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Measuring Up: Prototypes for Mathematics Assessment
appears in the intersection of the "Hog Game" task and Fractions and Decimals because, as the protorubric states, one effective approach to the question about competing strategies depends on calculating each player's expected score, and this will require work with fractions or decimals. Similarly, children might create fractions to knock down pins in the "Bowl-A-Fact" task, and fractions could arise as part of finding an average number of buttons per person in "How Many Buttons?'' Indeed, any sufficiently rich mathematical problem will allow for a variety of different approaches, and so the mathematics actually used may vary from one student to another. (On the surface, this appears to pose yet more difficulties for grading and judgment since student responses may be entirely satisfactory even while ignoring the skills supposedly being examined. Taking the broader view, however, the aim in these prototypes is to assess mathematical power, not individual specific skills.)
It is clear from the chart that the tasks have been designed so that each of them touches several of the NCTM Standards. (Note in particular that every task involves the four all-pervasive standards of problem-solving, communication, reasoning, and connections.) Of course it is a deliberate goal of these particular tasks to emphasize that mathematics is a connected and coherent discipline. Assessment tasks designed to involve many areas within mathematics will promote the parallel idea that instructional activities should also cross boundaries between topics.
Figure 1 also shows how the tasks are arrayed with respect to some aspects of mathematics that go beyond the NCTM Standards for K-4. Two of these — discrete mathematics and algebra — appear as components of standards in higher grades. Drawing attention to them here is meant only to suggest that