Key Tasks as a Means of Advancing Schoolchildren's Mathematical Knowledge When Learning Elementary Functions

The article presents the results of a pedagogical experiment on the study of the effectiveness of a diagnostic and technological approach for identifying key tasks of mathematical problem systems. The authors focus on the functional substantive line of studying mathematics-one of the main lines in the school course, since it helps to achieve not only subject but also meta-subject learning goals, as well as to develop schoolchildren's personalities. The authors note that many professors of schools and higher education institutions draw attention to schoolchildren's insufficient knowledge on the topic "Elementary Functions". Undoubtedly, modern professors must master information and communication technologies, innovative methods of developmental teaching so as to overcome this drawback. The aim of this research was to test the effectiveness of the above-mentioned authorial approach in order to advance the knowledge of Grade 10 schoolchildren on the topic "Applying Derivatives to Analyze Functions". After studying the topic, the schoolchildren had a test on it. The performance was graded on a scale from 1 to 5. The average score was 3.10. The tightness of links between the exercises of the system of mathematical problems was determined using the t-criterion, and five key tasks out of the fourteen proposed were singled out. The key tasks determine the successfulness of schoolchildren's learning; practicing various skills of solving key tasks significantly improves the results of solving other tasks of the test. An important component of the proposed approach is working out a logical structural scheme of elementary components of mathematical problems, which makes it possible to identify problematic issues common to all schoolchildren. In order to verify the importance of key tasks and to increase the level of schoolchildren's mathematical knowledge, a special tutorial on solving the singled-out key tasks was organized; it preceded the second test. As a result, in the second test, the schoolchildren's average score increased and reached 3.81. The use of the Student's criterion in the analysis of the results made it possible to draw the conclusion about the statistical significance of the differences in the average scores on the tasks of the test. The conducted pedagogical experiment showed the effectiveness of the suggested diagnostic and technological approach. Thus, the identification of key tasks and exercises in tests on algebra does not appear very time-consuming and makes it possible to significantly increase schoolchildren's mathematical knowledge.

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