Modeling Adaptive Success: A Discrete Hill-Type Hazard Approach in Education

Traditional models of student performance and item mastery often assume a constant probability of success across attempts-for example, the geometric distribution. Such models cannot capture the “aha” moment where performance transitions from repeated failure to rapid success. In this work, we adapt a discrete Hill-type hazard (equivalently, a log-logistic hazard function) to the problem of first success over repeated attempts in educational settings. We use this trial-indexed success probability, τ(n;h,K) = nh/(nh + Kh), as the discrete hazard of a nonhomogeneous geometric process governing the trial of first success. The resulting model-which we refer to as the Learner’s Tau framework-is governed by a steepness parameter h and a midpoint parameter K and is designed to capture adaptive success in systems where the probability of success increases over time, such as human learning, reinforcement learning, and task mastery. Wederive the probability mass function, analyze key mathematical properties, and introduce a set of interpretable summary quantities-including the discrete hazard function, a Difficulty Ratio K/h, an Early Success Probability (the CDF evaluated at the midpoint), the Mean Time to Mastery, and the Peak Effort Trial where effort-weighted success is maximized. Using numerical simulations and maximum likelihood estimation, we demonstrate that the model can recover underlying parameters and distinguish between qualitatively different learner profiles. Empirical validation using educational data from the Cognitive Tutor system demonstrates that the model captures learning dynamics significantly better than a memoryless geometric baseline for non-trivial tasks (∆AIC = 8.0 and 16.6 for two algebra skills), while appropriately deferring to the simpler model when ceiling effects preclude observable learning. Rather than proposing a brand-new functional form, this work adapts a classical Hill-type hazard to an explicitly discrete, learning-centric context and highlights its utility for modeling adaptive success trajectories in education.