Mathematical models of individualised learning based on decision theory

The 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