1 research outputs found
ΠΠ²ΡΠΎΠΌΠ°ΡΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½Π°Ρ ΠΎΠ±ΡΡΠ°ΡΡΠ°Ρ ΡΠΈΡΡΠ΅ΠΌΠ° Β«ΠΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°Β» (ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΠΎΡΠ³Π°Π½ΠΈΠ·Π°ΡΠΈΠΈ 1-ΠΉ ΡΠ°ΡΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠ°)
The issues of building an automated learning system βSetsβ which will allow students to master one of the important topics of the discipline βDiscrete Mathematicsβ and to develop logical and mathematical thinking in this direction are studied. The corresponding topic of the 1st part of the project includes materials related to the concept of a set, operations on sets, algebra of sets, proofs of statements for sets, and the derivation of formulas for the number of set elements. The system is based on a construction of the statements proof editor for a set and of the formulas derivation editor for the number of set elements, both editors are to be used for teaching. The first of these allows students to split the original statement into a number of simpler statements, taken together equivalent to the original statement, to choose a method of proving each simple statement and to conduct their step-by-step proof. The second editor allows (using the inclusion-exclusion principle and the formula of the number of complement elements) to derive a step-by-step formula for the number of set elements through the specified numbers of elements for sets from which the resulting set is constructed. An important part of the system is to monitor the correctness of all actions of students, and on this basis the entire learning system is developed. The logical supervision over the correctness of the selected action in the first editor is performed by a Boolean function created by the system and corresponding to this action and by checking it for identical truth. In the second editor, invariants such as characteristic strings of the set and of its number of elements are used for verification. The rest of the system is related to learning of set algebra and to preparation to editors usage. The main focus here is on the learning strategy in which testing the understanding of the learned material is rather rigorous and eliminating the random choice of answers. The division of the material into sections with verification of the success of teaching not only by tests, but also by exercises and tasks, allows students to master the complex logical and mathematical techniques of proving statements for sets and derivation of formulas for the number of set elements.ΠΡΡΠ»Π΅Π΄ΡΡΡΡΡ Π²ΠΎΠΏΡΠΎΡΡ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ Π°Π²ΡΠΎΠΌΠ°ΡΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠΉ ΠΎΠ±ΡΡΠ°ΡΡΠ΅ΠΉ ΡΠΈΡΡΠ΅ΠΌΡ Β«ΠΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°Β», ΠΊΠΎΡΠΎΡΠ°Ρ ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΡ ΡΡΠ°ΡΠ΅ΠΌΡΡΡ ΠΎΡΠ²ΠΎΠΈΡΡ ΠΎΠ΄Π½Ρ ΠΈΠ· Π²Π°ΠΆΠ½ΡΡ
ΡΠ΅ΠΌ Π΄ΠΈΡΡΠΈΠΏΠ»ΠΈΠ½Ρ Β«ΠΠΈΡΠΊΡΠ΅ΡΠ½Π°Ρ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΠΊΠ°Β» ΠΈ ΡΠ°Π·Π²ΠΈΡΡ Π»ΠΎΠ³ΠΈΠΊΠΎ-ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΌΡΡΠ»Π΅Π½ΠΈΠ΅ Π² ΡΡΠΎΠΌ Π½Π°ΠΏΡΠ°Π²Π»Π΅Π½ΠΈΠΈ. Π‘ΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠ°Ρ ΡΠ΅ΠΌΠ° 1-ΠΉ ΡΠ°ΡΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠ° Π²ΠΊΠ»ΡΡΠ°Π΅Ρ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π», ΡΠ²ΡΠ·Π°Π½Π½ΡΠΉ Ρ ΠΏΠΎΠ½ΡΡΠΈΠ΅ΠΌ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°, ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠΌΠΈ Π½Π°Π΄ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°ΠΌΠΈ, Π°Π»Π³Π΅Π±ΡΠΎΠΉ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ², Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π°ΠΌΠΈ ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΠΉ Π΄Π»Ρ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ², Π²ΡΠ²ΠΎΠ΄ΠΎΠΌ ΡΠΎΡΠΌΡΠ» Π΄Π»Ρ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°. Π ΠΎΡΠ½ΠΎΠ²Π΅ ΡΠΈΡΡΠ΅ΠΌΡ Π»Π΅ΠΆΠΈΡ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΠ΅ Ρ ΡΠ΅Π»ΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ Π΄Π»Ρ ΠΎΠ±ΡΡΠ΅Π½ΠΈΡ ΡΠ΅Π΄Π°ΠΊΡΠΎΡΠ° Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π° ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΠΉ Π΄Π»Ρ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° ΠΈ ΡΠ΅Π΄Π°ΠΊΡΠΎΡΠ° Π²ΡΠ²ΠΎΠ΄Π° ΡΠΎΡΠΌΡΠ» Π΄Π»Ρ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°. ΠΠ΅ΡΠ²ΡΠΉ ΠΈΠ· Π½ΠΈΡ
ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΡΠ΄Π΅Π½ΡΡ ΡΠ°Π·Π±ΠΈΡΡ ΠΈΡΡ
ΠΎΠ΄Π½ΠΎΠ΅ ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΠ΅ Π½Π° ΡΡΠ΄ Π±ΠΎΠ»Π΅Π΅ ΠΏΡΠΎΡΡΡΡ
ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΠΉ, Π² ΡΠΎΠ²ΠΎΠΊΡΠΏΠ½ΠΎΡΡΠΈ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΡΡ
ΠΈΡΡ
ΠΎΠ΄Π½ΠΎΠΌΡ ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΡ, Π²ΡΠ±ΡΠ°ΡΡ ΠΌΠ΅ΡΠΎΠ΄ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π° ΠΊΠ°ΠΆΠ΄ΠΎΠ³ΠΎ ΠΏΡΠΎΡΡΠΎΠ³ΠΎ ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΡ ΠΈ ΠΏΡΠΎΠ²Π΅ΡΡΠΈ ΠΈΡ
ΠΏΠΎΡΠ°Π³ΠΎΠ²ΠΎΠ΅ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎ. ΠΡΠΎΡΠΎΠΉ ΡΠ΅Π΄Π°ΠΊΡΠΎΡ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ ΡΠΎΡΠΌΡΠ»Ρ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΠΈ ΠΈΡΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΠΈ ΡΠΎΡΠΌΡΠ»Ρ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² Π΄ΠΎΠΏΠΎΠ»Π½Π΅Π½ΠΈΡ, Π²ΡΠ²Π΅ΡΡΠΈ ΠΏΠΎΡΠ°Π³ΠΎΠ²ΠΎ ΡΠΎΡΠΌΡΠ»Ρ Π΄Π»Ρ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° ΡΠ΅ΡΠ΅Π· Π·Π°Π΄Π°Π½Π½ΡΠ΅ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ², ΡΠ²ΡΠ·Π°Π½Π½ΡΡ
Ρ Π½ΠΈΠΌ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ². ΠΠ°ΠΆΠ½ΠΎΠΉ ΡΠ°ΡΡΡΡ ΡΠΈΡΡΠ΅ΠΌΡ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΠΎΡΡΠΈ Π²ΡΠ΅Ρ
Π΄Π΅ΠΉΡΡΠ²ΠΈΠΉ ΡΡΡΠ΄Π΅Π½ΡΠ°, ΠΈ Π½Π° ΡΡΠΎΠΉ ΠΎΡΠ½ΠΎΠ²Π΅ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠ°Π½Π° Π²ΡΡ ΡΠΈΡΡΠ΅ΠΌΠ° ΠΎΠ±ΡΡΠ΅Π½ΠΈΡ. ΠΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΠΎΡΡΠΈ Π²ΡΠ±ΡΠ°Π½Π½ΠΎΠ³ΠΎ Π΄Π΅ΠΉΡΡΠ²ΠΈΡ Π² ΠΏΠ΅ΡΠ²ΠΎΠΌ ΡΠ΅Π΄Π°ΠΊΡΠΎΡΠ΅ ΠΎΡΡΡΠ΅ΡΡΠ²Π»ΡΠ΅ΡΡΡ ΡΠΎΠ·Π΄Π°Π½ΠΈΠ΅ΠΌ ΡΠΈΡΡΠ΅ΠΌΠΎΠΉ Π±ΡΠ»Π΅Π²ΠΎΠΉ ΡΡΠ½ΠΊΡΠΈΠΈ, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠ΅ΠΉ ΡΡΠΎΠΌΡ Π΄Π΅ΠΉΡΡΠ²ΠΈΡ, ΠΈ ΠΏΡΠΎΠ²Π΅ΡΠΊΠΎΠΉ Π΅Π΅ Π½Π° ΡΠΎΠΆΠ΄Π΅ΡΡΠ²Π΅Π½Π½ΡΡ ΠΈΡΡΠΈΠ½Π½ΠΎΡΡΡ. ΠΠΎ Π²ΡΠΎΡΠΎΠΌ ΡΠ΅Π΄Π°ΠΊΡΠΎΡΠ΅ Π΄Π»Ρ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ°ΠΊΠΈΠ΅ ΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΡ, ΠΊΠ°ΠΊ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΡΡΠΎΠΊΠ° ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° ΠΈ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΡΡΠΎΠΊΠ° ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°. ΠΡΡΠ°Π»ΡΠ½Π°Ρ ΡΠ°ΡΡΡ ΡΠΈΡΡΠ΅ΠΌΡ ΡΠ²ΡΠ·Π°Π½Π° Ρ ΠΎΠ±ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π°Π»Π³Π΅Π±ΡΠ΅ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ² ΠΈ ΠΏΠΎΠ΄Π³ΠΎΡΠΎΠ²ΠΊΠ΅ ΠΊ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ ΡΠ΅Π΄Π°ΠΊΡΠΎΡΠΎΠ². ΠΡΠΈ ΡΡΠΎΠΌ ΠΎΡΠ½ΠΎΠ²Π½ΠΎΠ΅ Π²Π½ΠΈΠΌΠ°Π½ΠΈΠ΅ ΡΠ΄Π΅Π»ΡΠ΅ΡΡΡ ΡΡΡΠ°ΡΠ΅Π³ΠΈΠΈ ΠΎΠ±ΡΡΠ΅Π½ΠΈΡ, ΠΏΡΠΈ ΠΊΠΎΡΠΎΡΠΎΠΉ ΠΏΡΠΎΠ²Π΅ΡΠΊΠ° ΠΏΠΎΠ½ΠΈΠΌΠ°Π½ΠΈΡ ΡΡΠ²ΠΎΠ΅Π½Π½ΠΎΠ³ΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° ΡΠ²Π»ΡΠ΅ΡΡΡ Π΄ΠΎΠ²ΠΎΠ»ΡΠ½ΠΎ ΡΡΡΠΎΠ³ΠΎΠΉ, ΠΈΡΠΊΠ»ΡΡΠ°ΡΡΠ΅ΠΉ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΠΉ Π²ΡΠ±ΠΎΡ ΠΎΡΠ²Π΅ΡΠΎΠ². Π Π°Π·Π±ΠΈΠ΅Π½ΠΈΠ΅ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° Π½Π° ΡΠ΅ΠΊΡΠΈΠΈ Ρ ΠΊΠΎΠ½ΡΡΠΎΠ»Π΅ΠΌ ΡΡΠΏΠ΅ΡΠ½ΠΎΡΡΠΈ ΠΎΠ±ΡΡΠ΅Π½ΠΈΡ Π½Π΅ ΡΠΎΠ»ΡΠΊΠΎ ΡΠ΅ΡΡΠ°ΠΌΠΈ, Π½ΠΎ ΠΈ ΡΠΏΡΠ°ΠΆΠ½Π΅Π½ΠΈΡΠΌΠΈ ΠΈ Π·Π°Π΄Π°ΡΠ°ΠΌΠΈ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΡΠ΄Π΅Π½ΡΡ ΠΎΠ²Π»Π°Π΄Π΅ΡΡ ΡΠ»ΠΎΠΆΠ½ΡΠΌ Π»ΠΎΠ³ΠΈΠΊΠΎ-ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠΌ Π°ΠΏΠΏΠ°ΡΠ°ΡΠΎΠΌ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π° ΡΡΠ²Π΅ΡΠΆΠ΄Π΅Π½ΠΈΠΉ Π΄Π»Ρ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ² ΠΈ Π²ΡΠ²ΠΎΠ΄Π° ΡΠΎΡΠΌΡΠ»Ρ Π΄Π»Ρ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°