# 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.

# 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.

# If it’s true for other language…

…it’s probably true for number language too.

University of Pennsylvania researchers have studied the quality of parents’ speech to their toddlers, and its relationship to the children’s vocabulary later on. “Quality” of speech was measured by how well an adult observer could guess a common word uttered by the parent, when the observer could see the parent and child, but with sound muted.

In the researchers’ words:

Strikingly, this parent-input quality difference at child age 14–18 months [about 1.5 years] signiﬁcantly correlated with the children’s vocabularies at 54 mo [about 4.5 years].

The ways in which parents were talking to their children at age 1 had an effect on the number of words the children knew at age 4.

## The way you talk to children like this…

Tabitha at 21 months (a bit OLDER than the subjects in the beginning of the study)

…has a profound impact on what they know when they are like this…

Tabitha at 54 months, the age of the subjects at the end of the study. She is on her way to her first day of Pre-K.

It probably goes for number words and shapes, too.

So let’s get out there and talk math with those little ones!

Count stuff, use number words at every opportunity, point out and talk about shapes. Start them young. And if you haven’t started them young, start now.

It’ll be fun, I promise.

# Learning to count

I am fascinated by watching children learn to count. There are many surprising twists and turns kids take along the way.

Even more surprising, perhaps, is that what seem like crazy mistakes to us adults are completely sensible attempts at getting it right for kids.

For example…

• In English, the pattern that occurs in the teens is complicated. “Thirteen” doesn’t sound very much like what it is: three plus ten, while “Fourteen” does.
• Likewise, the names for the “decades”: “Twenty” means two tens and “Thirty” means three tens.
• But once you get to twenty-one, the pattern is regular until twenty-nine.
• We start counting at 1 (not 0), but we don’t start the decades at 21 or 31, so kids following the 1, 2, 3 pattern will often skip 20 and 30.

If you put all of this together, you might expect a typical young child who is counting “as high as I can” to:

1. Have trouble in the teens
2. Skip 20 in favor of 21
3. Have more success in the twenties than in the teens, and
4. End the count at or about twenty-nine (since the word thirty is not very predictable from the previous language patterns).

Here goes…

# A circular conversation

The following conversation took place about two years ago. It is probably the first one that made me realize how important it is to talk math with my kids. Near the beginning of the conversation I noticed myself making a choice between engaging her mind and moving on to other things.

That choice—and the knowledge needed to notice it, and to follow up on it—has become interesting for me. Through this website, I hope to share what I have learned about that, and to learn more through interaction with readers. So please send reports of your conversations to me. And get those questions to me, too. You can do both through the About/Contact page.

It’s Sunday morning. Summer has arrived. We are enjoying a beautiful morning on the front porch. I am finishing my coffee. Tabitha (four years old at the time) has finished her donut.

Tabitha: [four years old] Why don’t circles have tips?

Me: What do you mean?

T: Why don’t circles have tips?

Me: What do you mean by tips? What shapes do have tips?

T: Triangles and stars. Why don’t circles have tips?

Me: Well…that’s a good question. I guess that’s part of what makes them circles. If they had tips, they wouldn’t be circles.

T: But what if a circle did have a tip?

Me: Well, then it wouldn’t be a circle. I guess what makes a circle is that it’s round. If it had a tip it wouldn’t be round.

There is a pause, during which I realize that I have not really given Tabitha my all with that explanation.

Me: Do you want the real answer?

T: Yes.

Me: OK. Here’s the real answer. See this plate?

It’s circular. Its edge is a circle, right?

T: Some plates are shaped like a fishy.

Me: Right. Good.

But this one’s circular. There’s a point in the middle of the plate; that’s called the center. All the parts of the plate on the edge are the same distance from the center. If there were a tip, then the part at the end of the tip would be farther from the center than the other parts, so it couldn’t be a circle. What really makes a circle a circle is having all parts be the same distance from the center.

T: What if there were spines?

Me: What do you mean?

T: What if there were spines all around the circle?

Me: Well then the tips of the spines would be further from the center than the base of the spines, so it wouldn’t be a circle.

T: What if they were all around the circle?

Me: Still, there would be parts at the end and parts at the base.

Did you like getting the real answer? That answer about circles being round, that wasn’t really the real answer. Did you like the real one?

T: Yes.

There is a thoughtful pause.

T: What about carousels? They are circles and they have points.

Me: I don’t understand what you mean.

T: What about carousels? They are circles. They have horses on them; those are like tips.

Me: Oh. Right. The circle is just the edge of the carousel. The horses aren’t part of the circle.

T: Oh.

Me: What got you thinking about circles, anyway?

T: [points out the window]

Me: What are you pointing at?

T: [smiles]

Me: I don’t get it.

T: The tree!

T: The bark!

T: It looks like a circle.

Me: Do you mean if you cut the trunk, the bark around the edge would look like a circle?

T: Yes.

Me: And that circle would have tips?

T: Yes.

# So what do we learn?

There is a lot in this conversation. As is often the case, when the conversation began I had absolutely no idea what she was talking about. What in the world could she mean by “why don’t circles have tips?” I work each semester with college students planning to be elementary teachers. I preach to them the importance of patient listening and asking questions to better understand what their students are telling them.

This is a message I frequently need to take to heart.

Tabitha’s questions are about making a transition from what shapes look like to what makes them what they are. She seems to want to know what makes a circle a circle.

This takes place as she thinks about the cross section of the tree in our front yard.

She knows that this would look circular, but that it isn’t a circle. She identifies a property that the tree cross-section has that a circle does not-tips, or sharp points.

I started with a crummy answer. I basically told her that Circles don’t have tips because if they did they wouldn’t be circles. And I felt guilty right away.

So I offered her a real explanation. That explanation was based on the definition of a circle, which is The set of all points a fixed distance from a common point, called the center.

This explanation was one that the average parent may not have ready at hand, though. So what do you do if you don’t know why a circle has no tips (or whether a square counts as a rectangle, or whether it’s still a right triangle if it points to the left, or…)? You model good information-seeking skills. Try to agree on what the question is (What do you mean by tips? What shapes do have tips?) Then consult books and friends and neighbors. You must know someone who has taken high school geometry more recently than you have. Maybe you have an engineer in the family, or a math teacher up the block. Your library has a librarian. Any of these people would be delighted to help out a young child with a geometry question.

And now that you’re reading this blog? You’ve got a friend ready to help. Shoot a note through the About/Contact page; we’ll get you an answer ASAP.

# Starting the conversation

This conversation was Tabitha’s idea. The only thing I did here was listen and try to understand her questions.

We can all do that.

# Postscript

Take the time to read the comments. Other parents weigh in with some lovely ideas for additional directions one could take this conversation. The key is that there is not one right conversation to have with your kids. The key is to have that conversation by asking and listening.