Reflections
Although I worked with these students in a small group, I know from my own ex-
perience and through talking with other teachers that the teaching moves I’ve de-
scribed are applicable in large-group settings and, in fact, help all students learn
mathematics with understanding. For example, when teachers ask students to
share strategies, they often target a few that illustrate important features of the
mathematical operation and then draw out the connections among them. They
refer to prior solutions that students have used and contexts and representations
that are familiar to the group. By creating a mathematical culture in their class-
rooms, teachers can encourage students to take risks and use their errors and mis-
understandings as learning opportunities for themselves and the rest of the class.
At the beginning of this essay, I referred to Davel and his inability to reason
through a multiplication number problem (see page 5). Although he had made a
mistake, he could not analyze it because he had trusted a procedure that he
thought he remembered properly. Through the process we developed in our small
group, Davel began to see mathematics differently. He began to understand that
mathematics is about reasoning and making sense. The emphasis on making con-
nections to prior knowledge, among strategies and representations, explaining
and analyzing strategies, working through errors, and reflecting on their learning
helped Davel think through the problems that I posed. Despite his slower learn-
ing pace and his confusions, he showed he was capable of reasoning.
After students felt more comfortable interpreting and solving multiplication
problems, we began to work on division story problems. Students spontaneously
used multiplication combinations to solve them. After a few sessions, Davel
asked, “I don’t understand! Are we dividing or multiplying?” Davel wanted to un-
derstand, and his question led us to explore an important mathematical idea: the
inverse relationship between multiplication and division.
Are We Multiplying or Dividing?