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Problem solving is considered by many educators to be the main aim of mathematics. Mentioned already is the effect of mental calculation in developing innovative capabilities, and so this would appear to be an ideal way to develop problem solving skills.
It is peculiar to arithmetic that once we have knowledge of how to count, we could, unaided, construct the whole science. Many famous lightning calculators have, in the past, developed remarkable talents of this type without any formal mathematical training at all. Since also arithmetic develops naturally in an extremely varied manner, mental arithmetic offers enormous scope for many delightful problems ranging from the very easy to the very difficult. Thus problem solving has considerable scope and much to offer of educational value. The vast creative potential and speed of the mind cannot be fully utilized however if the emphasis is on mechanically recording the steps of a mental process.
Problem solving seems to arise in the space between between mathematical topics - when one topic has been mastered this is the ideal time to relate it to other areas of mathematics previously learned. This also provides coherence and unity in education. Even very young children enjoy the challenge of being thrown back on their own initiative by being asked a question slightly different than the ones they are familiar with or to relate their new understanding to knowledge previously acquired. And since arithmetical problems can be extremely simple it is possible to begin acquiring problem solving skills at an early age.
Push-button calculators and computer are in widespread use nowadays and play an important part in our lives. This will undoubtedly increase in the future as programs become more sophisticated and the speed of machines increases and their size and price decrease. Unfortunately this has lead to a reliance on the calculator or simple calculations: the student automatically reaches for his/her calculator as soon as he/she sees an addition or multiplication has to be done, finds 13x3, writes down the answer and then realizes that it was really obvious. Or worse still, in multiplying one third by 3 the students finds 1 divided by 3, writes down the answer, clears the display, enters 0.333, multiplies this by 3 and gets 0.999(and maybe then gives the answer as 0.9). Other examples might be given relating to lack of number sense but the point is that students who are encouraged to find for themselves the laws of numbers are very unlikely to make such mistakes.
This reliance on the calculator, to do a job which our mind is perfectly able to do, must lead also to certain loss of dignity and the opposite of the confidence creating effect of mental calculation. And what do we do if the machine breaks down or gets lost or the power source fails?
As calculators get more and more sophisticated they can do more and more complicated jobs: drawing graphs, solving equations and differentiating and integrating. Where will this end? Ultimately all mathematical processes which the mind is capable of could be "taught" to the calculator. This demonstrates that we do not practice only mathematics which the calculator cannot do but that we practice mathematics for its ability to develop the mind.
MENTAL CALCULATION IN EDUCATION
The introduction of the calculator into schools was originally justified by saying that the arithmetic processes of multiplication, division, etc. were complex, boring and time consuming and that time saved could be used on other mathematical activities. However(apart from the dangers of short-circuiting the foundations of mathematics) now, with the availability of FOCUS MENTAL ARITHMETIC it is clear that all multiplications, divisions, square roots, combined operations etc. can be found in one line using simple patterns, so that mental mathematics with all its advantages can be introduced into schools and become a major part of mathematics education. Noth that we expect the children to become calculating wizards (though some might) nor would we expect them to retain the calculating powers which they do gain.
Those who have taught mental mathematics will know the fun and amusement that it creates. When the student reaches for the calculator to find 13x3 when he knows he answer it is the calculator that is wasting the time because if because if it we not there he/she would put the answer straight down.
THE ART OF MENTAL CALCULATION
The mind operates extremely fast. Unfortunately most of us interfere with the operation of our mind: we don't trust it, we want to see and check every step it takes so that we can feel secure about the result it offers us. In insisting on seeing and checking everything we cannot take full advantage of this super fast action. But it would appear that the deeper levels of activity are faster, more efficient and require less effort. Some rapid mental calculators have spoken about the mental activity they are aware of during calculation. G.P. Bidder, a lightning calculator who spoke about his abilities at a special meeting of the institute of Civil Engineers in 1856 said:
I desire, as far as I can, to lay open my mind to you, and to exhibit the rapid evolutions which it undergoes in mental computation.
Furthermore these activities become increasingly automatic, effortless and unconcious in time: Bidder describes multiplying two figure numbers together:
in what appears to be merely an instant of time; and I can do any quantity of the same
sort of calculation without any labour; and can continue it for a long period.
TEACHING MENTAL ARITHMETIC
Some rapid mental calculators and educators have been aware of the possibility and advantages of teaching mental mathematics.
I have, for many years, entertained a strong conviction that mental arithmetic can be taught, as easily, if not with greater facility, than ordinary arithmetic and that it may be rendered to more useful purposes, than that of teaching by rule; that it may be taught in such a way as to strengthen the reasoning powers of the youthful mind; so to enlarge it, as to ennoble it and render it capable of embracing all knowledge . . .
These remarks by Bidder were followed later in his talk by specific teaching suggestions: that numbers should be taught before symbols, first counting, then arranging marbles into rectangles and so on. According to Scripture "Fuller, Ampere, Bidder Mondeux, Buxton, Gauss, Whately, Colburn and Safford (all rapid mental calculators) learned numbers and their values before figures, just as a child learns words and their meanings long before he can read". Thus the child would see the properties of numbers first hand. Bidder gives several other examples in geometry and arithmetic, his message being direct experience so that discovery invites further investigation, and that proof by observation should come first.
Bidder gave his first talk in 1856 but his suggestions which revolve around direct experience based on personal observation and experimentation sound very modern.
However modern educators have not yet fully taken up the idea about teaching mental arithmetic, and his suggestion to teach numbers before numerals is very interesting. Do we teach the symbols that represent numbers too early, thereby abstracting the number concept before real appreciation of the numbers themselves is crystallized?
In FOCUS MENTAL ARITHMETIC we use the natural properties of numbers to our advantage. It is nice to have a simple general method for, say, multiplication but, as in everyday life every problem is unique, so every multiplication problem is unique and it suggests its own unique solution. By using the natural properties of numbers we are taking the intelligent and realistic approach to mathematics and thereby acquire the same attitude to everyday problems.
The capacity of the young mind is often underestimated - children have a great clarity of mind and ability to hold and remember. They enjoy using these faculties and respond when asked during a lesson to practice what they have learned without the aid of pencil, paper, etc.
Mental mathematics is very easy to introduce into lessons: a mental arithmetic test of 10 or 20 sums at the beginning of a lesson settles a class, brings their mind into the realm of mathematics, and the challenge of solving a problem by mere mental arithmetic is very attractive to children (and adults too). The sums should cover as wide a range (some may be geometrical) as soon as possible. These mental arithmetic sums and problems will naturally evolve, from test to test, the pupils will not want to hear the same kind of problem once they have mastered it, and this will naturally lead the teacher to give harder problems of the same type, to invent variations and to enter new areas of mathematics. Later in a lesson, when pupils may be working on paper, they could be challenged to give some answers mentally, and sums of this type could then be introduced into later mental tests.