In an evaluation of a mathematics program's consistency with the Curriculum Standards, instruction and the environment in which it takes place should be examined, with special attention to-

• mathematical content and its treatment;
• relative emphases assigned to various topics and processes and the relationships among them;
• opportunities to learn;
• instructional resources and classroom climate;
• assessment methods and instruments;
• the articulation of instruction across grades.

Focus

How mathematics is taught is just as important as what is taught. Students' ability to reason, solve problems, and use mathematics to communicate their ideas will develop only if they actively and frequently engage in these processes. Whether students come to view mathematics as an integrated whole instead of a fragmented collection of arbitrary topics and whether they ultimately come to value mathematics will depend largely on how the subject is taught. Thus, the evaluation of a mathematics program must include an analysis of instruction as well as content.

Discussion

An evaluation of how mathematics is taught must consider how well the program is being implemented and how instruction can be modified to better meet its goals and objectives. Such an analysis requires the collection from a representative sample of teachers of various types of information, including, but not limited to, sustained classroom observations, interviews with teachers, teachers' written self-reports, questionnaires, and summaries of peer observations.

Each of the following sections offers guidance for gathering information to evaluate instruction. Both quantitative and qualitative information must be obtained.

How Content Is Treated. An evaluation of instruction should look first at the accuracy of the mathematics that is taught. In addition, an evaluation should determine whether that mathematical content is treated in a manner that is sensitive to the developmental level and mathematical maturity of the students. Instruction that is sensitive to the students' developmental level is set in realistic contexts, incorporates students' experiences, uses language that is suited to the maturity of students, allows sufficient time for students to construct meanings by exploring and investigating mathematical ideas, and offers students opportunities to discuss their ideas.

The Standards specifies reasonable expectations for students and offers suggestions for the treatment of topics at various grade levels. Students' needs and characteristics were primary considerations as the Standards was developed. Nonetheless, because the needs and characteristics of students vary greatly even at a single grade level, it is important to take these variables into account when examining instruction.

The Standards advocates students' active involvement in learning, a stance that has important implications for the way content is to be treated during instruction. Rather than a routine presentation of mathematical ideas in a polished, finished form for students to assimilate, instruction should provide frequent opportunities for students to generate, discuss, test, and apply mathematical ideas and verify their findings.

Such instructional approaches also have implications for the role of the teacher during instruction. When using a problem-solving approach in developing an idea, for example, teachers must encourage students to guess courageously. Teachers must be willing to entertain suggestions from students and suspend judgment about their ideas. Teachers should help students evaluate one another's suggestions and critically reflect on them by anticipating objections and consequences. Clearly, these activities require the teacher to assume a role very different from that of a directive authority. Classroom observations for the purpose of evaluating instruction should focus on the role of the teacher and the appropriateness of that role to the content and activities.

Topic Emphases and Relationships. The Standards proposes that instruction should emphasize interrelationships among mathematical ideas. Classroom observations should gather information about whether mathematics is portrayed as an integrated body of logically related topics as opposed to a collection of arbitrary rules that students must memorize. For example, the K-4 standards suggest ways in which development and extension of number concepts and operations can be integrated into the treatment of other topics, such as measurement, geometry, patterns, and graphs. The extent to which new ideas are presented as natural or logical extensions of ideas the students have already encountered should be a focal point of instructional evaluation.

The Standards calls for an instructional emphasis on building strong conceptual frameworks on which to base the development of skills. It also emphasizes the importance of multiple representations of a mathematical idea and the translation of an idea from one representational system to another. A documentation of the extent to which these and other instructional emphases in the Standards are implemented should be a major part of any serious evaluation.

Classroom observations should also document the extent to which instruction emphasizes the relationships among the various branches of mathematics and between mathematics and other areas of the school curriculum.

Opportunity to Learn. If instruction is to result in the student outcomes specified in the Standards, students need to have sufficient opportunities to learn the specified content. Thus, program evaluations must consider the amount of time actually devoted to mathematics instruction; an hour of mathematics each day at all grade levels is a reasonable expectation. The frequency of interruptions caused by school assemblies and other school projects must be considered in the evaluation to determine the actual time.

Further, evaluation should focus on the attention given at each grade level to the various branches of mathematics, such as geometry, measurement, statistics, probability, algebra, discrete mathematics, and calculus in light of the recommendations in the Standards.

Similarly, if students are to become competent in problem solving, reasoning, and communicating mathematically, instruction must allow them opportunities to engage actively and frequently in these processes. Classroom observations for evaluating instruction must pay special attention to the frequency and quality of instructional activities that afford students such opportunities.

Instructional Resources and Classroom Climate. The evaluation of instruction should determine the extent to which the classroom environment is conducive to the attainment of program goals and student outcomes. One indicator would be the availability and use of instructional resources, such as computers, calculators, courseware, and manipulative materials. In addition, evaluation should determine whether an intellectual "climate" exists, in which students' curiosity, openness, and spontaneity are encouraged.

Uses of Classroom Assessment Methods. The methods, instruments, and tasks used to assess students' learning should be consistent with the content taught and the emphases placed on various topics and processes. Evaluation Standards 1-10 present guidelines for assessing students' outcomes as well as for judging the adequacy of the instruments and tasks used in that assessment.

Articulation of Instruction. An evaluation should examine the extent to which the implemented program is articulated across grades. The degree of consistency between the Standards and instruction should be determined on the basis of data obtained from a number of classrooms in a given school. It is insufficient to determine that one or two classrooms exhibit this consistency. Within a given school, evaluation must obtain information about the extent to which teachers have opportunities to coordinate their instruction with the Standards across grades.