In an evaluation of a mathematics program's consistency with the Curriculum Standards, the examination of curriculum and instructional resources should focus on--

• goals, objectives, and mathematical content;
• relative emphases of various topics and processes and their relationships;
• instructional approaches and activities;
• articulation across grades;
• assessment methods and instruments;
• availability of technological tools and support materials.

Focus

What mathematics is taught in the classroom and how it is taught are strongly influenced by a district's curriculum and adopted materials. Therefore, a necessary first step in determining the extent to which a mathematics program meets the Standards is examining the curriculum and the materials used in the program.

The Standards offers a vision of, and a direction for, a mathematics curriculum but does not constitute a curriculum in itself. If a mathematics program is to be consistent with the Standards, its goals, objectives, mathematical content, and topic emphases should be compatible with the Standards' vision and intent. Likewise, the instructional approaches, materials, and activities specified in the curriculum should reflect the Standards' recommendations and be articulated across grade levels. In addition, the assessment methods and instruments should measure the student outcomes specified in the Standards.

Discussion

Achieving agreement between a curriculum and the Standards is a necessary first step in having a mathematics program that reflects the vision expressed by the Standards. However, such agreement is not a guarantee that the curriculum, as implemented, will fully comply with the Standards. There is a difference between a curriculum as specified in a plan or textbook and a curriculum as implemented in the classroom. A curriculum specifies goals, topics, sequences, instructional activities, and assessment methods and instruments. An implemented curriculum is what actually happens in the classroom.

A deep, thorough analysis is necessary to determine the extent to which a curriculum and its materials are compatible with the Standards. The Standards offers a framework for curriculum development but not a scope and sequence. Simply checking topics on a scope-and-sequence chart is insufficient to determine the extent to which a curriculum and its materials are compatible with the Standards. A comparative analysis must provide qualitative documentation of the degree of consistency between the Standards and the curriculum. Such results can then be used to make decisions about the adoption of materials and how the curriculum needs to be modified to be more consistent with the Standards.

Goals, Objectives, and Content. The curriculum should include the major goals of developing students' problem-solving, reasoning, and communication abilities. In line with the Standards, however, these processes should not be listed as topics or separate strands to be taught in isolation. Rather, the curriculum and the adopted materials must provide a means of, and directions for, ensuring that these processes are integrated across topics. A superficial analysis will not suffice; for example, to determine how well problem solving as a curricular goal is reflected in the materials, it is necessary to look beyond such surface characteristics as the number of problem-solving lessons in a chapter. If problem solving is limited to a few lessons in each chapter and not integrated into the development of content topics, then the materials fall short of the Standards' intent.

The evaluation of instructional materials should take into account the quality of activities, their intended use in instruction, and their frequency of use. For example, textbook series commonly claim that problem solving is integrated into their programs. This claim is based, in part, on the common use of problems as "lesson openers." Yet, the lesson development that follows is, at best, remotely connected to the problem itself and is often didactic and prescriptive in nature. Such lessons fail to capture the spirit of problem solving and can convey an incorrect message to teachers and students about what problem solving really means.

Emphases and Relationships. The Curriculum Standards prescribe that emphasis be given to the meaningful development of concepts, to the interrelationships or connections among topics, and to the application of mathematics to the solution of realistic problems. These emphases should be explicitly and clearly stated in curricular and instructional materials and reflected in the development of lessons and activities following each lesson. For example, the Standards points out the importance of achieving an appropriate balance between concept development and computational proficiency. An inordinate emphasis on skill proficiency at the expense of a strong conceptual framework to support such skills renders materials inappropriate.

When feasible, curricular materials should develop new topics or ideas as natural extensions or variations of ideas students already know, thus making connections among topics explicit. Alternative representations and meanings of a mathematical idea should be featured in lessons as well as in practice activities. In addition, materials should allow ample opportunities for students to apply the mathematics they have learned in realistic and meaningful situations.

Instructional Approaches and Activities. When materials are evaluated for use in the classroom, a primary consideration is whether they promote students' active involvement in learning mathematics. Approaches and activities that call for the investigation and exploration of ideas, problem solving, conjecturing and testing of conjectures, and verification should be integrated through all the lessons rather than included as occasional isolated features at the end of a chapter or unit. Furthermore, such activities should be appropriate as student assignments in that they offer directions and guidance on how to use such approaches and activities effectively.

Articulation across Grades. The curriculum and materials should provide for the natural and logical development of mathematical topics across grades. Topic development should follow a progression from building a foundation of conceptual understanding, to extending concepts and procedures, to formalizing and abstracting ideas. A program that is well articulated will include directions for introducing, extending, and formalizing what has been taught before, with emphasis on the connections among related topics. The materials should also give teachers of different grades ways to coordinate the development of topics across the curriculum.

Assessment Methods and Techniques. Comparison of a curriculum with the Standards should include a careful study of what students are expected to know. To achieve agreement between the two, the assessment methods and instruments specified in the curriculum must be aligned with the student outcomes set forth in the Standards (see Evaluation Standard 1 on alignment). Tasks should be broad in scope and tap the extent to which students have integrated their knowledge of concepts, procedures, and processes. In addition, assessment instruments should reflect the relative emphases of various topics and processes as specified in the Standards.

Technological Tools and Resources. The mathematics classroom envisioned in the Standards is one in which calculators, computers, courseware, and manipulative materials are readily available and regularly used in instruction. Although no rigid criteria exist for judging what constitutes adequate resources and equipment, a program evaluation should include information about the resources available so that funds can be allocated properly.