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.

Photo Feb 07, 10 06 30 AM

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.

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:

Legos

The dirty little secret about Legos is how very many pieces there are to be cleaned up after building. And how very ugly the clean-up battle can become.

I try to keep calm. I try to turn the clean-up battle into math talk from time to time.

Here is how it played out a while back, when Griffin was 7 years old (nearly 8).

I send Griffin upstairs to clean up a big Lego mess. A while later, he returns.

Me: Did you pick those Legos up?

Griffin (7, nearly 8 years old): Yes. Well…not all of them.

Me: What fraction did you pick up? More or less than half?

G: Three fourths. No. One fourth. No….I’d say one third.

One third is more than one fourth.

Me: How do you know that?

Griffin draws a picture like the one below.

legos.1

G: You see that it’s bigger.

Me: OK. Do want to hear how I know it’s bigger?

G: Yeah. How?

Me: If we had one big cookie and I had to share it with three people or share it with four people, I’d get more if I only shared with three.

G: Yeah.

Me: Which is bigger: three fourths, or two thirds?

He draws a picture like the one below (the original is lost to history).

legos.2

G: Three fourths is.

Me: How do you know?

G: [referring to his picture]: It would be here [indicating the heavy line pointing to the lower left] if it were two thirds, and there’s more shaded in, so three fourths is bigger.

So what do we learn?

It seems to be easier to engage children in comparisons than in precise computations. I have a lot more success with Which is more? questions than I do with Exactly how much/how many? questions. This is especially true with fractions, where precise computations are often quite tedious.

In any case, the simple follow up question, How do you know that? is a powerful one. That is where the conversation happens nearly every time.

At this point Griffin is accustomed to being asked this question, which means he is pretty good at doing so. It seems natural to talk about how he knows something. If you are just starting these conversations with your own child, expect to have to ask on 10 or 15 separate occasions before it starts to become natural for them. This is just like introducing a new food. Repeated exposure is key.

Starting the conversation

This is classic lemonade-from-lemons parenting. I knew Griffin would not have completed his assigned task. We could fight about it, or we could talk about it. I chose to talk about it, and save the power struggle for a bit later.

You can do the same. What fraction dressed are you? What fraction of your teeth did you brush? How much of your body was touched by soap? Et cetera. Have a little fun with the math first. Then make them finish the job.

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.

Guess the temperature

This post is from last year on my math teaching blog. Presently we (along with much of the American Midwest) are in the middle of a serious cold snap. So I have edited and remixed it for the Talking Math with Your Kids audience.

guess.the.temperature

This morning’s situation. Colder air is on the way.

Enjoy.

And stay warm.

Griffin (8 years old in this story) and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, in winter, this means we get to study both positive and negative numbers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Me: So what if it had been 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. Griffin’s silence seemed a clear sign that he was ready to move on.

So what do we learn?

Two things are important in this conversation: (1) Griffin’s solution method, and (2) the connection to subtraction.

Griffin’s solution method. Griffin’s strategy is a common one for children to invent. He uses zero as the boundary between positive and negative numbers. To compare how much bigger a positive number is than a negative number, you have to cross that boundary. You have to go past zero, so it just makes sense to him to divide the distance into two pieces—the part to get to zero, and the rest.

We live near a major road—Arcade Street. We often use it as a boundary for our neighborhood. So the local recreation center is three blocks on the other side of Arcade; plus the one block to get to Arcade. That’s four blocks total. Talking in this way about everyday navigation supports thinking about temperatures, which in turn support thinking about integers.

Connection to subtraction. From years of teaching middle school (and—to be honest—college), I know that subtracting integers is tough going. The rules for solving 3-^{-}2 don’t seem to connect to students’ experiences with numbers.

Notice how quickly Griffin connects the 10° and 3° situation to subtraction, while not seeing that subtraction applies to the 3° and -2° situation we started with. Perhaps my mentioning that these are the same will lay the groundwork for him noticing it in the future.

In the meantime, we learn that learning subtraction is a lot like learning division. In a recent post, I showed how Griffin thought differently about division depending on the numbers involved and on the context for thinking about it. Now we see that this is true about subtraction, too. 10-3? No problem. 3-(-2)? Problem.

Starting the conversation

Children—all children—develop mathematical models of their worlds before they study them in school. Parents have opportunities to support this through conversation.

Talk about landmarks, way stations on your journeys. Mileposts, subway stations, bus stops, blocks…these are all opportunities to help children build the mental models necessary to think about zero as an important landmark.

Talk about distances. How many blocks is an example from our family life. How many subway stops came up for girl I observed in New York City last fall. How many pages did we read in our book is another example where subtracting endpoints is helpful.

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.

Days to Christmas: Place value follow up

In yesterday’s post, I told of the Christmas Countdown cube calendar, and of how Tabitha (6 years old) changed my 06 days to 6 days by removing the leading zero. I challenged readers to consider her reasons for this.

Of course I asked her. I showed her the pictures I took and asked why she had taken off the zero. Here are the results of our conversation.

Me: I’m curious about why you took off the zero.

Tabitha (6 years old): Because there aren’t sixty days until Christmas.

A new piece of research has been making the rounds in the media recently. I will write about it at length soon, but for now you just need to know that the common headline is something like, “Young children know more about place value than most people assume they do.” In particular, the research looked carefully at the partial place value knowledge children have, rather than just calling wrong answers wrong.

Helping children develop partial knowledge into better knowledge through exploring and talking is the heart of the work here at Talking Math with Your Kids. Naturally I want to explore Tabitha’s knowledge here.

I get out a piece of paper, a pen and write some numbers, asking her to tell me what they are. I write down her responses verbatim so that I can share them with you.

25

Twenty-five

52

Fifty-two

A reasonable hypothesis for Tabitha saying that “06” meant “sixty” would be that she doesn’t pay attention to the order of the digits in a number. We now know this isn’t true.

More numbers follow.

60

Sixty

06

Sixty

Now things are getting interesting. There seems to be something special about that zero out front. We do some more.

600

Six hundred

006

Six

What? I was sure she was going to say six hundred for this one.

060

Six

Hmmm….

0

Zero

00

Zero

No hesitation on this one, which surprises me. I thought she might object to two zeroes.

3

Three

30

Thirty

030

Three

At least this one is consistent with 060.

I am beginning to wonder how she will work with numbers that have zeroes in the middle of them, such as 1002. And I am curious how she will work with numbers that use the same words, but in a different order. I think of 1002 and 2001 as a good example: one thousand-two and two thousand-one. So I build up to those.

2000

Two thousand

1000

A thousand

Uh oh. I didn’t expect this. I expected one thousand, not a thousand. I have to change my examples.

3000

Three thousand

3002

Three hundred two…er…three thousand two

2003

Two hundred three

Beautiful! This is what the researchers were saying—children have partial place value knowledge. What is going on here is this: 3002 looks like 300 with a 2 at the end. This is a very common error, and it represents trying to make sense of a complicated idea.

We do only one more; I can tell she is getting tired.

4030

Seven

I cannot tell if she is being silly or serious, trying hard or being clever.

Photo Dec 22, 12 35 43 PM

So what do we learn?

Just as with learning to speak a native (or foreign) language, learning about numbers is not an all-or-nothing proposition. Children have partial knowledge that is sometimes inconsistent. Tabitha self-corrected on 3002, but not on 2003. This conversation supports her thinking about the structure of numbers. She will think about it in the future and be more prepared to pay attention to it because we talked about it. When she wants to know more, she will ask.

Starting the conversation

Children learn about numbers and language in similar ways—through exposure in their everyday environments. We read to our children to enrich their language environment, and we can expose our children to numbers to enrich their math environment. Waffles, toys and calendars are all examples of everyday objects with mathematical structure that children can play with.

Have these things available. Ask about how your children play with them. Listen to their answers. Then ask follow up questions.

 

Days until Christmas

Each day, Griffin (9 years old) has taken great joy in setting the Countdown to Christmas cube calendar the kids recently received from my father and stepmother.

On Thursday, I was doing my end-of-semester grading from home and noticed after he had left that he had neglected to set it on his way to catch the bus to school. So I did.

christmas.06

My wife got Tabitha out the door later in the morning while I was at work in my basement office.

Later, I noticed that Tabitha (6 years old) had modified it.

christmas.6

Here I will pause to give readers a day or so to consider why. What might a 6-year-old have in mind that would cause her to remove the zero from in front of the six?

I asked her about it later on and we had a lovely conversation. I’ll report on that soon.

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.

Canned pumpkin

Fall baking in our house requires canned pumpkin. We were out so I asked for Tabitha’s help at the grocery store, where the pumpkin in on the bottom shelf.

Me: Put four of those bright orange cans of pumpkin in our cart, please.

Tabitha (6 years old): I don’t know if I can carry four.

Me: Do two, then two more.

Photo Nov 17, 10 15 20 AM

T: [With two cans of pumpkin in her hands] I know, because two plus two is four.

Me: Right. You could do three and one, I suppose.

T: OK. Give me one back.

She takes it, picks up two more from the shelf and brings the three cans over to me.

T: I did two and three.

Me: So we have five cans?

T: No! You gave me one back, remember?

Me: So two plus three minus one is four?

T: Yeah.

So what do we learn?

Decomposing numbers is fun.

We tend to think of 2+2 as something to do, and that the answer is 4. But in this case 4 is the thing to do, and 2+2 is one of several possible answers. When we think about different ways to make 4, we are decomposing 4.

Tabitha can keep track of the moves in our complicated decomposition at the end (You gave me one back, remember?) but she does not have practice with the math notation that captures all of these moves (Two plus three minus one is four). That is one of my roles in the conversation.

Starting the conversation

Tabitha gave me the ideal beginning to this conversation—she pointed out that there were too many cans for her to carry. It shouldn’t be difficult to put your own child in such a situation. The grocery store sells lots of things that children can carry a few of, but not a lot of: apples, oranges, cans of soup, etc. Picking up toys at the end of a play session at home or school, or books at the library—all of these are opportunities for you to name the number involved, then suggest a way to decompose it.