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.

Counting by fives

The time I spend doing the dishes is frequently productive. Tabitha (6 years old) or Griffin (9 years old) will often linger nearby playing and talking. Sometimes they talk to me. Sometimes they talk about math, as on a recent evening.

Tabitha (6 years old): 5, 10, 15, 20, 25, 30…

She continues until 170, at which point she becomes bored.

T: Dad! I can count by hundreds. Want to hear?

Me: Yes. Yes I do.

T: One hundred, two hundred, …

She continues to nine hundred, when she pauses. I wait a few beats to see what will happen. When nothing does, I ask.

Me: What comes next?

T: Ten hundred.

Then…

T: Hey Dad, when do you get to one thousand?

Me: Awesome. Ten hundred is one thousand. That’s the word for ten hundreds.

T: What comes next?

Me: One thousand one hundred.

T: One thousand two hundred, one thousand three hundred,…

She continues. She reaches one thousand nine hundred.

Long pause.

T: Two thousand. [She smiles broadly.]

Later that evening, she is doing her weekly homework, which involves counting by fives. It is not clear from the directions whether she is to start over for each kind of five, or whether she is to continue counting all the way down the page.

Tabitha decides to continue the counting, and this seems the more appropriate challenge for her.

She gets to 95.

T: How do you write one hundred?

Me: A one and two zeroes.

T: How do you write one hundred five?

Me: You tell me. You try, and I’ll tell you whether it’s right.

T: One-zero-five.

Me: How did you know that?

T: Replace the zero with a five. How do you write one hundred ten?

Me: What do you think?

T: One-one-zero.

So What Do We Learn?

The first activity was about number language. Tabitha was following the patterns in number language and puzzling over the places where the patterns break down. One hundred, two hundred, three hundred…how does one thousand pop out of this sequence?

The second activity was about writing numbers (or numeration). Tabitha had no trouble counting through one hundred five, but she wasn’t sure how to write it.

A common error is to write this: 1005. The thinking goes like this: One hundred is 100, so I’ll write that, then put a 5 at the end.

Starting the conversation

Any time a child asks if you want to hear her count, the correct answer is yes. Then listen. Listen for the challenging bits (the teens are difficult, as is anything following a ‘9’—such as 19 to 20, or 49 to 50). Be ready to help if they get stuck but also to let them think it through and try for themselves. Then talk about how they knew to do what they did. Talk about their thinking.

Post-Halloween Math Talk

Waiting for the school bus this morning, the two adults and three children discussed last night’s Halloween events.

The neighbor girl, W (9 years old), announced that her brother, E (six years old), had gotten 90 pieces of candy for his trick-or-treating efforts. Griffin (9 years old) announced his haul of 51 pieces.

Me: Did E count each Nerd as one?

Image from Wikipedia

W: Oooo…maybe he did!

P (who is W and E’s father): We were at a house last night that had a bowl with a Take one sign. E went up, then came back and announced that he had taken three.

We told him he had to put two back.

He smiled and said, It’s a package of three!

I love this boy!

I thought for a moment about how various Halloween candies are packaged.

Me:Whoppers?

P: Yeah.

Image courtesy of Free Photo of the Day

I am not proud that I know this sort of thing. But on the rare occasion that my extensive candy knowledge is useful, I am not going to hide it either.

So what do we learn?

We learn that there is always a follow-up question, and that the follow-up question can bring out fun stories and ideas.

The conversation could have died after E’s 90 and Griffin’s 51 pieces were announced. But I got fun stuff by asking exactly what was being counted.

We have had fun with the question of what counts as one before, when Tabitha and I talked about Eggo mini-wafflesfor example.

Starting the conversation

North American residents probably don’t need my help here. Your children probably know yesterday’s candy count cold. Ask whether the Nerds (or Whoppers or Smarties or…) count as one piece.

If Halloween isn’t a thing where you are, keep an eye and an ear open for when your children are counting things that are packaged in groups.

Doll years

Out of the blue on our recent camping trip, Tabitha had an announcement for me.

Tabitha (6 years old): I am 12 in doll years and Griffy is 16 in doll years.

Her brother Griffin is 9.

T: So how old are you in doll years, Daddy?

Me: Well, how do doll years work?

T: Well, I’m 12 and Griffy’s 16.

Me: Is it twice as old? Then I would be two times as old, so nearly 86.

My birthday is coming up next week. This has been a point of discussion around the house recently.

T: No! It’s 6 times!

Me: You’d be 36 then.

T: No. I am 12 in doll years.

Me: Oh! Six years older not 6 times as old!

T: Yeah.

Me: Then Griffy is 15, not 16. And I would be almost 49.

So What Do We Learn?

Children build lovely and complicated imaginary worlds. For a long time, Griffin and Tabitha would play “creatures” together. Whole societies of stuffed animals, dolls and plastic figurines rose and fell. These societies had celebrations and tragedies. There was Creature Christmas that could take place at any time of year. Also a Creature State Fair. Et cetera.

Combine this parallel creature/doll universe with learning about the passage of time and pretty soon doll years are going to pop up.

Griffin and I talked about tortoise years and dog years a while back. At the time, Griffin was 8. He was comparing life spans of tortoises to those of humans, as we do with dogs to generate the 7 dog years per year comparison that is commonly known.

Tabitha is firmly grounded in comparing by counting and addition, as is appropriate for a 6 year old. Somewhere between third and sixth grade, children transition from always comparing by addition and subtraction to being able to compare by multiplying and dividing. This difference is what Tabitha and I are discussing in this conversation. She says Six times but means Six more.

Starting the Conversation

Listen for the comparisons your children make. Here, Tabitha compared ages. But heights, dollar amounts, number of Tootsie Rolls in a candy dish, et cetera; all of these are possible comparisons that children will naturally make. Ask a follow-up question. How do you know? is a good place to start. What if? is a lovely follow up. For example, What if there were a newborn baby in our family; how old would it be in doll years?

A few quick conversations

Sometimes we just have very short conversations that are worth sharing here, but not worth their own full posts. Here are three such conversations…

In the middle

The main television in our house is in the parents’ bedroom. As a result, 99% of the kids’ TV watching takes place on our bed. The other day, Rachel (my wife) was lying on the bed using the remote to start an episode of The Brady Bunch (the kids’ latest obsession). Griffin was already on the bed. Tabitha entered the room. I was standing up at my dresser, folding clothes.

Tabitha (6 years old): I want to be in the middle!

Rachel: Don’t worry, I am about to get off the bed.

T: But then there will be no middle.

Rachel had misinterpreted Tabitha’s desire to be in the middle as a desire to be in the middle of the bed when in fact she wanted to be the middle person.

T: There is no middle with just 2.

Me: But with 3 there is?

T: Yes.

Me: What are some other numbers that don’t have a middle?

T: Four doesn’t….Five does…Six.

Me: Is there a middle with 0 people?

T: No!

And here, dear readers, I do not have notes about her answer. It was a few weeks back. I invite speculation in the comments, and I will ask her again about whether 1 has a middle.

Tabitha has homework this year; both reading and math. The math has been awfully simplistic given her present knowledge. Counting small numbers of cows in a picture, filling in numbers on a hundreds chart, that sort of thing.

She has declared that it is too easy.

Me: When you do math in school, is it different from this or a lot like what you have for homework?

Tabitha: It’s the same mostly.

Me: How do you feel about that?

T: OK.

Me: Which makes you think harder, school math or daddy math talks?

Related

Perhaps it was the same evening, perhaps a different one recently, Tabitha wrote the wrong number for the number of cows in the picture. She caught her error and expressed frustration at needing to erase.

In a desperate attempt to bring something new to this task, I told her not to be frustrated.

Me: One of my favorite things to do in class is to fix a number on the board without erasing. My very favorite example is turning a 2 into a 3.

I demonstrate. Tabitha is eager to try.

Me: Also, a 5 can become an 8 quite easily. And 7s are easy to turn into 9s.

She has a fine time practicing and dreaming up new number-fixing techniques.

A week or so later, she is filling out a hundreds grid and becomes very excited to do the 80s row. Each and every 8 begins as a 5 and is corrected. She is very pleased with herself.

The conversations we have with our children affect their thinking. Of course they have their own interests, but the conversations we initiate have an impact.

The New York Times’ Motherlode blog (subtitle, Adventures in Parenting—we’ll talk about the equating of parenting with mothers another time!) quoted a University of Delaware study a while back:

Even [when their children are] as young as 22 months, American parents draw boys’ attention to numerical concepts far more often than girls’. Indeed, parents speak to boys about number concepts twice as often as they do girls. For cardinal-numbers speech, in which a number is attached to an obvious noun reference — “Here are five raisins” or “Look at those two beds” — the difference was even larger. Mothers were three times more likely to use such formulations while talking to boys.

The researchers note that these differences are not intentional. They were observed in the course of free interactive play between mothers and their children.

The potential consequences are important. The researchers speculate in the abstract to their published research article:

Greater amounts of early number-related talk may promote familiarity and liking for mathematical concepts, which may influence later preferences and career choices. Additionally, the stereotype of male dominance in math may be so pervasive that culturally prescribed gender roles may be unintentionally reinforced to very young children.

So do Tabitha proud, OK? Go ahead and use the two of us as a model for talking math with your daughters.

Counting fingers

Counting fingers

A while back I met a mathematician. He is the husband of a colleague. He found my Talking Math with Your Kids project fascinating and asked repeatedly for additional examples of the conversations I have had with Griffin and Tabitha.

He referred to my work as brainwashing, using the term with great delight.

He shared a story of a young child who, when asked Do you have more fingers on your right hand, or on your left hand? responded without counting, but by matching the fingers thumb-to-thumb, index-to-index, et cetera.

The child invented one-to-one correspondence! my mathematician friend exclaimed with pleasure.

In a sense this is true.

There are things that we tell children. And there are ideas they have on their own, without knowing that anyone has had these ideas before. These really are inventions.

Children can invent more than we sometimes suppose they can.

In any case, this mathematician friend of mine was very curious to know what Tabitha would make of this story. I promised him I would ask. Here is what happened.

We were lying on the bed one evening, having just finished a book and with a few minutes left before beginning the remaining bedtime rituals.

Me: Tabitha, I want to ask you a question.

I told her that I had met a mathematician who was curious to know what she thought about something, and that this something had to do with an interesting answer that another child had once supplied to a question.

Me: The question asked of this child was, “Do you have more fingers on your left hand or on your right hand?”

Tabitha (six years old): That question doesn’t make any sense!

Me: But it’s the question that was asked.

T: But it doesn’t make any sense. Look.

[She counted the fingers first on her left hand, then on her right]

T: 1, 2, 3, 4, 5…1, 2, 3, 4, 5.

Me: So it’s the same on both hands.

T: Right, so the question doesn’t make any sense.

Me: OK. But that’s not how the child answered it. The child did this.

Above, you see what the child did originally.
Tabitha re-enacted it later for the purposes of this post. We regret any confusion.

Me: The question I want to ask you is, what do you think the child was thinking?

T: Oh, I know what she was thinking!

Me: Really?

T: Yeah. It’s the same. If they all touch it’s the same number.

Me: I wonder if that would work with toes.

Tabitha proceeded to demonstrate that it does in fact work with toes.

Me: Ha! I was thinking about comparing the fingers on one hand to the toes on one foot.

T: Well, it would be hard because the toes are all squished together.

We spend a few moments playing with our fingers and toes, trying to match them up, noting their relative cleanliness, and then we get on with the rest of our evening.

So what do we learn?

The technique of asking what a child thinks of an idea is a powerful one. I use it in class all the time: What do you think the person was thinking who got a different answer from you? How do you think Brianna knew to do that?

Asking children to evaluate and comment on the ideas of others helps them also to think about their own thinking.

The specific idea we discussed here is that of one-to-one correspondence. We discussed this in the recent conversation about holding hands at the farmers’ market.

Starting the conversation

This is an easy one. It doesn’t depend on your child providing an idea or knowing any particular fact of mathematics. Sometime soon, you will have a quiet moment together. Maybe it will be at the end of an all-out living room danceathon, or after reading a big pile of books. Tell your child about the mathematician’s question. Show your child the answer that so impressed the mathematician and ask, What do you think the child was thinking?

I had this same conversation with a highly precocious three-year old recently. She insisted that you needed to count the fingers in order to be sure. We had a fine time doing that. Tabitha was within earshot of the conversation with a wry smile.

Sharing pears

I took the kids camping this weekend. Jay Cooke State Park is lovely. I recommend a visit if you have never been.

We had the following conversation at the campsite on Saturday evening.

Me: Griff, we have two pears and three people. What should we do about that?

Griffin (9 years old): Cut them in 3 pieces.

Me: How much does each person get in that case?

G: One and a third, I think…No…I don’t know.

His attention returns to the campfire, which is of course endlessly fascinating. A minute later, I try again.

Me: Tabitha! Two pears, three people. I’m going to cut them in half. That’s four pieces. You each get one. I get two. Fair?

Tabitha (6 years old): NO!

She passes a few moments pondering while I take the pears over to the picnic table.

T: You should cut them in four pieces. Give each of 1, then put the other away for tomorrow.

I originally interpret this to mean cut each of the pears in four pieces. Afterwards I am not so sure what she meant. In any case, I did what I thought she suggested.

Me: Tabitha, I did most of what you said. We each got 2 of those pieces. There are 2 leftover. I’m going to cut those in half. We each get 1 of those littler pieces.

Now what should we do with this last little piece?

She thinks

T: Cut it!

Me: In how many pieces?

T: Three!

I do so and we admire our handiwork.

So What Do We Learn?

The first fraction most children encounter in a serious way is one-half, and this is nearly always through fair-sharing.

Two kids, one pear (or cookie, or cupcake, or doll…) this is where children begin to think about fractions. They do not think about the notation of fractions, but they think about the important idea of fractions: A thing can be cut into equal pieces, and there is a name for these pieces.

Sharing multiple things in a way that involves cutting—as 2 pears among 3 people—is much more challenging than sharing one thing.

Nevertheless, this experience of halving is so fundamental that children often want to solve everything by halving. As long as we are willing to cut halves in half, this can be a powerful way of working things out.

Griffin’s idea to cut each pear in three pieces, then, is relatively sophisticated. That he could not name the resulting share should be no surprise in a campground conversation where the cutting is all imaginary and unwritten.

That this did not immediately occur to the younger child is no surprise. Likewise the idea to save the extra piece for another time is typical of young children’s thinking about fair sharing.

Cutting that last piece into three was a triumphant idea for her.

See Extending Children’s Mathematics for lovely, detailed writing about these ideas.

Starting the Conversation

Buy two pears. Get yourself two children of different ages. Ask them how to share the pairs fairly. Do what they tell you to do and ask follow up questions such as Is this fair? Do we all have the same amount? How do we all have? and What should we do with these leftover pieces?

Jim Trelease’s The Read-Aloud Handbook is lovely and very helpful for parents wanting to immerse their children in the world of written and spoken language, stories and books.

I aspire to creating the math version of this; the Read-Aloud Handbook of Math in a sense.

Here is how he began.

The dearth of accessible material inspired him to write and self-publish the first edition of The Read-Aloud Handbook in 1979. “I self-published because I never thought any of the major publishers would be interested in it. At that point, ‘reading aloud’ was too simple and not painful enough to do the child any good. At least, that’s what many educators thought,” he says in hindsight. But that mindset would soon change.

His book is now in its seventh edition and has sold nearly 2 million copies.

Wish me luck, OK?

The Pumpkin Patch

On a family trip to a farm from which we have bought a tremendous amount of produce this year, Griffin and I were heading to the pumpkin patch.

We had already taken the wagon ride to the other pumpkin patch; where the pie pumpkins were grown. We had helped with the harvest and had chosen several to take home. Now we were on our way to the Jack-o-Lantern pumpkin patch.

Griffin [9 years old]: We have 5 pumpkins! Is that enough to make a pie?

Me: More than enough.

G: Enough to make 5 pies?

Me: Probably not.

G: How many pumpkins go into a pie, or how many pies do you get from a pumpkin?

Me: Hmmmm… I would say about $1\frac{1}{2}$ pumpkins make one pie.

NOTE: This was semi-truthful. I really have no idea how many typical pie pumpkins are needed to make a pumpkin pie. I was making what I felt to be a reasonable estimate. But at the same time, I was pretty pleased with the estimate and with the math that it might encourage Griffin to do.

G: Oh! So we could make … 3 … 3 plus $1\frac{1}{2}$$4\frac{1}{2}$ … three pies! And have half a pumpkin left over!

Me: Which is $\frac{1}{3}$ of a pie.

G: Right.

NOTE: I do not trust that he got that $\frac{1}{2}$ of a pumpkin makes $\frac{1}{3}$ of a pie given my estimate. He may have gotten it, and he may not have. The pumpkin patch was approaching so I let it slide.

G: Will we make three pies?

Me: No. I don’t think I’ll have the patience for that. But we can make one pie for sure.

So What Do We Learn?

Griffin is thinking about division when he figures out how many pies we can make from five pumpkins. Other similar sorts of division problems include, How many feet tall are you if you are 49 inches tall? and How many groups of four can we make in our classroom of 30 students? The pumpkin pie problem is challenging because it involves fractions.

One of the hardest parts of the thinking Griffin does here is keeping track of the units. As he counts up to $4\frac{1}{2}$, he is counting pumpkins. The first 3 he utters counts pumpkins. But at the same time, he is keeping track of a number of pies. That’s the final 3 he utters: 3 pies.

I play with that idea by referring to his $\frac{1}{2}$ of a pumpkin as $\frac{1}{3}$ of a pie. I understand that not every parent is ready to do this on the spot. Don’t worry about that. Griffin got enough thinking from the basic conversation; the rest is gravy (or maybe whipped cream?)

Starting the Conversation

This was a special opportunity. We had some pumpkins. Griffin wanted to make things with these pumpkins. I could involve fractions.

Other such opportunities could include bags of apples, cups of flour (a standard 5-pound bag of all-purpose flour has about 18 cups), et cetera. If your child doesn’t ask the how many pies (or batches, or cakes, or whatever) question, you can ask it. But don’t make it feel like a quiz. You can just say, I wonder how many pies we could make with what we have?