# Spirals

A few weeks back, this short cryptic video came to my attention thanks to the magic of Twitter.

Thanks to kids connect (@KinderFynes on Twitter)

For more than a year now, I have been posting links and other short bits on Twitter using the #tmwyk hashtag. In the last few months, it has gained momentum. A day rarely goes by without someone posting something interesting or delightful or surprising there.

But back to the video.

We get a very brief glimpse of a classroom of Kindergarteners on a walk. At the moment the video captures, they are trying to decide whether the object on the wall is, or is not, a spiral.

That image in the video was not a spiral because “Spirals are connected”.

So I drew this.

Griffin’s reply: That’s three things connected, not one thing.

So I drew this (sort of).

The part I actually drew was two disconnected spirals. He drew the short line segments on the ends.

Griffin: If you close them off like this, it’s an outline of a spiral.

Next I drew this.

I was wondering whether spirals needed to be roughly circular.

Griffin: In this one, you are looking at a spiral from its edge.

Finally, this one.

I cannot recall his response. We were on the porch on a warm lazy sunny spring morning at the end of a long long winter. We may have gotten distracted.

## So what do we learn

This is how I teach critical thinking. Not just at home, but in my work, too. Get the child to make a claim and to give a reason supporting it. Cook up a problematic example and ask for a new claim. Repeat. Quit before angering child.

WARNING: It is my experience with my own children—as well as with my students of all ages—that they learn these lessons well. This means that over time they begin to argue back intelligently, and that they begin to pick apart my own claims and arguments.

# 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.

# [Product review] Leap Pad Paint Bucket

Tabitha’s neighbor friend has a Leap Pad. Naturally it became a much hoped-for Christmas gift. She did receive one and has spent quite a bit of time with it.

I have no interest in reviewing the thing itself (although I will give you a heads-up that apps on this thing are expensive in comparison to iOS and Android! Holy buckets!)

The Leap Pad comes with a few standard apps. One of these is a drawing app, called Art Studio. Tabitha (6 years old) has drawn many pictures on it.

This is the sort of thing I’m talking about.

This is fine.

And I wondered whether I could get some math out of it.

See, there is a paint bucket tool in there. When you apply the paint bucket, the paint fills up your drawing, but it doesn’t go across lines you have already drawn. So if you draw a square, you can paint the inside of the square and the paint won’t leak out. Or paint the whole screen outside the square and the paint won’t leak in. Unless you leave a small hole, in which case, the whole screen gets painted because the paint leaks through the hole.

I showed this feature to Tabitha and proceeded to draw some complicated curves, asking her to guess where the paint would go. For example, I drew a spiral.

This was no problem for her.

I asked her how many colors we could use to paint some complicated curve pictures if we used a different color for each section of the drawing.

Again, no problem.

I had her draw pictures and make me guess.

Finally, after about five minutes of this, she announced, “Daddy! You’re not allowed to do math on this!”

I was busted. I had to take a time out and let her just play with her toy.

But then, going back and looking at her more recent art, I can see I got into her head.

Don’t worry, though. The horses are still making appearances.

So if you have a Leap Pad in the house, I gladly give two-hooves-up for math in the Art Studio!

# 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.

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:

T: Oh no.

Me: What?

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

She tries again, producing this:

T: 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.

And I start to share out the bottom row—half up, half down.

Tabitha: 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.

We 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.

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

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.

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”.

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.

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.