The purpose of this paper is to clarity a method for facilitating the appreciation of aesthetic qualities of mathematical objects with learners. For this purpose, the author conducted a case study about high school students’ mathematical problem–solving process based on a theoretical framework derived from the theory of aesthetics.
As a result, the effectiveness of the method was confirmed in both single-case studies. In other words, the validity of the theoretical hypothesis was supported by both single-case studies.
On the other hand, through detailed description and analysis of the cases, specific appreciation processes not mentioned in the theoretical hypotheses were described, and the following empirical compensations for theoretical hypotheses were made: In cases where the “equivalent relation” also has a role as the “essence”, intuition as the “essence” can be made through trial and error or can be hampered by objects of the “equivalent relation”. Also, in the same case, intuition of the “essence”, intuition of the “whole”, and feeling the “vastness” are phases that learners may pass back and forth complementarily. In response to these empirical compensations, the author improved the method for facilitating appreciation of aesthetic qualities of mathematical objects with learners.