The main purpose of this study was to test J. S. Bruner´s theory of representation and to find out the applicability of his theory of instruction. According to Bruner, any domain of knowledge can be represented in three ways: by a set of actions (enactive representation), by a set of summary images or graphics (iconic representation), and by a set of symbolic or logical propositions (symbolic representation). If information is to be learned effectively, it must be translated into learner´s ways of thinking and must be presented in proper sequences, that is from enactive to iconic and then to symbolic. In this study, a teat named "Test of Cognitive Representation" was constructed according to Bruner´s theory: (1) to compare the cognitive development of elementary school children in different grade levels and between boys and girls; (2) to compare the cognitive development of Junior high school students in 1-3 grade levels and between boys and girls; (3) to compare the cognitive development of gifted, normal, and mentally retarded children, in order to find out the differences of learning characteristics among these three populations; (4) to test the learning efficiency of different teaching strategies; and (5) to test the feasibility of acceleration of cognitive development. Five parts of experimental observations were undertaken. In part Ⅰ, it was found that there was significant difference between boys and girls on the cognitive development, and the enactive representational ability was the highest one in all grade levels of children, the "iconic representation" the next, and the "symbolic representation" the lowest. Besides, the higher grade the child went into, the more cognitive representational ability grew. In part Ⅱ, it was found that there was no significant difference in sex and grades on the congnitive development of Junior high students. In part Ⅲ, it showed that the cognitive development of the gifted was superior to that of the normal and that of the mentally retarded children. Furthermore, the gifted used more symbolic representation than normal and mentally retarded children did. In part Ⅳ, school children were randomly assigned to three groups, each group was taught to learn mathematics concepts by different teaching strategy. The results revealed that the group who was thaught by proper sequence, from enactive to iconic and then to symbolic, was superior to that of group who was taught to learn by symbolic representation directly. In part Ⅴ, the children of fourth grade were accelerated to learn mathematics concepts that were supposed to be difficult for them to learn. The results were compared against the control group of no accelerated learning. Evidence showed that the acceleration of cognitive development was feasible. All above these findings provide positive support for J. S. Bruner´s assumption that "……any subject can be taught effectively in some intellectually honest form to any child at any stage of development."