Abstract
The rapid transition to digital education has revealed the fundamental limitations of linear, «conveyor belt» platforms that offer identical content to all learners. Such systems are unable to take into account individual differences in initial training levels, cognitive styles, learning speeds, and content format preferences, which reduces the overall effectiveness of learning. This article addresses this problem by developing and describing a formalized mathematical model of an adaptive educational system. The proposed approach is based on a vector representation of both the learner and the learning materials in a single multidimensional feature space. The central element of the model is an interpretable utility function (U), which evaluates and selects the optimal next learning module for a specific student in real time. The function finds a balance between several pedagogically significant criteria: potential knowledge gain (G), cognitive load (E), and format compliance (M). A key feature is the built-in machine learning mechanism (online learning), which iteratively updates the learner's knowledge vector based on their results, allowing the system to continuously adapt. To validate the model, a numerical experiment was conducted, which confirmed that the system is capable of making non-trivial choices, finding the optimal balance between all factors, and correctly tracking the learner's progress. Thus, the proposed model represents a transparent and flexible framework for creating a new generation of intelligent learning systems capable of building truly personalized and effective educational trajectories.