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.

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.

milk.by.the.gallon

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!