This study explores how preservice teachers (PSTs) interpret and enact variation theory (VT) in practice when collaboratively planning, enacting, and reflecting on problem-solving instruction in a mathematics education course. Twenty-two first-year PSTs participated in a course where VT served as a theoretical framework for conceptualising learning as the discernment of critical aspects and as a didactic resource guiding instructional design. The empirical material includes audio-recorded collaborative planning and reflection sessions, video-recorded enactments of problem-solving activities, and written and oral reflections. VT was used analytically to examine how theoretical principles were enacted in practice, focusing on how PSTs identify critical aspects, operationalise patterns of variation and invariance, and use representations throughout the instructional cycle. The findings show that PSTs’ work is organised through three interrelated didactic practices, problem interpretation, representation, and justification, which shape how instructional purposes are articulated and translated into instructional design. Certain groups implemented VT systematically by aligning instructional purposes with coherent design. Other groups only partially implemented VT, resulting in discrepancies between their intended purposes and the enacted instruction. The collaboration and reflection during PSTs planning, enactment, and reflection supported negotiation, re-theorisation, and justification of instructional choices, and facilitated the translation of theoretical principles into practical application. This study contributes valuable insights to the body of research focused on theory-informed didactic practice, particularly in the realm of problem-solving instruction. 

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.Keywords Preservice teachers, Arithmetic word problems, Visualisation, Reflective practice, Variation theory, Sustainable problem-solvingPaper type Research article

This study is analytical research that examines a theoretical approach to transformative pedagogy (TP) and transformative learning (TL) in order to conduct a logical deductive construction of the transformative instruction model (TI model). The study aims to enhance the understanding and effectiveness of the interplay between teaching content and learning in the form of requires for the high-quality teaching of students in the cognitive development of mathematics. It also analyzes the conditions for transformative teaching and learning in a task-oriented context. The crucial psychological outcomes focus on the domains of TP and their interpretation within the context of Mezirow’s TL theory, emphasizing an instrumental, dialogic and self-reflective learning discourse, as well as meaning schemes in learning. The methodological approach is based on a systematic deductive analysis. The analytical results as a conceptualization and an analytical construction of a TI model related to transformative teaching and learning algebraic concepts. The TI model is the result of extending theoretical approaches to TP and TL, as well as logical connections to research about the formation of algebraic concepts. Benefits of the TI model is a new knowledge about an effective teaching of algebra, where students’ algebra learning, and knowledge development is crucial.

Preservice elementary teachers often face challenges in solving arithmetic word problems, revealing gaps in conceptual understanding and reliance on procedural methods. This study examines how preservice teachers’ reasoning relates to the emergence of disorienting dilemmas in written solutions to arithmetic word problems. By applying an analytical model that integrates reasoning attributes with the concepts of dilemmas and disorienting dilemmas, the study contributes to a more nuanced understanding of how cognitive and reflective processes are manifested in written mathematical work. Results from this study makes four key contributions: theoretically, by integrating disorienting dilemmas with a mathematics reasoning framework; empirically, by showing how specific reasoning attributes signal disorienting dilemmas; methodologically, by presenting a replicable four-step analysis model; and practically, by offering guidance for designing problems that support reflection and transformative learning in teacher education. 

While much has been written about transformational learning, the preservice teacher’s role in the process is undertheorized. This article develops the concept of the power of dilemmas as a theoretical construct for understanding transformative learning in mathematics teacher education. The framework views dilemmas not as obstacles but as generative mechanisms that stimulate reflection, variation, and theoretical integration. It explains how preservice teachers’ engagement with dilemmas—emerging from tensions between their practical experiences, didactical approaches, and theoretical perspectives—supports their progression across different levels of professional reasoning and understanding. Dilemmas generate variation by exposing differences in attitudes and interpretations. This variation encourages reflection that links their practical experiences with theoretical ideas. In this process, dilemmas serve as catalysts that connect theory and practice, supporting ongoing and meaningful professional development. The article also discusses how teacher education environments can be designed to capitalize on productive tensions and foster reflective dialogue. By using dilemmas as opportunities for noticing critical differences and rethinking established ideas, the framework offers a way to understand how transformative learning can develop in mathematics teacher education and how preservice teachers can build a more reflective, adaptable, and theoretically informed professional approach.

This study investigates how preservice teachers use pictures in their written solutions to arithmetic word problems and how these visualisations reflect their reasoning processes. A total of 144 written solutions from 36 preservice teachers were analysed. The findings reveal notable differences in the types of pictures used and the reasoning processes employed by the preservice teachers, depending on whether the problems involved fractions. In fraction-related problems, preservice teachers predominantly used representational, informational, and organisational pictures, which is lacking in non-fraction problems. Reasoning processes such as selecting and exploring were common across both problem types. However, attributes such as reconfiguring, encoding, and abstracting appeared more frequently in solutions to nonfraction problems. This result suggests that, although preservice teachers more often employ visual tools when solving fraction tasks, they do not fully exploit these tools to support deeper conceptual understanding or the problem-solving process. The study highlights the need to develop preservice teachers’ competencies in using visualisations to effectively represent the problem-solving process by reasoning about the mathematical concepts involved in problems. These findings provide concrete suggestions aimed at strengthening preservice teachers’ skills in using visualisations and reasoning, which equip them to better support students’ mathematical reasoning in their future classrooms. The study also contributes to a deeper understanding of how visual tools and reasoning interact in mathematics education, thereby addressing existing gaps in the research.

This study examines two students’ reasoning processes when solving a Calculus problem, focusingon their discourse characteristics and factors influencing narrative endorsement. Using videorecordings of paired discussions, the results show rituals, social acceptance, ambiguity,inconsistency, and the impact of mathematical representations as characteristics of the narratives.The results also indicate that rituals and ambiguities are key factors in determining whether thenarrative becomes endorsed. Additionally, ambiguity provides opportunities for the two students tothink critically and creatively, which can support their reasoning process.