Decisions concerning students' learning should be made on the basis of a convergence of information obtained from a variety of sources. These sources should encompass tasks that--

• demand different kinds of mathematical thinking;
• present the same mathematical concept or procedure in different contexts, formats, and problem situations.

Focus

The quality of judgments about students' knowledge depends on the consistency of the results obtained as well as on the alignment between the assessment tasks and the knowledge they purport to measure (see Evaluation Standard 1). Although written tests structured around a single correct answer can be reliable measures of performance, they offer little evidence of the kinds of thinking and understanding advocated in the Curriculum Standards.

When teachers find that students perform in consistent ways on various formatted tasks that demand a range of mathematical thinking or represent different aspects of mathematical thought, teachers can have confidence in the accuracy of their judgments. As defined here, mathematical knowledge reflects the integration of mathematical concepts and procedures in a coherent, meaningful structure.

To achieve this integration, the Standards advocate an inquiry approach to instruction, stressing problem solving as the medium through which mathematics is learned and applied. Assessment, in its turn, is conceived of as an integral element of instruction, a method of generating valuable information for both student and teacher. As instruction explores content through an array of problem-solving situations, so too must assessment. In addition, to achieve the full range of goals articulated in the Standards, a variety of assessment formats is necessary; they should include written and oral tests, observations, essays, and performance evaluations.

Discussion

Within the instructional context, teachers continually make informal judgments about their students' progress. Nonetheless, they often are reluctant to use these observations as the basis of important instructional decisions because of their potential subjectivity and unreliability. If we are to assess the types of mathematical thoughts and actions delineated in the Standards, the richer information that results from a variety of assessment methods is not only desirable but essential. For example, students who can easily find the area of rectangles when given linear measurements often are puzzled when presented with a grid and asked to find the area of an irregular plane figure. In such a situation, we might ask, How well do they understand the notion of area as a covering? How do they view their actions when they multiply the length and width?

A constant theme in the Evaluation Standards is the need for multiple sources of information. The tasks described here cover many aspects of a single concept, use procedures in many contexts, and require students to integrate knowledge, particularly through problem-solving activities. These types of assessment are intrinsically valuable because they extend the experiences of the student and offer important instructional feedback to the teacher. But they have an extrinsic value as well because they allow teachers to develop a more complete picture of students' performance and mathematical power to give to both students and their parents.

When a student performs similarly on many tasks, teachers can have confidence in their judgment of that student. Discrepancies in performance can also be useful because they can identify difficulties that might go undiscovered if an assessment is made with a single instrument. For example, a student might perform well on individual written assignments but be unwilling or unable to describe his or her problem-solving approach during group discussions. Another student might be able to apply rules in a familiar context but fail to recognize the appropriate procedure when the task is placed in an unfamiliar context. Such discrepancies should suggest other questions: Is the behavior of the first student the result of shyness, lack of confidence, or insufficient understanding? Does the second student experience difficulty with understanding the task or the underlying procedure? How can each student be helped?

The advantage of using several kinds of assessments, some of which are embedded in instruction, is that students' evolving understanding can be continuously monitored. The disadvantage is that such a procedure is perceived as cumbersome. Records of students' progress should be more than a set of numerical grades or checklists--they can include brief notes or samples of students' work. Such records are evidence of students' continued growth in understanding. Students should also maintain their own records. At all grades, students can keep portfolios of their work; in the higher grades, as they become more verbally fluent and reflective, they should be encouraged to keep a mathematics journal. These journals can contain goals, discoveries, thoughts, and observations, as well as descriptions of activities. Journals not only allow students to chart their progress in understanding but also act as a focus for discussion between student and teacher, thereby fostering communication about mathematics itself. For example, indications that a student understands rational numbers and the record to be kept for each such indication might include the following:

• Correctly answering test items on computing with fractions, percents, and decimals. (Record the percentage of correct items and note the range of topics tested.)
• Solving a homework problem situation by depicting graphically and in other forms how the speed of a car changes as it is driven through a city. (Record a score based on a four-point scale--four points for accurately drawing a graph that includes all the details of the change in speed, including stops, accelerations, and decelerations; three points if the general shape of the graph is given without the attention to detail; two points if the approach was correct with some errors in the details and shape of the graphs; one point if some notion of proportional thinking was evident but the graph does not represent the situation and details are missing; or no points for not trying.)
• Accurately representing, in a paper for civics class, the chances of winning a lottery and correctly using words like odds, proportion, and probability. (Student records in a log the mathematical concepts used in other subject areas.)
• Successfully using proportional scales and charts to format pages in print shop. (Write a short note of conversation with the printing teacher and put in the student's folder.)
• Being observed in class helping another student solve a percent problem. (Make a note of the use of percent and put in the student's folder.)
• Painting in graphics-art class an enlargement of a picture from a magazine. (File student's notes of what was done to ensure that proportions were correct.)

The learning of mathematics is a cumulative process that occurs as experiences contribute to understanding. A numerical score or grade assigned at a single point in time offers only a glimpse of students' knowledge. If the goal of assessment is a valid and reliable picture of students' understanding and achievement, evidence must come from a variety of sources. The assessment process described in this standard will produce a more complete and valid indicator of achievement than that possible from a single type of instrument.