The problem and the goal. The article focuses on the development of intellectual and mathematical abilities of schoolchildren that function within the digital educational environment. Two sides of the abilities being formed are highlighted: cognitive and emotional-value ones developed due to the dialogical format of discussing the search for solutions to complex mathematical problems. The purpose of the study is to present ways to motivate schoolchildren to solve complex mathematical problems using a dialogical learning format. Research tasks: 1) to make a connotation of cultural mediation in a pedagogical interpretation; 2) to consider an example of solving a complex mathematical problem with a dialogical discussion of the results of the search for solutions; 3) to present a variant of the discussion that determines the pedagogical result of cognitive search. Methodology. Analysis of psychological and pedagogical categories, interdisciplinary transfer of pedagogical psychology information, modeling of cognitive studies of mathematical content. Methods. Observation, questioning, comparison, deduction. Results. The study finds out that the motivation of students to solve complex mathematical problems is associated both with achieving the ultimate goal of getting an answer, and with the joy of dialogue with classmates in a digital educational environment. There have been cases when studying the cognitive situation in a digital educational environment, the motive was separated from the goal noted above. So, for some of the schoolchildren, the motive shifted to the very activity on the network, which brought didactic joy to the student who was searching for original geometric sketches both to solve specific problems, and to create a constructive dialogue with peers. Conclusion. The personality of a student participating in a collective search for a solution of a cognitive problem reveals the direct perception of the material in his insights, actions, and assumptions. Direct perception of the content of mathematical education is quite acceptable, however, such perception requires the student’s ability to apply patternfragments while solving the elementary/basic mathematical sketches. Indirect mathematical knowledge is also appropriate in certain situations, because the student’s own impulses of development are not always effective, but such insight impulses lead to a productive result when indirect knowledge is actualized for the student, and memorized formulas and theorems are included into the free educational action.

Keywords: intelligence, personality, digital educational environment, motive, cultural mediation