use the manipulatives for two reasons: (1) to represent their ideas in a concrete
model, and (2) to ground the discussion about the distributive property with ob-
jects they could move around as they were explaining their ideas. Julio and Davel
worked together. Alejandro worked with Lucía, and Fleurette worked with
Vanessa.
JULIO:[after building the arrays] See we have a 3 8 array and a 4 8 array
and when we put them together it makes a 7 8 array.
ALEJANDRO: The same with us. It is the same.
TEACHER: Yes, now we see it is the same. I wonder why it is the same to do
3 8, then 4 8, and add both products or just do 7 8?
LUCÍA: 3 and 4 make 7 but the 8 stays the same. [Lucía points to the arrays as
she moves the smaller arrays closer to each other to make a 7 8 array.]
TEACHER: Who understood what Lucía explained to us? [It was very clear to
me what Lucía said; however, I wasn’t sure everyone followed her and this was a
key idea I wanted more students to think about.]
FLEURETTE: I don’t understand what she said. Can you say it again?
TEACHER: Who understood that idea?... Go ahead, Alejandro.
ALEJANDRO: When you put together the 3 and the 4 [pointing to the dimen-
sions of the smaller arrays] you make 7 rows of 8 cookies.
FLEURETTE: Ah! Yes, I understand. It is the same! That is what we did.I could have stopped the exploration there. However, I wanted the children
to understand why they could either find the product of both trays at the same
time or calculate 1 tray at a time and add both products. I wanted them to have
both a conceptual as well as a procedural understanding. They had to articulate
that the number of groups (7 or 3 4) and the number of items in each group
(8) had not changed: “You make 7 rows of 8 cookies.” They had to explain the
distributive property of multiplication over addition, even though they didn’t
name it. I did not teach students a procedure but asked them to reason about an
important property of multiplication.
In addition to contributing to the class discussion, the conversation also served
as a reference for future class discussions, especially when students had to solve
problems involving multiplication facts that were hard for them to remember.
Thinking About What They Know Before Solving a New Problem
I once assumed that if I worked on an idea in one class, students would general-
ize the ideas right away and use them in other situations to solve a variety of prob-
lems. However, I have learned that students who struggle often have difficulty
applying knowledge from one situation to another. I try to support these students
Are We Multiplying or Dividing?