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Measuring Up: Prototypes for Mathematics Assessment an essential part of doing mathematics, for fourth graders as well as for everyone else. However, since the question produced virtually no information, it was dropped. One whole prototype was dropped entirely. It was a task on what is known as "Pick's Theorem" — which relates the area of a polygonal region on a geoboard to the number of nails on the boundary and in the interior of the region. The task was extremely open-ended and required extensive interaction between the teacher and individual students or small groups of students. Even if one assumed (as we do) that the teachers involved in the assessment are uniformly well versed in the subtleties of the underlying mathematics, there seemed to be no way of separating the effects of the teacher from the progress that individual students might make on the task. Perhaps such a task could be viewed as an assessment of the teacher-class unit, but in any case it seemed to be too problematic to include in this collection. The Format How we present the prototypes Each of the thirteen tasks is presented using the same outline. After the title, there is a suggested time allotment, which can vary from one to three class periods. This is followed by a suggested student social organization, although in many cases the task does not depend substantively on a particular form of grouping. Next comes the task itself. First there is a description of assumed background. In most cases this refers to specific aspects of the children's mathematical background, assuming — hypothetically, of course — that the children have had a K-4 education that fully meets the NCTM Standards. Second, there is a section on presenting the task, which details exactly what the teacher (or other assessor) should do. Finally, there is the student assessment activity. Very often this involves one or more sheets of paper on which students record their responses. (To reproduce these pages, which were scaled to the 7" × 10" page of this volume, the copying machine should be set to magnify them appropriately.)
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