takes practice. Although I consider the previous conversation very successful, the
initial conversation consisted of the students’ descriptions of their own methods,
the numbers and operations they used. Yet describing their own strategies was im-
portant in helping the students notice similarities. Using their descriptions, I
asked targeted questions to draw out their thinking about their strategies and the
similarities among them.
Again, these discussions are important not only for what they contribute
to one particular moment but for the references they establish for future dis-
cussions. I knew that after this class, some students would continue to use skip
counting, yet I could refer to the connection between skip counting and mul-
tiplication they had begun to make to help them further think about it. I
posted the chart with the different solutions and Alejandro’s notes for future
reference.
Making Connections Among Representations
Some students are able to connect the meaning of words in a multiplication story
problem with an array and a multiplication equation. For others, the multiplica-
tion story, the array, and the equation are not necessarily connected. Because
many students can make sense of multiplication through a real-life context, or
through a mathematical model like an array, it is important for them to see how
these different ways are related.
When I reintroduced arrays in a subsequent lesson, I began with the familiar
context of cookies on a cookie sheet. The students immediately described the trays
with cookies as follows: “In one tray, there are 3 rows of cookies and there are 8
cookies in each row. In the other tray, there are 4 rows of cookies and 8 cookies in
each row.”
They were able to use words to describe the number of groups (rows of cook-
ies) and the number of items in each group (cookies in each row). I then asked
them to use numbers to represent what they had just articulated in words. After
they wrote 3 8, I asked what the 3 and the 8 meant in the context of the cook-
ies. I wanted them to connect the multiplication expression to the context. I also
wanted them to connect the array model to the context and the multiplication
expression, so I asked them to build an array with snap cubes. Because the stu-
dents had already talked about what they had seen in the picture, they made the
connection between the cubes and the cookies easily: each snap cube stood for a
cookie and each row of cubes was a row of cookies. I purposefully asked students
to move from one representation to the other and to explain how they connected
to each other. This practice continued throughout the lessons, and became espe-
cially helpful in solving story problems.
Are We Multiplying or Dividing?