Assessment methods and instruments should be selected on the basis of--

• the type of information sought;
• the use to which the information will be put;
• the developmental level and maturity of the student.

The use of assessment data for purposes other than those intended is inappropriate.

Focus

The purpose of an assessment--to identify areas of difficulty for individual students, to gather data for instructional planning, to assign grades, or to evaluate a program--should dictate the kinds of questions asked, the methods employed, and the uses of the resulting information. When one type of measure is used in lieu of another, the information obtained is often invalid or useless. In addition, the methods used to gather information should be appropriate to the developmental level and maturity of the students. Table 3.1 outlines some common purposes of student evaluation. The purposes in the table overlap, and the questions are only examples.

Discussion

The assessment of student performance serves many purposes. For the student, assessment aids learning and measures mathematical knowledge and power. For the teacher, it provides information about how instruction should be modified and paced. For the administrator, it charts the effectiveness of a program. In addition, the general public expresses concern about academic achievement. Each of these groups asks different questions. Each needs different kinds of information. An assessment designed to answer one kind of question can misrepresent the answer to another. Although this caveat might seem obvious, the legitimate uses of an assessment often are not well understood or defined, as when, for example, standardized tests of general mathematical achievement are used for curriculum evaluation.

The information required by different audiences lies on a continuum from the most specific, such as measures of a curriculum objective, to indicators of general mathematics performance. Teachers who want to know how they can help Sandra or John understand fractions as parts of a region, for example, will learn best by questioning students about their perceptions of a specific concept. However, when the same teachers ask how well Sandra or John has understood the course material as a whole, they must cast a wider net; assessment methods limited to a single aspect of learning will not suffice. Students should have the opportunity to show how well they have integrated their knowledge by applying their learning in a larger context.

TABLE 3.1
PURPOSES AND METHODS OF ASSESSMENT

Purposes(examples of questions asked)	For Whose Use	Unit of Assessment	Type of Assessment	Assessment Methods
Diagnostic What does this student understand about the concept or procedure? What aspects of problem solving are causing difficulty? What accounts for this student's unwillingness to attempt new problems or see the application of previously learned materials?	Individual teacher Individual teacher	Individual student	Tasks that focus on a specific skill, type of procedure, concept, strategy, or a type of reasoning Each student evaluated	Observation Oral questions that ask students to explain their procedures Focused written tasks Directed test items
Instructional Feedback What do students know about the material presented? Can students apply their learning to new situations? Do students understand the connections among ideas? How shall I pace instruction? Does the class need more intensive review of more challenging material?	Individual teacher	Class	Tasks that require an integration of know- ledge Tasks that cover a range of skills, concepts, and procedures Tasks that require the application of learning to new contexts Problem solving and reasoning tasks Tasks that vary the format and context in which the material is presented Matrix sampling test situations	Written tests, including those that require differential methods for solutions to problems Class presentations Extended problem solving projects Observation of class discussion Take-home tests Homework, journals Group work and projects
Instructional Feedback What do students know about the material presented?   Can students apply their learning to new situations?   Do students understand the connections among ideas?   How shall I pace instruction?   Does the class need more intensive review or more challenging material?	Individual student Parents School	Individual student	Tasks that demand the integration of material that was taught Tasks that are intrinsically interesting and challenging to the student Tasks that require the student to structure the material and generate solutions, in the context of the real world, as well as in math- ematics	Extended problem solving projects   Papers or written arguments that demand thoughtful inquiry about a mathematical topic   Written tests that present problems with a range of difficulty based on expectations for course   Oral presentations
Generalized mathematical achievement How does the general mathematical capability of this student compare with others or with a national norm?	Parents Teachers Administrators	Individual student	Tasks organized in highly reliable tests designed for maximum discrim- ination among students	Standardized achievement tests
Program Evaluation How effective is this instructional program in achieving our goals for mathematical learning?	Teachers Administrators Other decision makers	Class School	Tasks that reflect the intent of the curriculum goals   Tasks that are aligned to the instruct- ional methods and content of the curriculum (see Standards 12 and 13)   Matrix sampling test situations	Student interviews   Performance tests   Criterion referenced tests   Observation of class discussions   Success of students who have completed the program

When assessments are used for grading or as summative indicators of achievement, a number of measures should be used (see Evaluation Standard 2). In addition to the more traditional types of written tests, students should be given tasks that are challenging and complex and that allow them to perform at their maximum level of ability. Not only should such tasks provide meaningful information to parents and school authorities, but they should be interesting and valuable experiences in their own right, giving the student a sense of accomplishment.

Criterion-referenced tests, often used for program evaluation, can cover a narrower range of content. Generally designed to measure the attainment of specific objectives, they are, in some sense, a collection of mini-tests that each focus on a relatively narrow area of achievement. When the objectives on these tests reflect the goals of the school or the program, the tests are a valid measure of effectiveness. However, such tests usually present material in only one format (written multiple choice), which limits what can be measured. Thus, other information is needed to confirm their results.

In contrast to assessments that yield useful curriculum-based information to the teacher and student, more general types of tests are relatively insensitive to individual curricula. The standardized achievement test is the most prominent example. Its purpose is to measure an individual student's relative position in a population. Consequently, this type of test must maximize individual differences among students while measuring the common elements of their instruction. As a result, it is less appropriate for measuring the effectiveness of any specific curriculum and is more likely to reflect a student's general achievement, background, and prior knowledge. Furthermore, because of their format, standardized norm-referenced tests have difficulty measuring the generation of ideas, the formulation of problems, and the flexibility to deal with mathematical problems that are not well structured (i.e., problems similar to those encountered in everyday life). For these reasons, they are inappropriate as the only measure of whether teaching and curricula reflect the spirit of the Standards.

Whatever the purpose of the assessment, the methods used should consider the characteristics of the students themselves. Students' mathematical and cognitive development is a gradual, cumulative process built on prior experience and understandings. This consideration is particularly important in the early years, when basic skills and concepts are initially encountered. For example, recording responses on op-scan sheets imposes an additional irrelevant demand on young children. The very results of written tests in the early grades can be suspect: when children use their developing reading and writing skills to process the mathematical content of questions, the results might be more reflective of achievement in these areas than of their understanding of mathematics. At this stage, when children's understanding is often closely tied to the use of physical materials, assessment tasks that allow them to use such materials are better indicators of learning.

Students differ in their perceptions and thinking styles. An assessment method that stresses only one kind of task or mode of response does not give an accurate indication of performance, nor does it allow students to show their individual capabilities. For example, a timed multiple-choice test that rewards the speedy recognition of a correct option can hamper the more thoughtful, reflective student, whereas unstructured problems can be difficult for students who have had little experience in exploring or generating ideas. An exclusive reliance on a single type of assessment can frustrate students, diminish their self-confidence, and make them feel anxious about, or antagonistic toward, mathematics.

Finally, problems that reflect the potential of some students can be enigmas to others, disqualifying them from making any kind of meaningful response. Prior knowledge, experience, and the opportunity to learn are important considerations in interpreting test results. A challenging task in problem solving for some students is an exercise in recall for others. In part, the solution to this dilemma lies in the types of tasks used in the assessment. The complex, multifaceted tasks advocated in these Evaluation Standards can be structured to allow students to answer at different levels of sophistication. If students are to perform at their maximum levels of ability, the measures by which they are judged should give them the opportunity and the encouragement to do so.

STUDENT ASSESSMENT

If a conversation is to be meaningful to both of its participants, each must listen to the other; in the absence of such mutual attention, the conversation becomes an exercise in futility. Similarly, the act of teaching should be founded on dialogues between teachers and students, each responding to the other on the basis of what has been said or done. Assessment refers to the process of trying to understand what meanings students assign to the ideas being covered in these dialogues; as such, it is an integral element of effective teaching. Periodic assessment provides the teacher with a basis for deciding what questions should be asked and what examples and illustrations should be used; ultimately, it offers a foundation for any meaningful dialogue between teacher and student.

The student-assessment standards describe what is to be observed and measured in the process of understanding what mathematics students know. Teachers drawing meaning from their interactions with students is central to this process. At this level the most important decisions about student learning are made. The general-assessment standards, offer principles for the student-assessment standards, but the latter are paramount for helping students acquire the knowledge of mathematics as described in the Curriculum Standards. Thus, the student-assessment standards are more specific and relevant to teachers.

Assessment must be more than testing; it must be a continuous, dynamic, and often informal process. It manifests itself in teachers' statements, for example, "Johnny seems to have difficulty in graphing equations" or "Wilma demonstrated a lot of insight in solving those addition and subtraction problems." Assessment is more than the establishment of definitive conclusions. Assessment is cyclic in nature, a process of observation, conjecture, and constant reformulation of judgments about students' understanding. Assessment should produce judgments that are evolutionary in nature, regardless of whether those judgments are based on classroom discourse or on the more formal aspects of testing that characterize nearly every instructional program.

Testing to assign grades is one of the most common forms of evaluation. But assessment is a much broader and basic task, one designed to determine what students know and how they think about mathematics. Assessment should produce a "biography" of students' learning, a basis for improving the quality of instruction. Indeed, assessment has no raison d'être unless it improves instruction.

Teaching is effective to the degree that it takes student thinking into consideration. Without ongoing communication, a teacher's instructional strategies can only randomly enhance learning. Consider the student who answers the following item incorrectly: See figure 3.0

Fig. 3.0

If the fractions represented by points C and D on this number line are multiplied, what point will best represent the product?

If the student answers "point A," is he or she confusing multiplication with subtraction? Similarly, if points E or F are selected, is multiplication being confused with addition? Or is the answer a manifestation of the general misbegotten rule "multiplication makes bigger"? Did the student assign values to C and D but incorrectly multiply the fractions? What representation or model of multiplication is the student using? Would the student identify the same point if presented with the product D x C instead of C x D? How would the student respond if specific fractions were assigned to C and D? Only through explicit and careful assessment of how a student does mathematics can instruction be tailored to individual needs, thereby enhancing a student's chances for success.

These seven student-assessment standards focus on assessing students' understanding of, and disposition toward, mathematics. Each presents a list of mathematical outcomes derived from the standard's central focus. These standards and their associated tasks can serve as heuristics for designing assessment systems that include instruments, procedures for aggregating data, and ways of record keeping that are comprehensive and that tap more than superficial understanding. It is reasonable--indeed, expected--that every lesson cover more than one aspect and that over a series of lessons most, if not all, aspects are addressed in both instruction and assessment. For example, whereas instruction in any particular procedure may not encompass all seven aspects of procedural knowledge, each of the seven aspects will be represented in both instruction and assessment over a series of lessons.