Help students construct a hub-and-spoke cognitive structure: a key pathway to reduce burden and improve quality in mathematics classroom teaching

The fragmented organization of mathematical knowledge may make it more difficult for students to understand knowledge structures and engage in mathematical reasoning. This study examined a “hub-and-spoke cognitive structure” teaching framework, aimed at promoting students’ formation of a more coherent and structured mathematical knowledge system by optimizing how instructional content is organized. The framework organizes related mathematical content around a generative core concept (the hub) and connects associated knowledge through consistent reasoning pathways (the spokes), thereby integrating knowledge construction and review into a recursive learning process in which prior and new knowledge interact. A two- round action research design was implemented in two parallel Grade 10 classes, using the vector representations of the four centers of a triangle (the centroid, circumcenter, incenter, and orthocenter) as the instructional content. The first round was conducted in Class A ( n = 61). Based on the problems identified through an immediate feedback questionnaire, student interviews, classroom observations, and students’ written mathematical reasoning, the instructional design was refined by strengthening the derivation of the foundational example, comparing alternative proof methods, optimizing inquiry tasks, reallocating instructional time, and providing differentiated scaffolding. The revised design was then implemented in another parallel class, Class B ( n = 60). The results showed that students in Class B gave higher ratings for the clarity of teacher explanations, acceptance of the instructional approach, and their self-perceived understanding and application of the content. Interview data and written reasoning further indicated that some students understood the reasoning pathways underlying the related concepts and attempted to apply them in subsequent tasks. Across the two action research rounds, the hub-and-spoke teaching framework provided a practical approach for teachers to organize related mathematical instructional content around core knowledge and to iteratively refine instruction in response to classroom feedback. The findings also provide exploratory classroom-level evidence of students moving from fragmented knowledge learning toward a more relational understanding of mathematical knowledge.