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Abstract: Currently, there is an active process of reform of the entire educational system. One of the major changes recorded in the regulations, is to strengthen the focus on the training of teaching and research and design activity of pupils and students. In this connection there is a necessity of formation of professional readiness of the future teachers of mathematics to the organization of teaching and research and design activity of students. The paper highlighted the main provisions of normative documents (federal state educational standards for general and higher education, teacher professional standard, program "Ural Engineering School), requiring modernization of training of the future mathematics teacher in the field of teaching and research and design activity of students. The structural components of the theoretical (the knowledge) and practical (skills needed) professional readiness of the teacher to the organization of teaching and research and design activity of students are identified. The types of tasks aimed at the formation of structural components of theoretical and practical readiness of the future teachers of mathematics to the specified activity are described. For each of the selected type of tasks the authors give examples of assignments for students, which can be used in the process of educational and extracurricular activities.