We all make mental calculations from time to time, though we may not always be aware of it. In deciding at exactly what moment and speed to venture across a busy road, for example, our mind judges continuously the positions and speeds of several vehicles and accurately finds the required gap in which to move forward. If our mind can make such complex judgements as this it is certainly able to manipulate a few figures. It is the cumbersome calculating devices we have probably been taught, which require pencil and paper or calculator to work out because of their difficulty, and a lack of encouragement for mental calculation which have prevented us from becoming efficient mental calculators.
This book demonstrates that this need not be so: mental calculation is easy and to be preferred to pencil and paper or calculator, and has many advantages over these calculating methods. This introduction describes these advantages and presents the case for mental calculation.
Most people would probably agree that mathematics holds a special position among subjects of study: that it possess qualities of absolute certainty and precision which cannot be attributed to any other subject. On the other hand however mathematics is seen as difficult and remote by most people: the same people who are also aware of its special absolute qualities. This situation has come about because mathematics education has not been effective enough in bringing out the real nature of mathematics. As young students we glimpse the beauty of mathematics but this is usually a passing phenomenon.
Though mathematics has applications at many levels it is primarily a mental subject. This being so it is likely that lack of mental calculation is partly responsible for the situation described above, and that a system of mental mathematics could provide students with a lasting link with the realm of mathematics and also engender a deeper understanding of the structure and processes of mathematics, as well as helping to develop other important personal qualities.
In Favour of Mental Mathematics
The following points outline the benefits available from a mental approach to mathematics.
1. Mental calculation sharpens the mind and increases mental agility and intelligence. This will be evident to anyone who has practised or taught mental calculation or who has seen its effects.
2. It enhances the precision of thought. Numbers and other mathematical objects are unbiased, giving only one correct answer to which everyone will agree: there is never a contradiction. This absolute precision is unique to mathematics, so dealing intimately with numbers as we do in mental calculation we cultivate fine and careful thinking.
3. Mental calculation leads naturally to the search for, and discernment of, constancy and law, which are very necessary attributes in a swiftly changing world. This point is expanded in the next section on mental algebra.
4. Our mind has the ability to retain several ideas at once so that they can be compared, combined and so on. This facility is enhanced by mental calculation as we practise holding the sum in the mind whilst operating with some of the figures.
5. Mental calculation improves the memory. Memory depreciates if it is not exercised. Short term, medium term and long term memory are all stimulated by mental calculation.
6. Because numbers are absolutely dependable and reliable, calculation promotes confidence. In particular, mental calculation creates confidence in oneself and in ones capabilities. To solve a problem, perhaps a difficult one, by mere mental arithmetic without having to rely on some artificial aid is a source of great satisfaction and encouragement.
7. Mental calculation is a delight to the mind: the intrinsic qualities, relationships and beauty of numbers and the way they create new numbers out of themselves is a source of great enjoyment.
8. Through mental calculation one becomes familiar with numbers and appreciates their various properties. This leads to a real understanding of number.
9. In calculating mentally the subtle properties of numbers and their relationships are appreciated much more readily than if the calculation was written down and thereby fixed. Thus mental calculation leads naturally to innovation and to the invention of new methods, thereby developing the student's natural creativity. This point is developed in the section on problem solving.
10. Practical uses of mental calculation are many, since we all need to make quick, on the spot, calculations from time to time.
Thus we see that mental calculation has so many advantages and really brings mathematics to life as well as providing motivation and strengthening and enlivening the mind. This is because numbers are mental concepts, they do not exist on paper. Our mind operates very fast and has a variety of operational properties. With proper training we can use these properties of the mind to our advantage.
This is not to say that pencil and paper or calculating instruments are to be totally avoided in mathematics: they certainly have their place, but mental calculation should, it is suggested, be the primary method of calculation.
In playing with numbers we find patterns. These patterns delight the mind because they indicate that some deeper, more general law has been found. And this means that we can use the law or pattern to our advantage. We may see that the square numbers
1 4 9 16 25 ...
increase by odd numbers, for example. This pattern may be seen later as part of a greater pattern. This process of generalising from more specific knowledge is mental algebra.
It is well known that algebraists use symbols, but each symbol and each algebraic statement expressed in symbols is just an idea: the letters are the means to convey or store the idea. So the mental calculator can do much advanced algebra, but by directly generalising from the numbers themselves: the algebraic terminology is not necessary for this. Of course, this can usefully be combined with formal algebra: the laws discovered mentally can be formulated in the usual algebraic way, and the students should be able to see an arithmetic technique which they know in an algebraic identity. In other words they should be able to translate between their mental generalisations and algebraic formulae. In this way the algebraic symbols would come to life instead, as is often the case nowadays, of the algebra being seen as totally alien and unintelligible.
Problem solving is considered by many educators to be the main aim of mathematics. Mention has already been made of 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 utilised however if the emphasis is on mechanically recording the steps of a mental process.
Problem solving seems to arise in the space between mathematical topics. When one topic has been mastered this is the ideal time to relate it to other areas of mathematics previously learnt. 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 computers 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 the machines increases and their size and price decrease. Unfortunately this has lead to a reliance on the calculator for simple calculations: the student automatically reaches for his calculator as soon as he sees an addition or multiplication has to be done, finds 13×3, writes down the answer and then realises that it was really obvious. Or worse still, in multiplying one third by 3 the student 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 discover for themselves the laws of number 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 a 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 Vedic Mathematics (see next section) 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. Not 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 13×3 when he knows the answer it is the calculator that is wasting the time because if it were not there he would put the answer straight down.
Vedic (pronounced Vaydik) Mathematics was reconstructed earlier this century from ancient Indian texts called the Vedas. This reconstruction is the work of Sri Bharati Krsna Tirthaji (1884-1960) whose book "Vedic Mathematics" is currently available.1
Vedic Mathematics provides a coherent structure for mathematics: the Vedic methods are beautifully interrelated and complementary. While modern mathematics is a hotch-potch of unrelated techniques, bewildering in their complexity, the Vedic system offers unifying and natural principles whose effect is to transform mathematics into an easy and delightful activity. To take a simple example, the general method of multiplication in the Vedic system enables us to give the product of two numbers in a single line from right to left or from left to right using a simple pattern. And this is easily reversed to provide simple one-line division. By contrast the modern methods of multiplication and division require many cumbersome steps and are far from being either simple or complementary. Furthermore, the Vedic system offers many methods of multiplication (as this book demonstrates), division etc. There are special methods which may be used for special types of sum, and there are general methods. This adds to the fun: instead of having just one method which must be applied we have a choice, we apply whichever method we like or think is easiest. It is the rigidity which has been erroneously ascribed to mathematics which is in part responsible for the low regard in which mathematics is held by many people. The Vedic system, with its mental approach and its variety promotes flexibility, innovation and creativity and brings mathematics to life. There is a certain amount of disagreement nowadays between those who believe in the traditional teaching methods in which the children are taught by the teacher and practice it, and modern practical methods of discovery. Both systems have their advantages. By taking a mental, and therefore a practical approach to mathematics and by teaching unifying principles the Vedic system reconciles these apparently opposing teaching styles.
The Vedic system rests on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These formulae are given in word form such as "On the Flag" and "Vertically and Crosswise". Being given in word form each of these formulae has a wide range of application and Sri Bharati Krsna Tirthaji says that they cover all branches of pure and applied mathematics. The Sanskrit word "Sutra" means "thread" and it appears that these sixteen formulae and the sub-formulae run like threads right through the whole of mathematics, giving a unification to the subject, or rather, showing the unity that is already there. This wholeness which is a feature of the Vedic system, and its use of pattern recognition, activates the right hemisphere of the brain, thereby enriching the student's practice of mathematics, instead of using only the analytical left hemisphere.
Careful study of these Vedic formulae shows that they may have deeper levels of meaning than the purely mathematical, and this could explain how it might be possible for them to have such a unifying effect in mathematics. It would not be relevant to go into details here, but to give some idea we may just take "On the Flag", the title of the first chapter of this book. It is a very common mental activity to hold an idea "on one side" whilst we briefly think about something else, then when we wish we can bring this idea back into our conscious mind. This is a natural mental function. The memory button on a calculator has the same function as do carry figures in a calculation. In fact all written mathematics is "held" on the paper for us so that we do not have to remember it. Other Sutras express other fundamental and natural functions such as succession, reversal, balance etc. So these Sutras could represent natural functions of mind, which we all use. If this were so the Vedic system would necessarily be the most efficient, easiest and most enjoyable mathematical system possible. Those who are familiar with the Vedic system will know that it certainly does manifest these qualities, and Vedic Mathematics has been called "Mathematics with Smiles" as a consequence. These formulae therefore also provide us with a useful way of classifying the various mental calculation techniques which are the subject of this book: each of the nine chapter titles is a Vedic Sutra. We may also mention that the formulae may be combined and applied consecutively or simultaneously.
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 lightening calculator who spoke about his abilities at a special meeting of the Institute of Civil Engineers in 18562 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 unconscious in time: Bidder describes multiplying 2-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.
E.W.Scripture,3 a psychologist and one of the first to make a study of calculating prodigies, refers to this unconscious activity: "after considerable practice I was able on the sight of two figures to add or subtract them before they attracted my full attention; in other words while they were yet in the field of consciousness they aroused the proper association and the result entered the focus of consciousness first".
F.D.Mitchell,4 who also made a study of mathematical prodigies, noted that "as the process gradually becomes more and more familiar and automatic, many of the intermediate steps of the computation may partly sink into the background of consciousness, perhaps even disappearing altogether from the field of attention".
The great mathematician Karl Friedrich Gauss was also a rapid mental calculator who was aware of, and also described, the unconscious but reliable process of calculation.
Truman Henry Safford was able, like many other rapid mental calculators, to cast his eye over long fences of 147 or 274 posts and give their number, and
Jedediah Buxton, in addition to his other extraordinary abilities in mental computation could accurately estimate large areas of land by walking over them.
Professor Aitken, a fairly recent lightning calculator, described to the Society of Engineers in 19545 (in a talk entitled "The Art of Mental Calculation") the mental activities he was aware of during calculations:
but mostly it was as though they (the numbers) were hidden under some medium, though being moved about with decisive exactness in regard to order and ranging..... I have noticed also at times that the mind has anticipated the will; I have had an answer before I even wished to do the calculation; I have checked it, and am always surprised that it is correct. This, I suppose (but the terminology may not be right), is the subconscious in action; I think it can be in action at different levels; and I believe that each of these levels has its own velocity, different from that of our ordinary waking time, in which our processes of thought are rather tardy.
These descriptions refer to activity at different levels of consciousness, and to the increased accuracy at deeper levels. We have all probably experienced simultaneous activities in our mind, but at different levels.
The art of mental calculation lies in using the natural propensities of the mind, or rather in allowing the mind to operate naturally. And it seems that the most natural activity is also the fastest, most accurate and most efficient.
Our mind can be programmed like a computer. We have all developed highly useful internal programmes which we can activate for walking, tying shoe laces, brushing teeth and so on: activities which were difficult to learn at first but which gradually became automatic. Our mind has assimilated each sequence of actions as a whole and assigned this to an unconscious area from which it may be recalled at will. Consequently we can, for example, walk, bounce a ball and carry on a conversation all at the same time. If we did not have this faculty we would be in serious trouble. By assimilating simple and pleasing mathematical techniques our mind can give us the result whenever we desire to use the technique: no force is necessary, only quiet observation.
Suppose you know the beautiful and simple "Vertical and Crosswise" multiplication method described in chapter 6, by which any numbers can be multiplied together in one line. Suppose that you are familiar with this and that you have two 2-figure numbers to multiply. The next step would be to start the calculation and most of us would jump in and start working. However, if you do not allow your mind to start acting that way you may find that you have the answer almost immediately. You may or may not see the "calculation" that preceded it. And this is not surprising: our mind has assimilated the complex techniques required for reading, articulating ideas into speech etc. so it can certainly apply the simple Vertical and Crosswise pattern to multiply the numbers and just give us the answer. Everyone is potentially a lightning calculator. Our problem is, of course, that we interfere: we want to DO the sum because this is what we have been taught. Perhaps any child who can acquire the skills of speech and writing, if placed in an environment with efficient mental calculators would become an efficient mental calculator. There is nothing really unusual about this, the ability to calculate effortlessly and its consequent advantages (noted earlier) could easily be developed in our educational system
The Unified Field
Let us take this one step further. In considering the shorter time interval between problem and solution at deeper levels of consciousness we may consider whether there is an ultimate level, at which presumably, results are instantaneous.
Theoretical physicists have been trying to reduce all the various forces of nature to a single "unified field", which unites them all. Maharishi Mahesh Yogi, founder of the Transcendental Meditation programme has offered the ingenious postulate that since the unified field is necessarily the ultimate level of existence it can be experienced subjectively by bringing ones awareness to its deepest level. This is achieved by Transcendental Meditation. Thus the ultimate reality which scientists arrive at objectively through theory and experiment, can also be directly experienced.
And we see a similar thing in calculation: that which can be solved by computer, pencil and paper etc. can also be solved directly, by the natural operation of the mind. Calculation by pencil and paper or calculator is the objective, external method, mental calculation is the subjective, internal method.
Mental calculation can be carried out at different levels: we can rigidly apply the steps mentally which we would expect to write down, or in a relaxed state we can allow the mind to operate naturally, this state being far more enjoyable, accurate and efficient. Perhaps at the ultimate level problem and solution would coincide. Our progress toward more and more efficient computation therefore depends on our ability to operate from deeper levels of our consciousness.
A considerable body of research (over 500 studies) shows that the practice of Transcendental Meditation brings the mind to a relaxed but alert state and also that by familiarising the mind with the process of integration through this practice creative insight is enhanced and deeper understanding is acquired.6
Teaching Mental Mathematics
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, Ampère, 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 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 his idea about teaching mental arithmetic, and his suggestion to teach numbers before numerals is very interesting. Do we teach the symbols which represent numbers too early, thereby abstracting the number concept before a real appreciation of the numbers themselves is crystallised?
In Vedic Mathematics 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 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 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 been learning 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 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.
About this Book
This book sets out to show something of the fun, variety and potency of mental mathematics. It also illustrates the system of Vedic Mathematics. If the sixteen formulae of Vedic Mathematics cover all of mathematics, all the various types of multiplication (for example) can be classified under these headings. The book deals mainly with multiplication but includes some addition, subtraction and division. Multiplication is considered, especially by mental calculators, to be the fundamental mental operation because unlike addition and subtraction it reveals the properties of numbers. All the great mental calculators were able to multiply large numbers together.
In the Vedic system only tables up to 5×5 are needed (although tables up to 10×10 is assumed here in chapters 1 to 3). It will be found however that the mental calculator naturally acquires higher products through practice. Bidder, by arranging shot in rectangles, taught himself the multiplication tables up to 10×10: "Beyond which I never went; it was all that I required".
Although some chapters refer to and use methods from previous chapters it will be possible for most people to read the book in any desired sequence. Algebraic proofs of the various techniques are given at the end of the book.
It is hoped that some pleasure will be obtained from the variety and beauty of the devices shown in this book and that it will encourage some to take up and teach mental mathematics and the remarkable system of Vedic Mathematics.
I would like to thank Ulf Linér for his encouragement during the writing of this book and for his many helpful suggestions. The quotations from Professor Aitken's address in the Introduction are with the kind permission of the Society of Engineers (see Reference 5), and the quotations at the beginning of the chapters are mainly from E.W.Scripture and F.D.Mitchell (References 3 and 4).