Math teacher seeks primary classrooms to talk shapes

I made a shapes book recently. The response has been a ton of fun. I have heard from parents of preschoolers, from primary grades teachers, upper elementary teachers and middle school teachers. All have reported having interesting and sometimes surprising conversations with their children and students.

I have heard from high school teachers, including a class studying set theory.

But I feel as though I am missing out on the fun. I want in.

If you teach kindergarten, first or second grade in the Twin Cities (MN) area, and you would be willing to have a visitor come talk geometry for a half hour in the coming months, shoot me a note, please. If you know such a teacher send them this post, please.

We’ll work out the details and I’ll see you in class!

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.

A month of math in March!

This is just a quick note to let everyone know that Talking Math with Your Kids is guest-posting on Tiggly’s blog throughout the month of March.

It’s March Mathness.

You celebrate that, right?

March.mathness

There have been two posts so far:

While you’re there, browse around a bit. The folks at Tiggly are super smart, sincere and hard working. They have lots of good resources.

Talking Math on Facebook

This is just a quick note to let you know that Talking Math with Your Kids now has a Facebook page. It will be a place for links and resources as I come across them, and typically before they get worked into posts here on the blog. It can also be a place for quick, informal discussion different from blog comments.

You are invited to join the fun over there! Just click on through.

whats.gnu

What’s Gnu? A Facebook page! That’s what’s gnu!
This image originally appeared in “Talking math with a word game“.

While we’re discussing social media, you can find me on Twitter, too. We like to use the hashtag #tmwyk over there.

The meanings of division

I was talking with Griffin one day when he was in third grade.

Me: Do you know what 12\div 2 is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about 26 \div 2?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade], so it was her magic number. We had 13 pairs of kids.

Me: What about 34 \div 2?

G: Well, 15 plus 15 is 30…so…19

My notes on the conversation at this point only have (back and forth), which indicates that there was probably some follow-up discussion in which we located and fixed his error. The details are lost to history.

Our conversation continued.

Me: So 12 \div 2 is 6 because 2 \times 6 is 12. What is 12 \div 1?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Me: What is 12 \div \frac{1}{2}?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and 12 \div 6 is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

So what do we learn?

Mathematical ideas have multiple interpretations which people encounter as they live their lives. As we learn more mathematics, we become better at connecting these different ways of thinking about ideas.

In this conversation, Griffin relies on three ways of thinking about division:

  1. A division fact is a different way of saying a multiplication fact. (12 \div\ 2 is 6 because 6 \times 2 is 12).
  2. Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students).
  3. Division tells us how many will be in each group if we make groups that are the same size. (When he was working on 34 \div 2, Griffin put 15 in each group to start off with.)

We were just talking for fun, not homework or the state test. So I wasn’t worried about his connecting those ways of thinking. I was just curious how he would apply them to some more challenging tasks, such as dividing by 1 or by a fraction.

I was surprised by how difficult 12 \div \frac{1}{2} was for Griffin. Not because it is an easy problem, but because he could have applied his how many of this are in that? idea, or his multiplication facts idea. But he did neither and reinterpreted the task as twelve divided by half-of-twelve.

I was also surprised at the length of the pause he took for 12 \div 1. It makes sense in retrospect. After all, are you really making groups if it’s just one group? I imagine he had to think that through, rather than the number relationships involved.

Starting the conversation

When the opportunity presents itself—when you and your child are not under homework stress, not rushing to get out the door or find the dog’s leash; when you happen to be talking about number anyway—ask follow up questions. Even a simple set of division problems got a lot of good thinking out of Griffin. Problems involving 1, 0 and \frac{1}{2} are especially challenging.

Vary the size of the numbers.

Don’t worry about whether the answers are right or wrong.

Keep asking How do you know? and listening to your child’s answer.

Offer a few ideas of your own.

Quit before anybody gets frustrated or bored.

Talking Math in Minnesota Parent

We are delighted to share a short feature in this month’s Minnesota Parent magazine featuring the “Real Dad” behind Talking Math with Your Kids. 

Share widely, please, and do give Minnesota Parent a bit of extra web traffic, won’t you?

Griffin and Tabitha thank you!

Notes on process

Michael asked in the comments recently:

One question: how are you preserving all these amazing conversations?

The answer is surprisingly low-tech.

See, I have this notebook.

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And it’s pretty much always with me and I use it to keep track of pretty much everything in my life. So whenever we have a talk, I write it down as soon as possible afterwards. Usually within minutes.

20130811-090224.jpg

My children are now accustomed to Daddy writing in his notebook as one of many needless delays in their lives. But then it takes them like 10 requests and 20 minutes to get their shoes on and out the door, so I figure we’re even at worst.

I have made a few videos of our math talks. But that requires set up which usually spoils the moment.

I have made a few audio recordings. But people listening to them have noticed that it’s frustrating not being able to see what we’re talking about.

I will share some of each of these in future posts, and you can hunt them down at my blog, Overthinking My Teaching if you need to see or hear Griffin or Tabitha talking math with me.

Also, I applied to be a Google Glass Explorer. I never heard back. It strikes me that the first-person view of a conversation could be powerful, and that Glass would mean no set up time. Maybe someday?

In the meantime, text and photos are our medium, and the details are captured in a notebook. Photos are mostly from an iPad, which I got from the good people at TED-Ed when I made that “One is one” video.

 

A few words on apps

The New York Times Bits Blog reports that the Campaign for a Commercial-Free Childhood has filed complaints with the Federal Trade Commission against Fisher-Price and Open Solutions for claims the latter organizations make on behalf of their mobile apps for infants. A quote:

[t]he programs could take time away from activities, like hands-on creative play or face-time with caring adults, that have proven benefits for infant learning.

I say, Take no chances!

Talk math with your kids!

If you don’t let a day go by without reading to them, then don’t let a day go by without asking, How many? or What shape?