In the article there are represented the results of theoretical and practical studies of the problem of using web-quest technologies in math training, directed on development of gifted schoolchildren. A theoretical model of web-quest technologies is represented in the methodological system of modelling research activities of a student. Methodical conditions of usage web-quest technologies in management of the process of student’s development are revealed. There are shown ties between a didactic object, modelling the process of giftedness development through training studies and task construction of a web-quest. An example of using of the developed approach in math training practice is supplied. It is proved, that web-quest, as a didactic object, is necessary to be viewed on four levels: aims, content, forms and means of teaching. In the process of forming student’s math research activities web-quest technology can have functions: to serve a means of stimulation of student’s search activity, to be a means of objective development of methods of research, to become one of the forms of displaying research training method. Depending on the training situation these functions can be introduced both totally and separately from each other. A task construction of the research content has the following form: orientation, creation of biblio-quest with useful material, trying, creation of hypothsis-quest, checking the hypothesis, introducing to the history of the question, creation of a report (a video-film, persentation, site and so on).