To teachers and parents: if your children have problems in mathematics, read this article, please. Maybe the cause of the difficulties is very simple.
For success in school mathematics it is necessary to master elementary mental computational skills at first. This statement is obvious not only for teachers. Everybody knows that addition and subtraction within the limits of 20, multiplication and division within the limits of 100 are the foundation of all next arithmetical and algebraic topics. But my practice shows that a level of the skills, which maybe is suitable for primary school, very often is insufficient for secondary school and not suitable for online universities and regular colleges.
During last twenty years I investigated why some pupils can not study mathematics successfully. Now I am sure the first of the causes is poor mental arithmetic. If elementary mental computational skills are not enough good, a pupil has no chances to understand and master more complicated topics.
How can we diagnose a lack of the skills? The answer seems very simple. Mental computations must be swift and errorless. We may say that the skills must be driven to automatism (the top quality of skills) which means quick and errorless mental implementation of the simple arithmetical operations.
Thus the computational speed is the first criterion of the automatism. Meanwhile an error may be caused not only by lack of skills. There are many other outside causes - a bad condition of a pupil, a brief distraction of attention and so on. Therefore a probability of occurrence of an error, which must be sufficiently low but not equal to zero, must be taken as the second criterion.
The results of my study allow determining permissible limits of the average time of implementation of one operation and relative frequency of occurrence of errors while a pupil implements a sequence of simple uniform operations. All pupils who had not reached the limits could not learn mathematics without big problems. They could not work at lessons of full value and do homework themselves. Their knowledge and skills were very bad. In contrary, in those cases when it was possible to improve their elementary mental computational skills, they began to make progress.
Here are two effective ways for improvement of the skills:
It must be noted that the level of elementary mental computational skills of actively working pupils only do not decreases in due course. If a pupil works passively at lessons and does not carry out home works himself/herself, the level decreases gradually. In some time it leads to difficulties in maths learning.
Thus a level of elementary mental computational skills is a good means of 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. It means that results of testing of the skills may be used for prediction of failure in school mathematics.