Simulation and Optimization of Review Intervals Based on the Mathematical Model of the Ebbinghaus Forgetting Curve
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Radhiatul Husna
Nabila Gusti Rohima
Alwan Ronan
Mursyid Nur Fahmi
Amma Liesvarastranta Haz
Evianita Dewi Fajrianti
Abstract
Humans naturally experience memory decay over time due to the brain’s limited capacity, a phenomenon first systematically quantified by Hermann Ebbinghaus in 1885 through the forgetting curve, which illustrates the exponential decline of retention in the absence of reinforcement. This curve demonstrates that newly acquired information fades rapidly unless reviewed, negatively affecting educational outcomes as students struggle to retain knowledge in the long term. Spaced repetition, involving scheduled review sessions, has emerged as an effective strategy to counteract forgetting; however, optimal review intervals are often determined intuitively rather than derived mathematically. This study aims to model memory retention dynamics using an extended Ebbinghaus forgetting curve formulated through a learning and forgetting differential equation model, estimate parameters from empirical data, and optimize review intervals to enhance long-term retention. Data were collected through questionnaires from 15 students in the 2023 Mathematics Study Program at Andalas University enrolled in Real Analysis I, yielding parameter estimates of learning rate (α = 0.70), forgetting rate (λ = 0.30), initial knowledge level (K(0) = 100%), and maximum knowledge capacity (K_max = 100%). The model was solved analytically, and numerical simulations compared three strategies: no review, random review, and optimal review at an 80% retention threshold. The optimal review time was found to be t = 1.098 days (approximately 26 hours and 32 minutes), corresponding to the point at which retention declines to 80%. Simulations showed that no review leads to near-zero retention over time, random review produces inconsistent improvements, and the optimal review strategy maintains retention above 80% efficiently. Overall, the mathematically derived optimal review strategy significantly outperforms alternative approaches, providing a personalized, evidence-based method to improve learning efficiency and long-term memory stability while demonstrating the value of integrating psychological memory theory with mathematical optimization for practical educational applications.
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Copyright (c) 2025 Radhiatul Husna, Nabila Gusti Rohima, Alwan Ronan, Mursyid Nur Fahmi, Amma Liesvarastranta Haz, Evianita Dewi Fajrianti
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