Counting in downtown Saint Paul

I had my first book event today—for How Many? at Subtext Books in downtown Saint Paul. Lovely people, great little independent bookstore. You should buy some books from them.

We had a small but loyal crowd that included a three-year old and an eight-year old. The three-year old was charming, as all three-year olds are, and today she answered all yes/no questions in the affirmative. She and I talked about shapes and eggs and money. It was good times.

But I really got to get into the head of the eight-year old.

We discussed the grapefruit page below, and the unsolved mystery of whether there are exactly six grapefruit—the ones we can see directly—or more than that with at least one hiding underneath, possibly reflected in the surface of the bowl.

6.jpg

We moved on to the next page, which is where the real fun began.

7

My eight-year old conversation partner looked carefully, thought for a while, and announced that there must have been more than six grapefruit on the previous page because there are more than six on this page.

I asked, “How do you know?” and it turns out he was visually pairing the grapefruit halves on this page. He used his fingers to show me the pairs he made, but he was having trouble keeping track of their number. So when he came out with more than six pairs of grapefruit halves on this page, that meant there must have been more grapefruit in the bowl.

We flipped pages back and forth several times while sorting this out, and he finally concluded that there were six grapefruit on both pages. Children rarely have math tasks that connect this way, but they expect that the tasks should connect. It was delightful to watch this expectation play out.

Next up was the avocado page.

10

He thought for a bit and decided there were “seven point five avocados”. I thought I knew how he knew—same as the grapefruit—but I asked to be sure, and I was wrong.

“Three fives is fifteen, and then divide by two.”

It took a few more exchanges to extract that dividing by two makes sense here because there should be half as many whole avocados as there are half-avocados. Of course this is brilliant and important mathematics, and it arose in the context of making sense of a meaningful counting situation. Also notable is that three fives was a fact he retrieved quickly while three fours (of grapefruit halves) did not seem to occur to him.

The lesson here is that children are brilliant. They build math out of their everyday experiences, and when you offer them opportunities they apply the math they know to make further sense of their worlds.

Another lesson is that my new book—titled How Many?—is out. The best price and free shipping are at Stenhouse.com. If you read it with children, please report back and maybe leave a review at Amazon.

 

Cocoa Puff or Cocoa Puffs: The language of nothing

In honor of Tabitha turning 11 this week, here’s a conversation from 6 years ago. 

We have a little family tradition. When we go grocery shopping the weekend before your birthday, you can choose one box of any cereal you want-no restrictions. In the weeks and months leading up to the grand event, much time is spent in the cereal aisle weighing the advantages of the various sugar-laden options.

The week before turning five, Tabitha nearly dropped the ball. She just grabbed the first box of anything at hand. I don’t remember what it was, but it seemed out of character for her. I reminded her of the cereals she had been coveting as recently as the previous week.

She went for the generic Cocoa Puffs.

I steered her towards the real deal. If you’re only gonna eat ’em once a year, you might as well have the sugar-addled bird bouncing off the box in front of you, right?

A sugar-addled bird

One morning shortly afterwards, we had this conversation:

Tabitha: Do I have Cocoa Puffs or Cocoa Puff in my hand?

Me: Well, you have four Cocoa Puffs.

T: [with only one in her hand now] Do I have Cocoa Puffs or Cocoa Puff?

Me: You have Cocoa Puff.

T: [huge smile] Right!

Me: [with empty hand displayed] Do I have Cocoa Puffs or Cocoa Puff in my hand?

T: [silent but smirking]

Me: Well…Is it Cocoa Puff or Cocoa Puffs?

T: [continued silence]

Me: I have zero…

T: [bigger smile]

Me: …Cocoa…

T: Puffs!

Me: Yeah. Isn’t that weird? If you have one, it’s Puff; if you have none it’s Puffs.

T: I knew that.

Me: Of course you did.

T: No! I knew that; I was showing you that [you had zero] by not saying anything-zero words!

So what do we learn?

Children listen carefully to language patterns. They do not learn a native language like a second language in school. The rules are not carefully explained to them one at a time.

Instead they listen, speak, get corrected, and try again. All of this can be tremendously fun for child and parent alike.

It is an odd quirk of English that zero is plural, grammatically speaking. We talk about having one child, but zero children. More commonly, we use no instead of zero, as in My neighbors have no children. The grammar is the same either way; saying My neighbors have no child sounds funny to our ears.

Starting the conversation

In discussing place value, zero is sometimes called a place holder. To understand that, children need to understand zero as a number. They need to understand that zero can legitimately answer the question, How many are there?

We talk a lot about zero in our house. You can too. Ask your children, “Would you rather have one cookie, two cookies or zero cookies?” Ask who has more of something, even when one of the people has none.

The Mad Hatter in Alice in Wonderland gives an example of this. The March Hare offers Alice “some more tea”. When Alice says she can’t possibly have more, since she hasn’t had any yet, the March Hare replies, “It’s very easy to take more than nothing.”

Another silly language game we play in our house is this. If you look in the pantry and see that there are three cookies left, you can report this in the following two ways: (1) “I checked the cookies; there are three left,” and (2) “There are three cookies.” If, however, there are no cookies in the pantry, these two ways of reporting the sad fact become: (1) “I checked the cookies; there are none left,” and (2) “There are none cookies.” We like to treat none as a number. There is no good reason for this; it is for personal amusement purposes only.

Postscript

Tabitha again chose Cocoa Puffs on this, the week of her eleventh birthday. She is enjoying them, but she has also stated the obvious—they look like rabbit poop.

Time Zones

Griffin is 13 years old and seems to be coming to the end of that early adolescent phase of rejecting everything those around him hold dear. Engaging him in math talk has taken more finesse in this phase of life.

Mostly it has involved giving him responsibility for things that involve making calculations. When he was little, we could talk collaboratively about how many tangerines are in a 3 pound bag and discuss whether this would be enough to last the family a week. Now I tend to put him in charge of getting enough tangerines to last us a week. He still has to do the same thinking, but he’s in charge.

This is not enough tangerines for a week at our house. (By the way, which is more?)

From time to time, though, we still put a mathematical idea up for discussion, and as he ages through adolescence, these conversations happen a bit more often. Yet he is still wary. Nevertheless, I persist.

We have been watching the Olympics, and we have wondered about which events are happening as we watch them, and which ones happened earlier (yet somehow happened “tomorrow”!)

Griffin was thinking about time zones, and about their implications for traveling as we wrapped up an evening this week, and made preparations for the next day.

Griffin (13 years old): So they’re 14 hours ahead of us?

Me: Yes.

G: You’d get a lot of jet lag, huh?

Me: Yeah. Maybe not as much as it looks like, though. Maybe it’s just 10 hours’ worth, going the other way.

There is a bit of a puzzled silence.

G: Wait. Really?

Me: Yeah. Well, plus a day.

G: Wait. Is this one of your mathy talks?

Me: Absolutely not.

If you’re reading this, Griff, I’m sorry (sort of). I am totally busted.

Me: Yeah. 14 hours ahead is the same as 10 hours behind, right? Just going the other way.

G: But the day would be wrong.

Me: Yeah. You have to add a day, but you don’t get jet lag because the day changes, you get jet lag because the time of day does.

G: Maybe.

He returns to packing his lunch. I go back to whatever I was doing. Putting turtles in boxes, probably.

A couple minutes later…

G: So the east coast is 23 hours behind us?

So What Do We Learn?

Keep trying. Opportunities to talk about numbers, shapes, and patterns present themselves. Seize them and do not stop. Ask questions, think out loud. Don’t worry about whether any particular conversation goes anywhere. Just keep at it.

Cold snap

Tabitha (9 years old) is keenly attuned to the temperatures these days, as subzero air temperatures or wind chills mean indoor recess. Being a child of great physical energy, indoor recess is not ideal.

photo-dec-15-9-05-57-am

We have an indoor/outdoor thermometer on our kitchen table, which she checks several times a day. Yesterday evening before doing the dishes together, she checks the thermometer.

Tabitha (9 years old): It’s 1 below.

Me: What was it this morning? Five degrees?

T: Four

Me: Crazy. So it’s colder now.

T: Yeah.

Me: How much colder?

T: Five below

Me: How do you know? Is it because 4 + 1, or did you count?

T: Neither.

Me: Oh! Now I have to hear it!

T: Well…Four minus four is zero, then it’s one less, so it’s five.

Me: So one more than four less…er…one less than four….no….

[we laugh]

T: It’s one more because it’s one less!

So what do we learn?

This conversation reminded me very much of a game I used to play with Griffin (who is now 12 years old) on cold winter mornings. In both cases, the children naturally developed a strategy using zero as a stopping point in making comparisons.

The thing I especially love about this story is that Tabitha expresses a complicated relationship that is crystal clear to her: “One more because it’s one less.” Expanded out, she’s saying that “The difference between -1 and 4 is bigger than the difference between 0 and 4—the difference is bigger by 1 because -1 is one unit further from 4 than 0 is.”

She can express this complicated idea because it is her own.

If I tried to tell her that this is how subtraction with negative numbers works, she would definitely pronounce my ideas confusing—whether they were expressed in the language of 9-year-olds or the language of mathematicians.

I cannot tell her these things and have them be meaningful. What I can do is ask how much colder it is now than it was this morning.

Starting the conversation

Move to Minnesota.

I’m kidding.

You can buy a Celsius thermometer, though.

You can make comparisons more generally, both asking your child how she knows, and talking about how you think about it. How many more full cups in the muffin tin than empty ones? How many more fork than spoons? How many more adults on the bus than children (or vice versa)? How many more quarters than dimes in the change bowl?

How Many? An invitation to #unitchat

Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn’t BelongI showed a way to make geometry playful. Now with How Many? I’m working on a way of making counting playful.

The idea has grown out of the TED-Ed video I did a while back, and the more I play with it, the more I see it in the world around me. My goal is to help parents, teachers, and especially children see it too.

Most counting tasks tell you what to count. Whether it’s Sandra Boynton’s adorable board book Doggies, or Greg Tang’s more sophisticated The Grapes of Math, the authors tell you what to count—or even count it for you.

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn’t specify what to count. Each image has many possibilities.

An example. How many?

shoes-box-open-2

Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you’ll see.

It’s important to say what you’re counting, and noticing new things to count will lead to new quantities.

Another example. How many?

2016-11-01-09-00-17

A few possibilities: 1, 2, 3, 4, 6, 12, 24, 36. What unit is each counting? Maybe you see fractions, too. 2/3, 4/6, 3/4, 1/12….others? What is the whole for each fraction? The number 3 shows up more than once—there are three unsliced pizzas, and there are also three types of pizza. Are there other numbers that count multiple units?

All of this leads to two specific invitations.

Let me come talk with your students.

(It turns out my schedule filled very quickly, and I’m no longer seeking new classrooms to visit right now—thanks to everyone for your support!)

If you are within an hour of the city of Saint Paul and work with children somewhere in the first through fourth grades, then invite me to come test drive some fun and challenging counting tasks with your students. I have set aside November 17 and 18 and hope to get into a variety of classrooms on those two days. Get in touch through the About/Contact page on this blog.

Join the fun on Twitter.

I’ve been using, and will continue to use and monitor, the hashtag #unitchat, for prompts and discussion of fun and ambiguous counting challenges. Post your thoughts, your own images, the observations of your own children or students, and I’ll do likewise.

How Many? A counting book will be published by Stenhouse late next year.

Birthday Chocolate

Today is my birthday. Griffin (12 years old) gave me three chocolate bars as a gift. He gave me candy because he is deeply aware of its value in life. He gave me dark chocolate because he knows it’s my preference.

He is frequently disturbed by how slowly I eat these gifts of candy he gives me.

Here’s how my after-work greeting went this evening.

Griffin (12 years old): Happy birthday, Dad.

Me: Thanks.

G: One thing I’ve noticed about you is that you eat the candy I give you incredibly slowly.

Me: I know. But actually I ate half of one today.

G: Half of a bar, or half of all the bars?

Me: Half of one bar. And then maybe I’ll have another half tomorrow.

G: Oh brother.

Me: And since I know 3 divided by 1/2 is 6…

G: You ate one-sixth of it.

Me: And it’ll last me 6 days.

Having arrived home a bit chilly and damp from the bike ride in the 45° rain, I went downstairs for a shower and he returned to his iPod.

So What Do We Learn?

I haven’t written a lot about this boy recently because he is in a phase of rejecting everything the adults around him care about. All adolescents go through some form of this. He is doing it with gusto.

In any case, the groundwork we’ve laid in the early years has paid off. When math is useful for his purposes, he will use it. Here, he wanted to prove his point that I am a painfully slow candy consumer. That made it important to clarify that I had not eaten half of my candy, but only half of one bar of candy.

We play around with units like this frequently. It has contributed to both children’s place value understanding, as well as their fraction work.

Starting the Conversation

Ask frequently about the units that are attached to the numbers in your lives. When you’re cooking, ask, Should we use 3 eggs or 3 dozen eggs? Ask about how many pieces of candy a pack of Whoppers is at Halloween.

Look at these pictures—one at a time—and ask How many? Challenge yourselves to find different numbers, and different units. (For example, there are 15 avocado halves, 7.5 avocados, 8 pits, 7 holes, and 1 cutting board).

 

Ceiling fan arithmetic

Summer has arrived in Minnesota, and that means we alternate between warm days where we open the windows and run the ceiling fan, and hot days where we close everything up and run the air conditioning (a luxury, btw, that our 1928-built home only got about five years back).

ceilingfan

Not our ceiling fan.
Image credit: Brian Snelson (CC-BY 2.0)

Tabitha is naturally curious about how the ceiling fan works. In case you don’t have experience with them, or yours works differently from ours, here are the basics: There is a switch on the wall—just like a light switch—that powers the fan. Then there is a chain hanging from the fan itself that affects the speed. There are four settings controlled by the chain—High, Medium, Low, and Off.

By the time this conversation takes place, Tabitha and I have already explored a variety of ceiling fan questions, such as If the fan is off, should you pull the chain to turn it on, or head over to the light switch? and How many times can I pull the chain before my parents tell me to stop playing with the fan?

On this day, I ask Tabitha to flip the wall switch to turn on the fan, which she does. She then starts to stand up on the couch to reach the chain. I ask why.

Tabitha (9 years old): I want to see if it’s on high.

Me: But how will you know? If you pull the chain it will slow down, but that’s what it always does. So how will you know whether it was on high to begin with?

T: Well, it doesn’t always slow down, otherwise how would it ever be on high?

So What Do We Learn?

There is some very deep math going on here.

Tabitha and I are playing with properties of modular arithmetic, but she (and you) don’t need to know the specifics. Things that go in cycles are all examples of this kind of math.

The classic example is time. You could say that later times have bigger numbers. 4 is later than 3; 12 is later than 9. This is just like my claim that every pull of the chain slows down the fan. Both of these claims are only sort of true. Three in the afternoon is later than 11 in the morning, despite having a smaller number. If the fan is on low and you pull the chain twice, it’ll be on high.

People study these things in great depth in the field of Modern Algebra, and the ideas are useful in all sorts of places.

Starting the Conversation

Play with a ceiling fan. Talk about staying up all night. Notice together that weird things happen when the fan is in the off position, and at midnight and noon. Wonder aloud whether 12 o’clock is like zero (and if not, what is?)

Play around with basic facts in this ceiling fan environment. If it’s on high, how many pulls to turn it off? If it’s on low, how many to get it to medium? I just pulled the chain three times and now it’s on low—where was it before? Etc. Challenge the child; have the child challenge you.

Stay cool!

Weighing onions

I have had several conversations with relatively new parents in which the question of how/whether to talk math with babies.

I always try to help such parents see math like they see reading. You read with your baby long before she knows what your words mean. An important reason to do so is to immerse the child in language. This is how she will learn language. Reading books increases the variety and quality of language the child is exposed to.

It’s the same with math. We can surround our children with number and shape long before they understand what these things mean. It is through this exposure that they learn.

For parents of children of all ages, this principle applies. Don’t worry about whether the child can get right answers; make a conscious effort to notice number and shape in your world together. It is through this exposure that they will learn.

To this end, Tabitha and I have been playing with the scales at the grocery store. Not the ones at the checkout; the ones in the produce department.

The other day we found a rather large onion.

Tabitha holding a large onion

Here she is holding the onion safely back at home.

Me: What do you think this weighs?

Tabitha (8 years old): Four pounds.

Me: Hmmm…I say a pound and a half.

T: Half a pound!

She is easily influenced. We put it on the scale. It’s a pound and a quarter. I celebrate my victory briefly.

Then Tabitha notices the bananas are nearby. There are several individual bananas lying loose. She grabs one and begins to put it on the scale.

Me: Wait! Not yet! Let’s guess what it weighs.

T: With the onion…two pounds.

We add it in and see that now it’s very close to one and a half pounds.

Pretty soon we are weighing bananas by the bunch and guessing whether an avocado is heavier than a banana.

We are surrounding ourselves with numbers and having a grand old time.

So What Do We Learn?

Immersing your child in numbers is low stakes and opportunities are everywhere. We grocery shop every week, but have only recently started playing with the scales. As a general principle, anytime you encounter a number in the company of your children, you can talk about it.

When the children are infants, they won’t participate. That’s OK. They’ll learn that numbers are things to talk about.

When the children are older, they’ll make wildly inaccurate guesses. That’s OK. They’re getting practice talking about numbers.

When the children are even older, they’ll start to turn their wildly inaccurate guesses into serious learning.

Along the way, they’ll initiate the conversations themselves because you will have taught them that numbers are things people talk about.

A tale of two conversations

Here are two conversations about hot chocolate.

The first one didn’t happen. The second one did. Read them both, then I’ll tell you about their meaning.

Both conversations begin on a cold November night in Minnesota. Unseasonably cold. Fourteen degrees, to be precise (–10 Celsius).

A cup of hot chocolate

Zero marshmallows for me on this cold night.

Tabitha (7 years old), Griffin (10 years old) and I get in the car to head for Tabitha’s basketball practice.

What might have been

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have marshmallows in it, it does.

T: I won’t have any marshmallows, then. So I can have some Jell-O.

Griffin agrees that this is the way to go, and the conversation moves on to other things.

What actually happened

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have two marshmallows in it, it does.

T: I’ll have zero marshmallows in mine, then, so I can have some Jell-O.

Griffin (10 years old): I’ll have one marshmallow, and a small serving of Jell-O. Wait, no! I know! I’ll cut a marshmallow in half!

I presume that this is in order to maximize his allowable Jell-O serving, while still retaining some marshmallow in his hot chocolate. It’s a scheme nearly as complicated as credit default swaps.

So what do we learn?

One small difference changed the course of the conversation—my use of a number word. I could have said, “It counts as dessert if you have marshmallows in it.” But I did say, “It counts as dessert if you have two marshmallows in it.”

Using numbers—two marshmallows instead of just marshmallows—invited the children to talk about numbers. It invited them to use numbers to maximize their benefit. It invited them to think about numbers.

This invitation is important.

A few years back, researchers paid careful attention to the ways preschool teachers talked with their students. Those teachers who used more number words and concepts as they talked with children stimulated greater growth in math than those who used less math talk.

This was not a study about math instruction; it was a study about the math language that these teachers used when they weren’t teaching math. “Yes, you three may help me.” versus “Yes, you may help me.” is the sort of difference that matters.

Using number words and math concepts in everyday speech invites children to notice and to think about number. That’s what Talking Math with Your Kids is all about.

Link to full study ($)

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.

ten hundred dora

 

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”.

ten hundred

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.