ability The capacity to do something; the power to perform. In this document, ability emphasizes developed powers to perform, which are influenced through educational experiences as well as by natural talents, aptitudes, or traits.

access Rights and means to approach or engage in with understanding. Assessments provide for equal access when they include tasks that are shown to be equally appropriate for all students, allow multiple approaches and strategies, and accept multiple justifiable responses. See open-ended questions.

achievement Successful accomplishment or attainment of educational goals.

activities The cognitive functioning or physical actions in which students are engaged, whether by assignment or on their own initiative.

adequacy Ability to satisfy the requirements of the intended purposes and their consequences.

assessment 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. Assessment is a term that has often been used interchangeably with the terms testing, measurement, and evaluation, or to distinguish between student assessment and program evaluation. In this document, assessment is used as defined above to emphasize understanding and description of both qualitative and quantitative evidence in making judgments and decisions. See evaluation, measure, test.

assessment standards Criteria for judging the quality of assessment practices, which embody a vision of assessment consistent with the Curriculum and the Teaching Standards derived from shared philosophies of mathematics, cognition, and learning.

authenticity The degree to which activities are faithful, comprehensive representations of the contexts and complexity found in important, real-life performances of adults that are nonroutine yet meaningful and engaging for students. Authentic activities are either replicas of or analogous to the kinds of problems faced by adult citizens or professionals in the field and are accompanied by the resources and opportunities for discussion, collaboration, revision, and justification typical of the production of quality adult performances (Wiggins 1993).

benchmarks Descriptions of student performances at various developmental levels that contribute to the achievement of performance standards.

bias Conditions in content, procedures, or interpretation of assessment information that favor one or more groups of participants over other groups.

coherence The quality of logical connection and orderly relationship of parts.

concepts General and fundamental ideas—for example, the ideas that are needed to guide reasoning, problem formulation, and problem solving in nonroutine situations.

consistency Compatibility or agreement among successive acts, ideas, or events.

context The circumstances or situation in which a mathematical problem occurs or in which mathematics can be applied.

credibility The quality of being plausible, believable, dependable, or worthy of confidence.

disposition Interest in, and appreciation for, mathematics; a tendency to think and act in positive ways; includes confidence, curiosity, perseverance, flexibility, inventiveness, and reflectivity in doing mathematics.

equal Having the same quantity, measure, value, privileges, status, or rights.

equality A state of uniformity in quantity, measure, value, privileges, status, or rights.

equitable assessment The degree to which the process of gathering evidence has provided opportunities equally appropriate for each student to demonstrate the valued thinking processes, knowledge, and skills that he or she has developed. Equitable assessment is not achieved by creating the same conditions for all students but rather by creating conditions that are appropriate to the same extent for each student.

equity The state or quality of being fair, just, equally appropriate for all students. See equitable assessment.

equivalent Equal in value or meaning, interchangeable, or having comparable effects.

evaluation The process of determining the worth of, or assigning a value to, something on the basis of careful examination and judgment. As used in this document, evaluation is one use of assessment information. See assessment.

framework An organizing system for, and arrangement of, the mathematical understanding, performances, and dispositions to be assessed, which will assist the planning of assessments.

generalizations Inferences or conclusions from many particulars of the evidence in hand, supported by a theory of the relationships between the particulars and the more general inferences or conclusions.

inferences Conclusions or assertions derived from evidence; deductions.

item A single, often decontextualized test question or problem.

judgments Authoritative estimates or opinions of quality, value, and other features, formed by distinguishing the relations among multiple sources of sound and reasonable evidence; formal decisions.

mathematical power "Mathematical power includes the ability to explore, conjecture, and reason logically; to solve nonroutine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity. Mathematical power also involves the development of personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions. Students’ flexibility, perseverance, interest, curiosity, and inventiveness also affect the realization of mathematical power" (NCTM 1991, p. 1).

measure To indicate how much of some specified, quantifiable unit is present; to assign numbers to variations in a quantifiable attribute or trait. For conditions that cannot be quantified with sufficient certainty and accuracy for the intended purposes, a description is more appropriate than measurement.

measurement-based assessment Procedures for (1) developing and selecting test items or assessment tasks by the degree to which they differentiate among examinees and for (2) administering and scoring such tests to provide statistically adequate scores for making valid generalizations regarding the psychological traits, attributes, and skills that the test items or tasks are designed to measure.

norm-referenced test A test that compares quantitative scores (such as the number of correct responses) to a normal distribution of such scores for the same age or grade. Such testing has long been used for ranking students (e.g., for allocating scarce resources) and sorting students (e.g., for forming homogeneous instructional groups).

open-ended questions Tasks that allow for various acceptable answers and for multiple approaches to an effective solution. Open-ended problems engage students in interesting situations and allow students at many levels of understanding to begin working on the problems, make their own assumptions, develop creative responses, and effectively communicate their solutions (Pandey 1991).

openness Accessibility; availability of information; receptiveness to discussion and participation; candidness; lack of secrecy.

opportunity to learn The degree to which a student has been exposed to the learning experiences needed to meet high academic standards, which is largely a function of the capacity and performance of the courses and schools the student has attended. Equitable opportunities to learn consist of equal chances for learning, with equally appropriate, favorable, or advantageous combinations of circumstances (i.e., opportunities to learn are equitable when they are responsive to the same extent to each student’s needs).

outcomes Learning, results, or consequences. Equal outcomes across broad classifications of students (such as gender, race, ethnicity, etc.) that should be unrelated to performance may be evidence of equity.

performance The carrying out or bringing to completion of a physical activity or production of some significance, which displays one’s knowledge and judgment while engaged in the task.

performance criterion, or standard A statement of expected performance quality that can be used to make judgments about performances that are central to the curriculum. A set of performance criteria, or standards, includes the nature of the evidence required and the quality of performance expected to demonstrate that a curriculum or content standard has been achieved. These statements often describe performances at one level, such as either adequate or exemplary, but may also describe a range of quality levels.

program evaluation The process of determining the effectiveness of an educational program in achieving its goals and, therefore, its value in comparison to its required resources.

quality Degree of excellence. The quality of assessment evidence is characterized primarily by the authenticity of the tasks, the reliability of the sample of evidence, and the credibility of the evidence for the intended purposes.

reliability The degree to which assessment evidence supports a clear, complete, and accurate understanding of the quality of an individual student’s performance across time, tasks, and scorers (i.e., credibility and representativeness of the evidence).

representativeness The degree to which assessment evidence represents the valued thinking processes, knowledge, and skills that the student has developed.

rubric A set of authoritative rules to give direction to the scoring of assessment tasks or activities. To be useful, a scoring rubric must be derived from careful analysis of existing performances of varying quality. A task-specific rubric describes levels of performance for a particular complex performance task and guides the scoring of that task consistent with relevant performance standards. (A task-specific rubric is more specific than a performance standard and can apply a performance standard to a particular context found in a performance task.) A general rubric is an outline for creating task-specific rubrics, or for guiding expert judgment, where task-specific scoring rules are internal to the scorer.

scoring Discriminating among performances according to differing levels of quality and assigning a descriptive label or number to the performance. In holistic scoring, the entire performance as a whole is considered, and one label or number is assigned. In analytic scoring, separate scores are assigned to fundamentally different dimensions of the performance.

skills Abilities to perform routine mathematical procedures, typically by computational or manipulatory methods.

standard A statement about what is valued that can be used for making a judgment of quality.

standardized test A test that is administered, scored, and interpreted in a consistent manner whenever, wherever, and to whomever it is given.

standard-referenced assessment An assessment that compares the quality of performances to relevant performance criteria or standards to make a determination of the degree to which the standards have been attained or to describe progress toward the attainment of the standards.

task An authoritatively specified or assigned, purposeful, contextualized activity.

technical Relating to formal, statistical determinations of the quality of numerical scores.

test "A measuring instrument for assessing and documenting student learning. The traditional test is a single-occasion, one-dimensional, timed exercise" (Hart 1994, p. 114). "A formal, systematic procedure for obtaining a sample of [students’] behavior; the results of a test are used to make generalizations about how [students] would have performed on similar but untested behaviors" (Airasian 1991, p. 440).

understanding The ability to employ knowledge "wisely, fluently, flexibly, and aptly in particular and diverse contexts" (Wiggins 1993, p. 207).

valid Justifiable, well grounded, sound; producing the desired results, efficacious; incontestable.

valid inferences Justifiable assertions and conclusions that lead to and support desirable results. Justification is made primarily on the quality of the evidence and its adequacy for the intended purposes and their consequences.

validity The degree to which an assessment provides information that is relevant and adequate for the intended purpose. The reasonableness, quality, and efficacy of an assessment for particular educational purposes, decisions, and consequences are important issues of validity.

worthwhile mathematical tasks Tasks "that engage students’ intellect; develop students’ mathematical understandings and skills; stimulate students to make connections and develop a coherent framework for mathematical ideas; call for problem formulation, problem solving, and mathematical reasoning; promote communication about mathematics; represent mathematics as an ongoing human activity; display sensitivity to, and draw on, students’ diverse background experiences and dispositions; promote the development of all students’ dispositions to do mathematics" (NCTM 1991, p. 25).