This article is about the problem of development of students’ research abilities in the process of teaching mathematics is considered. In the system of research abilities are highlighted: non-complexity of thinking; critical thinking; independence of thinking and ability to self-organization; multiplicity of thinking; prediction of thinking. As one of the types of research tasks it is suggested to consider task with parameters. The activities structure of the teacher of mathematics and students, aimed at acquiring students' skills of research activity, is proposed. It is specified: the level of methods’ mastery of solving problems with parameters is an indicator of awareness of knowledge corresponding to the main content lines of the school math course from the main sections of school mathematics, the level of development of thinking. In more detail, the problem with parameters on the topic "Equation and Inequality Index" is considered. Proposed author assignments (O. Chashechnikova) from algebra and geometry. Some results of experimental teaching of senior pupils and students are described in order to determine the impact on the development of students’ research abilities in solving tasks for research. Positive dynamics noted.