This time, Michele had a real purpose for counting, and she could connect
what she had been doing in class and in math group to accomplish her goal.
She even gave an explanation for why counting all the popcorn would not be
so difficult if we counted by 10s. She definitely understood the advantages of
counting by 10s. She was very excited and her excitement was so contagious
that we were all interested in knowing how many pieces of popcorn we had.
Michele gave each of us a few handfuls of popcorn, which we put in groups of
10 and counted. After each of us counted how many we had, Félix began to
count all the groups by 10s. We all joined in. There were so many popcorn ker-
nels (609) that children had to think about the number sequence in ways they
had not done before. What number do we say after 100? How do we write that
number? And after 200? If we have to count by 1s after 600, what do we say?
How do we write that number?
In her book The Having of Wonderful Ideas, Eleanor Duckworth (2006) writes:
“Learning in school need not, and should not, be different from children’s natu-
ral forms of learning about the world. We need only broaden and deepen their
scope by opening up parts of the world that children may not, on their own, have
thought of thinking about” (49). Counting collections of objects is something
children do: they count how many cards they have, how many pieces of candy,
how many coins in their collection, and, as in this example, how many popcorn
pieces. I was happy to see that the work we had been doing in class broadened
Michele’s experience of counting. Counting the popcorn presented a natural op-
portunity to count by groups other than 1. There were so many popcorn pieces
that counting by 1s would have been overwhelming. I had been prepared to ask
questions that would help students think about counting by 10s during the pop-
corn activity, but, thanks to Michele, I didn’t have to ask them. What better way
of demonstrating a new understanding could I have expected? I was so excited
about the results of this informal assessment opportunity!
I continued to provide other experiences to help Michele build on this im-
portant understanding and, as a result, she began to integrate counting by 10s
much more fluently to solve different problems. Earlier in the year I had intro-
duced the model of “towers of 10”—10 snap cubes joined together to make a tower
of 10—in order to model counting by 10s. Michele relied on the use of these tow-
ers to keep track of groups and elements in each group as she solved problems. A
few Fridays later, I bought 4 chocolate bars to share. I opened 1 bar, which had
accidentally broken in half. I asked children how many small squares were in half
of the chocolate bar. With no hesitation they said 6, because they were able to
subitize—quickly identify a small quantity visually without counting—3 small
rectangles on top and 3 on the bottom. They concluded that there were 12 small
rectangles in each bar because 6 6 12. Then the children had to figure out
LINKINGASSESSMENT ANDTEACHING