This study demonstrates that the validity and usefulness of mathematics achievement tests can be improved by defining psychologically meaningful subscores that yield differential relations with student, teacher, and school variables. The National Education Longitudinal Study of 1988 (NELS:88) 8th- and 10th-grade math tests were subjected to full information item factor analysis. Math knowledge and math reasoning factors were distinguished at both grade levels. Regression analyses showed that student attitudes, instructional variables, course, and program experiences related more to knowledge, whereas gender, socioeconomic status, and some ethnic differences related more to reasoning. Teacher emphasis on higher-order thinking, student use of home computers, and early experience with advanced mathematics courses related to both dimensions. It is recommended that national educational surveys use multidimensional achievement scores, not total scores alone. One figure and eight tables illustrate the analysis. (Contains 35 references.) (Author/SLD)