I was encouraged that Kendrick took another risk to express an idea, although it was
not correct. I could have told Kendrick that he was wrong, but I decided to see what
the other students were thinking so I could listen to what they were all understand-
ing. I chose instead to rephrase what he was saying so that in hearing his idea
expressed back to him, he could process for himself whether his idea was reasonable.
TEACHER: Because all of them are high numbers. So, OK, at least you’ve
given me some kind of a reason to work with. So that means to you, any
number that is low is not a square number? Is what you’re saying? So a square
number has to be a high number is what you’re saying? [Kendrick nods.] OK.
TEACHER: Sharonda, what do you think? Are these square numbers?
SHARONDA: I think one of them is a square number because I think 49 is a
square number.
TEACHER: And prove how you know that 49 is a square number.
SHARONDA: I know that because I remember once a lady told me that 7 was
a square number and like if you multiply 7, 7 times, you’ll get 49 or when
you count by 7s to like 1,000, 49 will be one of those numbers. That is how
I know that 49 is a square number.
TEACHER: Hmm... what do you think, Malia?
MALIA: I think all of them are square numbers because 25 is when you mul-
tiply 5 by 5, and when you multiply 9 by 9 you get 81, and when you multi-
ply 7 by 7 you get 49, and so like the number 5 and you multiply it again like
5 5 gives you a square number...
TEACHER: So what can you tell me about how to finda square number?
GIRLS: Ooh [raising their hands].
SHARONDA: Yeah. Like when we are looking for factors of a number... and
like 1 times 12 equals 12, and we’re like looking for numbers that are factors
of 12, or like sometimes when we are doing 25, and we do, like, 1 times 25
and we do 5, and we ask what can go with 5 and we say 5, and that makes
it a square number.
TEACHER: Hmm... anyone else want to add to that? Because these are all
great ideas, but I want to hear some more. Anyone else have something they
want to add?
CHELSEA: I think 25 is a square number like Malia because 5 is a square
number.
TEACHER:[to Sharonda] You think 5 is a square number too, and you think
7 is a square...
SHARONDA: I think all numbers are square numbers.
TEACHER: All numbers are square numbers, is what you’re saying?
SHARONDA: Yes, they are.Talking About Square Numbers