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.
If you want to know more about the implemented study, you can go to my site, Prevention of Failure in School Mathematics. You can find there a description of the study with some figures and diagrams, a test for diagnosis of elementary mental computational skills and a description of two effective ways for improvement of the skills:
- The testing tables which are an effective means for training work. Their using lets to bring up quickly the elementary mental computational skills to the level exceeding the calculated permissible limits of the considered parameters.
- The teaching computer program for improvement of elementary mental computational skills. It makes a diagnosis of a level of the skills; carries out the work on improvement of the skills; carries out control by a psycho-physical state of a pupil and by a level of permissible working load; allows controlling the results of working.
For confirming the influence of quality of elementary mental computational skills over success in school mathematics bad achieving pupils (from one to three years after the multiplication table was completely studied) were chosen. The work upon development of the skills, which level had been very bad, was carried out with each of them individually. In 85% of cases the level was brought to stable correspondence to the calculated values of parameters. The results exceed the limit values of the parameters significantly. After that a work upon main basic topics of school math (common fractions, algebraic equations and so on) was carried out with the pupils. The work was successful, and all of them had not big problems in their subsequent math's learning.
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.
MAIN INFERENCES
- Unsatisfactory development of the elementary mental computational skills adversely affects progress in school mathematics. The skills must be driven to automatism. That is the obligatory condition for success in school mathematics.
- A level of the skills may be determined by two parameters: average time of implementation of one operation and relative frequency of occurrence of errors.
- For testing it is handy to use the standard tables, for which the permissible limits of time and errors were calculated, as supplied on the web site...
Victor Guskov, PhD. Pedagogical Sciences, is a teacher of mathematics in Rumania. His site, Prevention of Failure in School Mathematics, offers free downloads of the mental math software used in the studies - which you can use yourself or with your kids.
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