Organisation and Integration of Procedural and Conceptual Knowledge in Geometric Reasoning across the van Hiele Levels

Abstract

This study examines how procedural and conceptual demands are organised across the five van Hiele levels of geometric  reasoning  and  develops  a  theoretical  account  of  their  relationship.  Using  qualitative  document  analysis  and directed content analysis, 25 representative descriptors of Visualization, Analysis, Abstraction, Deduction, and Rigor were synthesized  from  established  accounts  of  the  van  Hiele  framework  and  analysed  through  four  dimensions:  the  role  of procedural or operational activity, conceptual demand, the form of procedural-conceptual coordination, and the epistemic basis of reasoning. The analysis indicates a developmental transformation rather than a simple replacement of procedures by  concepts.  At  Visualization,  reasoning  is  mainly  perceptual-operational.  At  Analysis,  procedural  activity  supports attention  to  properties.  At  Abstraction,  actions,  representations,  and  informal  arguments  are  explicitly  coordinated  with relational  reasoning.  At  Deduction  and  Rigor,  procedural  activity remains  present  but  becomes  increasingly  embedded within  formal  conceptual  and  axiomatic  structures.  These  patterns  are  synthesized  as  the  Procedural-Conceptual Integration  Trajectory  (PCIT),  which  describes  progression  from  perceptual-operational  engagement  through  property-based and relational coordination to deductive and axiomatic regulation. The PCIT provides a conceptual bridge between van  Hiele  theory  and  research  on  conceptual-procedural  knowledge  and  offers  a  basis  for  designing  geometry  tasks  thatprogressively connect action, representation, explanation, justification, and proof.