The National Assessment of Educational Progress (NAEP) is a congressionally mandated survey of educational achievement of American students in a variety of curriculum areas and of changes in that achievement over time. The National Assessment Governing Board (NAGB) has established new standards for reporting the results that determined three achievement levels: basic, proficient, and advanced. The basic level denotes partial mastery of the knowledge and skills fundamental for proficient work at each grade. Proficient, the central level, represents solid academic performance and demonstrated competence over challenging subject matter. The advanced level signifies superior performance beyond proficient. This book, volume III of the Initial Performance Standards for the 1990 NAEP Mathematics Assessment, describes the process of how these three levels were determined. The chapters include: Chapter 1: Executive Summary; Chapter 2: Overview to the Achievement Level-Setting Process; Chapter 3: Achievement Levels Methodology: Phase 1; Chapter 4: Analysis of Achievement Level Ratings: Phase 1; Chapter 5: Conclusions and Recommendations: Phase 1; Chapter 6: The Replication/Validation Study: Phase 2; Chapter 7: Analysis of Achievement Level Ratings--Validation/Replication; Chapter 8: Additional Topics; and Chapter 9: Conclusions and Recommendations. The bulk of the document is contained in the appendices that follow. These 14 appendices provide a detailed account of the development and validation of the established levels in a series of 85 related tables and documents. (MDH)