Recent calls for mathematics instruction reform recommend that a problem-solving inquiry-oriented (PSIO) classroom environment be established. Students, however, may view mathematics as a static, ruled-based discipline that runs counter to the PSIO approach. A study examined the student/teacher interactions as seen from the viewpoint of five honors geometry students holding a rule-based viewpoint of mathematics as they participated in a PSIO approach to learning mathematics for an entire year. A case study methodology was employed to gain insight into students' beliefs, thinking, and behavior. Results are reported according to themes that emerged concerning the students' beliefs about understanding mathematics, learning priorities, learning strategies, and mathematical power. Findings indicated that rule-based learners in the PSIO environment: (1) chose to ignore opportunities to think about relationship and concepts in-depth; (2) used inappropriate cues based on emotion and external factors to decide when understanding is achieved; and (3) remain uninvolved in their learning and became more passive towards the end of the study. Appendices provide definitions, teacher observation checklists, and a student survey. (Contains over 100 references.) (MDH)