Mathematical models of individualised learning based on decision theory
Ivan VovchokThe study provided theoretical substantiation and development of a system of mathematical models for the individualisation of the educational process based on the integration of decision theory methods. The developed system of mathematical models is based on a metamodel that combines four mathematical paradigms through an interaction matrix, the elements of which are determined by the function of cognitive compatibility, temporal consistency and interaction efficiency. The introduction of the method of optimising partial trajectories, based on recursive updating of model parameters through the analysis of intermediate results, increased the accuracy of parameter settings and ensured smooth adaptation to the individual learning rate. The developed modification of the Bellman equation with the function of the complexity of the learning material made it possible to formalise the process of optimising long-term learning strategies by addressing individual cognitive characteristics. The analysis of the stochastic nature of the learning process through an extended transition matrix was used to mathematically describe the processes of forgetting and repeating the material using a system of differential equations with time-dependent coefficients that account for the intensity of learning and individual memory characteristics. The study of collaborative learning mechanisms using the gametheoretic approach revealed the synergistic effects of group learning through nonlinear functions of interaction between participants in the educational process and has allowed the development of methods for forming optimal learning groups based on individual goals. The proposed system of multidimensional evaluation, implemented through a composite objective function, covers a wide range of indicators from basic knowledge acquisition to the development of higher-order metacognitive skills, including cognitive, metacognitive and motivational components, which provides a reliable tool for assessing the stability of learning trajectories and determining the level of adaptability of the system to individual characteristics of students
References
[1] Alamri, H.A., Watson, S., & Watson, W. (2021). Learning technology models that support personalization within blended learning environments in higher education. TechTrends, 65, 62-78. doi: 10.1007/s11528-020-00530-3.
[2] Aydogan Yenmez, A., Erbas, A.K., Cakiroglu, E., Alacaci, C., & Cetinkaya, B. (2017). Developing teachers’ models for assessing students’ competence in mathematical modelling through lesson study. International Journal of Mathematical Education in Science and Technology, 48(6), 895-912. doi: 10.1080/0020739X.2017.1298854.
[3] Bernacki, M.L., Greene, M.J., & Lobczowski, N.G. (2021). A systematic review of research on personalized learning: Personalized by whom, to what, how, and for what purpose(s)? Educational Psychology Review, 33, 1675-1715. doi: 10.1007/s10648-021-09615-8.
[4] Bertsekas, D.P. (2022). Abstract dynamic programming (3rd ed). Belmont: Athena Scientific.
[5] Bora, A., & Ahmed, S. (2019). Mathematical modeling: An important tool for mathematics teaching. International Journal of Research and Analytical Reviews, 6(2), 252-256.
[6] Canco, I., Kruja, D., & Iancu, T. (2021). AHP, a reliable method for quality decision making: A case study in business. Sustainability, 13(24), article number 13932. doi: 10.3390/su132413932.
[7] Cappart, Q., Moisan, T., Rousseau, L., Prémont-Schwarz, I., & Cire, A.A. (2021). Combining reinforcement learning and constraint programming for combinatorial optimization. Proceedings of the AAAI Conference on Artificial Intelligence, 35(5), 3677-3687. doi: 10.1609/aaai.v35i5.16484.
[8] Chuang, S. (2021). The applications of constructivist learning theory and social learning theory on adult continuous development. Performance Improvement, 60(3), 6-14. doi: 10.1002/pfi.21963.
[9] De Maeyer, S., van den Bergh, H., Rymenans, R., Van Petegem, P., & Rijlaarsdam, G. (2010). Effectiveness criteria in school effectiveness studies: Further research on the choice for a multivariate model. Educational Research Review, 5(1), 81-96. doi: 10.1016/j.edurev.2009.09.001.
[10] Dogan, M.F. (2020). Evaluating pre-service teachers’ design of mathematical modelling tasks. International Journal of Innovation in Science and Mathematics Education, 28(1), 44-59. doi: 10.30722/IJISME.28.01.004.
[11] Eglington, L.G., & Pavlik, P.I. (2023). How to optimize student learning using student models that adapt rapidly to individual differences. International Journal of Artificial Intelligence in Education, 33, 497-518. doi: 10.1007/s40593022-00296-0.
[12] Jiang, P., & Wang, X. (2020). Preference cognitive diagnosis for student performance prediction. IEEE Access, 8, 219775-219787. doi: 10.1109/ACCESS.2020.3042775.
[13] Luan, H., & Tsai, C.-C. (2021). A review of using machine learning approaches for precision education. Educational Technology & Society, 24(1), 250-266.
[14] Lyon, J.A., & Magana, A.J. (2020). A review of mathematical modeling in engineering education. International Journal of Engineering Education, 36(1), 101-116.
[15] Maghsudi, S., Lan, A., Xu, J., & van Der Schaar, M. (2021). Personalized education in the artificial intelligence era: What to expect next. IEEE Signal Processing Magazine, 38(3), 37-50. doi: 10.1109/MSP.2021.3055032.
[16] Minn, S. (2022). AI-assisted knowledge assessment techniques for adaptive learning environments. Computers and Education: Artificial Intelligence, 3, article number 100050. doi: 10.1016/j.caeai.2022.100050.
[17] On evaluating curricular effectiveness: Judging the quality of K-12 mathematics evaluations. (2004). Washington: National Academies Press. doi: 10.17226/11025.
[18] Ouyang, F., & Jiao, P. (2021). Artificial intelligence in education: The three paradigms. Computers and Education: Artificial Intelligence, 2, article number 100020. doi: 10.1016/j.caeai.2021.100020.
[19] Peters, M.A.K. (2022). Confidence in decision-making. Oxford Research Encyclopedia of Neuroscience. doi: 10.1093/ acrefore/9780190264086.013.371.
[20] Peterson, J.C., Bourgin, D.D., Agrawal, M., Reichman, D., & Griffiths, T.L. (2021). Using large-scale experiments and machine learning to discover theories of human decision-making. Science, 372(6547), 1209-1214. doi: 10.1126/ science.abe2629.
[21] Rodemann, J., & Augustin, T. (2024). Imprecise Bayesian optimization. Knowledge-Based Systems, 300, article number 112186. doi: 10.1016/j.knosys.2024.112186.
[22] Taherdoost, H., & Madanchian, M. (2023). Multi-criteria decision making (MCDM) methods and concepts. Encyclopedia, 3(1), 77-87. doi: 10.3390/encyclopedia3010006.
[23] Tetzlaff, L., Schmiedek, F., & Brod, G. (2021). Developing personalized education: A dynamic framework. Educational Psychology Review, 33, 863-882. doi: 10.1007/s10648-020-09570-w.
[24] Tu, S., Frostig, R., & Soltanolkotabi, M. (2024). Learning from many trajectories. Journal of Machine Learning Research, 25, 1-109.
[25] Wang, X., Jin, Y., Schmitt, S., & Olhofer, M. (2023). Recent advances in Bayesian optimization. ACM Computing Surveys, 55(13s), article number 287. doi: 10.1145/3582078.
[26] Wu, H., & Noé, F. (2020). Variational approach for learning Markov processes from time series data. Journal of Nonlinear Science, 30, 23-66. doi: 10.1007/s00332-019-09567-y.
[27] Xing, W., Li, C., Chen, G., Huang, X., Chao, J., Massicotte, J., & Xie, C. (2021). Automatic assessment of students’ engineering design performance using a Bayesian network model. Journal of Educational Computing Research, 59(2), 230-256. doi: 10.1177/0735633120960422.
[28] Zhang, D., Chen, R.T., Liu, C.H., Courville, A., & Bengio, Y. (2023). Diffusion generative flow samplers: Improving learning signals through partial trajectory optimization. ArXiv. doi: 10.48550/arXiv.2310.02679.
[29] Zhang, L., Basham, J.D., & Yang, S. (2020). Understanding the implementation of personalized learning: A research synthesis. Educational Research Review, 31, article number 100339. doi: 10.1016/j.edurev.2020.100339.