We discuss theoretical issues connected with methodological construction of problem-solving tasks for development of polycode competence in the system of language tertiary education. As one of the salient genres of polycode texts we take British and American political cartoon and the process of its interpretation. Yet another object of discussion is formed by particular difficulties when we try to include this format into text analysis. It gives us grounds to single out key features of polycode texts, which make them different from traditional text format. Here we include lack of precise genre typology, combination of various semiotic codes, lacunas presence, clip character, specific nature of intertextual chains. We single out several stages in the process of polycode competence formation and development. In particular, problem-solving tasks aimed at identification and primary decoding of certain elements typical for any polycode text; problem-solving tasks aimed at interpretation of verbal and visual components; carrying out group project/group projects of scientific educational or scientific research, of cultural and discourse character; participation in culturally-oriented discussions; assessment of the level of student’s polycode competence formation when interpreting a certain genre of polycode text. In this article, we focus upon possible problem-solving tasks of the first two stages. As one of possible solutions, we suggest conceptual cognitive metaphor modelling and oppositions. The oppositions in this case are built on basic sociocultural values while metaphor modelling includes structural, orientational and ontological concepts, which can be further found out in various types of polycode texts. As possible advantages of this approach, we consider it giving ground for interdisciplinary connections in the course of professional foreign language training. It also enables to widen the scope of questions and topics discussed with students as well as introduce certain linguocultural and sociocultural phenomena. All this assists in building professional competence of a would-be specialist as far as foreign media and multimodal environment is concerned, including its destructive practices. We also describe results of the experimental work. As part of the experiment, we assessed student’s ability to recognise images, symbols, archetypes, linguocultural subjects and meanings, precedent phenomena, continuity of forms and structures, conceptual metaphors and models, conceptual oppositions, basic and transferred meanings, connotations.