was especially excited that he was following along with this discussion. Two
months ago, I would not have expected him to share as much! It was good that he
was able to apply his knowledge of multiplication facts to this context.
SHARONDA: I’ve got a question... How come you put that little 2 on the
corner of the 9?
MALIA: Because it means you’re multiplying that number by that same
number!
TEACHER: So what about 49?
SHARONDA: That’s 7 squared!
TEACHER: So are they all square numbers?
SHARONDA: No!
OTHERS: Yes! Yes!
SHARONDA: Oh, Yes!
TEACHER: Are we sure? I am hearing yes and no...
SHARONDA: Yes, we just proved it!Analyzing the Session
The talk we engaged in and the use of the array model helped the students clarify
their ideas. After each small-group session, I took mental notes about what each
student seemed to understand and where I might go in the next session. From
our interactions, I understood what they already knew, so I was able to call on
them in the large group when the discussion was at a point when I knew they
could contribute successfully. After this session, I knew that I might want to work
with Malia about what she noticed about the square numbers increasing (1, 4, 9,
16, 25). I knew I would follow up with Kendrick in a miniconference to review
why or why not a number is square. Sharonda did not seem sure of her under-
standing, so I needed to check in with her, too, to determine what she understood
or was still confused about. I was pleased, however, at her use of language when
she said, “We just proved it”—it gave me an inkling that she was beginning to
understand that mathematical talk involves explaining and justifying your ideas.
I continued to work with this same small group of students, and most appeared
to be making progress in understanding what problems are asking and expressing
their mathematical thinking. My goal was for them to take what they learned in
the small group and feel more and more confident about sharing in the large group.
Contributing to the Whole-Group Discussion
My goal for all of these students was that they fully participate in our math com-
munity. It is important to me that children are provided opportunities to articulate
Talking About Square Numbers