additional steps required to turn these outlines into fully detailed scoring rubrics and to refine the levels of response to each task.

One important purpose of creating a scoring rubric is to communicate to the *students* exactly what is expected of them. Embedded in our assumption that the students have had an exemplary mathematics education is an implication that appropriate standards have already been communicated to the students. Thus, for example, when a protorubric mentions a "clear explanation" or an "appropriate drawing," it is assumed that the children and the assessor share a common understanding of what these terms mean.

Another purpose to is to help the *teacher* interpret students' responses by specifying or clarifying the mathematical essence of the task — which aspects of the task are critical mathematically and which are not. These clarifications will be improved as tasks such as these are tested with larger numbers of students, particularly with those who have studied in a *Standards*-based curriculum.

Since this entire project has been undertaken in a context of mathematics education reform, an important question that naturally arises is the extent to which these prototypes reflect the spirit of the NCTM's *Curriculum and Evaluation Standards for School Mathematics*. Figure 1 suggests how these particular tasks relate to the content that the *Standards* calls for in grades K-4.

Having constructed this figure, we must emphasize how potentially dangerous such tables can be because they promote a "check-off" approach that conflicts with a truly integrated view of mathematics. Each "x" within the body of the table is merely shorthand for a detailed account of how the particular task exemplifies, or illustrates, or even extends the ideas within that particular standard.

In some cases, the "x" means only that the idea is possibly, but not necessarily, involved in the task. For example, an "x"

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Measuring Up: Prototypes for Mathematics Assessment
additional steps required to turn these outlines into fully detailed scoring rubrics and to refine the levels of response to each task.
One important purpose of creating a scoring rubric is to communicate to the students exactly what is expected of them. Embedded in our assumption that the students have had an exemplary mathematics education is an implication that appropriate standards have already been communicated to the students. Thus, for example, when a protorubric mentions a "clear explanation" or an "appropriate drawing," it is assumed that the children and the assessor share a common understanding of what these terms mean.
Another purpose to is to help the teacher interpret students' responses by specifying or clarifying the mathematical essence of the task — which aspects of the task are critical mathematically and which are not. These clarifications will be improved as tasks such as these are tested with larger numbers of students, particularly with those who have studied in a Standards-based curriculum.
The Standards
Are we measuring the right things?
Since this entire project has been undertaken in a context of mathematics education reform, an important question that naturally arises is the extent to which these prototypes reflect the spirit of the NCTM's Curriculum and Evaluation Standards for School Mathematics. Figure 1 suggests how these particular tasks relate to the content that the Standards calls for in grades K-4.
Having constructed this figure, we must emphasize how potentially dangerous such tables can be because they promote a "check-off" approach that conflicts with a truly integrated view of mathematics. Each "x" within the body of the table is merely shorthand for a detailed account of how the particular task exemplifies, or illustrates, or even extends the ideas within that particular standard.
In some cases, the "x" means only that the idea is possibly, but not necessarily, involved in the task. For example, an "x"

OCR for page 16

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

OCR for page 16

Measuring Up: Prototypes for Mathematics Assessment
Figure 1 NCTM Standards and Prototypes of Fourth-Grade Assessment Tasks
Hog Game
Towers
Bears
Lightning
Taxman
How Many Buttons?
Quilt Designer
Point of View
BowlAFact
Hexarights
Bridges
Checkers Tournament
Mystery Graphs
K-4 Standards
1.
Problem Solving
X
X
X
X
X
X
X
X
X
X
X
X
X
2.
Mathematics as Communication
X
X
X
X
X
X
X
X
X
X
X
X
X
3.
Mathematics as Reasoning
X
X
X
X
X
X
X
X
X
X
X
X
X
4.
Mathematical Connections
X
X
X
X
X
X
X
X
X
X
X
X
X
5.
Estimation
X
X
X
X
X
X
X
X
6.
Number Sense and Numeration
X
X
X
X
X
X
X
X
X
X
X
7.
Concepts of Whole Number Operations
X
X
X
X
X
X
X
X
X
8.
Whole Number Computation
X
X
X
X
X
X
X
X
X
X
9.
Geometry and Spatial Sense
X
X
X
X
X
X
10.
Measurement
X
X
X
X
X
X
11.
Statistics and Probability
X
X
X
X
X
X
X
12.
Fractions and Decimals
X
X
X
X
13.
Patterns and Relationships
X
X
X
X
X
X
X
X
X
X
X
X
X
*14.
Discrete Mathematics
X
X
X
X
*15.
Algebra
X
*16
Proof
X
X
X
X
X
*Note: these are not K-4 NCTM Standards.