# Ten hundred Doras

There was a while when Tabitha (five and six years old at the time) would try to get away without wearing underpants when she dressed herself. Those days are pretty much over, but I still like to make sure she has done the complete job, so I ask her from time to time.

Tabitha (7 years old): I’m dressed!

Me: Are you wearing underpants?

T: Yup—Dora the Explorer.

Don’t worry. The child is not wearing these in the picture.

Me: Nice. How do you feel about your Dora the Explorer underpants?

T: I don’t really like Dora that much, but I have a thousand of them.

Me: That’s a lot.

T: I counted them once.

Me: All one thousand?

T: No. I don’t really have a thousand. I don’t even know how to count to a thousand. Just to ten hundred.

I pause for a moment. Does she mean one-hundred-ten? Can’t be. She must know that one-hundred-eleven comes next.

Me: Ten hundred. You mean like after nine hundred is ten hundred?

T: Yeah. That’s as high as I know how to count. I don’t even know how many a thousand is.

Me: A thousand is ten hundred.

T: Oh. Cool.

A few minutes later, I get an idea. I wonder how she would write ten hundred. She needs to get out the door for school so I make it quick. I ask her to read some numbers out loud as I write them.

• 900
• 832
• 110
• 1000.

For that last one, she says one thousand.

I ask how she would write ten hundred.

She writes, “1000”.

T: It’s the same.

Me: Because I just told you that. Right. How would you have written ten hundred before I told you it was the same as one thousand?

She shrugs her shoulders. Drat. Moment lost. We talk about hundreds for a moment. One hundred, two hundred, etc. up to ten hundred.

Then I have one more.

Me: OK. Last one, then off to school. How would you read this one?

I write 10,000.

She looks for a moment. And thinks.

T: Ten….

More thinking.

T: Ten thousand?

High five!

I zip up her sweatshirt and send her out the door to catch her bus.

## So What Do We Learn?

A recent research article argued that children learn a lot about place value through everyday conversation, and that kindergarteners know a lot more about the structure of the number system than parents and kindergarten teachers (on average) think they do.

Here you can see that knowledge in action. Tabitha knows that 1000 is a big and important number. She knows the pattern that allows you to keep counting by hundreds. She has not put these two pieces together. A short conversation helped her put those two pieces together, and then to extend the pattern.

## Starting the conversation

This didn’t start out as a math talk. It began as a clothing inspection. But the opportunity presented itself. Listen for those times your children use numbers, and ask follow up questions about them. You won’t get this much learning out of every such conversation, but if even 10% of those opportunities turn into a little bit of learning, the interest compounds.

I promise you that.

# Tens again

Slowing down at the end of a long, active spring day. Stormy clouds are rolling in. Tabitha and I watch them together for a couple of minutes from the west-facing window at the top of our stairs.

I ask Tabitha if I can ask her a quick math question.

She consents.

Me: How many tens are in 32?

Tabitha (7 years old): Three.

Me: So quick! How do you know that?

T: 10, 20, 30. Easy.

A silent moment elapses.

T: And there’s 10 tens in a hundred.

Me: Yes. Lovely. So true.

How many tens are in 200, though?

T: Twenty.

Me: Whoa!

T: Yeah.

Another silent moment elapses.

T: Asking “How many tens are in 30?” is like asking “How many ones are in 2?”

Me: Wow. I had never thought of it like that. And is it also like asking “How many hundreds in 300?”

T: Except I don’t know that one.

Me: You don’t know how many hundreds are in 300?

T: No.

Me: Three.

T: Oh. I thought it was tens in 300.

## So what do we learn?

The power of silence and of conversations in quiet moments. Both times a silent moment elapsed in this conversation, Tabitha continued with an idea of her own. And both are gems.

And there’s 10 tens in a hundred. Many grade school worksheets have attested that there are 0 tens in 100, when what they really mean is that there is a 0 in the tens place in 100. We can do a lot more mathematics with the ten tens in 100 conception than we can with the 0 tens in 100 one.

Asking “How many tens are in 30?” is like asking “How many ones are in 2?” This right here is powerful stuff. For Tabitha, ten is such an important part of the structure of numbers that it behaves like one. Ten, for Tabitha, is a unit—a thing that you count.

If you are new to this blog (and many of you are—Welcome!), you may not have spent four minutes with video. Do so now, please. Consider it your Talking Math with Your Kids homework. It’ll be fun. Promise.

## Starting the conversation

Wait for a quiet moment. Ask for consent. Ask How many tens are in 32? Listen, follow up and allow a few moments of silence.

# 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.

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.

Sixty

## 06

Sixty

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

Six hundred

## 006

Six

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

Six

Hmmm….

Zero

## 00

Zero

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

Three

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.

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.

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.

## 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.

# 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.

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.

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.