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This report documents the process undertaken to produce scales in three cognitive domains: knowing, applying, and reasoning. Included are the final scales showing differences among countries, as well as within countries. TIMSS 2003 is the third and most recently completed round of IEA's Trends in International Mathematics and Science Study, a very ambitious series of international assessments carried out in countries around the world to measure trends in mathematics and science learning at the fourth and eighth grades. Forty-nine countries participated in TIMSS 2003, with 23 having participated in all three assessments and another 14 having participated in two rounds. For mathematics, the five content domains were number, algebra (called patterns and relationships at fourth grade), measurement, geometry, and data. Each content domain described the topic areas to be assessed within that domain, and each topic area was elaborated with objectives specific to the eighth and fourth grades. Four cognitive domains were described--Knowing Facts and Procedures, Using Concepts, Solving Routine Problems, and Reasoning--together with the skills and abilities making up each domain. This report contains four chapters: (1) The Developmental Project to Report TIMSS 2003 Mathematics Achievement in Cognitive Domains; (2) Mathematics Achievement in the Cognitive Domains at the Fourth and Eighth Grades; (3) Achievement by Gender in the Mathematics Cognitive Domains at the Fourth and Eighth Grades; and (4) Country by Country Profiles of Achievement in the Mathematics Cognitive Domains. Appended are: (1) Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades; (2) Overview of Procedures TIMSS 2003 Developmental Project; (3) Coverage of TIMSS 2003 Target Populations and Participation Rates; and (4) Percentiles and Standard Deviations of Mathematics Achievement in the Cognitive Domains. (Contains 23 exhibits.) [This document was also published by