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.

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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!

Multiplication and rectangles

I want to suggest a lovely post by somebody else.

It is written by a math teacher who converses with his niece (who is 7 years old) about rectangles and multiplication. As an example, the rectangle below shows that 6×3 is 18. Or is it that 3×6 is 18? That becomes the focus of part of the conversation.

This rectangle shows that 6x3 is 18. Or is it 3x6?

The girls’ parents look on as the discussion unfolds.

At one point, the math teacher stops the mother who is trying to intervene to help the child see that 4×3 is the same as 3×4. And this leads to the lovely sentence in the blog post:

I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.”

What he means by this is that children can learn from thinking about math, even if they don’t end up with the right answer, and even if they do not experience the full story (here, that multiplication is commutative, which means AxB=BxA for all possible numbers).

Another fabulous math teacher, Fawn Nguyen, told me, “I dare say that it’s not obvious to teachers also.”

Finally, non-math teacher parents may be interested to learn that—consistent with Fawn’s observation—a regular piece of feedback I get from math teachers on my writing here is how impressed they are by my ability to not worry about Tabitha and Griffin getting right answers.

Canned pumpkin

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

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

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

Me: Do two, then two more.

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

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

T: OK. Give me one back.

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

T: I did two and three.

Me: So we have five cans?

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

Me: So two plus three minus one is four?

T: Yeah.

So what do we learn?

Decomposing numbers is fun.

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

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

Starting the conversation

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

The Pumpkin Patch

On a family trip to a farm from which we have bought a tremendous amount of produce this year, Griffin and I were heading to the pumpkin patch.

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We had already taken the wagon ride to the other pumpkin patch; where the pie pumpkins were grown. We had helped with the harvest and had chosen several to take home. Now we were on our way to the Jack-o-Lantern pumpkin patch.

Griffin [9 years old]: We have 5 pumpkins! Is that enough to make a pie?

Me: More than enough.

G: Enough to make 5 pies?

Me: Probably not.

G: How many pumpkins go into a pie, or how many pies do you get from a pumpkin?

Me: Hmmmm… I would say about 1\frac{1}{2} pumpkins make one pie.

NOTE: This was semi-truthful. I really have no idea how many typical pie pumpkins are needed to make a pumpkin pie. I was making what I felt to be a reasonable estimate. But at the same time, I was pretty pleased with the estimate and with the math that it might encourage Griffin to do.

G: Oh! So we could make … 3 … 3 plus 1\frac{1}{2}4\frac{1}{2} … three pies! And have half a pumpkin left over!

Me: Which is \frac{1}{3} of a pie.

G: Right.

NOTE: I do not trust that he got that \frac{1}{2} of a pumpkin makes \frac{1}{3} of a pie given my estimate. He may have gotten it, and he may not have. The pumpkin patch was approaching so I let it slide.

G: Will we make three pies?

Me: No. I don’t think I’ll have the patience for that. But we can make one pie for sure.

So What Do We Learn?

Griffin is thinking about division when he figures out how many pies we can make from five pumpkins. Other similar sorts of division problems include, How many feet tall are you if you are 49 inches tall? and How many groups of four can we make in our classroom of 30 students? The pumpkin pie problem is challenging because it involves fractions.

One of the hardest parts of the thinking Griffin does here is keeping track of the units. As he counts up to 4\frac{1}{2}, he is counting pumpkins. The first 3 he utters counts pumpkins. But at the same time, he is keeping track of a number of pies. That’s the final 3 he utters: 3 pies.

I play with that idea by referring to his \frac{1}{2} of a pumpkin as \frac{1}{3} of a pie. I understand that not every parent is ready to do this on the spot. Don’t worry about that. Griffin got enough thinking from the basic conversation; the rest is gravy (or maybe whipped cream?)

Starting the Conversation

This was a special opportunity. We had some pumpkins. Griffin wanted to make things with these pumpkins. I could involve fractions.

Other such opportunities could include bags of apples, cups of flour (a standard 5-pound bag of all-purpose flour has about 18 cups), et cetera. If your child doesn’t ask the how many pies (or batches, or cakes, or whatever) question, you can ask it. But don’t make it feel like a quiz. You can just say, I wonder how many pies we could make with what we have?

A short waffles conversation

Those Eggo mini-waffles are paying off.

We had this conversation the other day…

Me: There you are, Tabitha. Two sets of waffles.

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Tabitha (six years old): That’s 7. No…8!

Me: [washing dishes with my back turned to her] Right. Two sets of four is eight.

T: That’s not how I know.

Me: You counted?

T: No.

Me: Oh. How did you know, then?

T: Three plus three is six. And there’s 2 more.

Me: [Big smile and thumbs up for encouragement]

So What Do We Learn?

I recently pushed Tabitha past the limits of her patience by asking about lobsters and half-lobsters. But in doing so, I was continuing to lay the groundwork—how she thinks about things is interesting to me. I want to know, I value and reward her thinking. So she talks about it.

When you consistently talk math with your kids, you will make progress. It may seem slow at times, but you’ll make progress.

Mathematically, there is something really wonderful going on here. She is trying to figure out 4+4, but it’s not a fact she has handy. So she thinks of 4 as 3+1.

Now it’s 3+1+3+1, which she rearranges as 3+3+1+1, which is the same as 6+2.

She uses a fact she knows (3+3) to find one she does not (4+4). This is an example of using derived facts, which Griffin did also in a recent conversation about the number of fives in an hour.

Starting the conversation

Listen for the times that children announce how many things there are. Ask them how they know.

Another example: Griffin had his ninth birthday party recently at a local swimming pool. The cake was provided; the high schooler who brought over the cake asked me his age and proceeded to count candles from the pack. It was hot; the candles must have slightly melted into the container because she was struggling and took a good minute or two to dislodge the candles, leaving them on the table before disappearing.

She had left eight candles behind.

For a nine-year old’s cake.

Needless to say, this was a topic of great conversation among the children present. Somehow Griffin didn’t notice. But his friend from up the block, W, did. She asked me, “Hey wait! Why are there only 8 candles?” I don’t know, I replied, but how did you know there were 8? She gave me a funny look. Did you count them one by one? “No,” she said, “by twos…2, 4…”

It is that easy.

You just have to put up with a few strange looks from children sometimes.

How many fives in an hour?

Our local public library has a summer reading incentive program. Children keep track of the amount of time they spend reading, and when they reach 20 hours they get a prize. Some of the prizes are good, including a ticket to the State Fair.

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To keep track of their time, children get a chart. The chart has 20 individual hours, each represented by an icon. Half of these are circular, suggesting clocks, and half are rectangular, suggesting books. Each icon is broken down into five minute intervals. We were driving home one June Sunday afternoon after picking up Griffin and Tabitha’s summer reading charts.

Me: Griff, each hour on your chart is broken up into 5-minute chunks, right?

Griffin (seven, nearly eight at the time): Yup.

Me: So how many of those chunks are there in an hour?

G: (long pause) Sixteen.

Me: Why sixteen?

G: Well, I thought of 5 minutes like a nickel, and there’s 20 nickels in a dollar.

Me: Wow.

G: So I minused four, because it’s four less.

Me: Right. 60 cents is 4 tens less than 100 cents, though. So I think we need to…

G: (interrupting) Oh! RIght! So…it’s twelve. Twelve fives in an hour.

Me: That’s some really good thinking there, buddy. I wouldn’t have thought to do it that way.

So what do we learn?

If you are new to thinking about people learning math, it may be surprising that asking children to explain their thinking aloud often leads them to correct their mistakes.

Math is very often portrayed as a subject where things are either right or wrong with no in-between. This is not a helpful image of the subject. Indeed, there are many shades between these two extremes. Sixteen was a wrong answer; there are not 16 fives in 60. But underneath that wrong answer is some pretty sophisticated thinking.

When we figure out some new answer based on one we already know, this is called using derived facts. It’s a very useful mental math strategy and it should be encouraged at every opportunity.

You can only encourage it if you know it is being used. And that’s another reason we need to ask about process. We want to know how kids are thinking so that we can help them make that thinking better.

Starting the conversation

While mental math strategies are becoming more explicit in schools, many parents today did not learn many such strategies when they were in school. The emphasis for many parents may have been on (1) memorization of facts, and (2) paper-and-pencil computation. Therefore you may not know very much about derived facts, or more likely, you don’t notice that you use them.

If you have ever thought “58+9 is 67 because 58+10 is 68, and 9 is one less,” you have used derived facts.

Whenever a computation of some kids comes up in daily life, ask your kids to talk through their thought process. Model your own thinking for your kids.

In short, make everyone’s thinking part of the number conversation.

You and they will get better at it as you keep at it.

Lobsters

Griffin turned 9 the other day. His birthday dinner request was lobster. We live in Minnesota, where lobsters are not native. They do not come cheap.

Tabitha, who is 6, announced in advance that she would not be having any lobster. My mother joined us for dinner. A grand time was had. Children held live lobsters. There were also clams, oysters and some of the season’s first corn on the cob. Having an August birthday must be great.

That night, the children were overstimulated and exhausted. This may not be the best time for math talk. But I could not resist the opportunity.

Tabitha and I were on her bed in the final stages of bedtime.

Me: Tabitha, how many lobsters were there today?

Tabitha: (6 years old) Two.

Me: You saw that I cut them in half, right?

T: Yes.

Me: How many half lobsters were there?

T: NO! I am not talking about that now!

Me: Oh come on; I promise it won’t be a big talk. How many half lobsters were there?

T: Four.

Of course I knew she knew this, and that my question wasn’t challenging for her. But I was desperate to know how she knew. Could she see them in her mind? Did she remember that everyone at the table, except her, had their own lobster with no leftovers? Had she counted them at dinner time? Did she know to double the number of lobsters (this seemed unlikely—she’s only six)?

I had to know.

Me: OK. Just one more question, I promise.

T: NO!

Me: Seriously. Just one last question, which is: How do you know?

T: I am not…

Me: Come on. You’re right and I just want to know how you thought about it.

T: 2 plus 2 is 4.

Me: Oh. Good.

I pause to ponder my next move.

Me: So I am just going to say something. You don’t have to respond.

T: Grrr….

Me: I think you were thinking that the first “2” was for the two halves of the first lobster, and that the second “2” was for the two halves of the second lobster. So 2+2 adds the parts of the two lobsters together. Is that right?

T: Yeah.

So What Do We Learn?

First of all, we learn that it is dangerous (but still possible) to talk math with overtired children.

Secondly, we learn that 2+2 is a strange math fact. Here’s why. Imagine there had been three lobsters. If my guess about Tabitha’s thinking was right, she would think 2+2+2 is 6. (Well, more likely in stages…2+2 is 4, then 4+2 is 6.)

But because I stated the meaning of her numbers I do not know that for sure. Maybe the first two was really for the 2 left-hand sides of the lobsters, while the second 2 was for the right-hand sides. In that case, she would answer the question about three lobsters with 3+3 (3 left-hand sides plus 3 right-hand sides). There is no way to know based only on the conversation we had.

But there was no way I was going to drag more out of her at the time. I’ll ask her sometime.

Finally, we have another demonstration of changing what we’re counting. In the Things that Come in Pairs conversation, we were grouping eyes and counting the number of pairs. In this conversation we are cutting lobsters in half and cutting the halves. Opposite process, same underlying idea.

Starting the Conversation

There are lots of opportunities to count both things and halves (or thirds, or fourths, or whatever). You do not need to buy lobsters. Pizzas, cookies, sandwiches and cakes are examples of things children experience in both their whole form and cut up.

Start with things that are either right in front of kids, or that have recently been. And start with the actual numbers. Then move to the what-if scenarios. Starting with things children have actually touched or seen makes things concrete and easier for many children to think about.

You can also talk about things you are about to cut. “You see these three grilled-cheese sandwiches? I am about to cut them in half. How many half sandwiches will we have after I do?”