Methods and tasks for assessing students' learning should be aligned with the curriculum's--

• goals, objectives, and mathematical content;
• relative emphases given to various topics and processes and their relationships;
• instructional approaches and activities, including the use of calculators, computers, and manipulatives.

Focus

The assessment of students' mathematics learning should enable educators to draw conclusions about their instructional needs, their progress in achieving the goals of the curriculum, and the effectiveness of a mathematics program. The degree to which meaningful inferences can be drawn from such an assessment depends on the degree to which the assessment methods and tasks are aligned or are in agreement with the curriculum. Little information is produced about students' mastery of curricular topics when the assessment methods and tasks do not reflect curricular goals, objectives, and content; the instructional emphases of the mathematics program; or how the material is taught.

When assessment instruments are aligned with the curriculum, the curriculum becomes the standard against which an assessment instrument should be judged. This alignment can be determined by examining the extent to which the instruments measure the content of the curriculum; are consistent with its instructional approaches, particularly the use of calculators, computers, and manipulatives; and cover the range of topics weighted according to the emphases of the curriculum.

Discussion

Alignment is a critical issue in the development or selection of assessment instruments or in the use of assessment data. Teachers, test developers, and administrators all must be concerned about the alignment of curricula and assessment, although their interpretations of assessment and its scope may vary. For teachers, curriculum can refer to the material covered in a chapter, unit, semester, or year; for administrators, it can be the content of the entire mathematics program; for test developers, it can refer to content that is common to the instructional programs of a number of school districts. Regardless of perspective, the same considerations are essential in determining the alignment of an assessment instrument with a curriculum.

The following discussion examines alignment with respect to the curricular aspects specified in these standards. Although the discussion focuses on assessments that are broad in scope, such as end-of-year or standardized tests, the issues are relevant to classroom assessment as well.

Goals, Objectives, and Mathematical Content. For assessment to be properly aligned, the set of tasks on the assessment instrument must reflect the goals, objectives, and breadth of topics specified in the curriculum. Ideally, all topics in the curriculum should be assessed. Assessment instruments aligned with the secondary school core curriculum, for example, will include tasks reflective of all the 9-12 standards.

To determine the alignment of an instrument with a curriculum, simply comparing the objectives and content of an assessment instrument with the objectives and content of the curriculum is insufficient. Individual items must be examined to determine the degree to which they measure the mathematical content that they purport to measure; that is, whether they are aligned in terms of content. For example, a student's capability of measuring length or distance using an appropriate instrument is not adequately assessed by items such as the one in figure 1.1.

Fig. 1.1. A poor task for assessing measurement

To assess this skill, information is needed about whether each student can select an appropriate measurement tool, use it correctly (i.e., align and iterate it if necessary), and read the result. This information is best obtained through tasks that require a student to think about what mathematics is needed, to select a measuring tool, and to make an actual measurement. Assessment can be based on the student's answers or on observations of actions in the task. Only in this manner can a student's growth in mathematical power be determined.

The format of an assessment is an important factor that should be considered in determining the alignment of items and content. Other areas that might require particular formats for assessment include communication (which may involve talking, listening, or writing), reasoning (which might involve justifying or explaining responses), problem solving (which might involve recording processes as well as results), and estimation.

Another factor that affects the alignment of items and content is whether students answer items correctly for the wrong reasons. A common example is given in figure 1.2.

Fig. 1.2. A poor task for assessing students' knowledge of perimeter

Students may correctly add the lengths of the sides because that is the most obvious way to use all the given information. Little or no knowledge of perimeter is required. An item that will give a better indication of students' knowledge of perimeter is this:

Draw a six-sided irregular polygon with a perimeter of 23 units. Show all dimensions.

Items should be judged against the mathematics as described in the curriculum. Thus, one must be alert to differences in the interpretation of topics in an assessment instrument and in a curriculum. For example, problem-solving items can be anything from simple one-step story problems to open-ended, multistep problems. A simple story problem can measure problem solving according to the criteria established by the developers of an assessment instrument yet not measure problem solving as defined in the curriculum. The interpretation of content is of particular importance in selecting an assessment instrument that has been constructed by others, such as a standardized test.

Relative Emphases, Processes, and Relationships. The degree of alignment also depends on the extent to which the assessment's relative emphases on various topics and processes reflect the curricular emphases. An assessment instrument that contains many computational items and relatively few problem-solving questions, for example, is poorly aligned with a curriculum that stresses problem solving and reasoning. Similarly, an assessment instrument highly aligned with a curriculum that emphasizes the integration of mathematical knowledge must contain tasks that require such integration. And, for a curriculum that stresses mathematical power, assessment must include tasks with nonunique solutions.

Instructional Approaches and Activities. A final consideration is the extent to which the assessment reflects important aspects of instruction, such as the use of manipulatives, calculators, and computers. When these materials are used during instruction, they should be available during assessment, as long as their use is consistent with the purpose of the assessment. For example, if students routinely use calculators for solving problems in class, they should also be able to use calculators during assessments of their problem-solving abilities. Similarly, if students' understandings are closely related to the use of physical materials, they should be allowed to use these materials to demonstrate their knowledge during an assessment.

To select an appropriate assessment instrument, one must consider more than whether calculators, computers, or manipulatives are permitted to be used. Test items must be appropriate for use with these materials. For example, a multiple-choice algebra test that gives all answers in radical form, for example, (2 + 3)/4), might not be as appropriate for students using calculators as a test that includes answers that are decimal approximations.