The Visualisation–Construction–Reasoning (VCR) Instructional Model for School Geometry: A Design Grammar for Moving from Seeing to Justifying
Keywords:
School Geometry, Geometric Visualisation, Geometric Construction, Mathematical Reasoning, Visualisation–Construction–Reasoning Model, VCR Instructional Model, Geometry Learning, Geometry Education, Deductive Reasoning, Geometric Thinking, Instructional Design, Mathematics Pedagogy, Inquiry-Based Learning, Mathematical Cognition, Spatial ReasoningAbstract
Geometry instruction often struggles to connect three essential forms of mathematical work: disciplined noticing of structure, constraint-based production of geometric objects, and warrant-based justification of claims. As a result, students may rely on diagram appearance rather than invariant properties, treat construction as routine drawing, and experience reasoning as a disconnected formal demand. This paper introduces the Visualisation–Construction–Reasoning (VCR) Instructional Model as a geometry-specific instructional design grammar intended to address that coordination problem. The paper positions VCR in relation to adjacent traditions in mathematics education, including representational perspectives, visualisation research, construction-oriented approaches, and scholarship on proof and justification. It then defines the model conceptually and operationally, articulates its programme theory, and specifies its design principles, task ecology, sequencing rules, predictable error patterns, and teacher role shifts. The paper further describes the implementation architecture of the model through an 8-week, 35-lesson, bilingual, multimodal lesson-plan system and illustrates its classroom functioning through three flagship exemplars: altitude in a triangle, angle relations in parallel lines cut by a transversal, and area derivation through decomposition. The article also frames VCR as a replicable instructional package supported by structured lesson-plan validation evidence from expert and school-teacher panels. The contribution of the paper lies in presenting VCR not as a generic teaching strategy, but as a coherent, operationally specified instructional framework that links visualisation, construction, and reasoning into a pathway from perceptual engagement to mathematical warrant. The paper concludes by discussing the conceptual, methodological, and practical significance of the model, together with its limitations and the research agenda it enables.
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