Results of cognitive diagnoses were validated using the rule-space model, and the usefulness of cognitive diagnosis for instruction was demonstrated. Through preliminary studies, 33 stages of knowledge were defined as cognitive diagnoses to be applied to results from a test of knowledge about fractions of seventh and eighth graders (initial test groups of 287 and 191). Results strongly indicate that the rule-space model can effectively diagnose students' knowledge states and can point out ways for remediating their errors quickly with minimum effort. It was also found that the designing of instructional units for remediation can be effectively guided by the rule-space model, because the determination of all possible ideal item-score patterns, given an incidence matrix, is based on a tree structure of cognitive attributes, knowledge states, and items. Fourteen tables and six figures present study findings. Appendix 1 contains the fraction addition problems, and Appendixes 2, 3, and 4 contain supplemental information about the analysis. (Author/SLD)