This study examined the effects of the learning hierarchy and reinforcement contingency on addition performance in retarded and non-retarded children. Eighty-eight retarded children selected from special classes and 88 non-retarded children selected from regular classes, both in elementary school level in Taipei City, were used in this experiment. Within each of these two IQ groups, the subjects were further divided according to learning hierarchy and reinforcement conditions into four subgroups (i.e., No overleaping-Reinforcement, No overleaping-No reinforcement, Overleaping-Reinforcement, and Overleaping-No reinforcement). 　　A hierarchical sequence of addition skills was developed for this study. The different levels of computational problems were selected from the hierarchical sequence of addition skills for each subject. The experiment was conducted with a small group (2 to 4 children with the same performance level) or with individual student in the subjects' school. Each subject participated in the experiment for three weeks, the first week was used for taking the baseline \and the following two weeks were used for the formal experiment. 　　The major findings of the study were as follows: 1) The retarded children showed a more inferior computational performance in terms of gain score, percent correct, correct rate and general error rate than did non-retarded children. 2) When the addition tasks were presented in a learning hierarchical sequence, the subjects not only easily acquired the new addition skills, but also performed the tasks quickly and accurately. It is worth pointing out that the lower percent correct and a higher error rate in the overleaping condition was more obvious for the retarded children. 3) Although the reinforcement contingency increased the subjects' computational proficiency (in terms of correct rate), it was not wholly effective in the acquisition situation in terms of percent correct.