that is not specifically included in these prototypes is that of the child talking individually to a teacher, explaining his or her solutions orally rather than in written form. Pilot testing of the tasks has shown that children who have not had considerable experience in organizing their thoughts on paper find it much easier to tell someone else what they are doing than it is to record it in writing. Teachers who use tasks like the ones in this collection for their own informal assessment of how children are progressing mathematically will want to supplement written responses with spoken ones. In fact, asking a child to explain a solution in two forms — spoken and written — can help the child to sharpen and deepen both responses.

These prototypes can be used either for informal classroom-based assessment by an individual teacher, or for more formal external assessment, although certain modifications may be necessary to make the tasks suitable for a given purpose. All of the prototypes call for responses that go well beyond simple numerical answers, and most require the student to explain underlying patterns, relationships, or reasoning. As a result, the same activities can be useful to an individual teacher as she or he tries to discern more deeply how students are progressing mathematically, and to a district to discern the effectiveness of its instruction.

As the NCTM Standards urge, assessment should be embedded in instruction, so that most children would not recognize the assessment activity as a "test." Even when certain tasks are used as part of a formal, external assessment, there should be some kind of instructional follow-up. As a routine part of classroom discourse, interesting problems should be revisited, extended, and generalized, whatever their original sources.

Increasingly, educators are recognizing the value of having children work together in groups. Certainly group work exemplifies the NCTM's goal of stressing mathematics as a means of communication. Some of the tasks in Measuring Up are designed to be carried out in small groups, while in other cases, small groups are certainly a reasonable option. Continuing experimentation will be required to determine how the children can best be grouped for assessment tasks like these, and how to weigh individual vs. group work in performance evaluation.



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