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Lecturer(s)
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Course content
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1. Repetition of mathematical concepts from previous studies. Logical concepts, sets, relations, maps, arrangements. 2. Set operations, unary, binary and ternary operations. Properties of binary operations, completeness, commutativity, associativity, existence of neutral and inverse elements of binary operation. 3. Algebraic structures with one operation. Classification of algebraic structure with one operation, groupoid, semigroup, monoid and group. Examples of binary operations from the curriculum of first degree mathematics. 4. Structures with two operations. Classification of algebraic structures with two operations, semicircle, circuit, field of integrity, solid. Examples of algebraic structures with two operations in the curriculum of first degree mathematics. 5. Axiomatic theory of natural numbers, the concept of axiom. Formal axioms of natural number theory. 6. Content axioms of natural number theory. Pean's axioms, the concept of follower and predecessor. 7. Proofs of theorems in natural number theory. Axiom of induction. The natural arrangement of the set of all natural numbers. 8. Divisibility of natural numbers. FRelation to be a divisor. Prime number and compound number. Decomposition of a number into the product of prime numbers. Hasse diagram. Eratosthenes sieve. Number of divisors of a number. 9. Division with remainder. Divisibility rules in decimal system. Rules of the last number, two digits and digit sum. 10. The greatest common divisor. The set of divisors of a natural number. Formalization of the search for the greatest common divisor. Euclidean algorithm. 11. Least common multiple. A set of multiples of a natural number. Formalization of the search for the least common multiple. 12. Linear indefinite equations. Diophantine equations. Solvability of the Diophantine equation in the field of integers. Solution depending on the parameter. Solution by reasoning and experiment. 13. Positional number systems. Numerical notation in history. Non-positional, combined and positional number systems. Writing a natural number in decimal and other number systems. Algorithm for converting decimal notation to notation in another number system. 14. Algorithms of addition, subtraction, multiplication and division in decimal and other number systems. Algorithm of addition, subtraction and multiplication below each other. Algorithm of long and short division.
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Learning activities and teaching methods
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unspecified, unspecified
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Learning outcomes
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The subject Arithmetic with Didactics I together with the subject Arithmetic with Didactics II in the summer semester of the 2nd year prepares students for their pedagogical work at the first stage of primary school in terms of acquiring mathematical knowledge, skills and didactic-methodological insight into the issue. In the lectures the student will obtain the necessary information in a comprehensive system, exercises are used to solve mathematical problems, experimentation, discovering mathematical contexts, creating mathematical problems and demonstrations of modern didactic methods. Increased attention is paid to the central concept of the whole subject, which is the concept of number. In accordance with the didactic constructivism to which the teaching of the subject applies, the knowledge bases are first created, on which the concept of number is gradually constructed during both semesters. Elements of mathematical discovery, research methods, etc. are used so that students not only build a comprehensive mathematical theory, but that the constructivist teaching style becomes a natural model for future pedagogical work of students in mathematics. The shown teaching methods will be supplemented by work with a textbook, didactic aids, computer technology, mathematical applications and extended by practical examples.
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Prerequisites
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unspecified
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Assessment methods and criteria
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unspecified
1. Active participation for min. 80 % exercise. 2. Successful completion of credit tests (success rate 50 %).
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Recommended literature
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Bělík M., Svoboda J. Peanova aritmetika přirozeného čísla, Pedagogická fakulta UJEP, Ústí n.L.,. 1998.
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Bělík M., Svoboda J. Binární operace pro studium učitelství 1. stupně základní ško-ly, Pedagogická fakulta UJEP, Ústí n.L.,. 1998.
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Drábek J. a kol. Základy elementární aritmetiky pro studium učitelství 1. st. ZŠ, SPN Praha. 1985.
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Kopka, J. Kapitoly o celých číslech, PF UJEP, 2004..
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Kopka J. Kapitoly o přirozených číslech, UJEP, Ústí nad Labem. Ústí nad Labem, 2003.
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