Mark Applebaum

DOI:10.63311/jksmed.2026.29.1.1 Vol.29(No.1) 1-19, 2026

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.

inquiry-based learning, problem posing, teacher education, computational thinking, Collatz conjecture

Fay Quiroz , Ali Bicer

DOI:10.63311/jksmed.2026.29.1.21 Vol.29(No.1) 21-52, 2026

Mathematical creativity instructional practices guide teachers to enable students to solve problems in the mathematics classroom through what some may consider “non-traditional” instruction, like problem-solving and problem-posing. Teachers with limited pedagogical content knowledge instruct students using mathematical creativity practices, need support from professional development and long-term coaching to convert their instructional practices to more creativity-directed ones. In this study, a mixed-methods explanatory-sequential design was employed. A quantitative group comparison analyzed survey results from K-8 classroom teachers (n=20) reflecting general and discipline-specific mathematical creativity instructional practices, followed by semi-structured teacher interviews and classroom observations. Results indicated teachers who received professional development on mathematical creativity followed by coaching increased their perceptions of mathematical creativity-directed practices. Interviews and classroom observations revealed two domino effects. The first domino effect held teachers accountable for trying new instructional practices, thus increasing their confidence and promoting leadership characteristics within the school. The second domino effect increased classroom discourse that built student confidence to share, leading to ‘in the moment’ instructional changes that addressed student misconceptions. The implication of this study is a need exists supporting teachers through professional development and coaching to implement cognitively challenging tasks and innovative instructional practices into their classrooms.

mathematical creativity, professional development, coaching, classroom instruction