Academic journal article Psychonomic Bulletin & Review

S-Shaped Learning Curves

Academic journal article Psychonomic Bulletin & Review

S-Shaped Learning Curves

Article excerpt

Published online: 25 September 2013

© Psychonomic Society, Inc. 2013

Abstract In this article, learning curves for foreign vocabulary words are investigated, distinguishing between a subject-specific learning rate and a material-specific parameter that is related to the complexity of the items, such as the number of syllables. Two experiments are described, one with Turkish words and one with Italian words. In both, S-shaped learning curves were observed, which were most obvious if the subjects were not very familiar with the materials and if they were slow learners. With prolonged learning, the S shapes disappeared. Three different mathematical functions are proposed to explain these S-shaped curves. A further analysis clarifies why S-shaped learning curves may go unnoticed in many experiments.

Keywords Learning curve · Mathematical model · Vocabulary learning · Power function · Exponential function

(ProQuest: ... denotes formulae omitted.)

Since Ebbinghaus (1885), many theorists have tried to char- acterize the shape of learning. Knowledge about the mathe- matical equation of the learning curve is important for theo- retical reasons-for example, to establish whether learning performance continues to improve at the same rate, or is instead constantly slowing down (Lewandowsky & Farrell, 2011). Knowing the precise shape of the learning curve also has practical advantages-for example, to optimize the learn- ing process in training situations. With computer-based learn- ing, it is possible to use a student's learning history to predict future learning performance, while the learning process itself may be optimized. For these purposes, neural network models or other detailed computational models-though insightful in their own right-are often too complex. The calculations involved in processing the learning histories of individual students would be too time-consuming. For such applications, a highly abstracted, concise mathematical model is preferable. Hence, the search for the learning curve equation continues.

I do not think, however, that it is theoretically fruitful to assume the existence of one principal equation that determines the shape of learning: "the learning curve," a hypothetical construct that would characterize learning at all levels, in all systems, across groups and individuals, and for all types of materials. The reason why I do not believe in a generally valid learning curve is that all of these mechanisms and processes may affect the shape of learning at a fundamental level. For example, taking the average of several learning curves of individual subjects or items sounds like a neutral operation, but it is not. Simple averaging may give rise to a power function for a group's performance, even if the individual learning curves are all exponential (see below for a brief discussion of the exponential and power function). Several studies have now established this both experimentally (Heathcote, Brown, & Mewhort, 2000), computationally (R. B. Anderson, 2001;R.B.Anderson&Tweney,1997;Brown & Heathcote, 2003), and mathematically (Murre & Chessa, 2011;Myung,Kim,&Pitt,2000). Rather than calling such averaged power functions "spurious" or "artifactual," we must recognize that averaging is an operation that affects the shape of learning. Instead of searching for globally applicable learn- ing functions, I propose to focus on limited domains of appli- cation with more explicit assumptions about the processes involved. I believe that this differentiation may result in more insightful analyses and more powerful results.

In this article, I will focus on learning verbal material, specif- ically foreign language vocabulary, and analyze how we can translate characteristics of the to-be-learned material and individ- ual subjects to parameters of the observed learning curve. I will investigate under what circumstances S-shaped curves emerge, what parameters characterize these, and which aspects of an experiment may influence them. …

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