Pedagogics

T.K. Ospanov, Cand. Sc., Professor of National pedagogical university after Abai, Republic of Kazakhstan

 

Algebra elements education technology in the elementary schools of the Republic of Kazakhstan.

 

Process of teaching a math to junior pupils in the Republic of Kazakhstan is implemented by a technology, which is focused towards formation of subject competence knowledge, skills and intellectual development of the person. It allows designing and realizing of a model of interrelations and interactions of the activity of a pupil and pupils adequately to the purpose and content of studying math in elementary schools.

Local technology of education of math, which is being realized in practice in the present time, inheres particular qualities, goal orientation, conceptual status, content of education, procedural characteristic.

Parameters of goal orientation of algebra elements education technology in the elementary schools are provided in the table and conditioned by the necessity to educate everyone according to the requirements of state standard for elementary math education.

 

Grades

Requirements to the mandatory level of education of pupils

1

-understands the meaning of words equality, inequality, valid and invalid equality, valid and invalid inequality, numerical and literal expression, equation;

-uses comparison symbols (>,<,=);

-distinguishes numerical and literal expressions, equations;

-knows the names of expressions connected with addition and subtraction;

-forms, reads, records and finds values of the simplest numerical and literal expressions;

-sums the simplest equations using a method of fitting (by trial and error).

2

-understands the meaning of terms: valid and invalid numerical equality, valid and invalid numerical inequality, equation as an equality, consisting of unknown, given as a letter; simplest equation and equation of more complicated structure;

-knows and uses the rules of orders of operation with expressions consisting of brackets;

-forms, reads and records: numerical equalities and inequalities; numerical and literal expressions, consisting of 1-2 operations with brackets and without brackets, finds their value;

-solves the simplest equations with additions and subtractions basing upon the properties of valid numerical equalities where addition and subtraction are inverse operations.

3

-knows and applies the rules of orders of operation with expressions consisting of brackets and without brackets having all arithmetic operations;

-forms, reads, records numerical equalities and numerical inequalities, numerical and literal expressions, consisting of all operations (max. three) in any combination with brackets and without brackets;

-solves the simplest equations with multiplication and division by method of selection and by method based on the properties of valid numerical equalities where multiplication and division are inverse operations.

-solves equations with more complicated structures where the component and result of operation is a numerical expression.

4

-understands the meanings of algebraic terms (valid and invalid numerical equality and inequality, numerical and literal expression, equation);

-understands the essence of algebraic way of solving the problems (by setting up an equation);

-knows and applies the rules of orders of operation with expressions consisting of brackets and without brackets, methods of solving of equations;

-forms, reads, records numerical equalities and inequalities, numerical and literal expressions with brackets and without brackets (consisting of max. 4 operations);

-calculates the value of numerical and literal expressions consisting of all operations (not more than three);

-solves the simplest equations and equations of more complicated structure where the component and result of operation is a numerical expression.

 

The followings are the priority conceptual provisions for selection and structuring of the content, procedural characteristic of researching of the present course of math:

- elements of algebra, grouped around the core of math arithmetic of integral nonnegative numbers and those integrated on the basis of realization of inter-subject relations, form the content of unite course of math in elementary schools;

- studying of algebra elements forms a practical fundament for studying a systematical course of algebra in the first grades at school and considers formation of simplest algebraic terms, including numeric equality and numerical inequality, numerical expression and literal expression, equation and methods of solving equations, solving problems by forming an equation;

- elements of algebra consist of two conditional groups, i.e. issues of mandatory level to be learnt by all pupils, and issues of possible level, which serve for widening, deepening of understanding of materials within the mandatory level and lay the foundation for further studying of important sections of algebra, have effect on children development, helps to digest the basic material more profoundly;

- the most important algebraic element is an understanding of equation and learning the methods of solving them, revealing the essence of algebraic (by setting up an equation) way of solving the problems, which is inducted with the examples of problems in two operations unlike traditional position. This allows motivating the necessity of induction of new method of problem solving dissimilar to arithmetical method;

- optimal method for realization of the inter-subject relation is a method of selection which fulfills an additional load training of calculative skills and knowledge, and for continuity method of solving which allows to algorithm the activity and to exclude reeducation in the following grades.

The peculiarities of the content of education of Algebra Elements are defined by the nomenclature of terms and methods of operations provided in the following table:

 

Grades

Content of course of math Elements of algebra

1

Comparison of two groups of subjects (greater, less, equal). Numerical equalities and inequalities. Forming, reading numerical equalities and inequalities. Expressions, value of expressions. Numerical and literal expressions. Sum and diff identities. Forming, reading, recording, comparison numerical and literal expressions, finding their values. Equation. Forming and solving simplest equations (by the method of fitting).

2

Order of arithmetical operations. Brackets. Forming, reading, recording, comparison of numerical and literal expressions consisting of 2 operations (addition, subtraction), finding their values. Forming and solving simplest equations (by the method of fitting and basing upon the properties of valid numerical equalities where addition and subtraction are inverse operations).

3

Order of arithmetical operations in the expressions with brackets and without them. Forming, reading, recording and comparison of simplest numerical and literal expressions with multiplication and division, and expressions of more complicated structures, finding their values. Forming and solving simplest equations (by the method of fitting and basing upon the properties of valid numerical equalities where multiplication and division are inverse operations) and equations of more complicated structures where component and result of operation is a numerical expression.

4

Forming, reading, recording, comparison, finding a value of numerical and literal expression of more complicated structure consisting of 3-4 operations. Simplest examples of simplication of expressions. Forming and solving of the simplest equations and equations with more complicated structures where a component and result of operation is a numerical expression. Equations with one radical, more than one radical and without radical s. Solving problems by setting up an equation.

Peculiarities of the procedural characteristics of algebra elements education technology in the elementary schools are defined by planning and realization of the types of activity adequately to the stages of consideration of materials of the present substantial and methodical course of math. For example:

- first stage acquaintance with the terms of numerical equality and numerical inequality basing on comparation of two groups of subjects at the beginning, and then the integers which characterize the amount of subjects, set the relations greater, less, equal and record results of comparation by the use of symbols (>,<,=), train skills of reading of obtained equalities and inequalities;

- second stage acquaintance with the simplest expressions, values of expressions, components of arithmetical operations where the terms are introduced;

- third stage training the skills to read, record, compare the numerical expressions and find their values, to use the rules on the order of doing arithmetical operations in the expressions with brackets and without brackets;

- fourth stage introduction of the term literal expression, formation of terms numerical expression and literal expression, development of skills to read, record, compare the numerical expressions and find their values, to use the rules on the order of doing arithmetical operations in the expressions with brackets and without brackets;

- fifth stage acquaintance with the equation-equality consisting of one letter with the goal to determine the unknown number which is provided as a letter; teaching how to solve equations by different methods, developing an understanding of the radical of equation and of the variety of methods of solving the equation (one radical, lots of radicals, without radicals);

- sixth stage introduction of the algebraic method of solving of problems (by setting up an equation), illustration of its advantages in comparison with the arithmetical method.

The technology of education of algebra elements has its own set of essential singularities. First of all, it is important to note that while educating the algebra elements the potential of the impact towards the development of numerical skills and abilities is realized in the optimal variant, as they are always considered along with the execution of arithmetical operations with numbers. Attention is also given to the exclusion of variant reading of terms, which allow to differentiate, for example, record of type 3+2 naming as a sum (analogous to diff, product, quotient) and an integer 5 obtained in the result of addition of two numbers not naming as a sum, but as a value of sum (analogous to the values of diff, product, quotient).

During the work with expressions high importance is attached to the rules on the order of execution of arithmetical operations the amount of which is reduced to two: operation, recorded in the brackets to be executed first, in the expressions without brackets multiplication and division to be performed first and then addition and subtraction in the same order as they are provided from the left to the right, which are introduced and applied.

Selection of methods of solving the equations is a question of principal, which first of all has to satisfy demands of elementary math education, and second of all has to realize the perspective and successive links in education of elementary and systematical courses of math. In this regard the term equation has to be introduced in the 1st grade as an equality comprising of literal expression (or a letter) with the problem to find the value of the letter, which transforms the equation into valid numerical equality. Such value of the letter is obtained through the method of selection (trial). This method is applied when calculations with numbers are restricted to the tabular cases or brought to the operations with bit numbers. If while solving of an equation, all operations with numbers are implemented in written, then in this case the application of selection method becomes difficult. In this case the equations are solved through the methods based on the effects of valid equalities and inversability of addition-subtraction and multiplication-division. For example, to solve the following equation x+265=582, we are isolating the unknown, subtracting from both parts of the equation 265, then we have x=582-265, in the result x=317. To solve the equation 1283-x=957, firstly we are coming to the equation x+957=1283 basing on the effect of inversability of addition and subtraction, then we isolate the unknown by subtracting 957 from both parts, and obtain the result x=326. Equations with multiplication and division are solved similarly.

Advantages of algebraic method in solving the problems in compare to the arithmetic method can be shown in the problems such as the following one: First milk can had a few litters of milk, second milk can had 10 litters. When 2 litters of milk were poured into the first milk can, and 3 litters were poured out from the second, the amount of milk in both milk cans became equal. How much milk was there in the first milk can?

Solving the problem through the known method children would reason and write:

1) 10-3=7 (litters) this amount of milk remained in the second milk can when 3 litters of milk were poured out of it.

2) 7-2=5 (litters) this amount of milk was in the first milk can because there will be the same amount of milk in the first milk can if to pour 2 litters, i.e. 7 litters.

However, it is difficult for pupils to grasp and imagine all links between the givens and unknown and many of them can not solve this problem, in particular by arithmetical method.

Solving the problem through the algebraic method children would reason and write:

The amount of milk in the first milk can is unknown, let there be x litters, but it is known that there were 10 litters of milk in the second milk can.

1) Amount of milk in the first milk can is (x+2) litters, because 2 litters were poured into it.

2) Amount of milk in the second milk can is (10-3) litters, because 3 litters were poured out of it.

3) It become same amount of milk in both milk cans, that is why x+2=10-3. Solving the equation we find that x=5.

Answer: There were 5 litters of milk in the first milk can.

Comparing these methods we understand that arithmetical method of solving of the present problem is irrational in compare to the algebraic method.

 

 

Bibliography cited

1. State standards of the primary education in the Republic of Kazakhstan, - Almaty, republic publishers cabinet of Kazakh academy of education after Y. Altynsarin 1998.

2. State obligatory standards of secondary basic education of the Republic of Kazakhstan. Primary basic education. Almaty: ROND, 2002.

3. Math Programs for 1-4 grades. Almaty: ROND, 2003.

4. Math: Book for 1, 2, 3, 4 grades of educational schools /T.K. Ospanov and others. Almaty: Atamura, 1998-2009.