It is important to consider children’s mistakes as learning opportunities. In
the past, I tried to explain students’ mistakes to them, hoping that they would un-
derstand and correct them. Most of the time all I got were blank stares. I was pre-
senting my own reasoning, which did not make sense to them. I have learned that
it is better and more lasting for students to develop the habit of reasoning through
their own work in order to identify and clarify their mistakes. I facilitate this rea-
soning through asking them to restate their process and asking specific questions
about their steps, if necessary. Of course, the mistakes will not disappear right
away, but the fact that students have already talked about mistakes makes it eas-
ier for students to revisit, identify, and correct their work in the future.
Reflecting on Their Learning
At different times during my work with this group of students, I asked them to re-
flect on what they did and did not understand about multiplication. For me, these
reflections served as tools for evaluating their knowledge and overall attitude to-
ward their learning. For my students, it served to articulate their understandings
and misunderstandings and in the process helped them to see themselves as
learners.
At first, most of the students’ reflections focused on what they did not un-
derstand. They were aware that they did not know how to solve difficult problems
and that they did not know all the multiplication facts. But after we had worked
together for awhile, their reflections began to focus more on what they did un-
derstand:
“Multiplication is like skip counting and the number gets bigger.” (Vanessa)
“If you have 13 12, you have to times the number you have that number of
times, you put the number again that many times.” (Lucía)
“Multiplying by 4 and counting by 4 is the same thing. If you have 9 4, it is
counting by 4 nine times.” (Davel and Alejandro)Vanessa, Davel, and Alejandro paid attention to the relationship of skip count-
ing and multiplication, and Lucía tried to articulate the idea of having many groups
of the same size. Her statement refers to the transition she was working on between
skip counting and multiplying: “You put the number again that many times.”
Toward the end of our work together, I asked the students to reflect on
whether or not they felt more comfortable with multiplication and division.
Their answers were all positive. This is a very important aspect of learning: to be-
come aware that what was difficult before is no longer so. This is especially im-
portant to address with students who struggle, as they have a hard time seeing
themselves as learners.
MAKINGMATHEMATICSEXPLICIT