On the form of learning curves.

The authors looked at hundreds of learning graphs from animal labs. They asked one simple question: what shape fits the curve when responses get faster with practice?

They tried straight lines, exponentials, and power functions. A power function with an exponent between 1 and 2 won every time.

Learning curves are not straight. They bend. The bend follows a power law: improvement slows as skill rises.

The same bend shows up whether the task is lever pressing, maze running, or key pecking.

DeVellis et al. (1979) moved the power law from learning to timing. They showed that post-reinforcement pauses under fixed-interval schedules follow the same bend. The curve is not just about getting better; it also describes how long animals wait.

McAuliffe et al. (2020) and László et al. (2013) extend the curve to kids with autism. Both teams found the power law still applies, but the exponent is larger, meaning learning is slower. The shape is the same; the speed is not.

White (1995), Malone (1999), and Flory et al. (1974) show the power law pops up everywhere: stimulus control decay, concurrent choice, and DRL timing. The 1962 paper gave the first clear rule; later work keeps showing the rule holds across new parts of behavior.

When you graph client data, expect a bend, not a straight line. If the curve is flat, check the exponent; a big exponent means the skill is still hard for the learner. Use the bend to set realistic mastery criteria and to explain progress to parents: improvement starts fast, then slows—this is normal, not failure.