The relevance of the problem under study is due to the fact that the analysis of the system of methodological training of future teachers of mathematics has shown that although in general it is aimed at forming students' professionally significant knowledge and skills, however, at none of its stages is the purposeful formation of research skills associated with the implementation in practice of problem-based teaching of mathematics to students. The purpose of the article is to identify the theoretical and methodological foundations of problem-solving problems in the pedagogical university as a means of forming the research skills of a future mathematics teacher in problem-based learning of students. Research methods: analysis of psychological-pedagogical and scientific-methodological literature, textbooks on mathematics for students of pedagogical universities; survey of teachers and students of pedagogical universities; study and generalization of school and university practice; analysis of their own experience in school and in pedagogical university; analysis of the experience of foreign schools in problem-based teaching of mathematics; various types of experiments to test the main provisions of the study. The article highlights the main blocks of research skills of future teachers of mathematics necessary for the implementation of problem-based learning and high school students developed by these units system problems and search problems, and methods of forming these skills in technicalities. The system of problem-search tasks in mathematics for students of pedagogical universities, examples of the organization of their educational and research activities at lectures, practical classes, which can be used by teachers of pedagogical universities, is presented and used.