A propos of nothing one day, I ask Griffin (9 years old at the time, finishing up fourth grade) a question.

Me: Griff, imagine you are baking cookies and you need $\frac{3}{4}$ cup of sugar, but you only have a $\frac{1}{2}$ cup measure. How would you get $\frac{3}{4}$ cup?

Griffin (9 years old): You put $\frac{1}{2}$ cup of whatever you’re measuring.

Me: Sugar.

G: Does it matter?

Me: No. I suppose not.

The conversation could end here and I would be delighted. But it does not end here.

G: You put that into the bowl, then you fill the cup halfway and put that in.

Me: And that’s $\frac{3}{4}$ cup?

G: Yes.

Me: How do you know?

G: Because $\frac{3}{4}$ is a half, and then half of a half.

Me: Yeah. That is what you just described. How do you know that that’s right?

G: Like a square. If you shade in half of it, and then half of what’s left, that’s the same as shading $\frac{3}{4}$ of it.

## So What Do We Learn?

One question division helps answer is how many of this are in that? My question of Griffin asked how many halves are in three-fourths? This is a division question.

Griffin may not know that it is a division question. That is fine. He is thinking about a specific example of how many of this are in that? This will lead to good things further down the line.

That he sees “sugar” as a non-essential detail of the story is lovely. This will serve him well.

Griffin’s mental image for this task is a common one. He can see three fourths of a square in his mind, and he can see that this is the same as one-and-a-half halves of a square.

Finally, we learn (because I am about to tell you) that this scenario could never really happen when baking in our home. I have an awesome set of measuring cups (pictured below): $\frac{1}{4}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, 1 and $1 \frac{1}{2}$. (A friend—and friend of the project—has pledged to donate her $\frac{1}{5}$ cup measure to the Talking Math with Your Kids cause.)

## Starting the Conversation

There are so many ways to raise the question how many of this are in that? Measure each other in inches, wonder how many feet tall that is. Count your quarters, wonder how many dollars that is. Repeat with nickels, or dimes. Bake a batch of cookies using only the $\frac{1}{2}$ cup measure.

# Math in the alphabet

The children attended a well-run chess day camp this summer. Good people running things; a warm and welcoming atmosphere. Lots of varied activities to keep kids’ bodies engaged as well as their minds.

Sadly, this takes place on the complete opposite end of the Metro area from where we live. We had to drive all the way across St Paul, Minneapolis and deep into St Louis Park during rush hour. Ugh.

This led, one day, to my trying to find a topic of conversation to keep at least one of the children occupied while we drove home. I recount for you this conversation below.

Me: Tabitha. Can I ask you a question?

Tabitha (7 years old): Sure.

Me: What letter comes before I in the alphabet?

T: H. That was kind of an easy question.

I love that she has turned into a critic. If I am not challenging her, she calls me on it.

What she has not seemed to notice yet is that these questions she deems easy are just my openers for the good stuff.

Me: Yeah. Here’s a harder one. What letter comes two letters before S?

There is a fairly long pause here. This is a harder question because of how most of us know the alphabet—forwards. If we want to know what is 2 less than 71, it is not so hard to count backwards. We have lots of experience counting backwards. But we don’t have so much experience saying the alphabet backwards, so we need to make up a strategy.

T: Q and R.

Me: Q is two letters before S, yes. Now you ask me one.

T: What letter comes after Z?

Brilliant. What a great question. I wish I had thought of it myself.

Me: Oooooo. Good one. I say A. I say it starts over.

T: Nope.

Griffin has been listening in but not participating. He sees his chance to get in on the action.

Griffin (9 years old): Negative A.

Me: Wouldn’t that be what comes before A?

G: No. It comes after Z. It’s negative A.

T: Nope. Not that either.

Me: OK, then. I am stumped.

T: Nothing.

Me: Huh?

T: Nothing. No letter comes after Z.

## So what do we learn?

This is a more sophisticated version of another mathy letters conversation I had with Tabitha a while back. Back then, we were trying to figure out which of two letters comes first in the alphabet. Here, we are more paying careful attention to precise placement (two letters before, not just before).

The other interesting thing going on is our three different ideas about what comes after the end.

My idea: After the end, we go back to the beginning, like the days of the week.

Tabitha’s idea: There is nothing after the end. It just ends.

Griffin’s idea: The end is like zero. When you get to the end, you repeat what you already had, only using negatives.

It is OK that we didn’t resolve who is right.

## Starting the conversation

About a year ago, I started making a habit of having the kids ask me the next question. I highly recommend it.

You know how your children are always testing the limits of rules in everyday life? Like you say, “Do not touch” and they see how close they can get their finger to the forbidden object without actually touching it? That is normal and necessary behavior on the part of children.

They will do it in the world of ideas, too. Tabitha did not choose “What letter comes after Z” at random. She chose it because she knew it would be interesting to talk about. It probably would not have occurred to me to ask it. Our conversation was richer because she did.

# Dots!

The New York Times published an article about Common Core homework this week.

As is going to be the case with a news article (in contrast to say, a post on a blog dedicated to children’s mathematical ideas), one can’t really learn any mathematics from the piece. One critique got hit twice, though—that children are being forced to draw lots of dots.

Here near the beginning of the piece:

Ms. Nelams said she did not recognize the approaches her children, ages 7 to 10, were being asked to use on math work sheets. They were frustrated by the pictures, dots and sheer number of steps needed to solve some problems.

And a bit later:

Her daughter, Anna Grace, 9, said she grew frustrated “having to draw all those little tiny dots.”

“Sometimes I had to draw 42 or 32 little dots, sometimes more.”

I have no interest in picking up political issues surrounding the Common Core State Standards on this blog.

But I do think a parent frustrated by all those dots deserves an explanation of what all those dots are for.

Before we begin, please be assured that there is absolutely no mention of dots in the Common Core. What is mentioned is the array. An array is a collection of things arranged in rows and columns. We have discussed arrays before here at Talking Math with Your Kids. They are very useful tools for representing an important meaning of multiplication—that multiplication is about some number of same sized-groups.

Arrays (with dots or other things) are useful tools for making these groups visible, either actually visible or visible in the mind.

So I asked Tabitha (7 years old) to draw some dots for me.

Me: Tabitha, [neighbor girl and best friend] wants to play. Before you go outside, can you draw that picture for me? Three rows of five dots.

Tabitha (7 years old): That’s easy! Fifteen.

She is probably counting by fives here. She completes her picture for me.

I know that neighbor girl is waiting. I decide to press my luck.

Me: What if it had been 3 rows of 6?

There is a long, thoughtful pause.

T: Eighteen!

Me: How did you know that?

She shrugs her shoulders. Now is not the time to force things. Neighbor girl is waiting. So I offer a strategy.

Me: Let me tell you how I think you might know it.

T: OK.

Me: Six is one more than five. So each row would have an extra dot. That’s 15 for the 3 rows of 5, and then 16, 17, 18.

T: [smiles] Yeah.

We share a high five and she is out the door for a morning of clubhouse shenanigans in the backyard.

Quick note: Tabitha does not let me get away with stating her strategies incorrectly. I have done this before—summarized how I think she is thinking—and when I get it wrong, she objects. I am glad about this.

## So what do we learn?

This is what those dots are for. They give us something we can talk about. Without those rows and columns, the conversation is so much more abstract. We were picturing those dots in our minds as we talked about counting them.

The three rows of five she drew gave us a jumping off point for imagining the three rows of six we discussed. Three groups of five now has a relationship for her to three groups of six.

More importantly, the strategy of finding new facts based on old facts (here that 3 groups of 6 is 18 based on knowing that 3 groups of 5 is 15), has been introduced explicitly. It is something we will talk about in the future, and something she will know to consider.

Without the array, it is not at all clear to me that she would have been able to know what 3 groups of 6 is. She could have drawn 3 unorganized groups of 6, I suppose, and counted them individually. But this is a much less sophisticated strategy, and she is ready for more than counting individual objects.

## Starting the conversation

Many children do not naturally see rows and columns. Given an array, they may haphazardly count the objects around the edge, then in the middle. This often leads to double counting and skipping things.

But even children who are very good at keeping track of their haphazard counting—and who can get correct counts every time—may not see the row and column structure of an array.

So put 15 pennies in 3 rows of 5. Have your child count them and notice whether she counts in rows and columns, or whether she counts in some less structured way. Model the counting yourself so that she can see an example of the rows and columns at work. Don’t worry if she doesn’t see the structure yet, but do make a note to do more of this kind of counting in the future—seeing the structure of an array is an important stepping stone to multiplication and to the measurement of area and perimeter.

Or just have her draw dots.

# Nights of camping

The following conversation took place in the run-up to our annual summer camping trip recently.

Rachel has no interest in camping, so this ritual is all mine. I started the little ones young with a one-night trip within an hour from home so that we could come home if it’s a total disaster. As they have aged and we have developed our routines, we have gone further afield, exploring wide-ranging Minnesota state parks for two-night stays. We added a weekend fall trip, too.

Last summer, the kids began to ask why “we only go for two nights”.

Ladies and gentlemen, when the kids ask that question, you know you’re doing it right.

So this summer we are expanding to three nights. Tabitha was thinking about that change the other day.

I am straightening some things on the front porch, sweeping and tidying. Not thinking about anything in particular.

Tabitha (7 years old): If we’re going for three nights, is that 2 days and 2 half-days?

Me: Yes.

A few seconds pass.

I realize that I have an opportunity here.

Me: How did you think about that?

T: Every night is a day, except the last one, when we go home.

Me: What if we went for a whole week’s worth of nights? What if we went camping for 7 nights?

T: Easy. Six days.

Me: And?

T: Two half-days.

Me: OK. Ready for a hard one?

T: Yeah!

Me: There are 365 days in a year. So what if we went camping for 365 nights?

T: [slowly] Three…hundred…sixty…four!

Me: Nice!

T: I can even do 400.

Me: You mean 400 nights of camping? You know how many days that would be?

T: Yeah.

Me: All right. Tell me.

She does.

Later, she is in the shower. I am not-so-closely supervising nearby. I get an idea.

Me: Tabitha, what if we wanted seven days of camping?

T: How many nights?

Me: Right.

T: Eight. Am I right?

Me: I can’t trick you at all, can I?

Again, a sign that things are going well. Contrast with her claim a couple years back, “Sometimes I don’t want to tell you about numbers because it’s just going to turn into a big Daddy math talk!”

I have to think hard to dig up something that will be more challenging for her.

Me: You want a hard one? A really hard one?

T: Yes!

Me: Last year, we went camping twice. Altogether, we camped 4 nights. How many days did we have?

T: Three…five…

It turns out that Griffin is lingering in hallway outside the bathroom. He chimes in.

Griffin (9 years old): Four.

Me: Two days, and four half-days.

G: Right. That’s four.

Me: But she’s thinking about it as four half-days, since they aren’t attached to each other. I can see an argument either way.

This summer’s trip was to Lake of the Woods in the far northern reaches of Minnesota.

## So what do we learn?

It may surprise some readers that I have filed this conversation under Algebra.

Like many of the other algebra posts, we are not using x or y, or making graphs or solving for variables. Instead we are thinking about a relationship, and about what that relationship looks like for a wide variety of numbers.

The relationship we are working with here is a simple one: one less. Tabitha had noticed that the number of full days we camp is one less than the number of nights we camp. She had even generalized the idea—notice that she didn’t count the days individually. She said, “Every night is a day, except the last one.” This answer doesn’t depend on any particular number of days; it works for all numbers of days.

What I did in this conversation was help her to apply this idea. By asking her about a wide range of numbers of days, she got to feel the power of her generalization. That is algebra.

The other important part here was continuing the conversation while she showered. Thinking in reverse is an important mathematical skill. We had started with how many days do we get with a certain number of nights? I moved us to how many nights do we need for a certain number of days? The fancy math word for the relationship between these two questions is inverse.

## Starting the conversation

Camping trips, vacations, trips to grandma’s house…these are all opportunities to have the conversation we had. If your child doesn’t ask about it, you can ask your child. We are going to grandma’s house for three nights—how many days will you have to play with your cousins while we’re there?

More generally, there are two Talking Math with Your Kids moves I want to emphasize.

1. It took me a moment to notice that Tabitha had offered me an opening for conversation. I was thinking about something else at the time. When I noticed it, I put those other thoughts aside to talk, ask and listen. That part of the conversation took probably 2 minutes. We can all spare 2 minutes to get our kids’ minds working. We just need to notice the opportunities.
2. I followed up later on. Following up is good for two reasons: It lets you and your child examine an idea more deeply, and it helps cement memory of the conversation. We remember something we revisit multiple times better than something we only think about once.

# Hints at Holiday

I told an abbreviated version of the following story on my math-teacher blog, where I used it to drive home a point to my colleagues. This version is for parents.

My wife had been out of town for several days. I was tired of doing all the cooking and dishes. It was a lovely Saturday evening at the end of a busy day.

It was time for nutrition lessons.

It was time to get dinner at Holiday.

Oh right, like you have never done this.

The constraint was this: The kids had to select something from each of the four major food groups (do not try to talk to me about that new food pyramid; I will not listen.) They needed a meat/protein, a fruit/vegetable, a dairy and a grain.

Griffin (9 years old): Do donuts count as a grain? They have a lot of flour in them.

Me: Scratch that. WHOLE grain. No. Donuts do not count as a grain.

It turns out that the whole grains are hardest to find.

At Holiday, you’re not going to do much better than tortilla chips, whole-grain wise.

As a mathy bonus, Griffin later noticed that the claim underneath the picture of the chip on the bag reads, Enlarged to show texture and detail, but that the image is the same size as the chip.

But back to our story.

Tabitha (7 years old) had brought along money to buy some hot Cheetos.

She was under the impression that they would cost $1.35, and she had her money ready. Five quarters, one dime. She even had me check that these coins totaled$1.35.

When she got to the front of the line, it turned out that they Cheetos cost $1.49. It would have been fun to talk about the difference in price here, and have her fish out the right amount to make up the difference. But there were people in line behind us. We needed to move this along. I told her to get two more dimes out of her coin purse and give them to the man. I intercepted the change so as not to give away the answer to the question I was about to ask, and we turned to leave. Me: You owed him 14 more cents and gave him 20. How much change should you have gotten back? Tabitha seemed confused by my question. It was not that she was unable to answer it; rather she did not understand the whole getting change thing. I made a mental note of this and pressed on. Me: You gave him 20 cents when you only owed him 14 cents. So you get some money back. How much should that be? Still nothing. It seemed the money/change/debt thing was getting in the way of thinking through this number relationship. So I switched tactics. By this time, we are outside, strolling slowly home. Me: How much more is 20 than 14? This question put her in a different frame of mind. She slowed down and looked dreamily into space. She was thinking. Tabitha (7 years old): Thirty-four? or maybe thirty-five? Ugh. Right answer, wrong relationship. I think she cued in on the more in that sentence. I tried one last time to trigger the thinking I know she can do. Me: Let’s try this. Fourteen plus something is 20. What is the something? There was a long, thoughtful pause. Griffin interrupted the pause. Griffin (9 years old): How old were you last year? T: Six! Me: Did you work that out, or did you say it because Griffy said it? T: Griffy. Griffin and I had talked about this before. But we talked about it again on the way home—about how it is important for Tabitha to have the opportunity to think things through for herself. I tried to anticipate his needs: (1) to demonstrate that he knows, and (2) to help his sister. If he needs to demonstrate that he knows, he can: • Say he knows but keep the answer to himself, • Write it down, • Ask if he can whisper it in my ear. If he honestly wants to help his sister, he can ask a question that will help her think. How old were you last year? does not help her to think about the relationship between 14 and 20. But How much more is twenty than fifteen? might help her think, because she has often counted by fives. ## So what do we learn? We learn that it is sometimes quite difficult to get the right question that will get a child to think. Context, time pressures, level of difficulty, mood, the presence of siblings…all of these things can conspire to cut off the thinking. But if you are persistent in the moment, you may get somewhere. And if you are doing this every day, you’ll eventually hit the sweet spot. Most of all, we learn that it is the thinking that matters, not getting the kid to say the right answer. ## Starting the conversation Persistence is key. I didn’t get where I wanted in this conversation. You won’t get there sometimes either. That’s OK. Ask your question, adjust it if necessary. Let it go if you need to. There’s always another day. # The equal sign It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems. But I am tired. Wiped out and exhausted. So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?) This plan worked beautifully for about five minutes. She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook. Me: What goes in the box? Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is… Hey! This doesn’t make any sense! Me: Why not? T: 8 plus 4 is something, then plus 5? Me: What does the equal sign mean? T: Is. Like 2 plus 2 is 4. Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1? T: No! I let it go and we move on with our evening. Later on, though, after putting on jammies but before toothbrushing, I follow up. Me: Tabitha, I want to ask you a follow up math question. T: OK. Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1? T: Yes! Of course! Easy! Me: Can I let you in on a little secret? T: A secret secret? Or not really a secret? Me: Not really a secret. But something you might not know. T: [rolls eyes] OK. Me: The equal sign means “is the same as”. T: Of course! I know that! Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1. T: Oh. ## So what do we learn? This is kind of a big deal. We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!) But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as. Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right? Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do. The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right. You can follow Tabitha’s and my adventures in equality in the coming weeks. ## Starting the conversation If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun! ## Postscript Coincidentally, a fourth-grade teacher wrote up his class’s explorations in equality today. If you’re interested in what this can look like in school (easily adaptable for homeschool), head on over. # 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. 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. # Bedtime In the thick of the holiday season last month, Griffin (9) turned his attention to the mythical midnight on New Year’s Eve. Griffin (9 years old): Can we stay up until midnight on New Year’s Eve? It is a nice touch that he includes his sister in this request. They may argue but they are a team. Me: No. G: We can when we’re 12. At one point a year or two ago, he forced me to commit to the age at which he will be allowed to stay up to welcome the new year. The future me may be angry about the commitment I made, but for now it is paying dividends. Me: Yes. G: When I’m 12, I’ll get to stay up a lot later every night, since my bedtime is a half hour later every year. I’ll be able to stay up until….10. Me: We’ll see about that. Griffin restates his rule. I am pretty sure that I never endorsed this rule, but he is committed to it. I figure that I may as well exploit it. Me: By your rule, what will your bedtime be when you’re 18? G: Twelve to eighteen….so…three more hours…One o’clock! Me: I should have asked about an odd number. How about when you’re 21? G: Two-thirty in the morning. Me: By your logic, what would have been your bedtime when you were 1? G: Well, at 8 it was 8, so…4 hours…Four o’clock! Me: I disagree. I don’t think it should be 4 hours from 8 to 1. Griffin defends his answer. G: It’s eight years, which is 4 hours. Me: I don’t think 1 is 8 years younger than 8. G: Oh. Yeah. Four thirty. Then when I was 3, it would have been six o’clock. He pauses thoughtfully. G: Did I go to bed at six when I was three? Me: I honestly don’t know. We make sure your bedtime matches your need for sleep. ## So what do we learn? This conversation exemplifies an important Talking Math with Your Kids principle—use whatever interests your children as an opportunity to talk about math. Another related principle is to use the conversation you don’t want to be having as a launching point for math talk. I assure you that I am a relatively normal parent; I dread discussions of bedtimes. There is so much opportunity for whining, wheedling and comparisons to classmates. I understand that these classmates are mythical, but my children assure me that they all stay up later. The key moment in this conversation was when I grabbed the bull (as it were) by the horns and asked him to apply his rule. What will your bedtime be when you are 18? That question got us talking about some good math, and it turned a potential power struggle into a fun conversation. Using a rule to make a prediction is an important aspect of algebra. Griffin’s half hour later for every year older rule is a wonderful example of rates. I asked him to predict both forwards (what will be your bedtime when you are 18?) and backwards (what was your bedtime when you were 1?) I missed two opportunities here. It is a good idea to ask questions that make kids think in the opposite direction. So I should have asked something like How old will you be when your bedtime is midnight? This would force Griffin to think bedtime-to-age when he has been thinking age-to-bedtime. The other missed opportunity is to play with the silliness of extending this rule too far. I could have asked, How old will you be when your bedtime is 8:00 p.m. again? If his bedtime keeps getting later, it’s going to come back around. Could he work out that this would require 48 years? Or would he reject the question as silly and put a limit on his rule? Either way, it’s a productive math talk to have. ## Starting the conversation Anytime your child wants to enter a negotiation, there is an opportunity to turn it into a math talk. How many M&Ms can I have for dessert? How many pumpkins can we buy? When can I stay up later? How many friends can I have at my sleepover? All of these and more are opportunities to ask what if questions involving rates, predictions, past and future quantities. So don’t dread the negotiations. Take advantage of them! # Milk by the gallon Milk has been on sale at our local gas station/convenience store. Griffin and I walked up there the other day to buy some milk. Two percent milk for the kids and me; skim for Mommy. Me: Griff, the milk we just bought was$5.50 for two gallons. How much was each gallon?

Griffin (9 years old): With tax included? Or not included? I don’t do tax problems.

Note: I weep for the loss of 4% and 5% sales tax rates. They were so easy to compute mentally, and such a nice introduction to the financial world for elementary age children. Minnesota’s sales tax rate is presently $6\frac{5}{8}$%. The city of Saint Paul tacks on another half percentage point. I don’t even bother computing sales tax mentally any more.

Me: No worries about taxes. There is no tax on milk.

G: OK. Two twenty-five. Er…no that’d be $4.50. So…$2.75!

Me: How did you do that?

G: Well, I thought it would be $2.25, but that’s half for$4.50, so there’s an extra dollar. So I split that dollar in half, which is 50 cents, put that with the $2.25, which is$2.75.

Me: Nice. I could see that thinking in your first answer; when you said $2.25. I was curious whether you used that first wrong answer or started over from scratch. When I thought about it, I did it differently. I thought that half of$5.00 is $2.50, then I need to add half of 50 cents. Same answer, though.$2.75.

## So what do we learn?

I called in to a Minnesota Public Radio program on math education last week. One of the pervasive questions in such conversations is about how kids are learning to do arithmetic in modern American schools, and it arose in this program.

The thinking Griffin is doing here is lovely, and modern math curriculum is trying to encourage more of it than in the past. He is splitting $5 \frac{1}{2}$ in half, and he is doing it mentally by thinking about the related multiplication facts.

This thinking is not closely related to the standard long division algorithm. One of the big challenges in school curriculum is relating mental math strategies such as Griffin’s to efficient algorithms that are more useful for complicated computations. I have a few resources parents may find helpful over at Sophia.org.

## Starting the conversation

Anytime you find yourself wondering about such things, ask your child to think along with you. I wanted to know whether the gas station price for a gallon of milk was a good one. This required me knowing what the price was for each gallon. Not a hard problem for me, but I had to think for a moment. So then I asked Griffin. Do the same at the grocery store, the convenience store, the hardware store; anyplace where things are priced in groups.

If your kid needs a challenge, ask about gasoline. I paid x for y gallons yesterday. How much per gallon? This one will likely require estimation skills!

# Sharing pears

I took the kids camping this weekend. Jay Cooke State Park is lovely. I recommend a visit if you have never been.

We had the following conversation at the campsite on Saturday evening.

Me: Griff, we have two pears and three people. What should we do about that?

Griffin (9 years old): Cut them in 3 pieces.

Me: How much does each person get in that case?

G: One and a third, I think…No…I don’t know.

His attention returns to the campfire, which is of course endlessly fascinating. A minute later, I try again.

Me: Tabitha! Two pears, three people. I’m going to cut them in half. That’s four pieces. You each get one. I get two. Fair?

Tabitha (6 years old): NO!

She passes a few moments pondering while I take the pears over to the picnic table.

T: You should cut them in four pieces. Give each of 1, then put the other away for tomorrow.

I originally interpret this to mean cut each of the pears in four pieces. Afterwards I am not so sure what she meant. In any case, I did what I thought she suggested.

Me: Tabitha, I did most of what you said. We each got 2 of those pieces. There are 2 leftover. I’m going to cut those in half. We each get 1 of those littler pieces.

Now what should we do with this last little piece?

She thinks

T: Cut it!

Me: In how many pieces?

T: Three!

I do so and we admire our handiwork.

## So What Do We Learn?

The first fraction most children encounter in a serious way is one-half, and this is nearly always through fair-sharing.

Two kids, one pear (or cookie, or cupcake, or doll…) this is where children begin to think about fractions. They do not think about the notation of fractions, but they think about the important idea of fractions: A thing can be cut into equal pieces, and there is a name for these pieces.

Sharing multiple things in a way that involves cutting—as 2 pears among 3 people—is much more challenging than sharing one thing.

Nevertheless, this experience of halving is so fundamental that children often want to solve everything by halving. As long as we are willing to cut halves in half, this can be a powerful way of working things out.

Griffin’s idea to cut each pear in three pieces, then, is relatively sophisticated. That he could not name the resulting share should be no surprise in a campground conversation where the cutting is all imaginary and unwritten.

That this did not immediately occur to the younger child is no surprise. Likewise the idea to save the extra piece for another time is typical of young children’s thinking about fair sharing.

Cutting that last piece into three was a triumphant idea for her.

See Extending Children’s Mathematics for lovely, detailed writing about these ideas.

## Starting the Conversation

Buy two pears. Get yourself two children of different ages. Ask them how to share the pairs fairly. Do what they tell you to do and ask follow up questions such as Is this fair? Do we all have the same amount? How do we all have? and What should we do with these leftover pieces?