From the beginning, I tried to help them take control of their own learning
by developing productive work habits. I made it clear that they would be required
to explain their ideas to the best of their ability, listen to other students’ ideas,
and ask questions if they didn’t understand what their classmates said. I also
wanted them to become aware of what they already knew and how that knowl-
edge could help them solve what they did not know.
We also had a conversation in which we agreed that we were all going to
work hard and not give up, even if the ideas seemed difficult. We talked about sit-
uations in our everyday lives when we didn’t know something and how we dealt
with them. The students offered examples from sports and explained how they
learned to play by making mistakes and watching others. They also talked about
practicing a skill repeatedly before they could do it well. We made the connec-
tion between these experiences outside of school and our math class. Although
students’ motivation to learn math or skateboarding may be different, I tried to
highlight the commonalities. Trying hard and not giving up rang true for these
children. In school, they had not always been expected to persist to learn, and this
was an obstacle at first. However, by putting the students’ thinking front and cen-
ter, over time they got the message that their thoughts and words mattered, and
they slowly began to get a sense of satisfaction from their work. Although these
conversations helped the students become more focused and engaged, I knew that
I needed to implement specific teaching strategies to increase their mathematical
content knowledge.
Teaching Moves
Elaborating Their Answers
After students gave me an answer, I often questioned them further. I tried to find
out what knowledge they were using to choose their strategies. I asked them
where each number came from as they went through the steps of their solution.
Sometimes I asked them how they knew their answer was reasonable.
The distributive property was one key idea I wanted these students to grapple
with. I decided to introduce this concept through a familiar context for the stu-
dents: cookies on cookie sheets. I adapted a picture from the book Amanda Bean’s
Amazing Dreamby Cindy Neuschwander (1998) in which I drew a rack with 2
cookie trays. The cookies were organized in a 3 8 array and a 4 8 array (see
Figure 1–1).
Modeling multiplication using an array of objects allows children to visual-
ize and make sense of multiplicative contexts. The context was designed to
highlight the distributive property, that is, that the problem could be solved by
Are We Multiplying or Dividing?