Deconstructing cognitive fractures in visual-to-formal proof transitions: A diagnostic audit of proof without words tasks

Authors

  • Syaiful Hadi Department of Mathematics Education, Sayyid Ali Rahmatullah State Islamic University of Tulungagung, Tulungagung, East Java, Indonesia https://orcid.org/0000-0002-5094-0098
  • Ahmad Fauzi Department of Mathematics Education, Sayyid Ali Rahmatullah State Islamic University of Tulungagung, Tulungagung, East Java, Indonesia
  • Ardiana Fatma Dewi Department of Mathematics Education, Sayyid Ali Rahmatullah State Islamic University of Tulungagung, Tulungagung, East Java, Indonesia
  • Cholis Sa’dijah Department of Mathematics Education, State University of Malang, Malang, East Java, Indonesia

DOI:

https://doi.org/10.30762/f_m.v9i2.8317

Keywords:

Cognitive Fractures, Proof Without Words, Proof Competence, Proving Phases, Diagnostic Audit

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.

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Published

15-09-2026

How to Cite

Hadi, S., Fauzi, A., Dewi, A. F., & Sa’dijah, C. (2026). Deconstructing cognitive fractures in visual-to-formal proof transitions: A diagnostic audit of proof without words tasks. Journal Focus Action of Research Mathematic (Factor M), 9(2), 283–302. https://doi.org/10.30762/f_m.v9i2.8317

Issue

Vol. 9 No. 2 (2026): December 2026 Article in Press

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Articles

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Copyright (c) 2026 Syaiful Hadi, Ahmad Fauzi, Ardiana Fatma Dewi, Cholis Sa’dijah

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