Recent reports on mathematics education reform have focused the attention of educational practitioners and policymakers on new goals for mathematics education and new descriptions of mathematical proficiency. QUASAR is a national project (Quantitative Understanding: Amplifying Student Achievement and Reasoning) designed to improve the mathematics instructional program for students attending middle schools, grades 6 through 8, in economically disadvantaged communities. QUASAR is a complex research study of educational change and improvement, in which a major effort will be made to study carefully different approaches to unblocking the path to mathematical power for poor students. Parallel goals for the study are: to ascertain conditions that appear conducive to mathematical success; to derive pedagogical principles for effective mathematics instruction for middle school students; to describe effective instructional programs that are adaptable to other schools; and to devise new assessment tools to measure growth in higher order thinking, reasoning, and communication as they relate to school mathematics. Included in this report are: (1) an introduction that describes the purpose, the rationale, and the goals of this project; (2) a discussion of the educational considerations and mathematical conceptualizations underlying the proposed methods of assessment for mathematical proficiency; (3) a discussion of construct-irrelevant test variance as a data-gathering consideration for the assessment of mathematical proficiency; (4) a discussion of the development of specifications for the assessment tasks in terms of focus and components; (5) a discussion of the specifications encompassing the scoring rubrics within the assessment procedures; and (6) a list of sample tasks and administrative information. (15 references) (Author/JJK)