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.

I decided to ask Griffin (9 years old) about this to see what his ideas would be.

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

So I drew this.

spiral.post.1

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

So I drew this (sort of).

spiral.post.2

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.

spiral.post.3

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.

spiral.post.4

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.

10-minute reading time

A while back, bedtime was spiraling out of control. The kids share a room; they would be wound up at bedtime and the transition to sleep was not happening smoothly. We had a big, big problem on our hands.

We solved the problem with 10-minute reading time. The kids have to be in their beds. We dim the lights. We set a timer for 10 minutes. It has to be quiet during that time. Then we turn out the lights, give them something to picture in their minds, and sleep comes more easily.

Complete transformation. It is awesome.

One night, Tabitha (5 years old at the time) wanted to color. We talked and agreed that she could do it “sometimes”. As is the nature of 5-year olds, she soon wanted to know the limits.

The following conversation took place on a Wednesday night.

Tabitha (five years old): I know I can’t color every night but can I tonight?

Me: Yes.

T: Then read, then color the next night?

Me: I don’t know. I think reading twice before the next color is better.

To be clear. It was not my intention to get into a math conversation at this point. I just wanted her to go to bed (Warning! Link Not Safe for Work, and Possibly Offensive to Sensitive Ears. But Funny).

No, this move on my part was truly about literacy, not math. I don’t want 10-minute reading time to turn into 10-minute coloring time. I really, really like the idea that books will become part of my kids’ independent bedtime routines.

But Tabitha loves to know the rules she’s playing by. And when those rules are based on numbers, they’re going to lead to math every time.

T: So read-read-color-read-read-color…like that?

Me: Right. That sounds like a good ratio.

T: Or read-read-read-read-color-color?

Whoa.

Couple things.

First of all, I used the term ratio with absolutely no expectation that she would process it, and I am quite sure that she did not. I have long been an advocate of using good vocabulary with my children—there is no shame in not knowing the meaning of a word, but also no sheltering them from the fact that these words exist. This is at least partly the source of their substantial vocabularies. But I do not believe she knows the word ratio.

Secondly, Tabitha’s reformulation of the 2:1 ratio as 4:2 blew me away. It nearly slipped past me without notice. I was focused on getting them to bed; we were in the truly final phase of that process. I had pretty much tuned her out.

But when I looked at her, I could see she was expecting a reply. She needed to know whether she could get two coloring nights in a row by doing four reading nights in a row.

So I replayed her question in my mind, counting the reads.

Me: Yes. That would be fine. You may do that.

So what do we learn?

Together with the recent Easter Candy conversation, this makes clear that young children are thinking about early ratio ideas.

Think about this for a moment. What is the same about these two sets of tiles?

tile.ratio.2

And what is the same about these two sets of tiles?

tile.ratio.3

In each cases, it is the ratio.

In the first case, there is one blue for every yellow up top and also down below. This is what L was working on when she offered a second chocolate egg to Tabitha.

In the second case, there are two blues for every yellow. This is what Tabitha was working on when she asked about read-read-read-read-color-color.

Traditionally we think about ratio as a sophisticated fractions topic that needs to wait for early adolescence. I certainly would not want to be held accountable for teaching 5-year olds ratio and the associated notation. But their everyday experiences allow for them to think about these ideas. As parents, we can keep an eye out for those opportunities and talk about them when they arise.

Starting the conversation

Both of these ratio conversations with 5-years olds have resulted from constraints. Five year olds love to test rules. “Yes” and “No” are inflexible and allow no wiggle room. This is sometimes desirable. Yes, you must leave now to get to school on timeNo you may not leave the house without pants. Yes you must look both ways before you cross the street.

But when we are guiding our children’s behavior, we would be wise to allow some space for the little ones to test the boundaries. Does it really matter—in the big picture—whether L eats one chocolate egg or two? Does it really matter whether Tabitha colors two nights in a row instead of reading? I say no. Neither of these things really matters.

What matters is that L does not continuously gorge on candy, and that Tabitha has some alone time with books. Constraints rather than absolute mandates seem to have encouraged mathematical thinking in both of these cases while also addressing the big picture.

Easter candy

Easter Sunday saw St Paul, Minnesota waking up to weather perfection. Sunshine, low seventies (Fahrenheit), cloudless sky. Truly amazing.

There was a loon on Lake Phalen!

This was the sort of April weather that brings Minnesotans out of their homes to rediscover their neighbors.

So it is with Griffin, Tabitha and me on this warm spring morning. We are enjoying the warm sunshine on our front steps when L (five year old girl), O (3 year old boy) and their mom come biking, biking and strolling (respectively) down the sidewalk.

L is on a neighborhood mission delivering handmade Easter greeting cards.

It turns out that she has also pocketed some goodies from her own Easter basket. While we chat, she pulls out a bag of Cadbury mini eggs. In case you are unfamiliar, these are the size of pebbles. They are chocolate inside with a crunchy candy shell. Like an oversized egg-shaped M&M. Each little bag contains about a dozen.

so.much.candy

So. Much. Candy.
This is likely a small fraction of the candy L has consumed by the time she stops to chat.

Anyway, mom notices the bag as soon as it emerges from L’s pocket. (Side note—mom is across the street! Holy SuperMom powers!) She warns L not to eat any more of these; arguing that L has had enough candy for one morning.

L (5 years old): Please?

Mom: If you give everybody one, you can have one.

L proceeds to cheerfully open the package, hand one to Tabitha (who eagerly and gratefully receives it), one to Griffin and one to me.

I begin to think about what question to ask to get some math talk going.

But L is ahead of me.

After enjoying both her egg and a long thoughtful pause, she pokes her finger back into the bag. She begins to rummage around and asks:

L: Tabitha, do you want a second one?

So what do we learn?

Children use math to their advantage.

L knew what mom meant. Mom had compromised on the candy, allowing her one piece. L knew that. And she knew that the process was repeatable.

One does not always mean one. One might be taken to mean each. “Each time you give everybody one, you can have one.” This is also a reasonable interpretation of mom’s words.

L was rule bending here. But she was also building the precursors of ratios. For every one you give a friend, you can have one. This is a ratio. Giving a friend two and having two fits this rule just as well as giving a friend one and having one. Ratios are one of the more challenging ideas behind multiplication and division relationships, and fractions.

What is maddening for parents is at the same time great thinking practice for children.

Starting the conversation

This was a brilliant compromise strategy on mom’s part. I doubt that she intended to encourage L to think proportionally, but that doesn’t matter. More likely, she was trying to encourage the admirable social skill of sharing. By including numbers in her compromise, she opened the door for L to think.

As I have mentioned before, anytime your child wants to open a negotiation, there is an opportunity for math talk. Sometimes we parents need to give a flat out yes or no. But when negotiations are feasible, we can get our children thinking.

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.

peep.compare.1.small

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.

peep.compare.4.small

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] The bathtub

Talking Math with Other People’s Kids month continues…

Today we pay tribute to the family bathtub, and its profound contribution to family math talk over the centuries.

Photo Feb 04, 9 13 03 PM

 

Don’t laugh! Is yours more perfect?

Dad and loyal reader Jon Hasenbank reports some math talk at bathtime with his own 5 year old son, whom we will call Isaiah.

Isaiah is in the bathtub, having a lovely time. He has stacked his bath-toy Elmo on top of his bath-toy Cookie Monster.

isaiah (5 years old): Look! His eyes are peeking out!

Dad: The water is almost over his head. I wonder if it’s deeper near your feet?

He did not report further details to me.

But he did demonstrate an important principle of talking math with your kids—It’s not a conversation until you, as a parent, participate. When Jon turned Isaiah’s observation into a wondering, he set the stage for some good math talk.

The bathtub is great for this!

Tabitha has complained about the depth of her bath in the past—always that it is not deep enough. “It’s not even one foot deep” she has wailed as her toes stick out of the water. “Is it one hand deep?” I have asked. And—as with Jon and Isaiah—we have been off and running on a lovely exploration of measurement.

 

February is “Talking Math with Other People’s Kids” month

You won’t be hearing much from Griffin and Tabitha this month. Instead, you’ll hear from other children and their parents who have talked math and have shared their conversations with me.

It will be a ton of fun to get a peek into these other households, and to see how frequently ideas and questions about number and shape come up in life with young children.

I would love to hear your own reports, and to gather a collection of stories representing diverse families, cultures, languages and experiences. Shoot me a note describing conversations you have participated in or witnessed. I’ll feature as many of them here as I can.

Let’s kick things off with an example of a father and his five-year old daughter, and how Twitter helped them talk a bit more math than they otherwise might have…

Andy is a dad in Minneapolis. Let’s call his daughter Martine. Andy tweeted me on Friday (January 31).

Here is how this sort of thing would go in our house.

Martine (5 years old): If tomorrow is February first, does that mean today is February 0th?

calendar

Dad: Yeah, I guess we could call it that. If we do, what would yesterday have been?

M: February negative one.

Dad: Oooo. Nice! And what about the day before that?

Et cetera. At some point, the conversation would go somewhere else. Or if she’s still interested, I might give it a twist with a question like this.

Dad: So if today is both January 31 and February 0, and if tomorrow is February 1, shouldn’t it also have a January name?

I would be probing Martine’s double-naming idea for each day. And then…

Dad: Hey! I know! If today is both a January and a February day, then tomorrow should be both a February and a March day, right? What is tomorrow’s date in March?

As I mentioned, the conversation may very well have broken down by this point. But these what if questions are the things that turn a cool observation into a conversation. That conversation is where we turn kids’ minds on.

Dad Chris Hunter suggested that first follow up question: What about yesterday?

Andy asked Martine about that on Saturday.

Martine said that the day before February 0 would be February negative 1. Andy reports—and this is important—never having explicitly discussed negative numbers with Martine. No number lines, no backwards counting past 0.

But surely they have talked about the weather. Below-zero temperatures have been as common as snowflakes in Minnesota this year. Talking about the weather may have planted the idea. Then the calendar was an opportunity to make a connection.

All of this leads to two important ideas about talking math with kids:

  1. It’s not a conversation until you, as a parent, participate. Martine noticed something (Jan. 31 could be Feb. 0). Andy turned it into a conversation when he asked about the previous day.
  2. These conversations are facilitated by availability of objects. Turning the calendar became a learning opportunity for Andy and Martine. No calendar, no conversation.

You can read our full Twitter conversation here.

And you can read about other conversations facilitated by objects in these previous posts:

Playlists

Parenting is a tremendous amount of work. Within that work are beautiful moments of love and joy. For Tabitha and me, these moments often involve music. We had an impromptu dance party in the kitchen the other night that began with my putting on some music to do dishes by.

When Griffin was born, I began maintaining playlists. Each year, I collect songs that the kids liked, or that I was listening to, or that reminded me of them in some way. Some years I remember to burn these to CDs to share with family members. But I never delete them.

That first playlist is titled “Griffin year 1”.

Do you see the math here?

Tabitha (5 years old at the time): Are you done with my year 5 playlist yet?

Me: Yes. I finished that when you turned 5. Now I’m working on your year 6 playlist; I’m collecting a bunch of songs during the year and it will be done on your birthday.

T: Why isn’t this my year 5 playlist?

Me: Good question. Well…your first playlist I started before you turned one…

T: When I was zero years old.

Me: Right. Then when you turned one, I started your year 2 playlist. That’s what it means to be 1 year old; that your first year is over and you’re in your second year.

So when will I work on your year 10 playlist?

T: When I’m 9.

Me: How do you know that?

T: I don’t know. I just do…

So you’re working on Griffy’s year 9 playlist now? [Her brother Griffin was 8 years old at the time.]

Me: Yes. Nice. I was just about to ask you that, but you thought about it on your own. Good thinking.

T: Will you still be working on them when I’m an adult?

Me: I would gladly still work on them when you’re an adult. I don’t know if you’ll want me to at that point, but if you do, I will.

T: Oh, I will. Hey! Can you play my favorite song about the flower?

And so began the dance party.

So what do we learn?

There is an important idea about counting and measuring here. During your first year, you are zero years old. Something that measures within the first inch on a ruler is zero inches long (plus a fraction).

This is not obvious by any means. If you have ever been frustrated by the fact that the 1900s were the 20th century, or that ours is the 21st, you understand the problem.

Starting the conversation

These are fun things to talk about. Almost always, going back to the beginning is helpful for making sense of things. So ask your child about 2014 being in the 21st century, and why they think that is.

Or maybe start making an annual playlist. You won’t regret it.