There are some discussions about the relationship between creativity and the aesthetic sensibility. It has been a characteristic of mathematician and mathematical gifted that the aesthetic sensibility has a generative characteristic. On the other hand, some studies demonstrated that the generative characteristic also was available even by the non-mathematician or non-mathematical gifted. The purpose of this paper is to clarify the process to produce qualitative differences in the generative characteristic of the aesthetic sensibility through the analysis of learners' problem solving. For this purpose, two pairs of high school students and one pair of college students were observed during problem solving. As a result, it was clarified that it is critical whether learners have their own goal other than the given one to evoke the generative characteristic. Moreover, it was suggested that the difference of one's own goal is associated with qualitative difference in the generative characteristic.