<![CDATA[General Background: Understanding the relationship between area and perimeter requires more than formula use; it demands conceptual reasoning and informal proving in elementary mathematics. Specific Background: Action proof with manipulative objects offers a concrete pathway for students to test conjectures, yet many struggle to form accurate mathematical conjectures during such activities. Knowledge Gap: Limited studies describe how counter-examples guide elementary students through stages of revising primitive conjectures into more comprehensive proofs. Aims: This study analyzes stages of students’ action proof when stimulated by counter-examples using manipulatives. Results: Both low- and high-ability students initially formed incorrect primitive conjectures; confronting counter-examples prompted revision, and re-examination led to a more comprehensive conjecture that increased area does not always change perimeter. Novelty: The study details a three-stage action proof process showing how counter-examples function as cognitive triggers for refining conjectures in elementary geometry. Implications: Findings inform instructional strategies for teaching informal proof and conceptual reasoning through manipulatives in elementary classrooms. Highlights: Counter-examples prompted revision of students’ primitive conjectures. Manipulatives supported concrete representation during informal proof stages. Re-examination led to a comprehensive conjecture about area–perimeter relations. Keywords: Counter-Example, Action Proof, Manipulative Objects, Area And Perimeter, Elementary Mathematics]]>
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