Lucía was able to identify what type of problem it was and which equation repre-
sented the problem. (See Figure 1–2.) She wrote 168 24. Then she decided to
solve it using multiplication.
When I asked Lucía to walk me through her work she said: “I want to get to
168.” I asked her to go over her work out loud and think about what each num-
ber meant:
LUCÍA:[pointing to 24 2] 24 are how many popsicles Ms. Melissa takes out
each time. And 2 [times 2]... , because... she took out 2 trays. That is 48
popsicles. And then 4 trays... Oops! I messed up!
TEACHER: What makes you say so?
LUCÍA: Because these [pointing to the circled numbers] are the trays. I did more
popsicles than 168.After reviewing her first attempt and reflecting on her error, she wrote:
24 5 120 popsicles
24 2 48 popsicles 120 48 168
7 groups
There are 7 groups.Lucía knew she could use multiplication to solve a division problem, but she
often did it mechanically, without thinking about the meaning of the numbers.
Here, she did connect the meaning of the numbers to the procedure and was able
to identify her mistake. Lucía was connecting procedural knowledge to concep-
tual knowledge.
Are We Multiplying or Dividing?Figure 1–2.