Mathematical Model Tracks How Calculus Knowledge Is Won, Lost, and Regained

Forgetting calculus may be mathematically inevitable, but a new theoretical study argues that how much knowledge survives depends on a measurable balance between forgetting and relearning. Writing in Frontiers in Applied Mathematics and Statistics, Patience Sumalia Asie of the Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development in Ghana has formulated a four-compartment dynamical system that treats conceptual learning in undergraduate calculus as a continuous cycle rather than a one-time event. The model divides a fixed population of learners into four states: students who have not yet acquired a concept, students actively learning it, students who have mastered and retained it, and students whose previously acquired understanding has decayed.

The framework borrows its structure from epidemiology, where compartmental models track how individuals move between states such as susceptible, infected, and recovered. Here, the same state-transition logic is repurposed with educational meanings. Students begin in the susceptible state and enter active learning through instructional exposure at a rate denoted α. From the active learning state, they achieve mastery and move into the retained-knowledge compartment at rate β. Retention, however, is not permanent: without reinforcement, mastered concepts decay at rate γ, moving learners into a forgotten state. Finally, students in the decay state can re-engage with the material and return to active learning at rate δ, closing the loop.

This closed cycle is the model’s central conceptual move. Earlier approaches, the paper argues, tend to examine acquisition and retention separately, or treat retention as a fixed endpoint of instruction. By contrast, the new formulation represents retention as a dynamic state that changes over time, allowing the rates of learning, mastery, forgetting, and relearning to be analyzed simultaneously within a single system of ordinary differential equations. The model is deliberately deterministic and linear, and the author is explicit that it is a theoretical representation of learning dynamics, not an empirically validated prediction of actual student behavior.

The mathematical analysis establishes several properties of the system. Because each compartment represents a population of students, all state variables must remain non-negative, and the paper verifies this positivity for each equation. The total learner population is conserved, so solutions remain on an invariant manifold where the four compartments sum to a fixed value. At equilibrium, the susceptible compartment empties entirely: for any positive learning rate, every student eventually leaves the initial state, and the population settles into a steady distribution among active learning, retained knowledge, and concept decay.

The stability analysis carries a notable correction. Recalculation of the full Jacobian matrix shows one eigenvalue equal to −α, one zero eigenvalue arising from the conservation of the total learner population, and two remaining eigenvalues determined by the quadratic λ² + (β + γ + δ)λ + (βγ + γδ + βδ) = 0. For positive transition rates, both coefficients of this quadratic are positive, so by the second-order Routh–Hurwitz criterion both roots have negative real parts. The equilibrium on the fixed-population manifold is therefore locally asymptotically stable for every positive α, β, γ, and δ. An earlier condition requiring the forgetting rate to be smaller than the relearning rate for stability is not needed and has been removed.

This correction reshapes the model’s educational interpretation. The quantity 𝓛 = δ/γ, the ratio of the relearning rate to the forgetting rate, is not a stability threshold akin to an epidemiological reproduction number. Instead, at equilibrium it equals the ratio of the retained-knowledge compartment to the concept-decay compartment. A value greater than one means retained knowledge outweighs concept decay in the long run; a value below one means the reverse. Mathematical stability concerns whether the system converges to equilibrium at all, while educational quality concerns how that stable equilibrium is distributed among the states — a distinction the author describes as dissolving a previously claimed paradox between mathematical and educational stability.

The equilibrium formulas also yield clear directional sensitivities. Increasing the forgetting rate γ lowers the equilibrium level of retained knowledge and raises the share of learners in concept decay. Increasing the relearning rate δ does the opposite, raising retained knowledge and shrinking the decay compartment. Increasing the mastery rate β reduces the long-run share of learners stuck in active learning and pushes more of the population into the downstream states. These analytical results are illustrated with numerical simulations showing the transient redistribution of learners among the four compartments: the susceptible class declines monotonically, the active learning class rises early and then settles, retained knowledge grows toward a dominant level, and the decay class approaches a smaller but non-zero share.

The simulations, however, come with an important caveat. The parameter values used are illustrative rather than empirically estimated; they were chosen to produce interpretable trajectories while satisfying positivity and conservation. The author states plainly that the simulations demonstrate the mathematical behavior of the proposed structure and should not be read as calibrated predictions for any particular calculus cohort. Empirical estimation of the four transition rates from longitudinal data is reserved for future work, and the model’s assumptions — constant transition rates, a homogeneous learner population, and no external influences — mark it as a deliberately simple baseline.

The paper also positions itself carefully against a body of related applied mathematics literature, including delayed non-linear disease models, stochastic epidemic formulations with non-standard finite-difference schemes, and fuzzy-parameter COVID-19 models. The author emphasizes that the present model is not a new numerical algorithm and makes no claim of computational superiority over those richer frameworks. Its novelty, instead, lies in the educational interpretation of a closed learning–mastery–forgetting–relearning cycle and in the exact derivation of how the transition rates determine the stable equilibrium composition of learners.

Even as a theoretical baseline, the model carries tentative classroom implications. Because reinforcement alters the equilibrium distribution rather than creating or destroying stability, practices such as spaced review, cumulative retrieval, formative assessment, and targeted remediation may plausibly increase effective relearning and reduce prolonged concept decay. The author is careful to frame these as hypotheses for empirical testing rather than causal prescriptions, since the parameters are theoretical. Future extensions could introduce learner heterogeneity, time-varying instructional rates, delay terms for spaced retrieval, stochastic transitions, or fuzzy uncertainty — but, the paper argues, such additions should be justified by educational mechanisms and data rather than mathematical complexity alone. For now, the model offers a transparent mathematical language for a familiar classroom experience: understanding that fades can, under the right balance of forgetting and relearning, be recovered.

Subject of Research: A compartmental dynamical system model of concept retention, forgetting, and relearning in undergraduate calculus

Article Title: A compartmental dynamical system model for concept retention, forgetting, and relearning in calculus education

Article References: Asie, P. S. (2026). A compartmental dynamical system model for concept retention, forgetting, and relearning in calculus education. Frontiers in Applied Mathematics and Statistics, 12, Article 1940045. https://doi.org/10.3389/fams.2026.1940045

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1940045

Keywords: calculus education, dynamical systems, mathematical modeling, concept retention, forgetting, relearning, compartmental models, stability analysis, knowledge decay, mathematics education, compartmental, dynamical