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All Journal Journal Focus Action of Research Mathematic (Factor M)
Syaiful Hadi
Department of Mathematics Education, Sayyid Ali Rahmatullah State Islamic University of Tulungagung, Tulungagung, East Java, Indonesia
Education Mathematics Other

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Deconstructing cognitive fractures in visual-to-formal proof transitions: A diagnostic audit of proof without words tasks Syaiful Hadi; Ahmad Fauzi; Ardiana Fatma Dewi; Cholis Sa’dijah
Journal Focus Action of Research Mathematic (Factor M) Vol. 9 No. 2 (2026): December 2026 Article in Press
Publisher : Universitas Islam Negeri (UIN) Syekh Wasil Kediri

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30762/f_m.v9i2.8317

Abstract

The transition from visual intuition to formal mathematical rigor presents a persistent epistemological challenge, particularly when students must bridge the silent gaps in proofs without words. This study deconstructs cognitive fractures in the visual-to-formal proof transition among 30 mathematics education students using a Proof Without Words (PWW) task based on the Pythagorean theorem. Using a qualitative exploratory approach and a diagnostic case study design, the study employs a cross-model clinical audit that integrates Heinze and Reiss’s proof competence framework with Boero’s proving phases. This dual framework enables microgenetic identification of failure mechanisms during proof reconstruction. Data were collected through written work, think-aloud protocols, and task-based interviews and analyzed using a comparative diagnostic lens. The findings reveal a procedural paradox: although 80% of students demonstrated high algebraic proficiency in phase 5, success fell to 30% in phase 3 and 23% in phase 4 because of deficits in strategic and methodological knowledge. We identified three cognitive deconstruction typologies: pathological (illusion of transparency), stable (epistemic anchoring), and autonomous (ontological shift). The findings suggest that the wordless nature of PWW tasks can suppress metacognitive monitoring, producing a vertical decoupling between algebraic execution and geometric justification. This study contributes a diagnostic blueprint for developing precision scaffolding to mitigate cognitive fractures in undergraduate proof instruction.
Co-Authors Ahmad Fauzi Ardiana Fatma Dewi Cholis Sa’dijah