In the modern world, science plays a key role in the development of society and solving socially significant problems. To become a scientist and contribute to the scientific community, it is necessary to have the skills of scientific reasoning. Scientific reasoning skills include the ability to analyze data, formulate hypotheses and conclusions, argue their thoughts, as well as generalize and interpret research results. To show research initiative due to highly developed forms of theoretical and critical thinking and mastery of methods of scientific cognition are characteristic of students, mainly high school students. Because research activities provide an opportunity for high school students to test their capabilities in various scientific and educational, cultural projects and areas of future professional activity.

The article discusses the organization of students' research activities, assesses the level of readiness of subject teachers for mentoring. The main results of the survey conducted among school students and interviews of teachers of mathematics in Shymkent are described. The conducted research allowed us to understand more deeply exactly what difficulties students face when organizing research activities. Such difficulties may include insufficient knowledge of research methods, lack of skills in working with information resources, as well as insufficient motivation and independence of students.

Finally, the article substantiates the need to develop a special methodology for identifying and supporting gifted students. The question of the development of research skills through the independent solution of various mathematical problems is touched upon. Every mathematical problem that goes beyond standard solutions is undoubtedly creative and exploratory. Moreover, mathematical problems can serve as a reference for research problems arising in various fields of science and life.