Do people learn skills smoothly or in stages?
Averaged practice data follow the classic smooth power law, but individual learners' curves are better described as a series of separate power-law stretches, where a switch often brings a brief dip followed by a higher plateau.
Source
Piecewise power laws in individual learning curves
Study at a glance
- Design
- Computational / modelling — Fitting of single versus piecewise power-law (and exponential) models to the first 500 plays of each user on four Lumosity training games, with model selection validated on simulated curves
- N
- N=25280 · 25,280 individual learning curves from 22,460 unique users who completed at least 500 plays of a task; demographic analyses used the subset who reported age, gender, education and country
- Population
- Adult users of the Lumosity online cognitive-training program, mostly female and mostly from the USA
- Outcome
- Variance in individual learning curves explained by single versus piecewise power-law models, the number and timing of transitions between pieces, and performance changes around transitions
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What they did
The authors took scores from four online training games (speed matching, memory updating, a flanker attention game and a word-stem fluency game) for users who had played a game at least 500 times. For each person's first 500 plays they fitted either one smooth power-law curve or a 'piecewise' curve made of several power-law segments joined at transition points, using a conservative model-selection method that they first tested on simulated data. They then examined what performance looked like around transitions and how the number of transitions related to age, gender, education and improvement.
What they found
The average learning curve for each game was fitted almost perfectly by a single power law, but for individuals the piecewise model fitted significantly better, explaining 90.74% of the learnable variance versus 86% for a single power law. Most curves had two or three pieces. At a transition, performance usually dropped briefly (78% of transitions) but the new segment overtook the old one within 50 plays in over 80% of cases, and later segments had higher ceilings. Older users showed fewer transitions, and the number of transitions was linked more to how much a person improved than to how good they were.
The limits
What it doesn't show
The data come from an uncontrolled commercial platform, so the authors cannot know what users were doing between plays, whether one account was shared by several people, or why transitions happened; strategy shifts are inferred, not observed. The piecewise model has many more parameters, and although the authors used penalties and a strict extra test, some transitions could still be overfitted noise or forgetting after breaks. The users are a self-selected, mostly older, female and US-based group of heavy players, so the findings may not describe learning in other people or in real-world skills. The study describes the shape of learning but does not test what causes transitions.
Key terms
- Power law of practice
- The classic finding that performance improves quickly at first and then more and more slowly with practice, following a power function.
- Piecewise power law
- A learning curve made up of several power-law segments joined at transition points, each segment possibly reflecting a different strategy.
- Averaging artifact
- A pattern that appears in group-averaged data but not in individuals, for example a smooth curve produced by averaging many curves with jumps at different times.
- Model selection (AIC)
- Methods that compare models by fit while penalising extra parameters, to avoid choosing an over-complex model that fits noise.
- Overfitting
- When a model is so flexible that it captures random noise in the data rather than the true underlying pattern.
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Quiz yourself
What pattern did averaged learning curves show?
Common questions
If the average curve is a perfect power law, why not just use that?
Because averaging many individual curves that each jump at different times can produce a smooth curve that no single person actually follows. To understand how individuals learn, you need to model individual curves.
Why would performance drop at a transition?
The authors suggest transitions often mark a switch to a new strategy, which is initially less practised and so slower or less accurate, but eventually outperforms the old one. Forgetting after a break is another possible contributor.
How did the authors guard against finding transitions that are not real?
They tested their model-selection method on simulated curves with known numbers of pieces, required segments to span at least 50 plays, removed outlier scores, and added a strict pairwise test that removed doubtful extra pieces.
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