Guess the temperature

Griffin 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, this means we get to study integers.

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.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 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. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

Another cool thing…

In our discussion the other day of whether the difference between van Hiele levels 0 and 1 is the use of official math vocabulary terms, or whether there is something more there, a student asked this (I am paraphrasing):

Is this like the CGI research, where children know things about addition and subtraction before they are taught in school?

Here’s what I love about this question:

1. It demonstrates the power of having these students for a second semester;
2. It demonstrates that students will—over time—make these ideas their own; if we keep at it, and build cases for the importance of the ideas in our courses, students will play with these ideas, and they will look for opportunities to apply and connect them; and
3. It made me stop and think. I’m still not 100% sure whether it’s like that or not. I have something new to think about.

Salami-slice math

I was making lunch for Tabitha (who had just turned 5) one day. It was one of her favorites-salami quesadilla (house specialty). She was impatient and hungry. She asked for some salami while she waited.

Me: How many slices do you want?

Tabitha: Four.

Me: I’m sorry to say, we only have three.

T: Oh. That’s one less than I wanted.

Me: That’s right. Do you want them?

T: Yes.

Me: (Giving Tabitha the remaining salami slices) So if three slices is one less than four, what would two less be?

T: Two.

Me: And what would three less be?

T: (Thinking) One.

Me: Nice. How did you know that?

T: I don’t know.

Me: What would four less than four be?

T: (Thinking) Zero!

Me: Nice.

T: Ask me another.

Me: What would five less than four be?

T: (Thinking) Zero?

Me: Interesting.

T: Do a different “less”. Like less than five or something.

There are two important strategies here. The first is turning Tabitha’s request for salami into a situation that involves numbers. She asked for salami to eat while waited for the main part of her lunch. I could have just given her a few pieces of salami, but that’s a lost opportunity. It’s an easy move to ask her how many slices she wants, and then to compare that to the number of slices we actually have.

The second important strategy is the What if? questioning style here. This is where the abstraction happens. When I ask her What would two less be? I am offering here the opportunity to think in terms of salami slices, or to just think about numbers. She has already imagined the slices so they are available as a tool. Or she can count numbers in her head.

Note Tabitha’s comfort with zero as a number. When I ask her what four less than four is, she comfortably answers zero. Notably, she has to think about it. She hasn’t developed the rule that any number minus itself is zero, although this conversation could certainly go that direction by asking about three less than three salami slices and five less than five salami slices, etc.

Tabitha is five years old. There is plenty of time for her to learn about negative numbers. I did her no harm asking What is five less than four? I didn’t expect her to be able to say negative one, but I was curious about what she would say. Remember that being interested in children’s ideas is a key to talking math with them.

It matters, though, that she didn’t just reverse the numbers and say “One”. The numbers 5 and 4 have very specific meanings for her, and she uses those meanings the best she can.

Kindergarten addition strategies

A few months back, Tabitha (5) wanted me to ask her some math class questions. This led to some disagreement between the two of us on what constituted a math class question.

On our way back from our fall camping trip last weekend, Tabitha announced that she knows what 8+8 is.

Me: Ok. What is 8+8?

Tabitha: [longish thinking pause, eyes gazing up and to the right] 16!

Me: Wow. I could really see you thinking that through. How did you know that?

Tabitha: I counted.

Me: OK. Good. But how did you keep track of how much to count?

Tabitha: huh?

Me: How did you know when to stop counting?

Tabitha: When I got to 8. That was 16.

Me: Right. But how did you know when you had counted eight times?

Tabitha: I just counted to 8. That was 16.

Me: Oh! I know! Was it like this? Eight…nine (that’s one), ten (that’s two)…like that until you got to sixteen (that’s eight)?

Tabitha: Yes.

Me: How did you know to do that? Did you learn to do that in Kindergarten? Or was it your own idea?

Tabitha: [long pause] OK. I admit it; it was my idea.

I have written before about how sticking with this stuff pays dividends eventually. Tabitha announced that she knows what 8+8 is, but what she really meant was that she knows how to find 8+8. Tabitha knows that talking math is not about telling answers, but about actively thinking things through. That doesn’t come for free in classrooms, and it doesn’t come for free at home. It is a result of lots of prep work and lots of encouragement and lots of conversations.

Returning to the theme of the knowledge required for me to have this conversation with Tabitha, though, I needed to know something about possible strategies, and I needed to have thought some of those strategies through for myself-by trying them out.

If the sum in question had been 8+2, I wouldn’t have questioned her claim that “I counted”. You can subitize two, so you can know that you have counted twice without having a separate system for keeping track.

But counting 8 times without having a way to ensure that you have done so? This seems pretty much impossible. As an adult, I know to stop at 16, but that’s only because I already know the sum. As I count 9, 10, 11, 12, I am conscious that the only way I know I have counted four times is because I know that 8+4 is 12. I’m not able to keep track of the number of counts past three.

I had watched her fingers; no motion there.

So there had to be something else going on. I was only able to press because I knew what she knew (how to count) and what she didn’t know (addition facts) and some possible ways that this knowledge could be used to find the sum of 8 and 8.

The more I talk with these kids, the more impressed I am with the work of CGI. Much of what I know in this area is the result of studying their work.

UPDATE: The original CGI link in that last paragraph went to a tag search of CGI on my blog. That was confusing. It has been replaced by a particular post. Here’s the original link. And here’s a link to CGI on Wikipedia.

Further adventures in Kindergarten fractions

I took the kids camping this past weekend. Fall along the Mississippi River, mid-70-degree days and 50-degree nights. Pretty much perfect. Having read information about our state park together earlier in the day—including the park’s acreage—Tabitha posed a question.

Tabitha (five and a half): How many acres or miles is our campsite?

Me: It’s only a small fraction of a square mile, but it’s about of an acre.

Tabitha: What’s a fraction?

Me: It’s like when you cut something up. It’s a number bigger than zero, but less than one.

Tabitha: Huh? That doesn’t make any sense!

Me: Well, let’s say you, Griffy and I had three s’mores, and we wanted to share them equally. We would each get one, right?

Tabitha: Yeah.

Me: But what if we only had 2?

Tabitha: Well, then you’d have to cut them in half.

Me: Right. So is a fraction

Tabitha: Oh.

[later]

Tabitha: But what’s the number? You said a fraction was a number bigger than zero, but less than 1.

Me: One-half is more than zero, but less than one.

Tabitha: Half of what?

Me: Half of anything is more than nothing, but less than the whole thing.

Tabitha: But what’s the number? A half isn’t a number!

I have been thinking about the moment when there is a choice to talk math with my kids. I have been trying to understand what I need to know in order to recognize that a choice exists and in order to pursue a mathematical conversation.

Fractions are tough because there really is a lot of specialized knowledge about how people learn them. I have been reading and teaching from the book Extending Children’s Mathematics over the last year or so. The authors make the argument that fair sharing is the best entry point for children’s sense-making about fractions. Not part-whole. Not number line. Fair sharing.

Notice how this plays out in my conversation with Tabitha. I start with part-whole, move to (arguably) number line and she protests that these ideas make no sense.

But as soon as I go with fair sharing, she’s on it. She gets that things sometimes need to be cut up in order to be shared equally.

She also understands—and this is crucial—that halves are meaningless without a referent whole. “Half of what?” is a brilliant and essential question.

So what did I need to know in order to pursue this conversation? I needed to know that there are multiple ways of thinking about fractions, and that fair sharing is going to be helpful for a young child to think about. And that part-whole and number line are going to be dead ends.

Tabitha learns from the conversation that fractions have to do with fair sharing. She doesn’t understand one-eighth—the fraction that initiated the conversation. She doesn’t understand anything more about the size of our campsite, nor about acres, miles or even square miles.

She learns that fractions have to do with sharing. That’s a pretty good Kindergarten-level idea, right there.