This paper introduces a new measure of educational inequalities based on cognitive achievement data, and uses it to examine achievement inequalities in mathematics between groups of students enrolled in basic education in Brazil. The groups of students are defined by their race, sex, socioeconomic status (SES), and region of residence. The Brazilian system of basic education currently produces poor results with respect to both quality and equity. The paper recommends that Brazil should work, concomitantly, to improve the achievement levels of its students and to close the cognitive gaps observed among different groups of students. Placing emphasis on just one of these goals is not an adequate public policy at this time. Proposals for carrying out these reforms can be classified as input or management strategies and are briefly discussed.