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#### Protorubric

General notes: For the purposes of this protorubric, questions 1 and 2 are not included, for the reasons cited above.

In questions 3 and 4, here are the possibilities: With 2, 3, and 6, one can knock down the pins 4, 5, 6, 7, and 9. With 1, 3, and 5, one can knock down the pins 1, 2, 3, 7, 8, 9, and 10. Hence it is possible to make a spare, but not a strike, with the given rolls.

In question 5, however, one can knock down all the pins with 1, 2, and 4, so it is possible to get a strike.

Notice that one can obtain useful diagnostic information from children's incorrect equations. Consider, for example, these three incorrect equations (which might be given on question 1 or 2):

The child who writes (a) may not understand how and when parentheses are used; the child who writes (b) may have a problem with basic number facts; he may think either that 6 ÷ 2 is 2 or that 3 + 4 is 6. The child who writes (c) may fully understand that 6 divided by 2 is 3, but be confused about the order in which the operands are conventionally written. Errors such as these can be useful clues in understanding children's mathematical thinking.

Characteristics of the high response:

A high response shows flexibility of thinking. The use of parentheses is accurate, and the numbers obtained are correct. A spare is obtained in questions 3 and 4, and a strike in question 5.

The highest level of response would knock down all five possible pins in question 3, even though pins 7 and 9 could also be knocked down in the second roll.

Note that the questions do not ask for multiple ways of knocking down pins. The fact that

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