on a chart that has 10,000 squares instead of your 1,000 book that has 1,000
squares?” The first response was, “There would be way more squares, like... maybe
at least more than 1,000 more squares.” Another student said, “We can figure out
how many more squares if we know how many thousands.” Another student men-
tioned that there would be 10 thousands because “that is what it means when we
say 10,000 out loud.” I gave examples of adding and subtracting multiples of 10,
100, and 1,000 on the 10,000 chart to see if they would use what they learned from
the 1,000 chart. For example, if they knew how the value changed with 386
100, could they similarly figure out how the value changed with 3,860 1,000 or
3,860100? After a series of examples, we discussed what digits changed as the
students added or subtracted multiples of 10, 100, and 1,000.
That day in class we made the 10,000 chart. Heather and the students in the
math club were able to complete the task independently. Their experience work-
ing with the 1,000 books helped them visualize the structure of 10,000. A few
days later, the class worked on the Changing Places activity again, this time on
the 10,000 chart. The students from the math club had the opportunity to share
their strategies with the class, extending their strategies from locating new num-
bers in the 1,000 book to the 10,000 chart. They were able to identify what dig-
its changed and explain why. The students in the math club had become com-
fortable and confident with these larger numbers as a result of:
- creating and reviewing the 1,000 book
- practicing adding and subtracting multiples of 10 and 100 through the
Changing Places game - discussing what they needed to pay attention to when playing the game
- constructing the 10,000 chart
- locating numbers by adding and subtracting multiples of 10, 100, and 1,000
I was especially pleased that Heather could successfully participate in this
activity.
Learning About Learning
As the year went on, I continued to see evidence of Heather’s progress. Heather vol-
unteered to share more frequently, realizing that she not only had an approach for
solving a problem but was also able to explain it. For example, during a math club dis-
cussion about Quick Images, Heather was very explicit in her description of how she
determined the number of dots shown in Figure 19–5 (Russell, Economopoulos,
Wittenberg, et al. 2008). Heather circled groups of 3 in the first arrangement
of 9 dots, and she was able to explain that she knew there were 9 because she
counted by 3s. She then explained that she knew how many were in the other
TAKINGRESPONSIBILITY FORLEARNING