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.

Letter squares for q, r and s

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.

Letter squares for w, x, y, z and three blank squares

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.

 

Pistachios

My father buys things in bulk. Not the bulk bin, dispense-a-little-bit-into-a-plastic-bag bulk. Costco bulk. Sam’s Club bulk.

The children and I spent some time with my father and stepmother (who are wonderful, loving people) at the Wisconsin Dells recently. We shared a rented condo. They brought bulk snacks.

Did you know that you can buy graham crackers in a container that holds four of the usual boxes of graham crackers?

What need does one family have with FOUR BOXES of graham crackers?

More to the point, they brought pistachios. I forget to check whether it was a three-pound bag or a four-pound bag but it was an awfully large bag of pistachios.

The image below is a small fraction of the total.

Sandwich bag stuffed full of pistachios in their shells.

While we were in the condo, Tabitha (7 years old) took her first interest in pistachios. Her brother Griffin (nearly 10 years old) has been a fiend for them for years. One day, Tabitha announced something to me.

Tabitha (7 years old): I threw out eight pistachio shells.

Me: And what do you learn from that?

T: I ate four pistachios.

Me: How do you know that?

T: Four plus four is eight.

Me: Nice. And five plus five?

T: Ten!

We carried on this vein for a little bit before we got distracted.

A couple days later, I was rushing around preparing for a work trip. Tabitha was again snacking on pistachios.

T: Is 13 an even number?

Me: No. Why do you want to know?

T: I must have counted my pistachio shells wrong. I must have missed one. So it’s 14.

Me: And what does that mean in terms of pistachios?

T: I ate 12. No. That can’t be right.

Me: Oh! I think I know how you got 12!

At this point, I was headed downstairs to get something to put in my suitcase. By the time I got back up, both of our minds were on to different things.

We never did get to a solution, nor did I find out how she got her wrong answer.

So what do we learn?

Tabitha is playing around with the every pistachio has two shells relationship. She is thinking about ratios: Two shells for every one pistachio.

A child does not need to have mastered multiplication, or fractions, or division to think about these things. I have written about ratio thinking from young children before. Ratios come naturally from repeating a process. Eating a pistachio produces two empty shells every time. Sharing candy produces one piece of candy every time. And so on.

Starting the conversation

In light of this, help your child notice for every relationships. There are four wheels for every car. There are four legs for every chair. There are two wings for every bird. Point these relationships out and have your child do the same. Consider the exceptions (have you ever seen a 3-legged chair?) Count up how many wheels there are on two cars, and on three cars.

Eat pistachios.

Postscript

I have two theories about her answer of 12 pistachios for 14 shells.

1. She tried to figure it out by thinking about 10 and 4. Half of 4 is 2. She added that back to the 10 and forgot that she still needed to find half of 10.

2. She subtracted 2 from 14.

I like theory 1 a LOT better than theory 2 because it matches the ways she has been thinking so far. Using subtraction seems unlikely when she knows this is a different sort of problem.

But of course I do not know for sure.

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.

Array of dots: 3 rows of 5.

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.

Then have your child put things in rows and columns.

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?

T: Ask me another!

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.

Griffin posing with an oversized walleye statue in Baudette, MN

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.

What makes a sandwich

The 3-year old daughter of fellow Minnesotan, fellow math teacher and friend Megan Schmidt made the following proclamation a couple weeks back.

This simple claim has led to lots of fun conversation. Let’s call the daughter veganmathpup (since she is the daughter of Twitter’s @Veganmathbeagle), or VMP for short. 

All discussions with VMP are filtered through her mom via Twitter. All discussions with my own children are my best recollections of the recent silliness.

Open faced sandwiches

Veganmathpup’s assertion boils down to this: A sandwich needs these things: (1) a slice of bread, (2) a filling, (3) another slice of bread. I wanted to know about open-faced sandwiches. Is an open-faced sandwich properly called a sandwich? VMP was silent on this matter. So I asked Tabitha.

Tabitha (7 years old): That counts as a half-sandwich…actually more than a half-sandwich.

So an open face sandwich is not actually a sandwich for Tabitha. This gave me a chance to introduce the term misnomer.

Cookies

A week or so later, VMP claimed that “2, 3, 4 or 5 cookies can make a sandwich”. This was a clear violation of the earlier rule here. Two cookies, no filling? How can this be a sandwich when “It takes three things to make a sandwich”?

So I asked about Oreos. Does VMP think of an Oreo as 1 cookie? 2 cookies? Most importantly, Is an Oreo a sandwich? Megan related the following conversation.

Megan: [Handing VMP an Oreo]  VMP, I have a question.  Is this a sandwich?

VMP (3 years old):  [Examining carefully] Um, no.  It’s not.

Me:  Why isn’t this a sandwich?

VMP:  It doesn’t have things, like a burger.

Me: [Handing her two Oreos stacked on top of one another] Is this a sandwich?

VMP:  [Examining even closer this time] No.  it doesn’t have stuff in it. It needs lots of stuff inside like a burger to be a sandwich.  I want a burger.  Let’s get one [face full of oreos]. We won’t tell Daddy.

So many follow up questions I was unable to ask here. Does a Double Stuf Oreo have enough stuff inside to count as a sandwich? What about a Mega Stuf Oreo? Close up of a Mega-Stuf Oreo.

A Mega Stuf Oreo contains approximately 3.1 times the Stuf of a regular Oreo.

Marshmallows

Then the plot thickened.

Megan went on to report that, even after opening the Oreo to demonstrate that there is a filling, VMP rejected the Oreo as a sandwich because the filling is white.

Allow me to summarize:

  • Three things are required for a sandwich.
  • Unless they are cookies, in which case you only need two.
  • An Oreo is one cookie, so is not a sandwich.
  • Even if you want to call the Oreo wafers cookies and the Stuf the filling an Oreo is still not a sandwich because the filling is white.
  • The filling in a sandwich is properly referred to as a burger.

I saw a flaw in the logic, though.

Three marshmallows: mini, regular and giant

Marshmallows are white.

I asked about this. Megan reported that their marshmallows are colored.

I HAVE BEEN FOILED BY A THREE YEAR OLD!

So what do we learn?

Children have ideas.

Children use their minds. They think about things.

We can contribute greatly to our children’s learning by probing those ideas.

Formulating precise definitions is an important part of doing mathematics. Sorting things into examples and non-examples is part of this process. It really doesn’t matter whether we are sorting shapes (square, not square) or food (sandwich, not sandwich). And when the child is three years old, it really doesn’t matter whether she is consistent in her sorting.

What matters is that she is thinking in this mathematical way.

Starting the conversation

You can do as I did. Tell your child that another child says it takes three things to make a sandwich. Ask your child whether she agrees. Then ask about open face sandwiches and about Oreos.

But the bigger picture is important here too. There is a useful habit to develop as a parent—ask follow up questions when your child makes proclamations.

Other conversations we have had in this vein include Spirals, Circles and Armholes.

Counting grapes

I am pretty sure I have mentioned this before, but one of my proudest achievements has been watching a “Talking Math with Your Kids” hashtag (#tmwyk) blossom on Twitter in the past few months. Now, on a nearly daily basis I (and you, if you join us over there) get to see conversational gems such as Kindergarten kids talking about Spirals and cool math prompts such as Counting Grapes.

Michael Fenton—a father and math teacher—sent this photograph into the #tmwyk world recently. Naturally, I had to talk with Tabitha and Griffin about it.

Two bowls—one with five grapes, one with eight half-grapes

The conversation with Tabitha (7 years old), I captured on video.

Here’s the transcript:

Me: Which one of these bowls has more grapes?

Tabitha: (7 years old): [points to a bowl, probably the one on the right but hard to tell] Obviously!

Me: What do you mean, ‘obviously’?

T: I mean look at this! One, two, three, four, do you mean halfs?

There is a thoughtful pause.

T: Actually…

She points to the bowl on the left.

T: Cause these are halves

Me: But how do you know that there’s more here than here?

T: Cause look.

She uses her thumb and finger to indicate that halves of grapes are getting put into pairs to make whole grapes.

T: One, two, three, four

Now she shifts to the bowl on the left and counts the whole grapes individually.

T: One, two, three, four, five.

So what do we learn?

The key moment is right here: I mean look at this! One, two, three, four, do you mean halfs? (This occurs 8 seconds into the video.)

That is when she notices—on her own—that half grapes are not worth the same as whole grapes. It is where she shifts her attention from items (of which there are 5 on the left and 8 on the right) to whole grapes (5 on the left, but only 4 on the right).

The rest is tidying up details. The learning happens in that one brief moment of insight.

Starting the conversation

Ask your own child this question when you have a spare moment. Don’t correct or interrupt. Just listen. Object if their explanations are incomplete, but otherwise just listen.

Technical notes (and acknowledgements and thanks)

This was our first video using Google Glass.

There will be many more, I am sure. I’ll write more about this in the future, and I am happy to discuss with any interested parties. (You can hit me through the About/Contact link here on the blog.)

In the meantime, I want to thank Go Kart Labs for their sponsorship and financial support. They funded most of the cost of my Google Glass through a generous donation. These folks are smart, kind and interested in the overall goal of the Talking Math with Your Kids project, which is developing a world full of intelligent, creative and curious citizens. Upstanding people who do beautiful web-design work here in Minnesota.

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.

Eight plus four equals box plus five

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.

[Product review] Leap Pad Paint Bucket

Tabitha’s neighbor friend has a Leap Pad. Naturally it became a much hoped-for Christmas gift. She did receive one and has spent quite a bit of time with it.

I have no interest in reviewing the thing itself (although I will give you a heads-up that apps on this thing are expensive in comparison to iOS and Android! Holy buckets!)

The Leap Pad comes with a few standard apps. One of these is a drawing app, called Art Studio. Tabitha (6 years old) has drawn many pictures on it.

lf_art_000003This is the sort of thing I’m talking about.

This is fine.

And I wondered whether I could get some math out of it.

See, there is a paint bucket tool in there. When you apply the paint bucket, the paint fills up your drawing, but it doesn’t go across lines you have already drawn. So if you draw a square, you can paint the inside of the square and the paint won’t leak out. Or paint the whole screen outside the square and the paint won’t leak in. Unless you leave a small hole, in which case, the whole screen gets painted because the paint leaks through the hole.

I showed this feature to Tabitha and proceeded to draw some complicated curves, asking her to guess where the paint would go. For example, I drew a spiral.

lf_art_000085

This was no problem for her.

I asked her how many colors we could use to paint some complicated curve pictures if we used a different color for each section of the drawing.

lf_art_000086

Again, no problem.

I had her draw pictures and make me guess.

Finally, after about five minutes of this, she announced, “Daddy! You’re not allowed to do math on this!”

I was busted. I had to take a time out and let her just play with her toy.

But then, going back and looking at her more recent art, I can see I got into her head.

Don’t worry, though. The horses are still making appearances.

lf_art_000070

 

So if you have a Leap Pad in the house, I gladly give two-hooves-up for math in the Art Studio!

 

 

More patterns on the multiplication machine

When we left off last week, I had challenged Tabitha to find a pattern on the multiplication machine so that there would be the same number of buttons up as down. This challenge followed up on her sophisticated argument that her down-up-down-up pattern yielded more downs than ups.

ups.and.downs

There are 81 buttons, so the task of evening out the ups and downs is not possible.

But Tabitha is 6 years old. She knows little about even and odd numbers. Searching for a way to share 81 things equally (between up and down in this case) is a good way to get her thinking about the idea.

You may recall that I had shooed Tabitha off to her bath on giving her this challenge. This is where our story picks up.

At the end of the bath, she puts on her jammies and announces…

Tabitha (6 years old): I know how!

She runs into the room to get the machine.

T: Now Daddy, I don’t know if this is going to work, so just keep your ideas to yourself.

This line is awesome, is it not?

I do as I am told.

She produces this:

Photo Nov 20, 9 41 52 PMT: Oh no.

Me: What?

T: These [she points to top and bottom rows] are both up.

She tries again, producing this:

Photo Nov 20, 9 42 20 PMT: Oh no. Still too many up.

At this point she gives me a look which I take to mean that I can have a try. So I go back to her first pattern.

Photo Nov 20, 9 41 52 PMAnd I start to share out the bottom row—half up, half down.

patterns.6Tabitha: But Daddy! That’s not a pattern!

So what do we learn?

The raw beauty of Tabitha’s line, “I don’t know if this is going to work, so just keep your ideas to yourself!” strikes every time I think about this conversation.

Children enjoy investigating their ideas. I have to work very hard to get many of my college students back to a mental place where they trust that they have mathematical ideas worth investigating.

The best thing a parent or teacher can do in this situation is be quiet and let the kid work it out.

Starting the conversation

As all interesting conversations do, this one had a trajectory. We started in one place (making fun patterns), focused our attention on one part of what we were doing (comparing the number of ups and downs) and finished off with a “what if” question (what if ups and downs were equal in number, what would that pattern look like?)

You can practice that with your child. It doesn’t matter whether any particular conversation goes anywhere (many of ours do not), eventually you’ll hit on something interesting to both of you and pretty soon you’ll notice that 10 or 15 minutes have gone by.

And then the next time will be easier. Soon it will be a habit.

 

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.