As you may recall, my students referred to contexts as a way to think through
an idea in previous lessons as well.
- During the initial cookie tray problem, Alejandro used a context to make
Lucía’s idea clearer for the rest of the students:
TEACHER: 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 [pointing to the arrays as she
moves the smaller arrays closer to each other to make a 7 8 array].
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. - When students wanted to figure out if they had to subtract 7 or 9 from
9 8 to solve 9 7, Julio clarified the situation by creating a context:
DAVEL: We have just solved 9 8, which is 72. So 9 7 is... 1 group less.
LUCÍA: Yes! Yes! 72 – 7?
JULIO: No wait! First we had 8 plates with 9 cookies, now we have 7 plates
with 9 cookies, so we have to take away 1 plate of 9 cookies.
Accessing contexts or models to think ideas through is a very important skill
in solving mathematical problems. Alejandro brought up a context to prove why
an idea worked. Julio recreated a context to solve a problem. Understanding how
different representations relate to each other helped these students pick the rep-
resentation that worked for them and that related to the initial numerical ex-
pression they had to think about. This is the kind of flexibility I want students to
develop: to identify the tools that help them make sense of mathematical prob-
lems. Once again, the initial work we did with the cookie trays worked as a
springboard to understand multiple representations.
Working Through Errors and Misunderstandings
As students began to solve division problems using multiplication, I noticed that
when the problems involved larger numbers, students lost track of the meaning of
the numbers. They could no longer keep straight which were the groups, the ele-
ments in each group, and the total number of things. Their mistakes revealed
what they were struggling with and I wanted them to consider their mistakes.
I presented the following problem:
Ms. Melissa was selling popsicles as part of a school fundraiser. There were 168 pop-
sicles in the freezer. Ms. Melissa took out 24 popsicles at a time so they would not
melt before they were sold. She sold all the popsicles that day. How many groups of
24 popsicles did she take out of the freezer?MAKINGMATHEMATICSEXPLICIT