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From page 41... ...
Some form of the content principle may have always implicitly guided assessment development, but in the past the notion of content has been construed in the narrow topiccoverage sense. Now content must be viewed much more broadly, incorporating the processes of mathematical thinking, the habits of mathematical problem solving, and an array of mathematical topics and applications, and this view must be made explicit.

From page 42... ...
Assessment developers need characterizations of the 42 important mathematical knowledge to be assessed that reflect both the necessary coverage of content and the interconnectedness of topics and process. Interesting assessment tasks that do not elicit important mathematical thinking and problem solving are of no use.

From page 43... ...
This change, along with the continued use of items requiring students to construct their own responses, has helped provide a new basis for the NAEP mathematics examination.4 One promising approach to assessment frameworks is being developed by the Balanced Assessment Project, which is a National Science Foundationsupported effort to create a set of assessment packages, at various grade levels, that provide students, teachers, and administrators with a fair and deep characterization of student CONTENT

From page 44... ...
One noteworthy feature of the framework from the Balanced Assessment Project is that it can be used at two different levels: at the level of the individual task and at the level of the assessment as a whole. When applied to an individual task, the framework can be used as more than a categorizing mechanism: it can be used to enrich or extend tasks by suggesting other thinking processes that might be involved, for example, or additional products that students might be asked to create.

From page 45... ...
As these and other efforts demonstrate, researchers are attempting to take account of the fact that assessment should do much more than test discrete procedural skills.6 The goal ought to be schemes for assessment that go beyond matrix classification to assessment that elicits student work on the meaning, process, and uses of mathematics. Although the goal is clearly defined, methods to achieve it are still being explored by researchers and practitioners alike.

From page 46... ...
Instruction is organized into coherent, manageable units consisting of investigations, problems, and other learning activities. Through the California Learning Assessment System, researchers at the state department of education are working to create new forms of assessment and new assessment tasks to match the curriculum framework.8 Further exploration is needed to learn more about the development and appropriate use of assessment frameworks in mathematics education.

From page 47... ...
TASKS REFLECTING MATHEMATICAL C O N N E C T ~ O N S Current mathematics education reform literature emphasizes the importance of the interconnections among mathematical topics and the connections of mathematics to other domains and disciplines. Much assessment tradition is based, however, on an atomistic approach that in practice, if not in theory, hides the connections among aspects of mathematics and between mathematics and other domains.

From page 48... ...
· C I inch: I mile Now suppose lightning strikes again at a different place. The person at A and the person at C both hear the thunder after the same amount of time.

From page 49... ...
The mathematics in the task may be rather trivial and therefore inappropriate. Test items that assess one isolated fragment of a student's mathematical knowledge may take very little time and may yield reliable scores when added together.

From page 50... ...
that before the assessment is administered.~4 Presumably the lesson reduces the variability among the students in their familiarity with everyone's prior the task setting. The same idea can be found in some of the assess ment prototypes in Measuring Up: Prototypes for Mathematics Assess knowledge of mend In one prototype, for instance, a script of a videotaped introduction was suggested;~5 playing such a videotape immediately the setting is the before students work on the assessment task helps to ensure that eve rYone is equally familiar with the underlying context.

From page 51... ...
Another assessment items that has been widely discussed also shows the need to clarify assumptions. In 1982, this item appeared in the third NAEP mathematics assessment: "An army bus holds 36 soldiers.

From page 52... ...
Such a task allows the student who may not be able to produce a general rule at least to demonstrate his or her understanding of the geometrical aspects of the situation. T A S K S R E Q U ~ R ~ N G C O M M U N ~ C A T ~ O N The vision of mathematics and mathematics assessment described in earlier chapters emphasizes communication as an critical feature.

From page 53... ...
Some assessments can be carried out as individual inter views. The example below describes a question on an oral examination for 1 7yearold Danish mathematics students in the Gymnasium.23 (The Gymnasium enrolls less than 30 percent of the age cohort, and not all Gymnasium students take mathematics.)

From page 54... ...
The need for careful use of language and notation in stating a task has long been a goal of assessment developers, although one not always successfully skill must achieved. be seen as differences in mathematical 54 The example below from the Second International Math ematics Study (SIMS)

From page 55... ...
To be novel, assessment tasks must not have been seen in advance. However, students should be exposed to a variety of novel problems in daily work to demonstrate to them that nonroutine problems are valued in mathematics education and that they should expect to see many such problems.

From page 56... ...
We can no longer rely on task developers with superficial understanding of mathematics to tasks develop assessment tasks that will elicit creative and novel math ematical thinking. · ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ SCORING NEW ASSESSMENTS The content principle also has implications for the math · ~ ematical expertise of those who score assessments and the scoring approaches that they use.

From page 57... ...
Multiple readings and moderation by more experienced graders help to hold the scores to the agreed standard.27 Together, graders create a shared understanding of the rubrics they are to use on the students' work. Examination boards in Britain follow a similar procedure in marking students' examination papers in subjects such as mathematics, except that a rubric is used along with sample examinations discussed be the proud to held examiners agree on marks.28 a ~to or The development of highquaky scoring guides to match new assessment is a fairly recent undertaking.

From page 58... ...
In contrast, QUASAR, a project to improve the mathematics instruction of middle school the system of students in economically disadvantaged communities,3' uses an approach that blends taskspecific rubrics with a more general scoring or the rubric, resulting in scoring in which mathematical knowledge, strategic knowledge, and communication are considered interrelated type of rubric components. These components are not rated separately but rather are to be considered in arriving at a holistic rating.32 Another used.

From page 59... ...
Student demonstrates a lack of proficiency Score Point = I The student provides a less than satisfactory response that only begins to answer the question, but fails to answer it completely, e.g., provides little or no appropriate explanation, draws diagrams which are unclear, exhibits little or no understanding of the question being asked, or makes major computational errors. Student demonstrates no proficiency Score Point = 0 The student provides an unsatisfactory response that answers the question inappropriately, e.g., uses algorithms which do not reflect any understanding of the question, makes drawings which are inappropriate to the question, provides a copy of the question without an appropriate answer, fails to provide any information which is appropriate to the question, or fails to attempt to answer the question.

From page 60... ...
With problem solving a main thrust of mathematics education, there is a place for both kinds of judgments. Some efforts are under way, for example, to establish iterative processes of assessment: Students work on tasks, handing it in to teachers to receive comments about their work in progress.

From page 61... ...
a student's performances on perhaps 20 assessment tasks are arrayed in such a way that overall achievement is readily apparent while at the same time some detailed diagnostic information is conveyed.34 NAEP developed an alternative approach to try to give .

From page 62... ...
Nonetheless, this push also is toward reporting methods forcing mathematics to tit assessment, that tell people directly about the important mathematics students have learned. This is the approach that NAEP takes when it illus trates what basic, proficient, and advanced mean by giving specific examples of tasks at these levels.

From page 63... ...
Romberg, "A Framework for the California Assessment Program to Report Students' Achievement in Mathematics," in Thomas A Romberg, ea., Mathematics Assessment and Evaluation: Imperatives for Mathematics Education (Albany, NY: State University of New York Press, 1992)

From page 64... ...
, 145166; Mary M Lindquist, "Reflections on the Mathematics Assessments of the National Assessment of Educational Progress," in Developments in School Mathematics Education Around the World, vol.

From page 65... ...
, 46. 30 Marilyn Rindfuss, ea., Integrated Assessment System: Mathematics Performance Assessment Tasks Scoring Guides (San Antonio, TX: The Psychological Corpo ration, 1991~.

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