questions that surfaced what they did understand and allowed them to grapple
with examples to build their understanding. I wanted them to know that knowing
mathematics entails being able to explain ideas. Another goal was to encourage
them to let me know when they are confused.
We had been talking as a class about square numbers. Although the students
in the small intervention group were able to name some square numbers, I was not
sure that they had an understanding of what a square number means or whether
they could use a visual representation to illustrate square numbers and represent
how square numbers increase. So I decided to focus on this during one of our
small-group sessions.
I began the small-group session with a question from the state assessment
about identifying which numbers are square. I wanted the students to practice
with these ideas but in a new context, so that they would make better sense of
them. I began by eliciting ideas they already had so we could build toward a deep-
er understanding.
MALIA: The numbers are 25, 81, and 49.
TEACHER: Dante, are they all square numbers?
MALIA: No...
TEACHER: Wait a minute; I want to hear from Dante. Which is a square
number, and how do you know? [Dante looks unsure.] And remember it’s OK
to say, “I’m not sure” or “I’m stuck.” Dante?
DANTE:[softly] I’m stuck.
TEACHER: You’re stuck. Thank you for telling us that you’re stuck. OK,
Kendrick what do you think? Are they all square numbers?
KENDRICK: Yes...
TEACHER: Can you prove it? [Kendrick mumbles.] Dante has already told us
that he is stuck, so if you’re stuck too, you can say so. It’s OK.
KENDRICK: I’m stuck.I was very pleased the children could express their confusion, especially
Kendrick. Kendrick is a very sweet, yet quiet boy who rarely contributes to our
group discussions, so to hear him voice something at all was exciting. I had also
been working with him in a small group after school where I encouraged him to
share some of his thoughts and even confusions, so I was pleased that his
courage had stretched to where he could voice something with a different group
of students.
TEACHER: What is it about those numbers that makes you think that they
are square numbers?
KENDRICK: Because all of them are high numbers.BUILDINGUNDERSTANDINGTHROUGHTALK