Category Archives: Talking math with your kids

Talking Math with Your Kids for Kindle!

Someday there will be a full-sized paper version of a Talking Math with Your Kids book (Hear that publishers? Wanna talk? You can find me at the About/Contact page.)

Until that day, there is now a mini-version (15,000 words; roughly three chapters, $4.99) available on Kindle (and readable on other devices with the Kindle app).

Tabitha is delighted by the news!

Tabitha is delighted by the news!

Go have a look, won’t you? Share widely and let me know what you think.

Table of contents:

  1. Introduction
  2. Counting and other adventures in number language
  3. Adding and subtracting: Two peas in a pod
  4. Conclusion
  5. References and further reading

The book is structured around conversations I have had with Griffin and Tabitha. About 1/3 of the conversations in the book have been previously documented here and/or on the new Talking Math with Your Kids site. The rest are new to readers.

There is lots of new content summarizing research in parent-friendly ways.

The impetus for getting this out now is this: funding my New York Times Schools for Tomorrow trip. I got partial funding from my college, but it’s an expensive conference. So I hacked a couple of chapters out of a draft I have been working on for quite a while now, tidied and edited them and voilá!

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Griffin and Tabitha are moving

omt.moving

For two years now, I have been documenting the mathematical conversations we have around the house, filed under the category, “Talking Math with Your Kids”.

I have long wanted to bring these conversations to the attention of non-mathy parents. But seriously, they want to wade through my musings on College Algebra, elementary teacher preparation and Khan Academy? I don’t think so.

So these conversations are moving.

To talkingmathwithkids.com

Come join us over there, won’t you?

More importantly, do us the favor of sharing the link with parents and caregivers of your favorite 0—10 year olds. Especially those who might be a little bit afraid of math.

In writing for parents, I’m fleshing out the conversations with information about kids’ mathematical development, and with ideas for starting similar conversations with their own kids.

We’ll review products that are targeted at kids and math, and share relevant research and news.

There’s even a contact page where parents can report conversations they have with their kids, and ask questions.

We’re just getting started, so give us a hand won’t you?

 

The oldest man in the world

Tabitha (six years old) and Griffin (on the cusp of nine) are attending a three-hour soccer camp in the neighborhood every afternoon this week. Furthermore, she has been begging to come to Tuesday night Ultimate Frisbee with me this summer. This week was the first opportunity for her to come along. It’s about a half hour ride up to Blaine so we had time to chat in the car.

Tabitha (6 years old): My group is 6 and under, but everyone is 6.

Me: Interesting.

T: Yeah. Six and under, so even a zero-year-old could play.

Me: I suppose so. But everyone is six, so there are no “unders”.

T: Daddy, everyone is under something.

Me: Huh?

T: Like you. You’re under 100.

Me: I suppose so. But then everyone you know is under 100.

T: Not the famous guy.

Me: What famous guy?

T: The oldest man in the world. He’s not under 100.

Me: No he’s not. But you don’t know him either.

T: Yes I do. I read about him in a book.

Feeling smug for having won this round, Tabitha sits in silence for a moment.

T: Are his mom and dad still alive?

Me: Whose mom and dad?

T: The oldest man in the world.

Me: Let’s see if you can work this out yourself.

T: Oh! They’re not alive.

Me: How do you know?

T: Well, his mom and dad are older than him. So if they were alive, they would be the oldest people in the world.

Pause.

T: Or, they could be alive, but younger than him.

Postscript

Quick plug: Tony Sanneh is evidently from Minnesota. He has a foundation that, among other things, offers free soccer camps in Minneapolis and St Paul recreation centers. They seem to be really positive, well run affairs drawing kids of diverse economic and cultural backgrounds. From what I can tell, they are doing lovely work that we should applaud.

Units, attributes and four-year olds

From mrdardy in the comments recently:

Slightly off topic, but I wanted to share a conversation with my soon to be four year old daughter from this past weekend, We were on a long car drive and she was asking how far we were from our hotel. I replied that we were twenty minutes away. Later in the pool she was jumping to me from the pool steps and commanding me to back up some. I asked her how far I should go and she told me to be five minutes away. I said “Do you mean five feet away?” and she replied, firmly, that she meant for me to be five minutes away. I am wrestling with whether I think this is just charming and (semi) clever on her part or whether I need to start answering her pleas in the car with distances. Curious to hear some ideas on this.

I am happy to weigh in here.

Anna Sfard describes knowledge as participation in a discourse and learning as changes in that participation. That is, we can measure whether someone knows something only to the extent that they can talk in ways that adhere to the norms of other knowledgeable people. And when these behaviors change to conform more closely to these norms, we can say that they are learning.

Nowhere is this more clearly demonstrated than in the learning of young children.

The four-year old in question here (let’s call her “Little Dardy”) is trying very hard to participate in conversations about measurement. Measurement, though, is a challenging and rich domain. 

mrdardy outlines two scenarios in which the concept of how far comes up for Little Dardy. It shouldn’t be at all surprising—considering Sfard’s model—that she answers a distance question in the same way her father had earlier on. She has taken his example in using units of time to discuss how far something is.

My approach would not be to avoid using units of time to answer the question how far? After all, people do this frequently; it is part of the discourse of measurement.

No, I would use this tension to encourage Little Dardy to think about the two attributes in question here: time and distance. It might go something like this…

Little Dardy: (four years old) Back up, Daddy!

Daddy: This far?

LD: More!

D: Here?

LD: More! You need to be five minutes away!

D: Do you mean five feet away?

LD: No! Five minutes!

D: OK. Tell me when I’m there. But then don’t jump right away; I want to ask you a question before you do. [Daddy backs up slowly…]

LD: OK! There!

D: Right. Here’s my question: Do you think it will take you five minutes to get to me from where you are?

LD: Yes.

D: Do you know how long five minutes is?

LD: That far.

D: No, no. Can you think of something we do together that takes five minutes?

LD: No.

D: It takes us about five minutes to read [INSERT TITLE OF FAVORITE PICTURE BOOK HERE] together. Do you think it will take that much time for you to get to me?

At this point, I have no idea how Little Dardy will respond (which is what fascinates me so much about talking math with kids). I do know that pretty soon, she is going to want to jump, and that whether that’s right away or after a few more exchanges doesn’t really matter.

What matters is that she’s been asked to think.

This line of discussion lays the foundation for thinking about distances, times and their relationships to each other. It supports Little Dardy’s attempts to participate in the discourse of measurement.

My recent conversation with Tabitha about the height of our hill was in a similar spirit; we worked on the meaning of height when she asked me to lie down on the hill.

How tall is the hill? [summer project]

Our house in St Paul sits on top of an odd hill; higher than others around it. Historical reasons for this are murky but it makes the place easy for guests to find. One of my least favorite tasks in all of my domestic life is mowing the hill.

20130724-133524.jpg

For a while now, the precise height of this hill has been the subject of family speculation. One recent lazy summer afternoon, Griffin (8 years old), Tabitha (6 years old) and I found ourselves hanging out on the hill with not much to do.

Me: How tall do you two think the hill is?

Tabitha (6 years old): Five feet.

Griffin (8 years old): I don’t know.

Me: How about this: Which do you think is taller; me or the hill?

T: The hill.

Me: Wait. I’m six feet tall. How can the hill be 5 feet tall AND taller than me?

G: You’re six feet, one inch.

Me: Right. Even so…

T: Oh. I don’t know how tall the hill is, but I think it’s taller than you.

Me: Why?

T: Lie down.

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T: See?

Me: Yeah, but just because it’s longer than me doesn’t mean it’s taller than me.

Tabitha seems puzzled by this distinction. Griffin is standing on the sidewalk at my feet.

Me: Look at Griffy’s eyes. Is he looking up or down at my eyes right now?

T: I can’t really tell.

I stand up, right next to Griffy, who cranes his neck back to look me in the eye.

Me: Now?

T: Ha!

I lie back down on the hill.

Me: So how come there’s a difference?

T: You’re lying down now, so that’s not really how tall you are.

Me: So how can we decide whether I am taller, or the hill is?

Nothing much occurs for the next minute or so. We are distracted by butterflies, the edible nature of clover flowers and other wonders of Minnesota’s too-short summers.

Me: Hey! Let’s try this. Tabitha, you go to the top of the hill.

She does, and she stands there, looking down on me with a self-satisfied smile on her face.

Me: OK. So you plus the hill are taller than I am. What about just the hill?

T: I don’t know.

Me: Lie down.

She does, although it takes a few tries to achieve the desired position by which she can look at me from roughly the level of the top of the hill.

Me: Are you looking up or down at me?

T: I can’t tell.

Griffin takes his turn at the top of the hill. He, too, is unsure.

Me: So how can we be sure?

T: You know, Daddy, I don’t really need to know this.

Me: You’re right. You don’t. Nor do I, really. But I have always been curious how tall the hill is. Aren’t you?

G: We could measure a step, then use the number of steps to figure out how tall it is.

I obtain a tape measure.

We determine that each step is 7 inches tall. We notice that the bottom step is shorter than the rest and measure it at 5 inches. Griffin laboriously counts the steps, finding that there are eight of them, plus the smaller one.

G: So what is that altogether?

Me: What? You can do this.

G: Do you know whether you are taller than the hill?

Me: Actually, yes I do, even though I don’t know exactly how tall the hill is.

G: If I figure it out, will tell me whether I’m right?

Me: Yes.

G: [Far too quickly for me to be convinced he has run any computations at all] OK. The hill is taller.

Me: How do you know?

G: Hey! You said you would tell me!

Me: That’s part of doing the math!

G: OK.

A long, thoughtful pause ensues.

G: Eight eights is 64, plus 5 is 69. So you are taller.

Me: But you need eight sevens, which is 56.

G: Oh. Right. Plus 5.

Me: Yes…?

G: Tell me.

Child, please.

Me: Seriously? You can do 56 plus 5.

G: 61.

Me: Yes, and I’m 73 inches tall.

Tabitha, despite her protestations about not needing to know, has been paying attention all along.

T: You’re taller than the hill?

Me: Yes. See? I told you it was interesting.

G: You knew you were taller?

Me: Yes. But I didn’t realize it was by a foot. I thought it would be only by a few inches.

G: How did you know?

Me: Because I look down—only slightly—but I look down at the top of the hill.

In a few days, we will return to the topic of the State Fair Giant Slide and see whether these techniques generalize in my children’s minds.

Incommensurate 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.

On seeing this claim, I thought for sure that we were gonna have a We mean four times, but say twice sort of a situation on our hands. So I bought some.

And then I asked Tabitha (6 years old) and Griffin (8 years old) 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”.

image

I wanted to know the source of that. I thought she might be making the classic linear v. area error (i.e. interpreting twice to mean twice the side length). So I asked.

She pointed to the uncovered part of the BIG Cheez-It and argued that this didn’t constitute another full regular Cheez-It. Score one point for argumentation, but minus one for spatial visualization.

A few minutes later, it was Griffin’s turn. He ran like a chipmunk with his two crackers into the dining room. Experiment over, right?

Nope.

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.

Sadly the cut outs are lost forever.

Sadly the cut outs are lost forever.

His conclusion: BIG Cheez-Its are almost but not quite twice the size of the regular Cheez-Its.

Volume perhaps?

Addendum 1

If the crackers are twice as big, but the mass of one serving is constant, and if one serving of regular Cheez-Its consists of 27 crackers, how many crackers should be in one serving of BIG Cheez-Its?

There are 14.

image

Addendum 2

If the area of a BIG Cheez-It is about twice the area of a regular Cheez-It (as Griffin confirmed), then the side lengths should be in a ratio of approx. 7:5 (a reasonable estimate of the square root of 2).

cheez.its

 

addendum 3

Notice the progression in the children’s strategies. The six-year old worked with the crackers. The eight-year old worked with representations of the crackers. Similar conclusions were reached; the child who worked with representations could manipulate those representations in order to achieve a greater degree of accuracy, and to investigate hypotheses that the child working concretely could not.

Neither child used tools to calculate areas.

Armholes (6-year old topology)

We were packing for a trip recently. I have developed a system for getting the kids packed. It is beautiful. Here’s how it works:

  1. Send kids to basement to get suitcases.
  2. Keep suitcases on first floor.
  3. Send kids upstairs to get one type of item at a time. E.g. Three pairs of underpants. Then three pairs of socks. Et cetera.
  4. Kids throw each type of item in the suitcase.
  5. Repeat steps 3 and 4 as often as necessary.
  6. Done.

Seriously. It’s awesome.

I made an observation with Tabitha partway through.

Me: Isn’t it strange how a pair of socks is two socks, but a pair of underpants is only one thing?

socks

Tabitha (six years old): Yeah. It should “a pair plus one” because there are three holes.

Me: Wow. I hadn’t thought of that. So how many holes does a shirt have?

T: Three….No four!

Me: How do you figure?

T: The one you put your head through, the arms, and the head hole.

If you are like me, you may be a bit behind the curve on her language here. “The one you put your head through” is the one that ends up at your waist once your shirt is on. I had to think about this for a moment.

A few days later, I was curious to probe her thinking a bit further. She was getting dressed (a process which is always slow, and occasionally very frustrating for the parents):

Me: Do you remember how you said a pair of underpants has three holes and a shirt has four?

T: Ha! Yeah!

Me: I was thinking about that and wondering whether there are any kinds of clothing that have one hole or two holes.

T: Socks have one hole!

Me: Oh. Nice. Sometimes Daddy’s socks have two holes, though.

T: Yeah. When they’re broken.

By this time, she finally has the underpants on and her pants are being slowly pulled on.

Me: Wait. You need socks!

She goes to her dresser and proceeds to sort through the very messy sock drawer.

T: There are no matches.

I find what appears to be two socks balled up together.

T: No! Those aren’t socks! Those are for putting over tights to keep your legs warm.

We look at each other.

Big smile.

TThose have two holes!