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.

Uncle Wiggily

Tabitha was 3\frac{1}{2}  years old when we were playing a game of Uncle Wiggily.

In case you are not familiar with the game, I’ll briefly describe it. Uncle Wiggily is a board game with 100 spaces along a twisty path. Players draw cards; each card has a number and a brief poem. Perils and bonuses are judiciously spaced along the path. Uncle Wiggily is approximately 10% more complicated than Candy Land (which is to say, not very complicated at all!)

Tabitha: (Drawing a card for her first turn-it’s an 8) Got one Daddy!

Me: Mmm-hmmm.

T: What is it?

Me: Can you guess? Look closely.

T: (Quickly and with a big, eager smile on her face) Six!

Me: Good guess. It’s eight.

T: Oh!

Me: Can you count to eight?

T: (Bouncing her piece along the path, ending near the henhouse on the farm-themed board) One, two, three, four, five, six, seven, eight. By the cluck-cluck house!

Me: My turn. (Drawing a card-it’s a 10) What card did I choose?

T: Ten!

Me: Good. (Testing a hypothesis, I skip eight as I count) One, two, three, four, five, six, seven, nine, ten.

T: (Oblivious) My turn.

So what do we learn?

Learning to count is messy. Many things we might expect to be true about how children learn to count are not true at all.

We might expect children to learn the numerals (8) at the same time that they learn the words (eight). They do not. Notice that Tabitha counted flawlessly to eight, but did not recognize the symbol “8”.

We might expect children to learn the numerals in order, with all multi-digit numbers coming only after mastering the single-digit numbers. They do not. Tabitha recognized “10” but not “8”.

When I counted to ten, I intentionally left out eight to see whether she would notice. She did not. She could count to eight, but didn’t notice when it got left out on the way to ten. Mathematics is logical and orderly. The ways children learn mathematics are not.

This conversation came from a short video I made one day. I watched this video a year and a half later, when Tabitha was 5. After watching it, I immediately went into the kitchen where she was having a snack and counted: 1, 2, 3, 4, 5, 6, 7, 9, 10. She smiled and asked Why did you do that? (referring to counting in her ear). Then, a moment later, she said, Hey! You skipped eight!

Starting the conversation

Play games that involve numbers. Uncle Wiggily is great. So is Chutes and Ladders. Or Hopscotch. Any game that involves counting and reading numerals will give you the chance to practice these early number ideas.

While you’re playing, ask your child what number he drew, and what number you drew. If he doesn’t know, have him guess. Don’t worry about precision or correctness. Model good counting for your child. Help him count out some of his turns and let him count incorrectly on others. Have fun and don’t worry too much if he gets bored before the game is finished.

Have fun with it. Whatever you need to do to stay engaged in a couple of rounds of Uncle Wiggily is worth the effort. You can see the effort I invested in keeping myself entertained; I formulated a hypothesis about whether she would notice my own incorrect counting and tested that hypothesis.

Don’t get carried away with the hypothesis testing, though. Children do need models of correct counting. They won’t be damaged by a few experiments, of course. But you don’t want to become an unreliable source of knowledge.