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Automated discovery of symbolic laws governing skill acquisition from naturally occurring data

Nature Computational Science volume 4, pages 334–345 (2024)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Skill acquisition is a key area of research in cognitive psychology as it encompasses multiple psychological processes. The laws discovered under experimental paradigms are controversial and lack generalizability. This paper aims to unearth the laws of skill learning from large-scale training log data. A two-stage algorithm was developed to tackle the issues of unobservable cognitive states and an algorithmic explosion in searching. A deep learning model is initially employed to determine the learner’s cognitive state and assess the feature importance. Symbolic regression algorithms are then used to parse the neural network model into algebraic equations. Experimental results show that the algorithm can accurately restore preset laws within a noise range in continuous feedback settings. When applied to Lumosity training data, the method outperforms traditional and recent models in fitness terms. The study reveals two new forms of skill acquisition laws and reaffirms some previous findings.

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Fig. 1: Overall model architecture diagram.
Fig. 2: Results of the simulated data experiment.
Fig. 3: The skill acquisition patterns discovered by the proposed method.

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Data availability

All data generated or analyzed during this study are available via GitHub at https://github.com/ccnu-mathits/ADM (ref. 58) and Zenodo at https://doi.org/10.5281/zenodo.10938670 (ref. 59). Source Data are provided with this paper.

Code availability

The source code of this study is freely available on Github at https://github.com/ccnu-mathits/ADM (ref. 58), and via Zenodo at https://doi.org/10.5281/zenodo.10938670 (ref. 59).

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Acknowledgements

This work was jointly supported by the National Science and Technology Major Project (grant no. 2022ZD0117103 to J.S. and Z.Y.), the National Natural Science Foundation of China (grant no. 62293554 to J.S. and Z.Y., 62107017 to X.S. and 62077021 to J.S.), the Higher Education Science Research Program of China Association of Higher Education (grant no. 23XXK0301 to J.S.), the China Postdoctoral Science Foundation (grant no. 2023T160256 to X.S.), Hubei Provincial Natural Science Foundation of China (grant no. 2023AFA020 to S.L. and 2022CFB414 to J.S.), and Fundamental Research Funds for the Central Universities (grant no. CCNU23XJ007 to X.S. and CCNU22LJ005 to S.L.).

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Authors and Affiliations

  1. National Engineering Research Center of Educational Big Data, Central China Normal University, Wuhan, China

    Sannyuya Liu, Qing Li, Xiaoxuan Shen, Jianwen Sun & Zongkai Yang

  2. Faculty of Artificial Intelligence in Education, Central China Normal University, Wuhan, China

    Sannyuya Liu, Qing Li, Xiaoxuan Shen, Jianwen Sun & Zongkai Yang

  3. National Engineering Research Center for E-learning, Central China Normal University, Wuhan, China

    Sannyuya Liu & Zongkai Yang

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  1. Sannyuya Liu
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Contributions

S.L. and X.S. conceptualized the work. S.L. and X.S. designed the methodology. Q.L. performed investigations and curated the data, whereas S.L., Q.L. and Z.Y. performed the data validation. J.S. and Z.Y. administered the project. J.S. and Z.Y. acquired funding. X.S., J.S. and Z.Y. supervised the project. S.L. and X.S. wrote the original draft, whereas Q.L. and J.S. reviewed and edited it.

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Correspondence to Xiaoxuan Shen, Jianwen Sun or Zongkai Yang.

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Nature Computational Science thanks Martijn Meeter, Konstantina Sokratous and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Ananya Rastogi, in collaboration with the Nature Computational Science team.

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Extended data

Extended Data Fig. 1 Partial results on the Lumosity dataset.

More specifically, (a) the change curve of mean fitting absolute error during the iteration process of the deep learning regressor; (b) the change curve of the value of the regularization term during the iteration processes; (c) the prediction error distribution of 1000 randomly selected records from the trained H1000+R1 model. The box plot displays the interquartile range (IQR) with the median line, while the whiskers extend to the minimum and maximum values or a multiple of the IQR from the quartiles. Outliers are depicted as individual points beyond the whiskers; (d) the proportion of the number of practice times for each skill to the total number of practice times; (e) the distribution of feature importance for each skill in H1000+R1.

Source data

Extended Data Fig. 2 Schematic representation of the simulation experiment.

(a) Flowchart depicting the process of generating simulated data and the data format. (b) Schematic diagram illustrating the evaluation procedure of the simulation experiment. (c) Sample representation of the simulated data.

Source data

Extended Data Fig. 3 The feature set utilized in the real-world dataset (Lumosity) experiment.

The feature set comprises three components: user (U), exercise (E), and scheduling (S). The features consist of two categories: discrete and continuous. Discrete features are encoded through one-hot encoding, while continuous features are encoded using real values. Game features are two-dimensional, consisting of skill and subskill, both of which are discrete features and characterize the relationship between the game and cognitive skills. Subskill is a subdivision feature of skill.

Source data

Extended Data Fig. 4 Analysis of the difference between the mastery level of skills calculated by the deep learning regressor and the discovered governing laws.

Here, we present the changes in skill proficiency at each time point during 2000 practice sessions for five learners. The blue line represents the mastery curve computed by the deep regressor, while the red line represents the curve computed by the discovered governing law. The governing law is the same for all learners, but due to differences in their choices and order of practice, there are variations in the independent variable of the governing law for each learner.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1 and 2.

Reporting Summary

Source data

Source Data Fig. 1

Image source data for Fig. 1.

Source Data Fig. 2

Numerical source data for Fig. 2.

Source Data Fig. 3

Numerical source data for Fig. 3.

Source Data Extended Data Fig. 1

Numerical source data for Extended Data Fig. 1.

Source Data Extended Data Fig. 2

Image source data for Extended Data Fig. 2.

Source Data Extended Data Fig. 3

Image source data for Extended Data Fig. 3.

Source Data Extended Data Fig. 4

Numerical source data for Extended Data Fig. 4.

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Liu, S., Li, Q., Shen, X. et al. Automated discovery of symbolic laws governing skill acquisition from naturally occurring data. Nat Comput Sci 4, 334–345 (2024). https://doi.org/10.1038/s43588-024-00629-0

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