Students' mastering the art of proof is one of the most important educational results of teaching Mathematics in the secondary school. The study shows that there are significant challenges, gaps, and weaknesses in the practice of teaching reasoning and proving. The goal of the research is to determine the way of using traditional teaching practices to present reasoning and proving, the way of organizing the review and mastery of the theorem proofs. In addition, the obvious and hidden factors affecting the students' productivity in these activities were identified. To study the problem, a survey that involved 136 Mathematics teachers across Ukraine was carried out. A standard close-ended questionnaire was developed. Factor analysis was used to analyze the results of the survey. The important factors determining the time efficiency when a teacher organizes teaching and learning of a theorem proof, were discovered. The analysis of the factors and numerical loadings of variables forming them, result in the conclusions presented in this paper.

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О бґрунтовано, що інтегрований підхід до навчання математики дасть змогу успішніше засвоїти навчальний матеріал, посилити прикладну його спрямованість, а отже, й ефективно формувати математичну та інші ключові компетентності. Рекомендовано методичні прийоми реалізації інтегрованого підходу до відбору змісту шкільних підручників з математики, зокрема: врахування особливостей навчальної діяльності сучасних учнів, укрупнення навчального матеріалу, групування задач із орієнтацією на застосування їх на практиці, виокремлення практичних ситуацій, для розв'язання яких використовуються певні математичні моделі, посилення міжпредметних і внутрішньопредметних зв'язків та ін.

This empirical study determines the obvious and hidden factors that affect the Ukrainian Mathematics teachers to use Project-Based Learning and Teaching (PBL and PBT) Mathematics. A survey of 126 practicing Mathematics teachers was conducted using a Standard closed ended questionnaire. The study confirms that modern Mathematics teachers believe that it is impossible to teach Mathematics for modern students without using PBL and PBT. On the one hand, teachers demonstrate their beliefs in these educational techniques. On the other hand, the survey reveals hidden factors that are inhibitory in the practice of PBL and PBT Mathematics. The study shows the following statistically confirmed factors in teachers' beliefs concerning the use of PBL and PBT Mathematics: Content-Technological Factor; Result & Age Factor; Factor of the Target Audience on the expediency of using PBL and PBT Mathematics; Teacher's Overload Factor; Praxeological Factor; Locality Factor; Teacher's Awareness Spectrum Factor. The previous assumption that the experience of a Mathematics teacher is an influential factor in the use of PBL and PBT has not been statistically confirmed. The results show that an influential factor is the Locality Factor. So, it's important if a teacher teaches Mathematics in a city or a village. Its origins (genesis and causes) and the consequences of this factor influence on educational processes in Ukraine need more detailed study.

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