invitation to participate. I was puzzled about why she wasn’t participating in our
math discussions. However, when the mathematics began to focus more specifi-
cally on counting and quantity, Kristen’s confusions about number began to sur-
face. I realized that her reluctance to participate in discussions probably stemmed
from the gaps she had in understanding the mathematics. As the rest of the class
was learning how to make choices from a list of activities and keep track of their
work, it was clear that Kristen required frequent check-ins to help her complete
tasks. After observing her struggles with numbers for the first month of school, I
began changing the size of the numbers for Kristen so that she was only dealing
with sums less than 20. She was more successful with smaller numbers. She was
able to model the actions in addition problems with these small numbers, usually
drawing tally marks, circles, or pictures to explain her thinking. She also success-
fully completed activities designed to support the learning of addition combina-
tions to 10 10.
Although she experienced some success with my accommodations, I noticed
that she was inconsistent in her counting. She tended to count quickly, and she
sometimes skipped objects or said more than one number per object. I reminded
her to slow down and helped her find ways to keep track of what had already been
counted and what remained to be counted.
As I collected additional samples of Kristen’s work, I saw more evidence of
her struggles with counting. For example, for an activity called Counting Strips
(Russell et al. 2008b), in which students write the numbers by 1s on a strip of
adding machine tape, Kristen consistently skipped over the multiples of 10 after
20 (28, 29, , 31, 32... 38, 39, , 41, 42).
The Magic Pot
Once we began working on story problems, Kristen continued to struggle, al-
though she did not ask for help. In mid-October, the class spent some time
learning about what it means to double a quantity. The context that we used to
introduce the idea of doubling was a Chinese folktale about a couple that found
a magic pot (Hong 1995). Any set of objects that was dropped into the pot
would be doubled, so that twice the amount of objects that went into the pot
would be taken out of the pot. For example, if a bag with 5 gold coins fell into
the pot, 2 bags with 5 coins each would be taken out of the pot. To reinforce
their understanding of the action of doubling, the students wrote and solved
their own magic pot riddles.
Toward the end of this series of activities, the children were asked to solve
a problem asking what would happen if our class of eighteen students fell into
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