Teaching of Algebra Teaching of Algebra in the Czech Republicin the Czech Republic

Jarmila NovotnáJarmila Novotná

Charles UnCharles University in Prague,iversity in Prague, Faculty of EducationFaculty of Education

Department of MDepartment of Matathhematiematics and cs and Mathematical Education Mathematical Education

e-mail: jarmila.novotna@pedf.cuni.cze-mail: jarmila.novotna@pedf.cuni.cz

Czech Educational SystemCzech Educational System

From the historyFrom 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

Teaching of algebra in the Teaching of algebra in the

Czech RepublicCzech Republic Pre-algebra and algebra in Pre-algebra and algebra in

the compulsory educationthe compulsory education

Three programmes:Three programmes: Basic SchoolBasic SchoolGeneral SchoolGeneral SchoolNational SchoolNational School

Teaching of algebra in the Teaching of algebra in the

Czech RepublicCzech Republic Specific aims of mathematicsSpecific aims of mathematics

Mathematics together with the Czech language form the Mathematics together with the Czech language form the educational infrastructure of the basic schooleducational infrastructure of the basic school. . Mathematics provides pupils with the knowledge and skills Mathematics provides pupils with the knowledge and skills necessary for everyday life and prepares the foundations necessary for everyday life and prepares the foundations for successful development through professional training for successful development through professional training and further study at upper secondary schools. It develops and further study at upper secondary schools. It develops pupils’ intellectual abilities, their memory, imagination, pupils’ intellectual abilities, their memory, imagination, creativity, abstract thinking and ability to reason logically. creativity, abstract thinking and ability to reason logically. At the same time it contributes to the development of At the same time it contributes to the development of personal qualities, such as patience, diligence, critical personal qualities, such as patience, diligence, critical thinking. thinking.

Knowledge and skills acquired in mathematics are the Knowledge and skills acquired in mathematics are the preconditions for successpreconditions for success in the sciences, economics, in the sciences, economics, technology and the use of computers.technology and the use of computers.

Teaching of algebra in the Teaching of algebra in the

Czech RepublicCzech Republic Specific aims of mathematicsSpecific aims of mathematics

Teaching of mathematics Teaching of mathematics supports pupils’ learning tosupports pupils’ learning to:: …… use variables, understand what they represent, solve use variables, understand what they represent, solve

equations and inequalities and use them when solving equations and inequalities and use them when solving word problemsword problems

record and express graphically relationships between record and express graphically relationships between quantitative phenomena in nature and in society and quantitative phenomena in nature and in society and work with certain functions when solving word work with certain functions when solving word problemsproblems

prove simple theorems and conclude logical results prove simple theorems and conclude logical results from given assumptionsfrom given assumptions

……

Teaching of algebra in the Teaching of algebra in the

Czech RepublicCzech Republic The structure The structure of the schools subject mathematicsof the schools subject mathematics

Main parts of mathematicsMain parts of mathematics ArithmeticArithmetic AlgebraAlgebra GeometryGeometry Applications of mathematics Applications of mathematics

The relation to other subjectsThe relation to other subjects

The language of algebraThe language of algebra

Algebraic cultureAlgebraic culture

Teaching of algebra in the Teaching of algebra in the

Czech RepublicCzech Republic Algebra in Standards Algebra in Standards for the compulsory levelfor the compulsory level

ExpressionsExpressions Polynomials; Polynomial fractionsPolynomials; Polynomial fractions EquationsEquations Equivalent equations; Linear equations; Equivalent equations; Linear equations;

Quadratic equations; Systems of linear Quadratic equations; Systems of linear equationsequations

FunctionsFunctions Coordinate system; Functions; Direct and Coordinate system; Functions; Direct and

indirect proportionality; Linear functions; indirect proportionality; Linear functions; Quadratic functions; Trigonometric functionsQuadratic functions; Trigonometric functions

Programme: Programme: Basic SchoolBasic School

ContextContext KnowledgeKnowledge SkillsSkills The student The student should be able should be able

toto

Programme: Programme: General School

Core of the learning programme

• Topic Subtopic (details)

Samples

Programme: Basic SchoolProgramme: Basic School• 5th gradeTables, graphs, diagrams

• 6th gradeConsolidation of knowledge and skills

• 7th gradeDirect and indirect proportions

• 8th gradePowers with natural exponentsExpressionsLinear equations

• 9th gradePolynomial functions, Solving equations with the unknown in the denominatorSystems of linear equations with two variablesFunctions

Programme: General SchoolProgramme: General School

• 6th gradeCore: Good mastering of numerical operations with natural and decimal; drawing Word problemsPre-algebra (expressions, variable, function and its graph, linear equations)

• 7th gradeCore: Dependences, functions, orientation towards applicationsLinear equations and linear functions

• 8th gradeCore: Good mastering algebraic language, numerical mathematics; geometrical constructionsCalculations with polynomials

• 9th gradeCore: Knowledge necessary for mathematical applicationsAlgebraic techniqueFunctions

Algebra in Standards for Algebra in Standards for the upper-secondary level (Gymnthe upper-secondary level (Gymnáziumázium))

Basic knowledge about sets and Basic knowledge about sets and propositionspropositions

ExpressionsExpressions Polynomials; Polynomial fractions; Expressions Polynomials; Polynomial fractions; Expressions

with powers and rootswith powers and roots Algebraic equations and inequalitiesAlgebraic equations and inequalities FunctionsFunctions Basic concepts; Rational functions;Basic concepts; Rational functions; Quadratic Quadratic

functions; Exponential and logarithmic functionsfunctions; Exponential and logarithmic functions Trigonometry; Combinatorics, probability Trigonometry; Combinatorics, probability

and statistics; Sequencesand statistics; Sequences

Briefly about teaching in the Soviet Briefly about teaching in the Soviet

UnionUnion

• Algebra started in the 6th grade (12-year old pupils)

• In the 6th, 7th and 8th grades, it covered:

Algebraic expressions, equations, inequalities, functions, sequences (arithmetical, geometrical, oscillating and Fibonacci)

• Two main approaches: calculative, scientific

ResultsEntrance exams for important Universities based on complicated manipulations with algebraic expressions and functions

Examples

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