This section describes how the standards just presented can be used to critique and improve assessment of the learning of mathematics. The examples and vignettes in this document portray some shifts in assessment practice and in the ways in which teachers and others can use the standards to judge the efficacy of mathematics assessments, improve their quality, and generate ideas for alternative ways of assessing mathematics learning. Except for those identified as adapted from research, the vignettes are fictional illustrations and, although drawn from experience, are not factual accounts.

The section organizes the diverse purposes for which mathematics assessments are made into four broad categories. Although there are many ways to categorize these purposes, the four selected represent primary areas for the reform of assessment practice. What distinguish each purpose are the objectives to be achieved and the results to be obtained from the assessment. Figure 2 depicts the four purposes (in the ellipse) and the actions (in the rectangles) that result from the use of assessment data in conjunction with each purpose.

Fig. 2. Four purposes of assessment and their results

What progress is each student making?

What are the appropriate instructional decisions?

One important purpose of assessment is monitoring students’ progress toward learning goals. After setting high expectations, evidence should be collected to provide each student and the teacher with feedback about progress toward those goals. The feedback is used in an ongoing effort to promote each student’s growth in mathematical power. Monitoring is seen as a continuous process. Sometimes the collection of evidence is informal and spontaneous, and sometimes it is formal. Hence, the results are provisional, yet they provide the rich diagnostic feedback important to each student. The basic question to be answered about students’ progress is, How is each student progressing in relation to the goals we have set and agreed on?

A second and related purpose of mathematics assessment is that of making instructional decisions. Teachers use evidence of students’ mathematical understanding, along with other evidence from the instructional process, to modify instruction so that it will better facilitate learning. The teacher is the primary assessor of the mathematics that students know and can do. The basic question teachers consider when using assessments undertaken for this purpose is, How can I use evidence about my students’ progress to make instructional decisions?

Have students reached their goals?

Is the program working?

Evaluating programs is a fourth purpose of mathematics assessment. Evidence of students’ performance, as well as other data, is used to make decisions about instructional programs so that all students are encouraged to meet high expectations in mathematics. The question being addressed is, How well is the mathematics program working in relation to goals and expectations for the students?

Regardless of the purpose for which they are conducted, all school mathematics assessments envisioned in this document share common features. The six standards apply to each type of assessment. However, the way in which a particular standard is applied in assessments carried out for different purposes may vary.

Several changes in assessment practices are imperative if the practices are to be consistent with curricular and instructional reform efforts. Many current practices furnish incomplete and sometimes biased information about students’ mathematical understanding. In each section that follows, three or more shifts in assessment practices are highlighted and illustrated with examples and vignettes. The shifts should be seen as components of needed changes in the total assessment system and not viewed just with respect to the purpose with which they are presented.

Assessment is the shared responsibility of all who are concerned with students’ learning of mathematics. The specific educational purposes for which assessments are made have been deliberately chosen to blur the distinction between assessments that are internal to the classroom and assessments that are external. Assessments for monitoring students’ progress, making instructional decisions, and evaluating student achievement have typically been the responsibility of classroom teachers, whereas assessments for evaluating programs have been carried out by agencies outside the classroom. The illustrations that follow suggest that assessments for all purposes need to become more open and collegial; that is, teachers need to be involved in the assessment process for all purposes. The primary responsibility for assessment may lie with specific people, depending on the purpose, but it must be a collaborative endeavor if it is to meet the six standards defined in this document.

Information gathered from any source about a student’s mathematical understanding is only a sample of the possible information about such understanding. Thus, questions about the sample’s representativeness, reliability, and validity must be of concern. Furthermore, for two of the four categories of purposes–monitoring students’ progress and evaluating students’ achievement–the sample is of each student’s performance, and the information is aggregated to make decisions about that student. For the other two purposes–making instructional decisions and evaluating programs–the sample represents performance, and the information, although derived from students’ performance, is aggregated across students to make other decisions.

All mathematics assessments involve the same four phases–plan the assessment, gather evidence, interpret the evidence, and use the results–although the aspects of each process that are most crucial may vary with the purpose (see fig. 3). The illustrations in the following sections indicate how the phases of the assessment process relate to assessment purposes and fulfill the intent of the six standards.

Fig. 3. Relationship between phases of assessment and assessment purposes

The discussion and illustrations of the assessment undertaken for each purpose demonstrate both the shifts in practice and the methods for applying the standards to accomplish those shifts. The discussion shows how the questions for reflection that follow each standard can be used to determine how well specific assessment activities and practices meet the standards under review. The illustrations suggest how those responsible for students’ learning of mathematics might begin collectively to apply the standards to specific assessments, thereby making mathematics assessment a more public, open, and collaborative enterprise.