The study involving 60 learning disabled (LD) and 30 normal achieving seventh through ninth graders was designed to identify adolescents homogeneously defined as exhibiting a "specific learning disability in arithmetic" and to determine if the cognitive processes (visual-spatial, visual-reasoning, and visual-memory) are related to the academic task failure exhibited by this population. The hypothesis was tested that there is no difference among three groups--specifically LD in arithmetic (SLD-ARITH), SLD in reading (SLD-READ), and average achievers (AVE-ACH). Ss were administered six cognitive tests including four subtests from the Woodcock-Johnson Psycho-Educational Battery, the Revised Test of Visual Retention, and Benton Copying and Benton Memory. Results indicated that there is a relationship between two of the major components in the LD definition, academic task failure, and specific cognitive abilities and that there is validity to the above two components of the LD definition when a very specific population of students disabled in arithmetic has been identified. Among findings were that the SLD-ARITH and SLD-READ groups were achieving high except in the designated area while the AVE-ACH group was achieving high in all areas; and that the SLD-ARITH Ss performed lowest on the two visual-spatial tasks and one reasoning task and lowest on concept formation. Tables with statistical data are included. (Author/SB)