mathematics might be like, and as examples of what today's goals for instruction should be like. In the meantime, teachers can use them as ideas for instructional activities for today. (A list of resources for teachers including the names and addresses of contacts in each state appears at the end of the volume.)
Another audience is the community of university-based educators who are responsible for the pre-service education of prospective teachers. They will find Measuring Up to be a source of ideas to use today for connecting the tenets of the mathematics education reform movement to classroom practice.
Finally, of course, the ultimate audience for these new assessment tasks and the ideas that underlie them is the elementary school children for whom the tasks were designed. The tasks provide good examples of challenging mathematical problems and situations that effective teachers can use even now as part of their instructional strategies. Today's children can begin to see the challenge in authentic mathematical problems even before tomorrow's tests give them an opportunity to demonstrate their accomplishments.
The Prototypes
What we have accomplished
Measuring Up contains thirteen assessment prototypes that exemplify changes called for in the Standards. In some cases the particular settings or contexts for the tasks are original, while in other cases some aspect of the task has appeared in another from previously.
The tasks in Measuring Up are intended for a largely hypothetical audience: fourth-grade children who have had a K-4 mathematics experience fully consonant with the NCTM Standards. Unfortunately, very few U.S. fourth graders have had the benefit of such an education. This is, of course, the whole point of the reform effort. One would not expect many of today's fourth graders to do very well on these tasks. Nonetheless the aim was to keep the tasks accessible to most of today's fourth graders; they should at least be able to understand what the tasks are about and become engaged in working on them.
Too often test questions and assessment tasks are presented solely in written form, which may be a burden for poor readers and for children whose first language is not English. Such children might not be able to respond to the tasks in a way that shows their true level of mathematical knowledge or skills. Many alternative presentations are possible: videotaped introduction; teacher-taught introduction; computer-based presentation; and presentation using manipulative materials. The prototypes illustrate each of these alternative modes of presentation, and two of the tasks are written in Spanish as well as in English.
Notwithstanding the possible variety in presentation, the prototypes in Measuring Up adhere to a certain uniformity of structure. Most are organized as a sequence of questions, often of increasing difficulty. On the one hand, this provides a structure around which the child's problem solving must be organized. On the other hand, this sequence of questions may suggest that the problem-poser, rather than the problem-solver, is in charge of the problem-solving process. Although other forms of organization are certainly possible, these prototypes provide sufficient imposed structure to help the mathematically less sophisticated student get started and show what he or she can do, while allowing plenty of open-ended space at the top to challenge the more advanced student. Even though the questions within a task often grow in difficulty, many of the tasks involve problem solving, reasoning, and communication right from the beginning. These are important aspects of mathematics for all children.
Just as the tasks are presented in several formats, so they are also designed to give children a chance to respond in a variety of modes — perhaps by constructing an object or by creating a pattern on a computer screen. One important response mode
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.
