Repealing the power law: the case for an exponential law of practice

Heathcote, Andrew; Brown, Scott; Mewhort, D. J. K. · 2000 · NOVA (University of Newcastle Australia)

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Summary

This paper challenges the widely accepted "Power Law of Practice," which posits that response time decreases as a power function of practice trials. The authors argue that the empirical support for the Power Law is flawed because it relies primarily on data averaged across subjects or conditions, rather than individual learning series. They contend that averaging introduces a systematic bias favoring the power function, whereas individual data often better fit an exponential function. This distinction is theoretically significant: an exponential function implies a constant relative learning rate, while a power function implies a slowing learning rate, affecting how skill acquisition models are constructed. To resolve this debate, the authors conducted a comprehensive survey analyzing 40 data sets comprising 7,910 learning series from 475 subjects across 24 experiments. These experiments covered various paradigms, including memory search, counting, mental arithmetic, visual search, and key sequence production. The study compared the fit of power, exponential, and a newly proposed "APEX" function (which nests both power and exponential forms) to unaveraged, individual-level data. The authors utilized ordinary least-squares minimization and addressed technical issues in previous research, such as the unfair comparison between three-parameter exponential and four-parameter general power functions, and biases introduced by logarithmic fitting methods. The results demonstrated that the exponential function provided a better fit than the power function in all analyzed unaveraged data sets. The APEX function consistently outperformed the power function, even when the latter included an extra parameter for pre-experimental practice. The authors found that averaging data artificially created the appearance of a power law, confirming that the preference for the power function in prior literature was likely an artifact of data aggregation. Furthermore, the superior fit of the APEX function suggested that while relative learning rates may decrease early in practice due to component transitions, they approach a non-zero asymptote, supporting an exponential underlying process. The significance of these findings lies in the potential to "repeal" the Power Law of Practice. The authors conclude that the exponential or APEX function is the more accurate description of skill acquisition. This shift has profound implications for cognitive theories of learning, particularly those built to account for power-law dynamics, such as ACT, instance theory, and chunking models. The study urges researchers to analyze individual learning curves rather than averaged data to avoid misleading conclusions about the mechanisms of practice effects.

Key finding

The exponential function fits individual learning data better than the power function, and averaging data creates a systematic bias that incorrectly favors the power function.

Methodology

survey

Sample size: 475

Provenance

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Topics

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• automaticity skill acquisition