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Brief ideas about teaching algebra in the Soviet Union 1. Algebra started in the 6th grade (12-year old pupils) 2. In the 6th, 7th and 8th grades, it covered: Algebraic expressions, equations, inequalities, functions, sequences (arithmetical, geometrical, oscillating and Fibonacci) 3. There were two main approaches to teaching algebra (and not only algebra): Calculative approach (“For the broad population”) Much attention paid to “calculative techniques”, not restricted only on simple cases: Substituting ab a b Examples: ... ; substitute a 2, b 1 3 2 ( a b) ( a b) 2 a x a b b ; substitute x a ab x b By substituting y 4 x , solve the equation 5 4 x x 0 . Manipulating expressions, often based on formulas for: ( a b) 2 , ( a b)3 , a 2 b 2 , a 3 b 3 , a 4 b 4 , ( a b ) 2 , … Finding ranges of definition of algebraic expressions and functions c 9 Examples: Find the range of the expression c 1 5 x Find the range of the function y 2 x 2 x 80 Use of prescribed, “ready-made” algorithms Underestimating of pupils ability to work independently and creatively Scientific approach (“For the elite”) Less drill Introducing complicated concepts and relationships Big attention paid to the connections between algebra and geometry (e.g. via parametric situations) b c Work with parametric situations Example: Find all relationships among various values: V a d U a Presentation of several solving strategies Preparing several advanced mathematical ideas (e.g. limits, reducible and irreducible polynomials, …) Required rather big amount of home work Example: Solving the equation |1 |1 x || 1 2 Calculative approach: Only one solving strategy - (a) the use of intervals 1 1 In ,0 : |1 |1 x || x, x , x 2 2 1 In (0, 1): |1 |1 x || x, x 2 1 3 In (1, 2): |1 |1 x || 2 x, 2 x , x 2 2 1 5 In (2, ): |1 |1 x || x 2, x 2 , x 2 2 Scientific approach: Three solving strategies – (a) the use of intervals, (b) graphs of functions, (c) geometrical (b) y 1 x y 1 x y |1 x | y 1 x y 1 |1 x | y |1 x | y |1 x | y 1 x y |1 |1 x || y |1 |1 x || 1 ½ 0 1 1 3 y y 2 2 2 1 1 3 |1 x | x x 2 2 2 3 1 5 |1 x | x x 2 2 2 (c) | a b | is the distance of a and b; |1 y | 1 2 1 3 2 Results Entrance exams for important Universities based on complicated manipulations with algebraic expressions and functions Examples : 1977 2 y 3 2 x 2 3x 3 0, Solve the system of equations 2 z 3 2 y 2 3 y 3 0, 2 x3 2 z 2 3 z 3 0. 1978 1 a , Find all a for which the system of inequalities a 1 has a solution. 3x 2 10 xy 5 y 2 2 x 2 2 xy 7 y 2 Solve all whole numbers solving the equation cos( (3 x 9 x 2 160 x 800)) 1. 8 1983 Solve the equation cos( (3 x 9 x 2 160 x 800)) 1. 8 The idea of “a big nation” can be clearly seen – only about 1.5 % of students understood algebra, but these were excellent