One essential role of social robots is supporting human mental health by interaction with people. In this study, we focused on making people's moods more positive through conversations about their problems as our first step to achieving a robot that cares about mental health. We employed the column method, typical stress coping technique in Japan, and designed conversational contents for a robot. We implemented conversational functions based on the column method for a social robot as well as a self-schema estimation function using conversational data, and proposed conversational strategies to support awareness of their self-schemas and automatic thoughts, which are related to mental health support. We experimentally evaluated our system's effectiveness and found that participants who used it with our proposed conversational strategies made more self-disclosures and experienced less anger than those who did not use our proposed conversational strategies. Unfortunately, the strategies did not significantly increase the performance of the self-schema estimation function.