# Young children are more logical than you think they are

The kids (Griffin—12 years old—and Tabitha—10 years old) and I went on our annual summer camping trip recently.

This year it was Glacial Lakes State Park in western Minnesota.

While there, we visited the swimming beach where a family that included a 3-year-old girl was also enjoying the beautiful warm day.

There were minnows swimming in the shallow water, which the 3-year-old badly wanted to capture with her net. When she stood still, the fish would approach cautiously, but every time she moved, the fish quickly bolted.

Girl (three years old): I want to catch them, Mommy! Why do they swim away?

Mom: Maybe they’re scared.

[Thoughtful pause]

Girl: But I am not a monster! I am not a monster, Mommy.

Over the next several minutes, she repeated her monster claims a number of times. Eventually—as will happen with 3-year-olds—her attention shifted to other things and the swimming and splashing continued.

## So What Do We Learn?

Young children can think logically. This runs counter to some assumptions we make about three-year-olds, but it is true.

Here is the logical truth this girl understood:

The only thing to fear is monsters.

Fish fear me.

Therefore, I am a monster.

And deeper yet, she understood what logicians call the contrapositive.

The only thing to fear is monsters.

I am not a monster.

Therefore, the fish do not fear me (and so I can catch them).

This child was not expressing horror at being considered a monster. Rather she was a little frustrated that the fish were not behaving according to the logic she knew to be true. Or perhaps she was a little frustrated with the inadequate (and illogical) explanation her mother had provided.

In any case, her logic was perfect.

We don’t really expect this of three-year-olds but we should. Just as we don’t really expect rich place value ideas in kindergartners, but we should. If we keep our ears and eyes open, we’ll see it and hear it and be able to support its growth.

# Let the children play

Talking Math with Your Kids has been on something of a summer hiatus as I’ve geared up for Math On-A-Stick at the Minnesota State Fair. It has been a wild ride.

I have spent the last four days playing and talking math with kids of all ages for eleven hours a day.

My number one message coming out of this work is Let the children play.

Have a peek at our flickr photo albums to see what’s been going on. Here’s a sample (Thanks to Kaytee Reid for sharing these beautiful images).

I have been paying close attention to how children behave in this space we’ve built. I’ll just write about the plastic eggs today, but they stand in as an example for all of our activities.

When children come to the egg table at Math On-A-Stick, they know right away what to do. There are plastic eggs, and there are large empty egg cartons. The eggs go in the cartons. No one needs to give them instructions. (This is by design, by the way.)

A typical three- or four-year old will fill the cartons haphazardly. She won’t be concerned with the order she fills it, nor with the colors she uses, nor anything else. She’ll just put eggs into the carton one at a time in a seemingly random order.

But when that kid plays a second or third time, emptying and filling her egg carton—without being told to do so—she usually begins to see new possibilities. After five or ten minutes of playing eggs, this child is filling the carton in rows or columns. Or she’s making patterns such as pink-yellow, pink-yellow… Or she’s counting the eggs as she puts them in the carton. Or she’s orienting all of the eggs so they are pointy-side up.

The longer the child plays, the richer the mathematical activity she engages in. This is because the materials themselves have math built into them. The rows and columns of the egg crate; the colors and shape of the eggs; the fact that the eggs can separate into halves—all of these are mathematical features that kids notice and begin to play with as they spend time at the table.

We have seen four-year-olds spend an hour playing with the eggs.

I have observed that the children who receive the least instruction from parents, volunteers, or me are the most likely to persist. These are the children who will spend 20 minutes or more exploring the possibilities in the eggs.

The children who receive instructions from adults are least likely to persist. When a parent or volunteer says, “Make a pattern,” kids are likely to do one of two things:

1. Make a pattern, quit, and move to something else
2. Stop playing without making a pattern

We adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas.

But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own.

Here are some ideas children have explored in the last few days. I look forward to the next week’s worth of wonder. (Photos all shared by visitor and volunteers through Twitter and Intagram—handles are in the image titles. Many thanks to all for your generous sharing.)

# What makes a sandwich

The 3-year old daughter of fellow Minnesotan, fellow math teacher and friend Megan Schmidt made the following proclamation a couple weeks back.

This simple claim has led to lots of fun conversation. Let’s call the daughter veganmathpup (since she is the daughter of Twitter’s @Veganmathbeagle), or VMP for short.

All discussions with VMP are filtered through her mom via Twitter. All discussions with my own children are my best recollections of the recent silliness.

## Open faced sandwiches

Veganmathpup’s assertion boils down to this: A sandwich needs these things: (1) a slice of bread, (2) a filling, (3) another slice of bread. I wanted to know about open-faced sandwiches. Is an open-faced sandwich properly called a sandwich? VMP was silent on this matter. So I asked Tabitha.

Tabitha (7 years old): That counts as a half-sandwich…actually more than a half-sandwich.

So an open face sandwich is not actually a sandwich for Tabitha. This gave me a chance to introduce the term misnomer.

A week or so later, VMP claimed that “2, 3, 4 or 5 cookies can make a sandwich”. This was a clear violation of the earlier rule here. Two cookies, no filling? How can this be a sandwich when “It takes three things to make a sandwich”?

So I asked about Oreos. Does VMP think of an Oreo as 1 cookie? 2 cookies? Most importantly, Is an Oreo a sandwich? Megan related the following conversation.

Megan: [Handing VMP an Oreo]  VMP, I have a question.  Is this a sandwich?

VMP (3 years old):  [Examining carefully] Um, no.  It’s not.

Me:  Why isn’t this a sandwich?

VMP:  It doesn’t have things, like a burger.

Me: [Handing her two Oreos stacked on top of one another] Is this a sandwich?

VMP:  [Examining even closer this time] No.  it doesn’t have stuff in it. It needs lots of stuff inside like a burger to be a sandwich.  I want a burger.  Let’s get one [face full of oreos]. We won’t tell Daddy.

So many follow up questions I was unable to ask here. Does a Double Stuf Oreo have enough stuff inside to count as a sandwich? What about a Mega Stuf Oreo?

A Mega Stuf Oreo contains approximately 3.1 times the Stuf of a regular Oreo.

## Marshmallows

Then the plot thickened.

Megan went on to report that, even after opening the Oreo to demonstrate that there is a filling, VMP rejected the Oreo as a sandwich because the filling is white.

Allow me to summarize:

• Three things are required for a sandwich.
• Unless they are cookies, in which case you only need two.
• An Oreo is one cookie, so is not a sandwich.
• Even if you want to call the Oreo wafers cookies and the Stuf the filling an Oreo is still not a sandwich because the filling is white.
• The filling in a sandwich is properly referred to as a burger.

I saw a flaw in the logic, though.

Marshmallows are white.

I HAVE BEEN FOILED BY A THREE YEAR OLD!

## So what do we learn?

Children have ideas.

Children use their minds. They think about things.

We can contribute greatly to our children’s learning by probing those ideas.

Formulating precise definitions is an important part of doing mathematics. Sorting things into examples and non-examples is part of this process. It really doesn’t matter whether we are sorting shapes (square, not square) or food (sandwich, not sandwich). And when the child is three years old, it really doesn’t matter whether she is consistent in her sorting.

What matters is that she is thinking in this mathematical way.

## Starting the conversation

You can do as I did. Tell your child that another child says it takes three things to make a sandwich. Ask your child whether she agrees. Then ask about open face sandwiches and about Oreos.

But the bigger picture is important here too. There is a useful habit to develop as a parent—ask follow up questions when your child makes proclamations.

Other conversations we have had in this vein include Spirals, Circles and Armholes.

# Peeps

This is one of my favorite tasks in recent years. The idea is that we will compare two sets of Peeps. Are there more of one color or the other?

There is so much fun to be had counting Peeps. Now that Valentine’s Day is past, Peeps (a common Easter candy) are back in stores in much of the U.S. So here we go…

In the spirit of Talking Math with Other People’s Kids Month, I report to you conversations other people had about one of these photos, as well as one Tabitha and I had. This is truly, though, a task for all ages.

## Comparisons

Each of these conversations stems from this photograph.

### Liam

Kelly Darke reports this conversation with Liam, who was 3 at the time.

Kelly: Which box has more, the pink or the purple?

Liam (3 years old): Pink.

Kelly: Why?

Liam: Because I like pink.

Kelly presses on with the other photos. Liam offers a color preference each time; sometimes preferring pink and sometimes preferring purple.

This is fine. Liam is clearly not interested—or not ready—to make numerical comparisons. He is enjoying having a talk with Mom about comparisons. Another time, he’ll be ready. In the meantime, he has the idea that comparing collections of things is something people talk about. This increases the chances that he will think about comparing collections of things.

### “Brandon”

Luke Walsh reports the following conversation with his five-year-old son, whom we will call Brandon.

Luke: Are there more pink Peeps, or purple ones?

Brandon (5 years old): The purple is more because it is taller and they ate less.

Notice the difference between a 3 year old and a 5 year old. The 5 year old is using evidence.

Brandon has two arguments here. “Taller” is not a valid one. You could have one column of three Peeps and the taller argument would give you the wrong answer. It is more sophisticated than “I like pink Peeps” but it’s not really right. This is how ideas develop, though. Height is easy to observe, and it corresponds pretty well to size and age when comparing people. So it is commonly applied to quantities, too. As usual, this partially correct answer can lead to more discussion. Luke could ask, Will the taller arrangement always have more Peeps?

“They ate less” is insightful. Brandon seems to notice that the two boxes started with the same number of Peeps, and that if more have been eaten from one box, there are fewer left. The natural follow-up question here is, How do you know fewer purple Peeps have been eaten? and then Why does fewer purple Peeps being eaten mean there are more purple Peeps?

### Tabitha

Tabitha, who was barely six years old at the time, used Brandon’s first line of thinking.

Me: Which are there more of in this picture? Purple Peeps or pink?

Tabitha (6 years old): Purple.

Me: How do you know?

T: It goes all the way to the top.

A follow up task helped to push her thinking a little bit.

T: Purple.

Me: But they both go to the top in this one.

T: This one (purple) has full rows, and this one (pink) has holes.

I have used these Peeps photos to encourage discussions of number with fifth graders, with undergraduate education majors, and with middle school math teachers. Good times for all. With the older ones—and in a large group setting—we strive not to mention the actual number of either color of Peeps, and we strive to have multiple ways to describe how we know which is more.

You can download a complete set of four comparison photos by clicking on this link [.zip]. You can also just click on the photos below to enlarge them. Your choice. Either way, they are free for you to use to encourage math talk. Please report back what you learn.

# Book shopping

Math teacher mom (and long ago former student of mine), Megan Schmidt sent in the following report for Talking Math with Other People’s Kids month…

Her husband (who is not a math teacher) and three-year-old daughter—we’ll call her Marian—are playing “store”. Marian is trading coins and marbles for books and blankets.

Marian (3 and a half years old): I want to buy a book for mommy to read.

Dad: Pick one and I’ll tell you how much it costs.

M (grabbing a small book from her book shelf): This one is new. Mommy wants to read it to me.

Dad: That one will be 3 silver coins.

M: 1, 2, 3. Now I want this one (picks a bigger book)

Dad: How much do you think this one should cost?

M: 5 coins!

Dad: How come this one is three (pointing at the small book) and this one is 5? (pointing at the larger book)

M: This book is large, the other is medium.

Megan writes that Marian is quoting Dad here and that Marian’s fondness for the number five may have more to do with her response here than a certainty that five is more than three.

## So what do we learn?

Trading stuff is a fun game to play.

You don’t need all the fancy store equipment. A few coins and a few valued objects (here books) and you’re good to go.

There is so much opportunity to mention, discuss and ask about numbers. Fun, fun, fun.

While the idea that 5 is more than 3 is not at all beyond the grasp of a three-year old, I do love Megan’s tentative attitude here. It certainly is possible that Marian considers five more valuable than any number—that the large book should cost five coins because five is the best number, even if the medium book costs 23 coins.

## Starting the conversation

A beautiful part of this conversation is when Dad asks Marian, How much do you think this one should cost?

This question invites Marian to think about and to discuss numbers. It’s lovely, easy to do and is very low risk for both child and parent. It is low risk because there is no wrong answer. Marian is free to set her own price, but thinking about what that price ought to be engages her mind in a deeper way than does simply counting out the coins.

Don’t get me wrong: counting out the coins is a lovely activity too. But How much do you think this one should cost? is a brilliant conversational move that got even more thinking from a three-year old.

# 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.

# Uncle Wiggily

Tabitha was $3\frac{1}{2}$  years old when we were playing a game of Uncle Wiggily.

In case you are not familiar with the game, I’ll briefly describe it. Uncle Wiggily is a board game with 100 spaces along a twisty path. Players draw cards; each card has a number and a brief poem. Perils and bonuses are judiciously spaced along the path. Uncle Wiggily is approximately 10% more complicated than Candy Land (which is to say, not very complicated at all!)

Tabitha: (Drawing a card for her first turn-it’s an 8) Got one Daddy!

Me: Mmm-hmmm.

T: What is it?

Me: Can you guess? Look closely.

T: (Quickly and with a big, eager smile on her face) Six!

Me: Good guess. It’s eight.

T: Oh!

Me: Can you count to eight?

T: (Bouncing her piece along the path, ending near the henhouse on the farm-themed board) One, two, three, four, five, six, seven, eight. By the cluck-cluck house!

Me: My turn. (Drawing a card-it’s a 10) What card did I choose?

T: Ten!

Me: Good. (Testing a hypothesis, I skip eight as I count) One, two, three, four, five, six, seven, nine, ten.

T: (Oblivious) My turn.

# So what do we learn?

Learning to count is messy. Many things we might expect to be true about how children learn to count are not true at all.

We might expect children to learn the numerals (8) at the same time that they learn the words (eight). They do not. Notice that Tabitha counted flawlessly to eight, but did not recognize the symbol “8”.

We might expect children to learn the numerals in order, with all multi-digit numbers coming only after mastering the single-digit numbers. They do not. Tabitha recognized “10” but not “8”.

When I counted to ten, I intentionally left out eight to see whether she would notice. She did not. She could count to eight, but didn’t notice when it got left out on the way to ten. Mathematics is logical and orderly. The ways children learn mathematics are not.

This conversation came from a short video I made one day. I watched this video a year and a half later, when Tabitha was 5. After watching it, I immediately went into the kitchen where she was having a snack and counted: 1, 2, 3, 4, 5, 6, 7, 9, 10. She smiled and asked Why did you do that? (referring to counting in her ear). Then, a moment later, she said, Hey! You skipped eight!

# Starting the conversation

Play games that involve numbers. Uncle Wiggily is great. So is Chutes and Ladders. Or Hopscotch. Any game that involves counting and reading numerals will give you the chance to practice these early number ideas.

While you’re playing, ask your child what number he drew, and what number you drew. If he doesn’t know, have him guess. Don’t worry about precision or correctness. Model good counting for your child. Help him count out some of his turns and let him count incorrectly on others. Have fun and don’t worry too much if he gets bored before the game is finished.

Have fun with it. Whatever you need to do to stay engaged in a couple of rounds of Uncle Wiggily is worth the effort. You can see the effort I invested in keeping myself entertained; I formulated a hypothesis about whether she would notice my own incorrect counting and tested that hypothesis.

Don’t get carried away with the hypothesis testing, though. Children do need models of correct counting. They won’t be damaged by a few experiments, of course. But you don’t want to become an unreliable source of knowledge.