1. Teaching of Algebra in the Czech Republic Jarmila Novotná Jarmila Novotná Charles University in Prague, Charles University in Prague, Faculty of Education Faculty of Education Department of Mathematics and Mathematical Education Department of Mathematics and Mathematical Education e-mail: jarmila.novotna@pedf.cuni.cz e-mail: jarmila.novotna@pedf.cuni.cz

2. Czech Educational System From the history tradition of education in Czech history (population groups which would not achieve any education in other countries were often educated, e.g. Hussite women in the 15th century), the general literacy in the 1930s was of higher standards than were common in the rest of Europe Charles University was established in 1348 (the first European University east of Germany) Comenius - 17th century compulsory six-year school attendance was enacted in 1774 influence of the Soviet tradition, from which schools were only freed after 1989

4. Teaching of algebra in the Czech Republic Pre-algebra and algebra in the compulsory education Three programmes: Basic School General School National School

5. Teaching of algebra in the Czech Republic Specific aims of mathematics Mathematics together with the Czech language form the educational infrastructure of the basic school. Mathematics provides pupils with the knowledge and skills necessary for everyday life and prepares the foundations for successful development through professional training and further study at upper secondary schools. It develops pupils’ intellectual abilities, their memory, imagination, creativity, abstract thinking and ability to reason logically. At the same time it contributes to the development of personal qualities, such as patience, diligence, critical thinking. Mathematics together with the Czech language form the educational infrastructure of the basic school. Mathematics provides pupils with the knowledge and skills necessary for everyday life and prepares the foundations for successful development through professional training and further study at upper secondary schools. It develops pupils’ intellectual abilities, their memory, imagination, creativity, abstract thinking and ability to reason logically. At the same time it contributes to the development of personal qualities, such as patience, diligence, critical thinking. Knowledge and skills acquired in mathematics are the preconditions for success in the sciences, economics, technology and the use of computers. Knowledge and skills acquired in mathematics are the preconditions for success in the sciences, economics, technology and the use of computers.

6. Teaching of algebra in the Czech Republic Specific aims of mathematics Teaching of mathematics supports pupils’ learning to: … use variables, understand what they represent, solve equations and inequalities and use them when solving word problems use variables, understand what they represent, solve equations and inequalities and use them when solving word problems record and express graphically relationships between quantitative phenomena in nature and in society and work with certain functions when solving word problems record and express graphically relationships between quantitative phenomena in nature and in society and work with certain functions when solving word problems prove simple theorems and conclude logical results from given assumptions prove simple theorems and conclude logical results from given assumptions …

7. Teaching of algebra in the Czech Republic The structure of the schools subject mathematics Main parts of mathematics Arithmetic Arithmetic Algebra Algebra Geometry Geometry Applications of mathematics Applications of mathematics The relation to other subjects The language of algebra Algebraic culture

8. Teaching of algebra in the Czech Republic Algebra in Standards for the compulsory level Expressions Expressions Polynomials; Polynomial fractions Polynomials; Polynomial fractions Equations Equations Equivalent equations; Linear equations; Quadratic equations; Systems of linear equations Equivalent equations; Linear equations; Quadratic equations; Systems of linear equations Functions Functions Coordinate system; Functions; Direct and indirect proportionality; Linear functions; Quadratic functions; Trigonometric functions Coordinate system; Functions; Direct and indirect proportionality; Linear functions; Quadratic functions; Trigonometric functions

9. Programme: Basic School Basic SchoolBasic School ContextKnowledgeSkills The student should be able to Programme: General School General SchoolGeneral School Core of the learning programme Topic Subtopic (details) Samples

10. Programme: Basic School 5 th grade Tables, graphs, diagrams 6 th grade Consolidation of knowledge and skills 7 th grade Direct and indirect proportions 8 th grade Powers with natural exponents Expressions Linear equations 9 th grade Polynomial functions, Solving equations with the unknown in the denominator Systems of linear equations with two variables Functions

11. Programme: General School 6 th grade Core: Good mastering of numerical operations with natural and decimal; drawing Word problems Pre-algebra (expressions, variable, function and its graph, linear equations) 7 th grade Core: Dependences, functions, orientation towards applications Linear equations and linear functions 8 th grade Core: Good mastering algebraic language, numerical mathematics; geometrical constructions Calculations with polynomials 9 th grade Core: Knowledge necessary for mathematical applications Algebraic technique Functions

12. Algebra in Standards for the upper-secondary level (Gymnázium) Basic knowledge about sets and propositions Basic knowledge about sets and propositions Expressions Expressions Polynomials; Polynomial fractions; Expressions with powers and roots Polynomials; Polynomial fractions; Expressions with powers and roots Algebraic equations and inequalities Algebraic equations and inequalities Functions Functions Basic concepts; Rational functions; Quadratic functions; Exponential and logarithmic functions Basic concepts; Rational functions; Quadratic functions; Exponential and logarithmic functions Trigonometry; Combinatorics, probability and statistics; Sequences Trigonometry; Combinatorics, probability and statistics; Sequences

13. Briefly about teaching in the Soviet Union Algebra started in the 6 th grade (12-year old pupils) In the 6 th, 7 th and 8 th grades, it covered: Algebraic expressions, equations, inequalities, functions, sequences (arithmetical, geometrical, oscillating and Fibonacci) Two main approaches: calculative, scientific

18. Results Entrance exams for important Universities based on complicated manipulations with algebraic expressions and functions Examples

19.  The idea of “a big nation” can be clearly seen – only about 1.5 % of students understood algebra, but these were excellent