Visualisation as a didactical and reflective tool in preservice teachers' work with arithmetic word problems

Purpose As visualisation becomes increasingly important in mathematics education, preservice teachers (PSTs) must develop awareness of the didactical decisions involved in formulating and solving arithmetic word problems (AWPs). This study examines how their use of external visualisations in planning and teaching AWPs supports sustainable problem-solving strategies and shapes their reflective practice. Design/methodology/approach Integrating variation theory and reflection theory, the study conceptualises critical aspects and patterns of variation as tools for making problem structures visible. Twenty-two first-year PSTs worked collaboratively in six groups within simulated teaching scenarios. Data from lesson plans, recorded discussions and enactments, and reflective reports were analysed using a combined deductive–inductive approach. Findings The results show that PSTs used diverse visualisations, such as bar models, ratio schemas, pie charts, and drawings, to make problem structures and relationships visible. These supported reflection-on-, in-, and for-action across planning, enactment, and future-oriented evaluation. The use of visualisations became productive sites for collaborative reflection and informed instructional decisions. Sustainability in problem-solving did not emerge automatically but depended on structured opportunities for critique, comparison, and iterative refinement. Research limitations/implications Small sample and simulated contexts limit generalisability but indicate visualisations support PSTs' reasoning. Practical implications The findings have practical implications for mathematics teacher education, showing how visualisations can support conceptual understanding and teaching skills. The study concludes that using visualisations as a reflective tool links theory and practice, and helps PSTs to develop sustainable problem-solving strategies. Social implications Promotes equitable, concept-focused mathematics teaching. Originality/value This study demonstrates the dual role of visualisations as instructional tools and reflective resources in mathematics teacher education. It introduces a functional-analytic, theory-informed framework that links visualisation, reflection, and the development of sustainable problem-solving strategies.