Living learning trajectory based on reduced first-order approximation models in an adaptive english language learning platform

Abstract

The article presents the concept of an intelligent platform employing adaptive algorithms for personalized English language learning. To formalize its operation, the study introduces mathematical formulations expressed as reduced first-order approximation models, which ensure conceptual clarity, generalization, and avoidance of unnecessary mathematical complexity while adequately reflecting the dynamics of the learning process. The system architecture is examined, comprising user, adaptive learning, assessment, recommendation, analytics, vocabulary monitoring, and grammar modules. For each component, a simplified mathematical model is provided to describe parameter evolution in discrete time, enabling real-time adjustment of the individual “living learning trajectory.” This approach supports dynamic content personalization, result prediction, and continuous improvement of language competencies. Contemporary AI-based platforms (Grammarly, Duolingo, ELSA Speak, Google Assistant, VoiceThread AI) are analyzed to identify their strengths and limitations for integration into adaptive learning systems. The developed model incorporates behavioral, cognitive, and emotional parameters of users, forming an effective interactive environment for developing reading, writing, pronunciation, and speaking skills. The results confirm that the integration of adaptive algorithms with artificial intelligence technologies substantially enhances learning efficiency, supports motivation and self-regulation, and contributes to the formation of stable language competencies.