In some countries, including England, France, Hong Kong, Singapore, and the Netherlands, there are permanent national centers or institutes that conduct multi-year research and curriculum development efforts in school mathematics. In the United States, the government has funded both a research center for mathematics learning at a single institution and projects to develop materials for teaching and learning mathematics at a number of other institutions.33 Typically, the curriculum development programs have required, as part of the project, both pilot testing of the materials while they are under development and the collection of evidence on the effectiveness of the materials, once developed. In some cases, the evaluation studies have been only perfunctory and the evidence gathered of poor quality. In others the support has resulted in sustained research-based curriculum development that systematically uses evidence as to what U.S. students can learn.34 Such a development program can be interactive, with improved learning materials yielding improved student learning that, in turn, yields improved and even-more-ambitious learning materials.

Developing teachers’ capacity to acquire and use good instructional materials is also a problem. Textbook selection processes can be overwhelming. Committee members usually do not have time to examine carefully the continuity of treatment of topics or the depth and clarity of the conceptual development facilitated by the materials. Instead, their focus is often on superficial features such as the appearance of the materials and whether all goals on a checklist are addressed. The problems created by checklists are especially keen in states and local districts with large numbers of specified special criteria. Failure to meet even a few of these criteria can eliminate an otherwise strong program.35

The methods used in the United States in the twentieth century for producing school mathematics textbooks and for choosing which textbooks and other materials to use are not sufficient for the goals of the twenty-first century. The nation must develop a greater capacity for producing high-quality materials and for using effectively those that are produced. In subsequent chapters, we cite research on children’s learning that can guide that production and use.


In general, assessments of children’s mathematics learning fall into two categories: internal and external. Internal assessments are those used by teachers in monitoring and evaluating their students’ progress and in making

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