Important constraints in teaching foreign languages

The article analyzes the didactic constraints and methodological decisions that determine the effectiveness of teaching foreign languages to students of mathematical specialties in higher education institutions, in particular at Kyiv National Taras Shevchenko University. The focus is placed on the problem of selecting, structuring and pedagogically supporting instructional materials within ESP courses where linguistic training is combined with professional content. The growing importance of international academic mobility, access to foreign-language research sources and participation in professional communication increases the need for accurate command of terminology as well as the ability to interpret logical proofs, graphical models and symbolic structures. The study generalizes contemporary approaches to materials development and demonstrates that real classroom practice is shaped by time limitations, different levels of students' preparedness, the complexity of mathematical discourse and the necessity to develop receptive and productive skills simultaneously. The paper considers ways of organizing a corpus of texts according to rhetorical functions, topics and sources, characterizes the principles of using authentic materials, and describes mechanisms for integrating lexical, grammatical and professional components. Special attention is paid to exercise planning, the development of contextual guessing strategies, the use of monolingual dictionaries, and the formation of argumentation and critical evaluation skills. The methodological potential of audio-visual tools, group discussions, project work and meetings with field experts is outlined. Approaches to testing and multi-level course evaluation through continuous student feedback are presented. The feasibility of systematic materials design based on theoretical validity, practical applicability and teachers' professional growth is substantiated. The proposed vision emphasizes that instructional resources should stimulate cognitive activity, provide opportunities to use language in professional situations and foster understanding of scientific reasoning. Implementation of these principles creates conditions for sustainable motivation of future mathematicians and for their integration into the international academic environment.