How to Improve Simple Mental Computational Skills (Part 1)
Jul 28, 2009 Study Skills 1961 Views
In my previous article “Can We Improve Simple Mental Computational Skills?” I described the results of my work on improving these skills with a class of the fifth grade. After the publication I began to get letters with one question – how can we do such a work? At this time I’ll try to detail my methods.
Both for determination of a level of simple mental computational skills and for improving them I use tables including 64 uniform elementary operations. Each table contains operations on one of the arithmetical rules – addition, subtraction, multiplication or division (addition and subtraction within the limits of 20, multiplication and division within the limits of 100). I name these tables stochastic because the sequences of addends, subtrahends etc. were chosen by chance. The accidental selection of required numbers simulates spontaneous appearance of corresponding operations in ordinary computations. The big amount of uniform operations brings about transformation of quantity into quality finally. Offer pupils to fill in the tables at class (it must takes up not more than 12 minutes for one table) and at home. Gradually, one after another, they will begin to calculate correctly and quickly. At the grades after the fourth there is no need in supplementary exercises. But earlier necessity may appear to teach some pupils to add or subtract in some cases. For example, 8 + 7 = 8 + 2 + 5 = 10 + 5 = 15; 12 – 8 = 12 – 2 – 6 = 10 – 6 = 4 etc.
You can download free specimens of the tables at my site Prevention of Failure in School Mathematics (Improvement of Elementary Computational Skills, Tables). There are two possibilities for getting them. The first is a computer program for printing tables (10 versions for each arithmetic operation), which contains all necessary information for their using. The second is a folder with Word’s files, which contains 20 versions of the tables for each arithmetic operation and the brief description of their using. I transferred the tables to Word’s format specially, as some teachers had problems with the program. Certainly I tried to modify and simplify the program too (at present, for example, there is no need in entering code), but I think there are many people which prefer good old text files to any programming stuff.
So you can download and print the tables. Pupils fill in them in written form. The time from the start point to the finish point must be measured. Therefore you need a more or less good timer. When we make a diagnostics of quality of the simple mental computational skills, we must pay attention not only to correctness but to swiftness of computations too. It is a very important criterion, but we often underrate its significance. Slow mental computations are one of possible causes of failure in understanding more complicated operations: reducing to a common denominator, operations with brackets and similar terms, solving simple equations.
If you work with one pupil, it is not difficult to measure time. If you work with a class, the work has to be started simultaneously for all pupils. When a pupil gives back a table, you must fix the time and write down it. As a rule I approximate results to five seconds (of course in favor of a pupil). It is handy when several pupils bring their tables at the same time.
It is necessary to note that you must carry out training for the first test. All pupils must understand how to fill in the tables and get accustomed to implementation of big series of elementary operations. Do not hurry pupils while working – rush can increase number of errors. An optimum time will be reached by even rate of work. Each pupil will choose the suitable for him/her speed of working after a preliminary training.
Two criteria are used for estimation of a level of simple mental computations – total running time and number of occurred errors. I calculated the permissible maximum values of these parameters for the stochastic tables included 64 uniform elementary operations. If you want to know how I did it, you can find the description at my site. For example, the following maximum values were obtained for the pupils finished primary school recently (the multiplication table had been completely learnt a year and a half ago):
Addition – 8 minutes 40 seconds and not more than 3 errors.
Subtraction – 8 minutes 55 seconds and not more than 4 errors.
Multiplication – 7 minutes 10 seconds and not more than 3 errors.
Division – 6 minutes 30 seconds and not more than 3 errors.
My method of determination of maximum values gives approximation with surplus only. Therefore the permissible limits of time and errors may be considered as sufficiently mild demands. My practice shows that these limits of the parameters may be overcome significantly. Usually it happens within the limits from three to eight cycles (a cycle - one table at class, one table at home). But sometimes it will require much more efforts to this effect. For example I can describe the most difficult case from my practice occurred last year. In September my friends asked me to help Irena K., a pupil of the fifth grade (right after primary school). The first test showed that her elementary mental computational skills were very bad.
Addition – 9 minutes 50 seconds, 3 errors.
Subtraction – 11 minutes 5 seconds, 1 error.
Multiplication – 18 minutes 50 seconds, 17 errors.
At that time I did not even think about division because of awful multiplication skills. I met with Irena twice during weekends. Every time she filled in the tables on four operations of arithmetic. Furthermore I gave her the tables as homework. In total Irena filled more than 35 tables on addition, 45 tables on subtraction, 70 tables on multiplication and 60 tables on division. For short I'll list only the results on multiplication to show progress (W - the worst result; B - the best result).
Oct.: W – 11 minutes 20 seconds, 14 errors; B – 7 minutes 15 seconds, 1 error.
Nov.: W – 9 minutes 55 seconds, 3 errors; B – 6 minutes 50 seconds, 0 errors.
Dec.: W – 4 minutes 40 seconds, 3 errors; B – 3 minutes 20 seconds, 1 error.
Jan.: W – 6 minutes 40 seconds, 2 errors; B – 2 minutes 30 seconds, 0 errors.
Feb.: W – 3 minutes 10 seconds, 1 error; B – 2 minutes 20 seconds, 0 errors.
Now you can see the results obtained in March and compare them with the results of the first test.
Addition – 2 minutes 15 seconds, 0 errors.
Subtraction – 2 minutes 35 seconds, 0 errors.
Multiplication – 2 minutes 15 seconds, 0 errors.
Division – 2 minutes, 0 errors.
In conclusion it is necessary to notice that good and even excellent elementary computational skills are only the first required condition for good progress in school math. Its implementing creates a base for solving other problems, but it does not guarantee automatic removal of these problems.