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After the first part of this article had been published, I began to get letters with another question. Is a need of improving the skills of simple mental computations so important, that we must spend time on filling in tables or on any other exercises, while we have not enough time for other topics of curriculum? My previous articles (Elementary Mental Computational Skills and Success in School Math, Prognosis of Failure in School Math) contain the detailed answer. Now I want to cite one more reason.  

If we grudge the time for improvement of simple mental computational skills, we will waste much more time when teaching other math topics. Let us consider addition, subtraction, multiplication and division of the numbers expressed by several figures. Every such operation calls for carrying-out many simple mental operations. For example, it is necessary to carry out 14 simple mental operations to multiply 587 by 96. My studies show that pupils spend about 8 seconds in the average on each operation. We can reduce the running time of one operation by training to 2.5 seconds in 95% cases. As a result we will have much more time for development of more complicated skills. If we take into account that all topics of arithmetic and algebra require simple mental computations, the profit becomes evident.

The stochastic tables described in the first part of this article may be used both for individual work with one pupil and for work with a class. Furthermore for individual work you can use two computer programs, which you can download free at my site Prevention of Failure in School Mathematics (references – Simple Test and Improvement of Simple Mental Computational Skills). Now you will find there renovated versions. I tried to finish with errors and difficulties which prevented effective work with the applications (at present, for example, there is no need in entering code). You can test whether the modifications are sufficient or not.

The first of these programs is a simple computer test for diagnostics of a level of simple mental computational skills. It can be used both to clearing up whether a pupil’s skills need to be improved and to make sure that using of the stochastic tables turned out successful. If your pupils have learnt the multiplication table already, offer them this simple test and you will see that many of them will not pass it. A pupil must implement a sequence of 64 simple operations (addition, subtraction, multiplication and division in disorder) not only nearly error-free but quickly also.

The second program is designed to improvement of simple mental computational skills. The application is intended for individual work with one pupil. Therefore it is necessary to load a private database for each pupil. You may begin to use the application after the multiplication table and the corresponding cases of division have been learnt completely. It will be very useful to repeat the work every year during next five years.

The program performs the next functions: 1) diagnoses quality of elementary mental computational skills; 2) carries out the work on improvement of elementary computational skills; 3) watches the psycho-physical state of a pupil and a level of permissible educational load; 4) allows overseeing all results of the working.

During each testing a level of simple mental computational skills is studying. Two criteria are used for the ascertainment of it: number of errors and an average running time of one operation. The speed of mental computations is one of the two criteria of automatism – the top quality of skills. While a minimal number of errors is permissible (an error may be caused not only by lack of knowledge), a testing will end in failure because of slowness even if there are no mistakes. The values of parameters using in the testing were figured out experimentally in accordance with pupils’ age (during five years after the multiplication table had been completely studied).

Initial results are the basis for the following work. They are used for the examination of pupil’s psycho-physical state (warming-up) and for forming the tasks for improvement of his/her skills (working upon errors). To accelerate progress in mental computations, you may print the tasks for working upon errors after each session. The pupil must do the tasks in written form before the next session. The program stops the work with the pupil when his/her computational skills meet the established demands. Number of required sessions depends on pupil's grounding in math and mental faculties.