LIMIAR
A mathematical model built around a deceptively simple question: when is the best moment to stop a study session? I borrow the Marginal Value Theorem from behavioral ecology — the theory that predicts when a foraging animal should leave a food patch — and treat studying like foraging. You keep gaining correct answers, but at a slowing rate as fatigue sets in. The decisive move is the denominator: the model maximizes correct answers per total cycle time, study plus rest, which is what gives the problem a genuine interior optimum instead of a trivial one.
The rule the model turns on
Treat a study session like foraging in a patch. Let S(t) be the correct answers accumulated after studying for a time t, and let x be the rest planned after it. Instead of maximizing raw output, the model maximizes output per total cycle time, S(t) / (t + x). Taking the derivative and setting it to zero gives the stopping rule:
At the optimum, the instantaneous rate of correct answers equals the average rate over the whole cycle:
S′(t*) = S(t*) / (t* + x).
Geometrically, that is the steepest line drawn from the point (−x, 0) up to a tangent with the accumulation curve — which is exactly the 1976 diagram Eric Charnov drew for the Marginal Value Theorem, with the planned rest x playing the role of travel time between patches.
What I am testing
- Predicts a falsifiable rule: the longer the planned rest, the longer the optimal session should be — independent of any particular numbers.
- Its geometry reproduces the classic 1976 Charnov diagram, with rest taking the role of travel time between patches.
- Open question I am working on: does the optimal threshold keep shrinking cycle after cycle, or settle into a steady state?
The instrument
A model of when to stop studying is worthless without data on how correct answers actually accumulate over time. So I built the tool that collects it: you load a list of exercises, set your study time, rest time and number of cycles, and answer questions one at a time while it records every timestamp and response time. At the end it produces a spreadsheet and fits the accumulation curve — giving back the parameters that feed the model.
This is live research in progress, not a finished result. The model, its assumptions and its open questions are still moving — which is the honest state of any real project at this stage.