The purpose of the research is to explore second graders' concept of number development and quantitative reasoning. For this purpose, there were two stages of trials for the children. The first trial was concrete objects. After three months, the children participated in the second trial of half concrete objects. Since understanding the process of solving problems in children is necessary, the researcher observed how children used strategies to discriminate numerosity. After that, the researcher interviewed some students. The research sample came from second graders in Taiwan and Hawaii elementary schools. Twenty students participated in each place. In addition, it was observed whether numerical discrimination strategy in children is the same as in adults. The sample included 20 adults. The result of the research showed that Taiwanese and Hawaiian children reached up to a 90% rate of correct answers in the trial of real objects. However, in the trial of half concrete objects, the rate of correct answers was down to 65%, especially when children compared two quantities of 5:6 ratios. As for strategy, the strategies of the children who gave correct answers were the same as the adults who gave completely correct answers. Because their discrimination was not influenced by distance effect, they could judge the numbers correctly. However, the children who got 90% correct answers were impacted by distance effect, but not influenced by the size of the object. The research suggests that teachers give children real objects to estimate or discriminate the quantities of numbers in their lives. (Contains 2 tables.)