero from in front of the six?

I asked her about it later on and we had a lovely conversation. I’ll report on that soon.

Short notes

I passed a lovely morning with some math teacher friends at the Museum of Math in New York City today. Sadly, I did not have Griffin or Tabitha along. We would have had a ball.

I have three observations…

One. There is lots for kids to do there. It is truly a bonanza of mathematical conversation starters. I left wanting to be hired on as “Docent for Talking Math with Your Kids”. This is because the math does not smack you in the face. Instead, the math in the exhibits tends to surface in the process of playing, experimenting and paying very careful attention to what is going on.

In short, you want to talk about these exhibits and you want to linger.

Two. Many of the exhibits are designed in ways that allow kids to play and to do math beyond the intended activity. The exhibit below, for example, kept two little boys (each 5 or 6 years old) busy for 20 minutes although the original activity was way beyond them.

The first five minutes, they were sharing each of the shapes equally, making sure each of the two boys had the same number of squares, and the same number of triangles, et cetera. The next 10 minutes, they spent arranging the shapes on the screen, marveling at the things this provoked on the screen beneath.

It was lovely.

It could have been a bummer, though. If someone had insisted on doing the intended activity, these kids could not have done it. If the shapes had been electronic instead of concrete, the kids would not have been able to play with them in such creative ways.

Well done, Museum of Math!

Three. On the subway train back to my hotel, I noticed a little girl—probably 4 years old—holding up four fingers as she sat between her mother and brother.

I moved to be in listening range.

It turns out, she had been told they would be on the train for five stops. She had been putting up one additional finger as the train left each stop.

Sister (4 years old): [She is holding up four fingers as the train enters the station] This is our stop!

Brother (8 years old): No. The sign says there’s one more stop.

Sister: But you said five stops.

Mom: One more, sweetie.

The train stops. People get off and on. The doors close. The train starts up again.

The girl keeps exactly four fingers up the whole time.

Canned pumpkin

Fall baking in our house requires canned pumpkin. We were out so I asked for Tabitha’s help at the grocery store, where the pumpkin in on the bottom shelf.

Me: Put four of those bright orange cans of pumpkin in our cart, please.

Tabitha (6 years old): I don’t know if I can carry four.

Me: Do two, then two more.

Photo Nov 17, 10 15 20 AM

T: [With two cans of pumpkin in her hands] I know, because two plus two is four.

Me: Right. You could do three and one, I suppose.

T: OK. Give me one back.

She takes it, picks up two more from the shelf and brings the three cans over to me.

T: I did two and three.

Me: So we have five cans?

T: No! You gave me one back, remember?

Me: So two plus three minus one is four?

T: Yeah.

So what do we learn?

Decomposing numbers is fun.

We tend to think of 2+2 as something to do, and that the answer is 4. But in this case 4 is the thing to do, and 2+2 is one of several possible answers. When we think about different ways to make 4, we are decomposing 4.

Tabitha can keep track of the moves in our complicated decomposition at the end (You gave me one back, remember?) but she does not have practice with the math notation that captures all of these moves (Two plus three minus one is four). That is one of my roles in the conversation.

Starting the conversation

Tabitha gave me the ideal beginning to this conversation—she pointed out that there were too many cans for her to carry. It shouldn’t be difficult to put your own child in such a situation. The grocery store sells lots of things that children can carry a few of, but not a lot of: apples, oranges, cans of soup, etc. Picking up toys at the end of a play session at home or school, or books at the library—all of these are opportunities for you to name the number involved, then suggest a way to decompose it.

Multiplication Machine [Product review]

We have in our house a thing called the “Multiplication Machine”. It is a flash-cardy sort of thing. The multiplication facts are written on top of a 9×9 array of spring-loaded buttons. You press one and the button pops up, revealing the product. You can buy such a thing at your nearest teacher supply store. Ours came from Lakeshore Learning.

Talking Math with Your Kids is dedicated to helping parents and other caregivers to identify the mathematical opportunities afforded by everyday life, so we will not discuss here the traditional, intended use of this product (which is drilling and reviewing multiplication facts).

No, I want to talk about this thing as a toy.

Tabitha was hard at work pressing buttons on the Multiplication Machine the other evening. When I peeked in on her, I saw a scene that looked an awful lot like the one below.

patterns.3

She was playing with the arrangement of up and down buttons, not with the multiplication facts written on them. Patterns are tons of fun. So I went with it.

We developed the up, up, up, down pattern. We went across each row from left to right, top row to bottom row, as you would read a book.

Up Up Up Down, etc.

We developed its opposite—Down, down, down, up.

Down down down up.

We developed the Up, up, up, up, down pattern. This proved much more difficult for Tabitha, as she could not subitize the four ups. She counted them on her fingers, which she also needed for pressing buttons. She worked it out, though.

Up up up up down.

Before executing this last one, we noticed the right-to-left diagonals we had gotten from the Up, up, up, down pattern and predicted what we thought would happen when three ups became four. She correctly predicted the left-to-right diagonals, but I do not know why she predicted this.

We have not yet investigated the down-down-up pattern together, but I suspect she will get a kick out of it.

Down down up.

There are many more cool patterns to play with here. A few ideas that I am sure we’ll explore in the coming weeks:

  • What will happen with lots of different combinations of ups and downs?
  • What if we do columns instead of rows?
  • Are there any patterns where you cannot tell whether the person did rows or columns?
  • What if we follow a path back and forth across the rows, instead of starting at the left-hand side of each row?
  • What if go right-to-left? Or bottom-to-top?

The biggest number

I do not recall the beginning of this conversation, but I do recall that we were eating pizza at the dinner table when Tabitha anticipated my turn in the dicussion.

Tabitha (6 years old): I know what you’re going to say, Daddy. “Counting never ends.”

Me: I suppose that sounds like something I would say, yes.

T: What’s the biggest number, though? Googolplex?

Quick tutorial. A “googol”—spelled that way—refers to this number: 10^{100}, or “a one followed by a hundred zeroes”.

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

It is, of course, a very big number. Far too big to be practical in any meaningful sense. The very idea of such a large number having a name is fascinating to children. Most children (in my experience) encounter one googol in their social interactions with other children. The googol does not appear in the Common Core State Standards.

A “googolplex” is 10^{10^{100}}, or 10^{googol} or “a one followed by a googol zeroes”. You cannot write this number out in standard form.

You may Google googol for lots of interesting characterizations of how extremely silly this very large number is.

For example, you will not live for one googol seconds.

Indeed, the universe has not existed for one googol seconds (not even by the greatest estimates of its age—not even close).

You get the idea.

Me: Well, like you said I would say, counting never ends, so no googolplex is not the biggest number.

T: If you counted by 10,000 could you ever get to googolplex in your life?

Me: No.

T: If you counted by 11,000?

Me: No.

T: 12,000? 13,000?

Me: No. Even if you counted by googol, you couldn’t get to googolplex in your lifetime.

T: Well, what if you counted by googolplex?

Me: Well sure. It would the start of your count, wouldn’t it?

She decides to demonstrate this (Side note, we have been counting by various numbers of late).

T: Googolplex.

She smiles broadly, congratulating herself for successfully counting to what she has perceived to be the largest number.

We discuss further the existence of a largest number. Then Tabitha makes a claim that takes us in a different direction.

T: Eventually, numbers just go back to the beginning.

Me: So if you keep counting, you get to zero?

T: No.

Me: One?

T: No, Daddy! Don’t you remember there are numbers before zero?

So what do we learn?

Big numbers are fun. Boy howdy are big numbers fun. Children love to talk about the biggest number, and whether one exists. There is all kinds of lovely thinking going on when they ask these kinds of questions.

Talking about big numbers often leads to talking about infinity. If there is no biggest number, it is because numbers go on forever. The only thing Tabitha has experience with that goes on forever is a loop. She drew on that loop metaphor in imagining that numbers go back to the beginning eventually.

Starting the conversation

Listen for the biggest number talk. It often surfaces when children are comparing their athletic prowess (I can jump 2 sidewalk squares! I can jump 100 sidewalk squares! Pretty soon, someone is claiming to be able to jump googol or infinity sidewalk squares.)

When it surfaces, support it. Play and explore with your child. Answer questions. Ask questions. Talk about it and have fun. Look stuff up together when the questions go past your own knowledge. Shoot me a question here at Talking Math with Your Kids if I can answer any of those for you.