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 1^{st} 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.

1. State standards
of the primary education in the Republic of Kazakhstan, - Almaty, republic
publisher’s 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.