At regular intervals, students' work is examined, summarized, and reported. Reports thus compiled are designed to indicate each student's mathematical accomplishments at the time. Typically, these formal reports are based on a teacher's judgments about the student's demonstrated understanding or on scores from examinations. Examples are narrative reports of progress, checklists of course criteria achieved, grades on report cards, and scores on comprehensive examinations. The basic question is, How does each student's understanding at this time compare with the goals he or she was expected to have achieved?

Evaluations of students' achievement at particular times have several characteristics. They are summative in nature, are usually designed to communicate to audiences beyond the classroom, and are often used to make important educational decisions for the students (e.g., admission, placement, certification). Because of the importance of such decisions for each student, the inferences made from summary evaluations and the way those summaries are created are the focus of this section. However, conscious and careful consideration must be given to the relationship of all six Assessment Standards to the evaluation of students' achievement.

Note that many assessments for the purpose of evaluating students' achievement are closely related to a previous purpose, "Monitoring Students' Progress," in that both deal with information about individual students that is drawn from their work. In fact, much of the same data can be, and often is, used for both purposes. During the course of instruction, teachers monitor each student's progress toward specified mathematical goals and provide feedback that will help each student accomplish those goals. When teachers evaluate a student's achievement, they make judgments about each student's understanding of mathematics at a particular time with respect to the specific knowledge and performance criteria associated with those goals, and they formally summarize and report on the student's progress.

For assessment to be consistent with the reform vision of school mathematics and these Assessment Standards, several shifts in the way student achievement is evaluated are warranted. Four related shifts in practice are of particular importance:

• Toward comparing students' performance with performance criteria and away from comparing student with student
  
  
• Toward assessing students' growth in mathematical power and away from assessing students' knowledge of specific facts and isolated skills
  
  
• Toward certification based on balanced, multiple sources of information and away from relying on only a few, narrowly conceived sources of evidence about student learning
  
  
• Toward profiles of achievement based on public criteria and away from single letter grades based on variable or nonpublic criteria

Comparing Students' Work With Performance Criteria

In North America, grading practices have followed certain customs and traditions for many years. Beginning at some point in elementary school and continuing through high school, teachers have reported their evaluations of student work by awarding a letter grade, usually A, B, C, D, or F; writing a number between 1 and 100; or using a scale from unsatisfactory to excellent. There are certain assumptions that the public makes about this practice. One is that, in general, grades are based on the comparison of students with other students, so that the distribution of grades should follow an expected pattern: for example, there should be more Cs than any other grade and fewer As and Fs. However, this assumption is both outdated and counterproductive. Grades do not have to be the result of comparisons among students.

Example: Rethinking the Meaning of Grades

An alternative way to conceive of grades is to measure students' achievement against performance criteria. Using performance criteria as the basis for a student's grade puts the emphasis on the development of each student's mathematical understanding rather than on competition among students.