Cryptarithmetic as an Epistemic Task: Revealing Elementary Students’ Numerical Reasoning Under Constraint-Based Problem Solving

Abstract

Elementary mathematics learning rarely provides opportunities to observe how students reason numerically when solutions cannot be obtained through routine procedures. This study conceptualizes cryptarithmetic as an epistemic task that reveals elementary students’ numerical reasoning under constraint-based problem-solving conditions. Rather than evaluating correctness, the study focuses on how students coordinate numerical constraints, justify decisions, and regulate reasoning in the presence of interacting constraints. Using a mixed-method sequential explanatory design, the study involved 27 fifth-grade students and 4 mathematics teachers from an Indonesian elementary school. Students engaged with a cryptarithmetic task set designed to externalize numerical reasoning, followed by interviews with students and teachers. Quantitative performance patterns were interpreted as epistemic signals, while qualitative data illuminated reasoning processes, epistemic breakdowns, and mediation across perspectives. The findings reveal substantial variation in students’ numerical reasoning orientations, indicating identifiable epistemic thresholds as constraint interaction increases. A key result shows that three-digit addition produced greater epistemic difficulty than two-digit multiplication, demonstrating that constraint density, rather than operation type, governs reasoning complexity. Teachers interpreted cryptarithmetic as a diagnostic task for reasoning, whereas students experienced it as an epistemically demanding challenge, revealing both convergence and divergence across perspectives. These results demonstrate that cryptarithmetic functions as a powerful epistemic task that makes numerical reasoning visible through patterns of coherence, breakdown, and regulation, offering a theoretically grounded lens for studying constraint-based reasoning in elementary mathematics.