Making eight

I am writing a book. In the process of doing this, I come across homework assignments that parents find frustrating, and that they share on social media. These almost always get me thinking, and they frequently lead to math talks with my children.

This past weekend was one such instance.

Worksheet: "The whole is 8. One part is 8. What is the other part?"

Talking Math with Your Kids is not a place to hash out the details of whether this is a well written question, or whether this was an appropriate homework assignment for this child. We can discuss that on Twitter if you like, or through my About/Contact page.

Talking Math with Your Kids is about taking opportunities to have math conversations with our children. In that spirit, I share the conversation we had in our house.

Out of the blue, I asked Tabitha (7 years old) if I could ask her a math question. It was maybe Saturday afternoon. We had nothing special going on.

Me: Tabitha, can I ask you a math question?

Tabitha (7 years old): Yes.

Me: If I have eight things, and seven of them are in one hand, how many are in the other?

T: That’s not even a math question! That’s too easy!

Me: OK. But will you answer it anyway?

T: One.

Me: OK. What if I had five in one hand?

T: And you still had 8?

Me: Yeah.

She spent a few moments thinking.

T: Three.

I had a couple other questions, which I asked and she answered. The next day, I realized that I didn’t know how she knew that second one.

She was getting ready to brush her teeth on Sunday evening when I asked whether she remembered the previous day’s conversation. She did.

Me: How did you know it was three?

T: I counted.

Me: Like this? Five, then six, seven, eight?

T: Yeah. And that’s three. But actually, I kind of already had it memorized.

Me: Oh yeah? How did you memorize it?

T: Huh?

Me: Did you try to memorize it? When I want to memorize a phone number because someone told it to me and I don’t have a pen handy, I say it over and over to myself. Did you do that with 5 + 3?

T: No! I just have counted it out a lot of times.

Now, I should also mention that I asked Tabitha, If I had 8 things, and 8 of them were in one hand, how many would be in the other? She replied Zero without much hesitation. This If I have this many in one hand, how many are in the other formulation is probably less clumsy than the If this is one part, what is the other part? formulation on the original worksheet. But the intention is the same.

So What Do We Learn?

The kind of problem Tabitha and I were working with is called Part-Part-Whole. For young children, this is different from the standard “takeaway” problem because there is no “taking away”. I didn’t eat, lose, destroy or give away any of my eight things in these problems—I just have some in one hand and some in the other.

Because Part-Part-Whole involves a different way of thinking, it’s a good idea to practice some of these problems. It helps children to build a better understanding of addition and subtraction relationships if they see all the various ways these relationships appear in their worlds.

Tabitha herself pointed out an important principle of Talking Math with Your Kids: Many things that you hope to remember, you can remember by encountering them frequently. Tabitha has never sat down with flash cards to memorize her single-digit addition facts. Yet she is in second grade and is starting to feel confident with them.

She and I talked about familiarity—how maybe learning 5 + 3 is a little like learning the name of someone you see in your neighborhood. You don’t recognize the person as being the same person the first few times you see them. But eventually, if you see them frequently enough, you do recognize them, and you might introduce yourself. Pretty soon, you know their name. And if you just can’t seem to remember it? That’s when it’s time to drill yourself. That’s when you repeat the name over and over and over.

Starting the Conversation

Ask the questions I did. This is an easy conversation to have. If your child isn’t confident with addition and subtraction facts, ask about six in one hand instead of jumping to five in one hand. 

More broadly, look for Part-Part-Whole opportunities to talk about. This is an important interpretation of subtraction, and one that is often neglected. Examples include apples (Our fruit bowl has 8 apples—5 are red, how many are green?), pets (There are 8 pets on our block—5 are cats, the rest are dogs. How many dogs?), et cetera.

Guess the temperature

This post is from last year on my math teaching blog. Presently we (along with much of the American Midwest) are in the middle of a serious cold snap. So I have edited and remixed it for the Talking Math with Your Kids audience.

guess.the.temperature

This morning’s situation. Colder air is on the way.

Enjoy.

And stay warm.

Griffin (8 years old in this story) and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, in winter, this means we get to study both positive and negative numbers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Me: So what if it had been 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. Griffin’s silence seemed a clear sign that he was ready to move on.

So what do we learn?

Two things are important in this conversation: (1) Griffin’s solution method, and (2) the connection to subtraction.

Griffin’s solution method. Griffin’s strategy is a common one for children to invent. He uses zero as the boundary between positive and negative numbers. To compare how much bigger a positive number is than a negative number, you have to cross that boundary. You have to go past zero, so it just makes sense to him to divide the distance into two pieces—the part to get to zero, and the rest.

We live near a major road—Arcade Street. We often use it as a boundary for our neighborhood. So the local recreation center is three blocks on the other side of Arcade; plus the one block to get to Arcade. That’s four blocks total. Talking in this way about everyday navigation supports thinking about temperatures, which in turn support thinking about integers.

Connection to subtraction. From years of teaching middle school (and—to be honest—college), I know that subtracting integers is tough going. The rules for solving 3-^{-}2 don’t seem to connect to students’ experiences with numbers.

Notice how quickly Griffin connects the 10° and 3° situation to subtraction, while not seeing that subtraction applies to the 3° and -2° situation we started with. Perhaps my mentioning that these are the same will lay the groundwork for him noticing it in the future.

In the meantime, we learn that learning subtraction is a lot like learning division. In a recent post, I showed how Griffin thought differently about division depending on the numbers involved and on the context for thinking about it. Now we see that this is true about subtraction, too. 10-3? No problem. 3-(-2)? Problem.

Starting the conversation

Children—all children—develop mathematical models of their worlds before they study them in school. Parents have opportunities to support this through conversation.

Talk about landmarks, way stations on your journeys. Mileposts, subway stations, bus stops, blocks…these are all opportunities to help children build the mental models necessary to think about zero as an important landmark.

Talk about distances. How many blocks is an example from our family life. How many subway stops came up for girl I observed in New York City last fall. How many pages did we read in our book is another example where subtracting endpoints is helpful.