Developing students’ capacity to reason mathematically is a key aim of mathematics teaching. However, how such reasoning is supported remains complex. This study examines teacher-student interaction and students’ opportunities for mathematical reasoning in a Thinking Classroom context. While Thinking Classroom is designed to increase student engagement and mathematical thinking through substantial changes to classroom organisation, relatively little is known about how teacher-student interaction unfolds within this instructional context. The study draws on 34 video recordings from five upper secondary mathematics classrooms implementing the first three practices of the teaching approach Thinking Classroom, which concern task selection, random group formation, and work at vertical non-permanent surfaces. A thematic analysis of 124 teacher-student interactions identified four recurring patterns of interaction. Drawing on selected concepts from a complex dynamic systems perspective, these patterns are interpreted as relatively stable. The analysis further examines the students' opportunities for reasoning within these patterns. Opportunities for mathematical reasoning differed across the four patterns and often remained brief, even in interactions where reasoning was most visible. The findings suggest that, in the observed classrooms implementing the first three practices of Thinking Classroom, opportunities for sustained reasoning within teacher-student interaction were limited. The present study highlights the challenges involved in supporting mathematical reasoning and the importance of teachers’ interactional choices in creating and sustaining opportunities for reasoning.
This study explored sixth-grade children’s adaptability of strategy use in fraction comparison, focusing on the effects of problem characteristics (e.g. specific denominator-numerator relationships) and cognitive demand (e.g. required cognitive resources). In Experiment 1, participants solved eleven types of fraction comparison problems, nine of which were designed to align with nine specific strategies. Experiment 2 investigated cognitive demand through two manipulations: (a) comparing problems with two-digit fractions versus single-digit fractions, and (b) providing half of the participants with scratch paper to reduce cognitive load. Results of Experiment 1 showed that sixth-grade children could use various strategies and select contextually appropriate strategies based on problem characteristics. Experiment 2 demonstrated that higher cognitive demand (e.g. two-digit fractions) reduced the use of Intermediate Step strategies, likely due to their greater resource intensity. However, scratch paper provision did not significantly alter strategy choice. These findings suggest that sixth-grade children adaptively use different strategies in fraction comparison and adjust their strategy choice by considering problem characteristics and cognitive demand. This suggests teachers to adjust their teaching methods on children’s fraction learning and problem solving by placing emphasis on strategy use.
This study investigates how Korean immigrant parents draw on translanguaging practices to mediate mathematical meaning with their elementary-aged children during home mathematics interaction. Using video-based, ethnographically informed observations of six sessions across three parent-child dyads, the analysis examines how parents coordinated Korean and English alongside gesture, diagrams, and tools to make mathematical relationships more visible, interpretable, and open to negotiation in real time. In this study, translanguaging is conceptualized as the epistemic mechanism through which mathematical meaning is mediated. The findings suggest that parents used translanguaging to diagnose procedural and structural mismatches, make unfamiliar terminology more accessible, bridge semantic and structural relationships across concepts, reorganize algebraic representations, and coordinate geometric action through multimodal guidance. The most salient form of this mechanism appeared in diagnostic interactions, where bilingual contrast and multimodal coordination made otherwise implicit mismatches visible and available for repair. The study contributes to translanguaging research in mathematics by respecifying translanguaging not only as a resource for participation but as an epistemic mechanism through which mathematical structure is made visible and available for reorganization. The study also suggests that Korean-English interaction provides an analytically productive context for examining how typologically associated linguistic resources become interactionally available in mathematical activity.
Engagement has been shown to influence students’ understanding and achievement, yet little is known about how engagement with statements relates to students’ understanding of corresponding proofs. In this paper, we take a step towards addressing this gap by investigating whether engaging with statements is associated with better understanding of proofs by contradiction and contraposition. Eighty-seven Indonesian undergraduate mathematics education students, organized in four groups, participated in the study. Two groups (n = 43) were randomly selected to first engage with the statements—reflecting on their meaning, judging their truth, and attempting to prove or disprove them—before responding to an instrument designed to measure their understanding of given proofs for the same statements; the other two groups (n = 44) proceeded directly to the instrument. There were six statements overall, three proved by contradiction and three by contraposition. We found that students who engaged with the statements achieved significantly higher mean scores on measures of proof understanding than those who did not, with a medium effect size. Regression analyses confirmed engagement as a significant predictor of students’ understanding for both proof methods, and individual semi-structured interviews with 12 participants provided qualitative support for these findings. Overall, our findings identify engagement with statements as a meaningful factor in students’ proof understanding and suggests that selectively incorporating an engagement stage before introducing key proofs may enhance undergraduate mathematics instruction beyond the typical Definition-Theorem-Proof (DTP) approach.
