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?
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?
Me: (Giving Tabitha the remaining salami slices) So if three slices is one less than four, what would two less be?
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!
T: Ask me another.
Me: What would five less than four be?
T: (Thinking) Zero?
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.
Just be glad she hasn’t reached the age when her response will be, “For Pete’s sake, dad: can’t you just give me the salamis slices without turning this into a math lesson?”
Ha! I sat down to talk with Tabitha about a drawing she spent several days making, which involved some patterning work. Griffin noted, “You’re still writing that book, aren’t you?”
I love your “talks with Tabitha” posts! They seem to really get at the different ways we come to know things mathematically. This line stuck with me: “Remember that being interested children’s ideas is a key to talking math with them.” This should be a mantra for any teacher of mathematics.