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.

10-minute reading time

A while back, bedtime was spiraling out of control. The kids share a room; they would be wound up at bedtime and the transition to sleep was not happening smoothly. We had a big, big problem on our hands.

We solved the problem with 10-minute reading time. The kids have to be in their beds. We dim the lights. We set a timer for 10 minutes. It has to be quiet during that time. Then we turn out the lights, give them something to picture in their minds, and sleep comes more easily.

Complete transformation. It is awesome.

One night, Tabitha (5 years old at the time) wanted to color. We talked and agreed that she could do it “sometimes”. As is the nature of 5-year olds, she soon wanted to know the limits.

The following conversation took place on a Wednesday night.

Tabitha (five years old): I know I can’t color every night but can I tonight?

Me: Yes.

T: Then read, then color the next night?

Me: I don’t know. I think reading twice before the next color is better.

To be clear. It was not my intention to get into a math conversation at this point. I just wanted her to go to bed (Warning! Link Not Safe for Work, and Possibly Offensive to Sensitive Ears. But Funny).

No, this move on my part was truly about literacy, not math. I don’t want 10-minute reading time to turn into 10-minute coloring time. I really, really like the idea that books will become part of my kids’ independent bedtime routines.

But Tabitha loves to know the rules she’s playing by. And when those rules are based on numbers, they’re going to lead to math every time.

T: So read-read-color-read-read-color…like that?

Me: Right. That sounds like a good ratio.

T: Or read-read-read-read-color-color?

Whoa.

Couple things.

First of all, I used the term ratio with absolutely no expectation that she would process it, and I am quite sure that she did not. I have long been an advocate of using good vocabulary with my children—there is no shame in not knowing the meaning of a word, but also no sheltering them from the fact that these words exist. This is at least partly the source of their substantial vocabularies. But I do not believe she knows the word ratio.

Secondly, Tabitha’s reformulation of the 2:1 ratio as 4:2 blew me away. It nearly slipped past me without notice. I was focused on getting them to bed; we were in the truly final phase of that process. I had pretty much tuned her out.

But when I looked at her, I could see she was expecting a reply. She needed to know whether she could get two coloring nights in a row by doing four reading nights in a row.

So I replayed her question in my mind, counting the reads.

Me: Yes. That would be fine. You may do that.

So what do we learn?

Together with the recent Easter Candy conversation, this makes clear that young children are thinking about early ratio ideas.

Think about this for a moment. What is the same about these two sets of tiles?

tile.ratio.2

And what is the same about these two sets of tiles?

tile.ratio.3

In each cases, it is the ratio.

In the first case, there is one blue for every yellow up top and also down below. This is what L was working on when she offered a second chocolate egg to Tabitha.

In the second case, there are two blues for every yellow. This is what Tabitha was working on when she asked about read-read-read-read-color-color.

Traditionally we think about ratio as a sophisticated fractions topic that needs to wait for early adolescence. I certainly would not want to be held accountable for teaching 5-year olds ratio and the associated notation. But their everyday experiences allow for them to think about these ideas. As parents, we can keep an eye out for those opportunities and talk about them when they arise.

Starting the conversation

Both of these ratio conversations with 5-years olds have resulted from constraints. Five year olds love to test rules. “Yes” and “No” are inflexible and allow no wiggle room. This is sometimes desirable. Yes, you must leave now to get to school on timeNo you may not leave the house without pants. Yes you must look both ways before you cross the street.

But when we are guiding our children’s behavior, we would be wise to allow some space for the little ones to test the boundaries. Does it really matter—in the big picture—whether L eats one chocolate egg or two? Does it really matter whether Tabitha colors two nights in a row instead of reading? I say no. Neither of these things really matters.

What matters is that L does not continuously gorge on candy, and that Tabitha has some alone time with books. Constraints rather than absolute mandates seem to have encouraged mathematical thinking in both of these cases while also addressing the big picture.

Easter candy

Easter Sunday saw St Paul, Minnesota waking up to weather perfection. Sunshine, low seventies (Fahrenheit), cloudless sky. Truly amazing.

There was a loon on Lake Phalen!

This was the sort of April weather that brings Minnesotans out of their homes to rediscover their neighbors.

So it is with Griffin, Tabitha and me on this warm spring morning. We are enjoying the warm sunshine on our front steps when L (five year old girl), O (3 year old boy) and their mom come biking, biking and strolling (respectively) down the sidewalk.

L is on a neighborhood mission delivering handmade Easter greeting cards.

It turns out that she has also pocketed some goodies from her own Easter basket. While we chat, she pulls out a bag of Cadbury mini eggs. In case you are unfamiliar, these are the size of pebbles. They are chocolate inside with a crunchy candy shell. Like an oversized egg-shaped M&M. Each little bag contains about a dozen.

so.much.candy

So. Much. Candy.
This is likely a small fraction of the candy L has consumed by the time she stops to chat.

Anyway, mom notices the bag as soon as it emerges from L’s pocket. (Side note—mom is across the street! Holy SuperMom powers!) She warns L not to eat any more of these; arguing that L has had enough candy for one morning.

L (5 years old): Please?

Mom: If you give everybody one, you can have one.

L proceeds to cheerfully open the package, hand one to Tabitha (who eagerly and gratefully receives it), one to Griffin and one to me.

I begin to think about what question to ask to get some math talk going.

But L is ahead of me.

After enjoying both her egg and a long thoughtful pause, she pokes her finger back into the bag. She begins to rummage around and asks:

L: Tabitha, do you want a second one?

So what do we learn?

Children use math to their advantage.

L knew what mom meant. Mom had compromised on the candy, allowing her one piece. L knew that. And she knew that the process was repeatable.

One does not always mean one. One might be taken to mean each. “Each time you give everybody one, you can have one.” This is also a reasonable interpretation of mom’s words.

L was rule bending here. But she was also building the precursors of ratios. For every one you give a friend, you can have one. This is a ratio. Giving a friend two and having two fits this rule just as well as giving a friend one and having one. Ratios are one of the more challenging ideas behind multiplication and division relationships, and fractions.

What is maddening for parents is at the same time great thinking practice for children.

Starting the conversation

This was a brilliant compromise strategy on mom’s part. I doubt that she intended to encourage L to think proportionally, but that doesn’t matter. More likely, she was trying to encourage the admirable social skill of sharing. By including numbers in her compromise, she opened the door for L to think.

As I have mentioned before, anytime your child wants to open a negotiation, there is an opportunity for math talk. Sometimes we parents need to give a flat out yes or no. But when negotiations are feasible, we can get our children thinking.

Doll years

Out of the blue on our recent camping trip, Tabitha had an announcement for me.

Tabitha (6 years old): I am 12 in doll years and Griffy is 16 in doll years.

Her brother Griffin is 9.

T: So how old are you in doll years, Daddy?

Me: Well, how do doll years work?

Photo Oct 12, 1 34 29 PM

T: Well, I’m 12 and Griffy’s 16.

Me: Is it twice as old? Then I would be two times as old, so nearly 86.

My birthday is coming up next week. This has been a point of discussion around the house recently.

T: No! It’s 6 times!

Me: You’d be 36 then.

T: No. I am 12 in doll years.

Me: Oh! Six years older not 6 times as old!

T: Yeah.

Me: Then Griffy is 15, not 16. And I would be almost 49.

So What Do We Learn?

Children build lovely and complicated imaginary worlds. For a long time, Griffin and Tabitha would play “creatures” together. Whole societies of stuffed animals, dolls and plastic figurines rose and fell. These societies had celebrations and tragedies. There was Creature Christmas that could take place at any time of year. Also a Creature State Fair. Et cetera.

Combine this parallel creature/doll universe with learning about the passage of time and pretty soon doll years are going to pop up.

Griffin and I talked about tortoise years and dog years a while back. At the time, Griffin was 8. He was comparing life spans of tortoises to those of humans, as we do with dogs to generate the 7 dog years per year comparison that is commonly known.

Tabitha is firmly grounded in comparing by counting and addition, as is appropriate for a 6 year old. Somewhere between third and sixth grade, children transition from always comparing by addition and subtraction to being able to compare by multiplying and dividing. This difference is what Tabitha and I are discussing in this conversation. She says Six times but means Six more.

Starting the Conversation

Listen for the comparisons your children make. Here, Tabitha compared ages. But heights, dollar amounts, number of Tootsie Rolls in a candy dish, et cetera; all of these are possible comparisons that children will naturally make. Ask a follow-up question. How do you know? is a good place to start. What if? is a lovely follow up. For example, What if there were a newborn baby in our family; how old would it be in doll years?