Learning to do mathematics as inquiry: A pre-service teacher’s journey with the Collatz conjecture variant

Preparing mathematics teachers for genuine inquiry requires opportunities to pose, scope, and justify their own questions within carefully designed learning environments.This design-based case study examines how one preservice teacher, enrolled in a semester-long Mathematics Investigation course, developed an inquiry project around a principled variation of the Collatz conjecture.Rather than attempting the open problem itself, she constructed a bounded-yet-open inquiry by modifying the rule from 3n+1 to 3n-1 and combining lightweight computation with iterative conjecturing.Data sources included proposals, investigation logs, mentoring notes, presentations, a final report, and a reflective memo.Findings trace a shift from "exercise-solver" activity toward investigator-like practices, as the preservice teacher moved from empirical checking to constructing warrants that could travel beyond individual cases.Across episodes, she expanded her problem-posing repertoire from isolated modifications to family-level conjectures, demonstrating increased creativity in fluency, flexibility, and organization.Instructional design features, principled task variation, computational tools positioned as epistemic supports, mentoring that pressed for form rather than direction, and structured reflection, sustained cognitive demand while preserving learner agency and consolidating an inquiry-as-stance identity.The mathematical outcomes of the 3n-1 system (e.g., cycles and attractors) functioned pedagogically as datasets for reasoning rather than as contributions to number theory.The study contributes design principles for inquiry-rich teacher education, suggesting how compact, intuition-violating problems and coherent instructional scaffolds can cultivate dispositions of curiosity, justification, and agency that preservice teachers can transfer into their future classrooms.