ideas, the contexts, or the representations and establish connections among them.
Sylvia, on the other hand, had not thought about how the strategies she learned
from repeatedly playing the games could help her solve other number problems.
From my ongoing observations of Sylvia, I was able to determine when it was ap-
propriate to push her thinking in that direction. It was not by telling her what num-
bers to put together or what procedure to use, but by reminding her about a partic-
ularly relevant game and asking her if she could apply what she used there to solve
the number problem. I knew she first had to be very familiar with the game to draw
from those experiences to solve number problems. That is why I want my students
to play the same games over and over again. The more familiar they are with the
games, the more knowledge they have and can apply to different contexts.
As Sylvia found strategies that worked for her, she found herself doing what
many other students in the class had been doing. She grouped all the 10s first, and
made a new group of 10 with the loose cubes as she modeled the problem. Then
she computed the total. The towers of 10 offered her the possibility to think about
the numbers in a more flexible way: as composed of 10s and 1s, which in turn she
could decompose and recompose to make the calculation easier. She proceeded
slowly, and had a cautious sense of satisfaction, almost as if she couldn’t believe
her eyes. This sense of accomplishment and of understanding numbers in a new
way motivated her to continue solving a difficult problem. She began to see her-
self as a learner, someone who can have fun doing math. There is a point when
math becomes fun, even for students who have been struggling, because they
come to understand how numbers work.
Reflections
The work I did with this group of second-grade students made me reflect on im-
portant components of how to support students who are struggling. First, I need
to be clear about the mathematical ideas underlying the activities. In this case, I
wanted children to understand the idea of unitizing and, as a consequence, place
value: composing and decomposing numbers by 10s and 1s. Second, I need to de-
velop a picture of the strengths and weaknesses of the students and determine
mathematical goals by writing and reflecting on what mathematical ideas chil-
dren are using and connecting and which ones they are not connecting. Third, I
need to give students repeated practice with meaningful games and activities that
allow them to engage with the mathematical ideas. I can never assume that pro-
viding one kind of manipulative or representation in one class is enough to help
the children solidify their understanding. Finally, as students engage in their
work, I have to pose questions to students to help them make the connections
they need to have a flexible understanding of mathematics andsolve problems
LINKINGASSESSMENT ANDTEACHING