The opportunities education provides to develop the student's ability to use cognitive skills to understand and solve problems whose solution is not obvious and the student's willingness to engage in problem solving as a constructive and thinking citizen are essential for the realization of the approach from transversal competences to transversal personality. The study puts an emphasis on exploring the conditions for the development of transversal competence for solving educational-and-cognitive problems that is portable through different learning contents, different activities and ages. The highlights of pupils' cognitive development have been used as a basis for turning them into subjects of problem-based training aiming at the development of transversal competence to solve problems. The here developed task system for solving integrated problems in science education allows diagnosing the level of competence of cognitive-learning problem solving as transversal. Achieving this result in school education is directly linked to the teacher's competence to design a proactive educational environment i.e. to the requirements for the training of pedagogical specialists in existing conditions.