The article presents the results of the first series of practical research within the scope of an adopted program of pilot testing on "The Technology of Formation of Exploratory Competence in Future Social Teachers within the System of University Education." A set of questionnaires offered to students made it possible to identify the initial condition of formedness of exploratory competence of future social teachers, which helps to choose correctly the most efficient methods and techniques for its proper formation. It was found out that the students interviewed lack cognitive and professional motives, and do not have a clear idea about scientific activities and their place in a social teacher occupation. However, a considerable amount of respondents have positive attitude to exploratory activity, which means a material opportunity for fostering exploratory competence in future social teachers within the system of university education.