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• #### Credits 165-166

which values there are no people with exactly that number of buttons, and what (if anything) seems typical of their class.

Student assessment activity: The assessment should be done on the day after the activity just described. You will need the class data from the previous day, and pencil and paper for each student. Of course, they should have access to calculators.

Tell students they will be working in small groups to estimate the number of buttons there are in the whole school. They can refer to the class data from the previous session. Suggest that they may want to consider questions such as: "Is it likely that other classes in the school will have a similar number of buttons?" "Do younger students have fewer or more buttons?" "What about older students?" "How should teachers or other adults in the school be included in the estimate?'' These questions should be written on the board. They can contribute greatly to the depth and sophistication of the children's responses, so students should have equal access to them.

Each pair of students then works together to solve this problem, as described above. They should be encouraged to record their strategies, including the numbers they chose and why they chose them.

#### Rationale for the mathematics education community

This item pushes the curriculum to include work with large numbers and with a complex situation in which estimation is a legitimate mathematical process—not simply a prelude to finding the "real" answer. Further, it connects a range of mathematical ideas—multiplicative relationships (e.g., an average of about 6 buttons per student, with 22 students per classroom), estimation, averages ("I decided there are about 22 students in each classroom."), and beginning ideas about sampling. Use of the calculator is incorporated in a natural way.

The problem is accessible: Almost any child can arrive at some solution, but the possible solutions span a broad spectrum of depth and complexity. The problem provides all students with the opportunity to think about and discuss which factors to take into account — whether or not to include adults' buttons or clothes in the closet, and so forth. One strength of a task like this is that its context is immediate and tangible.

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