It is obvious that for success in school mathematics it is necessary to master elementary mental computational skills at first - addition and subtraction within the limits of 20, multiplication and division within the limits of 100.
In spite of this there are kids in third, fourth, and fifth grade who cannot - without a calculator - add 8 to 5, subtract 7 from 12,multiply 7 by 8, divide 54 by 9 and so on.
Just those very pupils have considerable difficulties while learning the other basic topics of arithmetic and algebra.
They cannot master well operations with two and three digit numbers, common fractions, negative numbers, like terms, brackets, simple equations etc.
Even calculators cannot help them.
For confirming the influence of quality of elementary mental computational skills over success in school mathematics I had decided to investigate a level of the skills of pupils in graduation-classes of primary school (the multiplication table had been completely learnt a year and a half ago) and their achievements in algebra three years later.
The study was carried out in period from 1994 to 2004 years.
The sample includes 403 pupils.
Standard tables including 64 elementary operations in addition, subtraction, multiplication and division were used for determination of the level of the skills.
You can see specimens of the tables at my site Prevention of Failure in School Mathematics (Improvement of Elementary Computational Skills, Tables).
The pupils completed the tables in written form.
The time from the starting point to the finishing point was measured.
For estimation of the level two criteria were used - a total time of completion of a table and a number of occurred errors.
Statistical analysis of the results had shown that there is possible to single out four uniform groups in the sample: 1) Pupils completed the tables quickly and made not more than 3 mistakes.
2) Pupils completed the tables significantly slower than the pupils from the first group but made not more than 1 mistake.
3) Pupils completed the tables at middle pace and made 3-6 mistakes.
4) Either a time or a number of errors, or both together were evidently "good-for-nothing".
Statistical analysis of the parameters of the first groups (separately for addition, subtraction, multiplication and division) confirmed (significance at 5% level) their internal uniformity: distribution in time is normal, and distribution in errors submits to Poisson's law.
In different cases these groups included from 50% to 70% of corresponding samples.
According to the criteria of rejecting extreme values they were sufficiently good isolated (significance at 5% level) from the other parts of the samples.
In three years after the testing the current results in algebra (of the pupils which had been tested) during five months was studied.
It turned out that all pupils from the first group had no problems, 87% of the pupils from the second and third groups had big problems, and the other pupils' results were, to put it mildly, very poor.
Therefore it was confirmed that unsatisfactory development of the elementary mental computational skills in primary school was the first cause of difficulties in math's learning.
It means that results of testing of the skills may be used for prognosis of failure in school mathematics.
It is evident that the population of pupils with satisfactory skills may be represented only by the pupils from the first group.
That's why the parameters of the first group were used as the basis for computing limits of values of the considered parameters.
In view of the fact that sample statistics only approximately estimate parameters of population, the top border under which 90% of samples means lie was considered as the starting-point (the sample mean of satisfactory achieving pupils), and only the 99th percentile of thus achieved normal distribution was taken as the limit for time and the last value of thus achieved Poisson's distribution which probability is not less than 0.
01 was taken as the limit for number of errors.
This method of determination of limit values gives approximation with surplus only.
Therefore the permissible limits of time and errors may be considered as sufficiently mild demands.
The limit values were calculated for the standard tables including 64 similar elementary operations.
For final-year pupils of primary school (the multiplication table had been completely learnt a year and a half ago) the next limit values had been obtained: Addition - 11 minutes 02 seconds and not more than 4 errors.
Subtraction - 11 minutes 35 seconds and not more than 4 errors.
Multiplication - 9 minutes 46 seconds and not more than 3 errors.
Division - 9 minutes 06 seconds and not more than 3 errors.
This level of elementary mental computational skills is a good means for determination of pupil's preparedness for successful studies.
The limit values of the considered parameters define the first threshold of school math's learning ability.
The pupils who have not crossed this threshold are doomed to poor progress.
The prediction of their failure in math will be right in about 95 cases of 100.
It must be noted in conclusion that the level of elementary mental computational skills of actively working pupils only do not decreases in due course.
If a pupil uses a calculator instead of mental computations, works passively at lessons and does not carry out home works himself/herself, then the level decreases gradually.
In certain cases it leads to difficulties in math's learning.
In spite of this there are kids in third, fourth, and fifth grade who cannot - without a calculator - add 8 to 5, subtract 7 from 12,multiply 7 by 8, divide 54 by 9 and so on.
Just those very pupils have considerable difficulties while learning the other basic topics of arithmetic and algebra.
They cannot master well operations with two and three digit numbers, common fractions, negative numbers, like terms, brackets, simple equations etc.
Even calculators cannot help them.
For confirming the influence of quality of elementary mental computational skills over success in school mathematics I had decided to investigate a level of the skills of pupils in graduation-classes of primary school (the multiplication table had been completely learnt a year and a half ago) and their achievements in algebra three years later.
The study was carried out in period from 1994 to 2004 years.
The sample includes 403 pupils.
Standard tables including 64 elementary operations in addition, subtraction, multiplication and division were used for determination of the level of the skills.
You can see specimens of the tables at my site Prevention of Failure in School Mathematics (Improvement of Elementary Computational Skills, Tables).
The pupils completed the tables in written form.
The time from the starting point to the finishing point was measured.
For estimation of the level two criteria were used - a total time of completion of a table and a number of occurred errors.
Statistical analysis of the results had shown that there is possible to single out four uniform groups in the sample: 1) Pupils completed the tables quickly and made not more than 3 mistakes.
2) Pupils completed the tables significantly slower than the pupils from the first group but made not more than 1 mistake.
3) Pupils completed the tables at middle pace and made 3-6 mistakes.
4) Either a time or a number of errors, or both together were evidently "good-for-nothing".
Statistical analysis of the parameters of the first groups (separately for addition, subtraction, multiplication and division) confirmed (significance at 5% level) their internal uniformity: distribution in time is normal, and distribution in errors submits to Poisson's law.
In different cases these groups included from 50% to 70% of corresponding samples.
According to the criteria of rejecting extreme values they were sufficiently good isolated (significance at 5% level) from the other parts of the samples.
In three years after the testing the current results in algebra (of the pupils which had been tested) during five months was studied.
It turned out that all pupils from the first group had no problems, 87% of the pupils from the second and third groups had big problems, and the other pupils' results were, to put it mildly, very poor.
Therefore it was confirmed that unsatisfactory development of the elementary mental computational skills in primary school was the first cause of difficulties in math's learning.
It means that results of testing of the skills may be used for prognosis of failure in school mathematics.
It is evident that the population of pupils with satisfactory skills may be represented only by the pupils from the first group.
That's why the parameters of the first group were used as the basis for computing limits of values of the considered parameters.
In view of the fact that sample statistics only approximately estimate parameters of population, the top border under which 90% of samples means lie was considered as the starting-point (the sample mean of satisfactory achieving pupils), and only the 99th percentile of thus achieved normal distribution was taken as the limit for time and the last value of thus achieved Poisson's distribution which probability is not less than 0.
01 was taken as the limit for number of errors.
This method of determination of limit values gives approximation with surplus only.
Therefore the permissible limits of time and errors may be considered as sufficiently mild demands.
The limit values were calculated for the standard tables including 64 similar elementary operations.
For final-year pupils of primary school (the multiplication table had been completely learnt a year and a half ago) the next limit values had been obtained: Addition - 11 minutes 02 seconds and not more than 4 errors.
Subtraction - 11 minutes 35 seconds and not more than 4 errors.
Multiplication - 9 minutes 46 seconds and not more than 3 errors.
Division - 9 minutes 06 seconds and not more than 3 errors.
This level of elementary mental computational skills is a good means for determination of pupil's preparedness for successful studies.
The limit values of the considered parameters define the first threshold of school math's learning ability.
The pupils who have not crossed this threshold are doomed to poor progress.
The prediction of their failure in math will be right in about 95 cases of 100.
It must be noted in conclusion that the level of elementary mental computational skills of actively working pupils only do not decreases in due course.
If a pupil uses a calculator instead of mental computations, works passively at lessons and does not carry out home works himself/herself, then the level decreases gradually.
In certain cases it leads to difficulties in math's learning.
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