Assessment needs to reflect the reform vision of school mathematics.

Teachers are in the best position to judge students’ progress.

The Assessment Standards for School Mathematics has been produced by the National Council of Teachers of Mathematics (NCTM) because we believe new assessment strategies and practices need to be developed that will enable teachers and others to assess students’ performance in a manner that reflects the NCTM’s reform vision for school mathematics. Our vision includes the mathematics we expect students to know and be able to use, the way they have learned it, and how their progress is to be assessed. For school assessment practices to inform educators as they progress toward this vision, it is essential that we move away from the "rank order of achievement" approach in assessment toward an approach that is philosophically consistent with NCTM's vision of school mathematics and classroom instruction.

At present, a new approach to assessment is evolving in many schools and classrooms. Instead of assuming that the purpose of assessment is to rank students on a particular trait, the new approach assumes that high public expectations can be set that every student can strive for and achieve, that different performances can and will meet agreed-on expectations, and that teachers can be fair and consistent judges of diverse student performances. Setting high expectations and striving to achieve them are quite different from comparing students with one another and indicating where each student ranks. A constant theme of this document is that decisions regarding students’ achievement should be made on the basis of a convergence of information from a variety of balanced and equitable sources. Furthermore, much of the information needs to be derived by teachers during the process of instruction. Teachers are the persons who are in the best position to judge the development of students’ progress and, hence, must be considered the primary assessors of students. However, depending on the purpose, there are other assessors, such as learners who assess their own progress.

All students are capable of learning mathematics, and their learning can be assessed.

Assessment should be a means of fostering growth toward high expectations.

These Assessment Standards have been designed to expand on and complement the NCTM's Evaluation Standards.

NCTM's earlier Standards documents–Curriculum and Evaluation Standards for School Mathematics and Professional Standards for Teaching Mathematics–presented a vision of appropriate mathematical goals for all students. This vision, based on the assumption that all students are capable of learning mathematics, is at the heart of the NCTM’s reform efforts. In the past, school mathematics was organized and taught, and students’ performance assessed, in a way that underestimated the mathematical capability of most students and perpetuated costly myths about students’ ability and effort. Too often, tests designed for other purposes have been used unintentionally as filters that deny underrepresented groups access to the further study of mathematics. Today, the mathematical development of each child in a diverse multicultural society must be valued. Assessment procedures must no longer be used to deny students the opportunity to learn important mathematics. Instead, assessment should be a means of fostering growth toward high expectations. To do otherwise represents a waste of human potential.

These Assessment Standards have been designed to expand on and complement, not replace, the NCTM’s Evaluation Standards. The Evaluation Standards proposed that

• student assessment be aligned with, and integral to, instruction;
  
  
• multiple sources of assessment information be used;
  
  
• assessment methods be appropriate for their purposes;
  
  
• all aspects of mathematical knowledge and its connections be assessed;
  
  
• instruction and curriculum be considered equally in judging the quality of a program.

These Assessment Standards establish additional criteria for student assessment and program evaluation and elaborate the vision of assessment that was described in the Evaluation Standards.

Shift in content:

Toward a rich variety of mathematical topics and problem situations.
Away from just arithmetic

• A shift in the mathematical content that students are expected to learn. This shift reflects the belief that all students need the opportunity to develop an understanding of algebra, geometry, trigonometry, statistics, probability, discrete mathematics, and even calculus. This vision implies that high expectations need to be publicly set for all students. Thus, the vision of a reform curriculum is toward a balanced variety of rich problem situations that encourage students to make connections among the various mathematical topics and that reflect cultural diversity. This shift is away from considering arithmetic proficiency as sufficient mathematics for most students.

Shift in learning:

Toward investigating problems. Away from memorizing and repeating

• A shift in the vision of learning mathematics toward investigating, formulating, representing, reasoning, and applying a variety of strategies to the solution of problems–then reflecting on these uses of mathematics–and away from being shown or told, memorizing, and repeating. This represents a shift from mechanical to cognitive work and also assumes the acquisition of a healthy disposition toward mathematics. Furthermore, cognitive work for all students is culturally dependent because students bring to each lesson their past experiences and the diverse facets of their cultural identities. Thus, instruction that capitalizes, and builds, on what students bring to a problem situation can motivate them to struggle with, and make sense of, the problem and share their thinking with classmates.

Shift in teaching:

Toward questioning and listening. Away from telling

• A shift in the role of teachers toward "questioning and listening" as their classrooms become stimulating intellectual learning communities and away from "telling" students what to do. High expectations, challenging work, strong effort, mutual respect, and assistance in supporting the achievement of all students characterize exemplary classrooms in NCTM’s vision

Shift in evaluation:

Toward evidence from several sources judged by teachers.
Away from a single test judged externally

Shift in expectations:

Toward using concepts and procedures to solve problems.
Away from just mastering isolated concepts and procedures

• A shift in the vision of evaluation toward a system based on evidence from multiple sources and away from relying on evidence from a single test as well as a shift toward relying on the professional judgments of teachers and away from using only externally derived evidence.

In NCTM’s Standards documents, the phrase mathematical power has been used to capture the shift in expectations for all students. The shift is toward understanding concepts and skills; drawing on mathematical concepts and skills when confronted with both routine and nonroutine problems; communicating effectively about the strategies, reasoning, and results of mathematical investigations; and becoming confident in using mathematics to make sense of real-life situations. It is away from mastering a large collection of concepts and skills in a particular order. In this document, the terms know, know how, be able to do, and disposition toward are used as indicators for the complexity of mathematical power.

Many schools and teachers have responded enthusiastically to the Curriculum and Evaluation Standards and the Professional Teaching Standards by changing both the mathematical content of their courses and the way in which the content is taught. As schools and teachers change their practices, they face the dilemma that the result of their efforts to meet new goals may not be supported by traditional assessment practices because such practices are inconsistent with these new views of mathematics and how learning progresses. The Assessment Standards presented in this document reflect the values and goals associated with the type of assessment system that must be achieved if the reforms envisioned in the teaching and learning of mathematics are to become a reality. We challenge teachers, school district personnel, and state or provincial officials to read and reflect on these standards, critically examine their own current assessment systems, and then work to develop new systems that are consistent with the reform vision.

The Assessment Standards is a guide, not a "how to" document.

However, this document is not meant to be construed as a "how to" document. Consistent with the philosophy underlying the earlier Standards publications, the ideas presented here are intended to guide teachers and others as they examine current assessment practices and plan new assessment systems. As with the previous Standards publications, NCTM will continue to develop companion publications to respond to specific assessment issues.

Mathematics Assessment

Assessment is the process of gathering evidence about a student's knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes.

Evaluation refers to the process of determining the worth of, or assigning a value to, something on the basis of careful examination and judgment.

In this document, assessment is defined as the process of gathering evidence about a student's knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes. (Important terms used in these pages are defined in the Glossary; definitions have been provided to clarify terms that may have other colloquial meanings or technical meaning in the assessment literature.)

Furthermore, by evaluation we mean the process of determining the worth of, or assigning a value to, something on the basis of careful examination and judgment. The term evaluation as used in this document refers to one use of assessment information. The focus on gathering evidence and making inferences emphasizes that assessment is a process of describing what mathematics students know and can do.

There are many audiences for, or consumers of, assessment data as well as different purposes for assessments. For example, every student is well aware of being assessed while in school. In any given mathematics class, some student is bound to ask, "Will this be on the test?" Students have learned that what is assessed and how it is assessed reflect what educators value. It is only reasonable that students should know how they are to be assessed, what mathematics they will be expected to do, the criteria for judging their performance, and the consequences of the assessment, and they have the right to receive timely feedback on their performance. Teachers’ responsibilities, in large part, involve judging students’ performance during the classroom lesson, students’ progress throughout a unit of instruction, and students’ knowledge and competence at various points during a semester or year. Also, the public has a need and right to know how well students are performing in schools. Information on students’ performance is used to indicate the comparative status of students, schools, districts, and even states and provinces. Summary performance data are also used to evaluate students’ achievement, to evaluate programs, and to make policy decisions. Judgments of students’ progress, instructional decisions, and evaluations of instructional programs must be based on reasonable interpretations of high-quality, relevant evidence.

Assessment involves several interrelated, but nonsequential, phases.

The assessment process can be thought of as four interrelated phases that highlight principal points at which critical decisions need to be made. Figure 1 shows the four phases–plan the assessment, gather evidence, interpret the evidence, and use the results. The division is arbitrary, however, and makes the process seem more orderly than it actually is. In practice, the phases are interactive, and the distinctions between them are blurred. Assessment does not proceed through them in a neat, linear fashion.

Fig.1. Four phases of assessment

The phases should also not be seen as necessarily sequential. Rather, they should serve as markers to be used as a guide. Each phase of the assessment process can be characterized by the decisions and actions that occur within that phase as follows:

The four phases of assessment can be characterized by the decisions and actions that occur during them.

• What purpose does the assessment serve?
  
  
• What framework is used to give focus and balance to the activities?
  
  
• What methods are used for gathering and interpreting evidence?
  
  
• What criteria are used for judging performances on activities?
  
  
• What formats are used for summarizing judgments and reporting results?

• How are activities and tasks created or selected?
  
  
• How are procedures selected for engaging students in the activities?
  
  
• How are methods for creating and preserving evidence of the performances to be judged?

• How is the quality of the evidence determined?
  
  
• How is an understanding of the performances to be inferred from the evidence?
  
  
• What specific criteria are applied to judge the performances?
  
  
• Have the criteria been applied appropriately?
  
  
• How will the judgments be summarized as results?

• How will the results be reported?
  
  
• How should inferences from the results be made?
  
  
• What action will be taken based on the inferences?
  
  
• How can it be ensured that these results will be incorporated in subsequent instruction and assessment?

An assessor might be a student assessing his or her own progress in learning probability, a teacher developing assessment activities for a unit on decimals, a district committee choosing among performance assessments in algebra, a state or provincial mathematics supervisor analyzing data from a collection of student portfolios, or a major test publisher developing an instrument to evaluate a course on applications of mathematics. Whatever the assessment, whoever the assessor, the assessment process itself includes some combination of decisions and actions drawn from the four phases. The phrase assessment process and the diagram in figure 1 are used in this document to emphasize the complex process that underlies the purposes for which assessments are done, the decisions made by the assessors, and the standards to which assessments are held. This vision of assessment should apply to any assessment purpose.

This document presents six assessment standards that are the criteria to be used for judging assessment practices. To illustrate their use, four general categories of educational purposes for which evidence about student performance is commonly gathered are presented. The document concludes with a section entitled "What’s Next?" In that section, suggestions are made about using the document, and the reform vision of assessment is summarized. Teachers and other assessors are challenged to examine their current practices, to recognize–and, if necessary, create–the conditions needed for implementing these six standards, and to build new assessment systems consistent with this vision.

Those who serve on school or district-wide mathematics committees will find this document a useful tool in redrafting assessments.

In order to develop mathematical power in all students, assessment needs to support the continued mathematics learning of each student.

This document can be used by classroom teachers, curriculum directors, and department chairs to guide classroom, school, and district assessment practices; by policymakers and administrators to generate and support needed improvements in mathematical assessments; and by state or provincial and commercial test publishers to align their assessment systems with those of schools or districts in order to provide more useful data to students, teachers, parents, and the public at large. In particular, those who serve on school or district-wide mathematics committees will find this document to be a useful tool in redrafting assessments that align with the vision of curricula and teaching outlined in NCTM’s previous Standards documents.

Finally, we suggest that all of us continue to consider, study, and work to apply the ideas presented in the three NCTM Standards documents, addenda, and other resources in our effort to examine current curriculum, instructional, and assessment practices. By doing so, we can continue to develop new assessment systems that support the changes in content and instruction envisioned by NCTM.

In order to develop mathematical power in all students, assessment needs to support the continued mathematics learning of each student. This is the central goal of assessment in school mathematics. In our view, assessment occurs at the intersection of important mathematics content, teaching practices, and student learning. Assessment that embodies the vision of the six standards presented here will be a dynamic process that informs teachers, students, and others and supports each student’s continuing growth in mathematical power.