The purposes of this study were (1) to expore the differences between the copnitive adilities in normal children and gifted children, and (2) to discuss the possibility of the acceleration in the gifted. There were two experiments in this study: (1) 224 subjects (112 normal and 112 gifted) in grades three to six were selected from three elementary schools in Taipei. Each subject was tested through a self-compiled test with ten questions three items in each. Each question was presented in the order of symbolic item, pictoral item, and then manually operational real materials, in order to prevent learning experiencne bias. (2) 192 gifted children in grades 3 to 6 were selected from three elementary schools in Taipei. They were randomly assigned three groups, each group was presented was presented a kind of material, (enactive, iconic or symbolic representation). The results were as follows: (1) The cognitive abilities of the gifted were superior to those of the normal children. (2) The cognitive developmental ability of enactive representation was found to be the highest one in all grade levels of gifted and normal children, the iconic representation, the next, and the symbolic representation, the lowest. Furthermore, the gifted used more symbolic representation the normal children. (3) It is possible to accelerate the cognitive ability of the gifted children The more enactive representation was used, the higher score gifted went into.