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. In this study, a test named "Test of Cognitive Representation" was constructed according to Bruner´s theory (1) to compare the cognitive development of Junior High school students in different grades, sexes and ability groups, (2) to explore the relationships between the "Test of Cognitive Representation" and the Piagetian type of cognitive development test, (3) to test the learning efficiency of different teaching strategies; and (4) to test the feasibility of acceleration of cognitive development. The main findings were as follows: (1) There were signifcant differences among grades, sexes and ability groups in abilities of symbolic representation and iconic representation. (2) There were significant correlation between the "Test of Cognitive Representation" and the Piagetian type of cognitive development test. The Test of Cognitive Repersentation was proved to be a valid cognitive developmental instrument. (3) There were significant differences among different teaching strategy groups. The group who was taught by proper sequence, from enactive to iconic and then to symbolic, was superior to those who were taught to learn by symbolic representation or by enactive representation only. (4) 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 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."