Tabitha on the well ordering of adjectives

Tabitha and Griffin enjoy each other’s company. She is 5; he is 8. They play well together and are quite close. One of their favorite pastimes is called creatures. They have built an elaborate society of small (mostly) plastic animals. Some creatures have farms; others are farmed. There are towns. There is currency. As near as I can tell, dogs are at the top of the food chain and all manner of poultry at the bottom.

The net result of all of this activity—from the adult perspective—is that each session of creatures makes a tremendous mess of the upstairs hallway and of their shared bedroom. There was once a creature diaspora which resulted in settlements on the first floor. We were stepping on small plastic animals for days so this is no longer allowed.

Tonight Griffin was out of the house when I had Tabitha choose the part of the clean up she wished to do. She chose to pick up the small plastic creatures, leaving the blocks for her brother on his return.

She was feeling sassy.

The definition of small plastic creatures was asked about again and again. Does this count? (Holding a rubber snake.) Does this count? (Displaying a wooden lion.) Et cetera.

Finally, she honed in on small.

Tabitha: Which is smaller, small or little?

Me: Hmmm…I don’t know. Like if you introduced me to your small friend and your little friend, who would be smaller? I think maybe they’re synonyms; they just mean the same thing.

T: No. Little is smaller, I think.

Me: I know tiny is smaller than little. And teeny tiny is smaller than that.

T: Here’s how it is. Teeny tiny is smaller than tiny. Tiny is smaller than little. Little is smaller than small. Small is smaller than medium. Medium is smaller than big. Big is smaller than giant.

Me: But giant isn’t smaller than anything, is it?

T: Uh huh! Giant is smaller than gigantic. Gigantic is smaller than humungous. Humungous is smaller than 11 daddies.

Me: So is tiny smaller than humungous?

T: Of course! Way smaller!

Me: How do you know that?

T: Because of the order.

Me: OK. Where does large fit?

T: [thinking] Wait. I’ll do it all over again with large in it.

[most of the list deleted to save the reader’s frazzled nerves]…is smaller than big. Big is smaller than large. Large is smaller than giant…Humungous is smaller than 11 daddies. Eleven daddies is smaller than 12 daddies. Twelve daddies is smaller than 500 daddies. Then that would never end.

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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.

Compare and contrast

Digging deep in the comments to bring out some interesting observations.

Chris Hunter reports the two following interactions with his 7-year old daughter:

First

My daughter comes home from school and tells me she hates math. She completed 13 questions on “The Mad Minute” and compares herself to her best friend who completed all 30.

Second

Bedtime conversation:

Daughter: Dad, did you know that in some countries girls aren’t allowed to go to school? They won’t even get to know what 9 plus 9 is.

Me: Do you know 9 plus 9?

D: No, but I know 10 plus 10 is 20. We do that one a lot.

Me: Okay, so what’s 9 plus 9?

D: 18.

Me: How’d you get that?

D: I counted two down. They make JUST THE GIRLS stay home and do chores ALL DAY.

And Steve Prosser reports a friend’s progress:

third

[She is a] very bright girl, who because she was home schooled never did much math. Finger-counting still in fifth grade. Hated math. Believed she was terrible at it. After just over 1,000 flashes, she “GOT IT”. She was not bad at math, but had never done enough to internalize the basics. Now she loves math (her favorite subject), and is beginning to excel at it.