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## Chapter 3:Assessment and Opportunity to Perform

One of the most striking aspects of task development is just how hard students find many tasks that are designed to assess conceptual understanding or problem solving. Time and again when tasks were piloted in classrooms—tasks that appeared to provide students the opportunity to show what they know—The tasks were for some reason inaccessible for most students. One explanation for this result is that many of the tasks may not closely resemble those that students are accustomed to completing in class.

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Keeping Score: Assessment in Practice Figure 7. Hang Glider Reprinted with permission from the Balanced Assessment project, University of California, Berkely. Finding a second equation relating l and w is much more difficult. One way to do this is to use the diagram to find a relationship between w and l: pq2 + ps2 = qs2, pq = ps = l, and qs = w. So, l2 + l2 = w2. Then w = 2l.

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Keeping Score: Assessment in Practice Finally, w = 2l can be substituted into the equation wl/2 = 130, giving 260 = 2l2. Solving this gives l 13.5. Clearly, the solution to this portion of the task involves highly non-trivial conceptual and manipulative demands. The length of qr still needs to be determined. This can be done using the law of sines: sin 45°/qr = sin 67.5°/13.5. But the student has no chance of reaching this point without sufficient success on Question 1 and Question 2, to be able to draw upon those solutions to set up the equation that acts as the springboard to Question 3. Figure 8. Hang Glider solution This task was piloted with 184 high school students. Using a four-point scoring rubric that defines a score of '1' as little or no success, just 17 students in the entire pilot group were able to achieve a score of '2' or higher. Not a single student was able to fully accomplish the task, and just one was able provide a response that could be marked as "ready for revision." It is difficult to make reasonable inferences about the specific nature of the obstacles that stand between the students and success on this difficult task. Is the obstacle that the students were unable to formulate successful approaches to the problem? Or was it that the students were unable to handle the total skill and concept demands? As it was given, Hang Glider indicated neither what students know and can do nor what students do not know and cannot do. Hang Glider demands that students make very high-level use of mathematical ideas. The data suggest that only the most talented of students will have enough experience to access these concepts and to use them in the sophisticated way that Hang Glider demands. In other words, Hang Glider is a task that asks students to make strategic use of concepts that are, for the majority of tenth grade students, not fully integrated into the students' existing conceptual frameworks (Hiebert & Carpenter, 1992). Our developmental experience shows that when students work simultaneously at the cutting edge of both their strategic domain

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Keeping Score: Assessment in Practice Figure 10. Student solution to Paper Cups Reprinted with permission from New StandardsTM. For more information contact National Center on Education and the Economy, 202-783-3668 or www.ncee.org. as H = 20 + 5(x-1), where H represented the total height of the task, and x represented the number of containers in the stack. When students produce a formula of this type, it is clear that they have successfully navigated what we refer to as the x-1 aspect of this family of tasks. This is an aspect of Shopping Carts that very few students manage to process correctly. The mistake that occurs when students do not successfully navigate the x-1 aspect of Storage Containers is usually expressed as follows: H = 20 + 5x, where H represents the total height of the task, and x represents the number of containers in the stack. In contrast to both Shopping Carts and Storage Containers, however, many students working on Paper Cups will immediately decompose the cup into the following two parts, which they sometimes label as the body and the brim, as illustrated in the student solution in Figure 10. This decomposition enables many students to create the required formula directly in terms of the height of body of one cup plus the height of x brims, as illustrated in the remainder of this student's response (Figure 11). Clearly, the structure of the cup lends itself to this decomposition, which enables students to finesse the x-1 aspect of the task. The specific features of the cup reduce the conceptual demands of the problem. We say this because the specific features of the cup enable students to deal with x lips rather than x-1 cups, and dealing with

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Keeping Score: Assessment in Practice Figure 11. Student solution (cont.) Reprinted with permission from New StandardsTM. For more information contact National Center on Education and the Economy, 202-783-3668 or www.ncee.org. x is less sophisticated than dealing with x-1. Put in another way, the contextual factors associated with the cup provide greater opportunity for students to perform. Paper Cups emerges, therefore, as a task that has a relatively high strategic hurdle, is appropriately challenging, and yet can be presented without relying on any directive questioning. It is a type of task that can he quite beneficial to use with students who are not accustomed to solving context-based problems. It also is a good introduction to non-routine tasks because it may be solved with lower levels of tenacity, it encourages perseverance, and it enables students to show what they can do rather than what they cannot do. These findings have important implications for the model of assessment that is advanced in Chapter 2, which recommends separating assessment of mathematical skills, conceptual understanding, and problem solving. In this family of problem-solving tasks that involved stacks, students showed the most success when the conceptual demand of the task was reduced. Therefore, task developers should take care that the conceptual demands of a problem-solving task do not prevent students from showcasing their problem-solving capabilities.

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