Some thoughts on education and mathematics
Energy then is morally neutral, It is good depending on the direction it takes.
Our task is to provide legitimate outlets for it.
Only by providing satisfactory activities for citizens can delinquency, vandalism, crime and violence become exceptional
rather than normal
The effectiveness of education depends on adaptability to individual differences.
Rigid syllabuses and lock-step teaching ignore the real forces in the mind that make for development and learning
and the resentment of the gifted when drill continues after a skill has been learned is very often evident. The
problem is how to get practice of necessary skills without causing such resentment, a problem often neglected by
some modern mathematics' reformers.
The appeal of arithmetic to infants is usually self-evident and recognising unusual mathematical maturity is not
difficult. The unjustified fears of some educationists about allowing children to forge ahead, needs discussion
and recognition of the need for young mathematicians to work in depth and at speed.
My concern has always been to promote joy in mathematics for all-ability pupils, and the driving force has been
a feeling that must be shared by anyone who has derived great pleasure from any subject - intense regret that the
subject in question is so often presented in a way that produces dislike, worry and a sense of failure.
t seems to me that it is not necessary to have a separate theory for the gifted;
a correct approach to them follows naturally from a sound philosophy of education in general.
An excellent foundation for such a philosophy can be found in William Blake's poem, 'The Marriage of Heaven and
Hell'. In this poem, of course, Eternity, Heaven and Hell have nothing to do with life after death. Eternity is
Blake's symbol for what to-day would be called the unconscious, and Heaven and Hell are forces within the unconscious
mind. Heaven stands for the forces of reason, of restraint. Hell stands for Energy - incidentally a much better
word than instinct, libido and other modern labels. Why Energy should be equated with Hell is clear enough when
we consider the evil caused throughout history by the misdirected energy of human beings. But Blake insists that
this same Energy is also the creative force behind everything positive and good. The business of Reason is therefore
not to dam Energy back but to guide it towards satisfactoy expression. As Blake puts it, 'Reason is the bound or outward circumference of Energy'
The words 'outward circumference' suggest the image of hosepipe - Energy surging up from its source and reasorn
like a hand holding the hose and directing the jet.
Energy then is morally neutral, it is good or bad depending on the direction it takes. Our task is to provide legitimate
outlets for it. The provision of such outlets is important in three ways - for society, for the individual and
for the learning of subjects. For any society that wishes to remain civilised this is the highest priority; only
by providing satisfactory activities for all citizens can delinquency, vandalism, crime and violence become exceptional
rather than normal. This is a priority of which our dominant institutions seem totally unaware.
For the individual, finding a satisfactory outlet for energy means a sense of fulfilment
and escape from frustration. For the learning of subjects it is the driving force without which little will be
To a conventional nineteenth century educationist this last contention would have seemed ludicrous. As it was then
seen, adults had to work in factories at jobs they did not particularly enjoy, and in the same way schools were
places where teachers made children do things the children did not want to do. It would have seemed fantastic to
suggest that children should derive the same kind of satisfaction from their work that artists, musicians or poets
did from the highest types of creative activity.
Anyone who has ever been gripped by enthusiasm for a subject or a hobby knows the infinite difference between the
way the mind works in such a situation and the way it works when we are engaged in something that does not appeal
to us. This difference can be observed even in the 'good pupil' who is anxious to please but is not attracted to
the subject itself or its presentation. Our unconscious energies are not at our beck and call.
The evidence for this is seen everyday, but we do not take it seriously. You are troubled; friends say, 'Don't
worry'. This is asking you to perform the impossible; you cannot stop worrying by an act of conscious will. Indeed,
it is probably easier to stop someone else worrying than to stop yourself; you may introduce some topic for discussion
in which your friend is keenly interested or get some intense physical activity started. To stop yourself worrying
you have to do the kind of thing you would prescribe for your friend. In short, Energy can be redirected but only
if we have a realistic appreciation of its mechanism.
What applies to worry holds for many other mental factors - laughter for instance. It is no good my saying, 'I
am going to tell you a joke and I want you to try hard to laugh at it'. Laughter is something that happens to us
like falling in love - or becoming interested in a subject.
Accordingly, it is quite futile to expect effective work at the conscious level if there is no force acting lower
down to produce it. The first duty of a teacher is not to talk but to listen; to try to understand the direction
the energies of each pupil are taking and not to expect activity in places that Energy has yet to reach.
These considerations are particularly important when we are dealing with the gifted for in them the drive to develop
in a particular way may be particularly strong, particularly tenacious, particularly inflexible. The life of Charles
Darwin is an outstanding example of this. From a very early age he showed great interest and ability in nature
study, which then was not recognized as a school subject. In school and university he was regarded by his father,
by his teachers and apparently by himself, as a failure. He spent much time talking to pigeon fanciers, an admirable
preparation for his later work on natural selection, though it is impossible to believe that he realized how wisely
his underground controls were directing him. He simply knew the compulsion was there. Perhaps the first question
about the gifted in the minds of educational administrators is 'Do we
need to make any special provision for the gifted? Are they not so clever that they will work out their own salvation?' The life of Darwin indicates clearly the damage that can be done if the curriculum is
unduly narrow and inflexible. There can also be damage if the rate of progression is rigidly laid down; I found
striking evidence of this in the pre-Sputnik California of 1957.
At that time the lock-step was the prevailing fashion in American schools, except of course where an enlightened
teacher made special arrangements. All pupils were expected to work through the same book at the same rate. There
were historical reasons why the arithmetic textbooks had a rather strange composition. In the formative period
of American education, classes contained immigrants from many countries with many different languages; the primary
teachers had the responsibility for teaching them the American language and instilling a sense of their new nationality.
These were substantial tasks and it was not unreasonable to allow four years for their completion. In this way
it came about that an arithmetic syllabus, that by itself would have fitted comfortably into four years, was extended
to eight. What no one seemed to consider was that a time would come when children would have absorbed both the
language and the national sentiments of the U.S.A. before they came to school. The result was a 4-year intellectual
vacuum in the arithmetic curriculum. The Grade 8 (age 13+) textbook was the worst; it was entitled Making sure of Arithmetic, which I took to mean
that the youngsters would not meet any new idea in that year. Bill Glenn, a mathematics supervisor in California
took me on a visit to a school. We found a boy working outside the Principal's office. Why was he there? They had
to put him outside the class room; he was totally unmanageable. Bill Glenn told me such situations were quite normal.
He talked to such boys and had usually found them well above average intelligence. He summed up his experience
by saying, 'Grades 5 to (ages 10 + to 13+) are the grades in which the
superior student becomes a superior delinquent'.
A few years ago I was told of a situation in London which showed the importance of thinking in terms of a student's
energies rather than a prescribed timetable. A girl was causing great trouble in a secondary school; among other
things she was swearing at the Vice-Principal. I asked what she was interested in and was told she had a passionate
interest in the French language. Suppose then I am teaching a mathematics class to which this girl comes. What
is my duty? To set her some problems in trigonometry? Surely not; this would leave the situation totally unchanged.
The first thing to do is to convey to her that I accept her need to develop in a certain way. For a few lessons
I would let her work at French in any way she wanted to do. Later, as confidence grew, I might ask her to translate
into English some parts of a French book on mathematics or the history of mathematics, for the benefit of other
students in the school. Gradually some of her feeling about French might spread to mathematics.
This situation interested me because, in a milder way, I had experienced the same problem with the subjects reversed.
I was not violent about it, but from the age of 8 to 16 I worked ineffectively at French, in a way that caused
contempt in my teachers and embarrassment to myself. The trouble was probably due in part to the fact that I had
been introduced to French at the age of 8 by a teacher who left me completely cold. It was not until I went to
university and had to read mathematics in French that I found to my surprise that I could do it and that I enjoyed
it. Since that time I have of my own free will studied a number of other languages, a thing that would have seemed
totally incredible to me in my early youth. At the age of 8, mathematics already had an intense aura of romance
for me. I used to go round second-hand bookshops looking for mathematical textbooks that I could afford. If a teacher
of genius had realized what was happening inside me and had told me that there were important books on mathematics
available only in French, or even better had lent me a book about mathematics in French, my whole attitude to the
subject would have changed overnight.
This indicates the kind of approach that is necessary if interest is to flow from one subject to another. Of course,
it is not always desirable or necessary to make such a change. it is important, however, to examine with the utmost
logical rigour, whether the subject you happen to teach is relevant to the life-goals of the student concerned.
Generally speaking, you are not likely to do much harm if you show a student the interest of some topic that previously
had been regarded as useless or unpleasant.
The question of drill
There is no doubt that practice is necessary if mathematical skills are to be developed. The question is how is
such practice to be brought about? The commonest complaint from the abler pupils is that if they finish a set of
exercises ahead of the rest of the class they are simply given more of the same kind, which they find very boring.
The best way to avoid boredom is to find interesting problems in which the old routines are used incidentally.
Supplementary work should either contain problems of this kind or else introduce new ideas and methods. These new
ideas do not necessarily have to be anything extraordinary. Recently two boys complained to me of boredom with
Algebra exercises in class. I gave them questions, of which involved solving simultaneous linear equations while
others led to solving quadratics by completing the square. These surely are trite enough, so long as they were
learning the procedures the boys found the work quite interesting. Their teacher agreed to let them work at such
problems when they had completed the normal class work.
Routines, even quite commonplace ones, are not herently dull so long as they retain the quality of novelty.
A few years ago the question of drill was at the centre controversies about the mathematics curriculum. In the
past, examiners found it easier to test a candidate's ability to perform some standard process than to test understanding
of it and awareness of its purpose. An appreciable number of teachers mistakenly decided that the best way to beat
the examiners was to subject pupils to long, repetitive, soul-destroying drill. Reformers, rightly enough, sought
ways of ending this and, as often happens, went too far. In Britain the most widely used new curriculum, The Schools
Mathematical Project (S.M.P.), essentially abandoned all attempt to teach algebraic manipulation. Teachers had
already been thrown off balance by some of the more extreme propoganda for 'Modern Mathematics' by certain pure
mathematicians who seemed to believe that the entire population was preparing itself for advanced work in topology.
Some of the reformers believed that if algebra was Dt taught in the earlier years it could easily be picked p at
the age of 16. This was a psychological error of the rst magnitude as was illustrated by an experience I had ith
gifted students. A small group of these, aged 15 ~ars, began to come to our home on Saturday rornings and continued
doing this throughout the year ntil they left secondary school. They went to a omprehensive school that gave every
encouragement bright students to forge ahead. They had already ome knowledge of calculus. However, they lived in
the ounty of Cambridge, which had wholeheartedly dopted the S.M.P. books and curriculum. In spite of ieir brilliance
they were completely paralyzed and norked at a painfully slow pace whenever anything wolving the manipulation of
symbols was required. hey had interest in sciences for which algebraic xpressions were fundamental and I tried
to find uestions that would interest them and would lead to lgebraic work. For instance, one of them was president
(the school astronomical society and we considered he design of a reflecting telescope. Suppose it has the hape
of the parabola y = x2 if light
falls on it parallel to he y-axis, at what point will it be brought to a focus? Over the years we worked in a variety
of ways, sometimes together, sometimes on different problems, sometimes one member having the responsibility for
reading a book and guiding the others through it. Various topics were covered - calculus, partial differentiation,
differential equations, Fourier series, complex variable, polyhedra, groups, quantum theory - and gradually their
facility in the incidental calculations grew. But it would have been much easier for them if they had a stimulating
introduction to algebra at the age of nine and had regularly worked interesting problems fom then on.
Co-ordinate geometry, incidentally, is an excellent subject for such work. It gives practice in algebra without
the boredom of drill and it calls for analytical thinking to express in algebraic terms a situation specified geometrically.
A variety of possible attacks have to be considered and evaluated, and good training in problem solving is necessary.
Often, too, there is an artistic satisfaction in the resulting diagrams. The students were intrigued by the shapes
of the curves with equations such as:
x5 + y5- 5x2y = 0 and y2
+ 2x3y + x7= 0, which they sketched with the help of Newton's
Diagram of Squares. These curves have loops which suggest some strange apparatus for an alchemist.
My group also did some introductory work on more recent topics such as metric spaces (including Hilbert space),
functional analysis and topology. It is clearly desirable that students should have a window open towards the mathematics
of the present century. What is quite wrong is to create among teachers and the wider public the impression which
some advocates of 'modern mathematics' gave, especially in U.S.A., that everything known before 1900 was obsolete.
In fact, the most fruitful results are obtained by the skilful combination of the new knowledge and the old.
Mathematics in the early years
G. H. Hardy wrote that a good mathematical discovery is simple, surprising and fruitful. In these terms it is easy
to see the fascination of mathematics once a learner has penetrated some way into the subject. There are striking
patterns, unexpected coincidences, proofs that give sudden insight into why something happens, and so forth. But
an obsessional interest in mathematics can arise long before this stage. Recently I worked with a 6-year-old boy
who was causing some tension in the family circle; he would wake up his father about 6 am. and demand long lists
of 'sums' to do. It would be of interest to analyze the appeal of mathematics at this stage, when arithmetic consists
largely of standard processes. It seems to be an intensified form of something all children feel. Somewhere in
The Gateways of Learning
Margaret Drummond tries to discourage primary teachers from bringing fairy stories into arithmetic to make it interesting.
For young children, she says, numbers are more interesting than fairies. To some extent we can account for this.
Much of adult life consists in the acceptance of illogical compromises and children prefer to be logical. If 'houses'
is the plural of 'house', surely 'mouses' should be the plural of 'mouse' - but it is not. Numbers do not play
tricks of this kind; they are law-abiding and reliable.
Children who retain this sense of delight are continually on the look-out for arithmetical
information. They are pleased if they come across the fact that a number must be divisible by 9 if its digits sum
to a multiple of 9, or that you can tell whether a number divides by 8 be examining the final three digits. They
carry out little explorations in arithmetic and remember the results, so that numbers acquire separate individuality.
They may remember that 111 is 3 x 37, because this result is mildly striking, or that 1024 is what you get if you
start with 1 and multiply it by 2 ten times. Their extra familiarity with numbers appears when they are allowed
to choose the complexity of a task. If you are doing a trick and ask children to 'think
of a number' the majority will choose a number less than ten; once, however,
I was doing this and a child chose 512. Quite recently I gave a gifted 7-year-old a programmable calculator to
work with while I attended to older, but less arithmetically minded, children. It was programmed to give a test
in mental arithmetic, with subtraction or addition. To begin with the user had to indicate the largest number that
might appear in the question. The boy chose 500.
Once in Toronto the question arose, in a class that had never met multiplication tables, of how many eights there
were in 32. The following remarkable answer was given; 'There are two-and-a-half
eights in 20 and there are one-and-a-half eights in 12, so there must be four eights in 32'.
An example of the eagerness with which new ideas are picked up was given by the early-rising 6-year-old I mentioned
earlier. One day, completely out of the blue, he volunteered, 'I know
what 91 - 1000 is'. He gave the correct answer. I asked, 'How did you learn about negative numbers?' He
answered, 'From the weather forecast'.
Enrichment and 'acceleration'
Often I have found among those who are concerned to help the gifted a great fear of covering some topic that is
in the syllabus for some future school year. The idea is that to do so is storing up future boredom for the student
in question. This shows a complete misunderstanding of the situation. It is clear from the examples above that
we are dealing with the activity of the child. A school may help this activity, or hinder it, or in extreme cases destroy it, but the basic process
is the child's own activity. The rate of progress that should result has nothing to do with syllabuses, or lesson
plans, or rigid schemes of any kind.
This is clear enough from examples within the experience of all of us, but is even
clearer if we look at the childhood of the really great. As I mentioned in connection with my translation from
the German of F. W. Ostwalds's writings on the gifted  Emmy Noether at the age of four pointed to four rows of brass nails in a chair, each row
containing 12 nails, and said, 'There must be 48 nails here'. K. F. Gauss had learned to read when he was two years' old and apparently picked up for
himself what numerals stood for. Before his third birthday he pointed out an error in his father's accounts. It
would be interesting to know how the opponents of 'acceleration' would have kept Gauss occupied until he found
himself in a class where numbers were added, and what they would have had Emmy Noether do until she found herself
in a class officially entitled to pass beyond the multiplication tables.
It is impossible to arrive at correct conclusions if one starts off by thinking of education as something done
to a child by a teacher. The excellent book Mathematics in Primaty Schools  emphasises that children learn by their own activity;
our concern is not expressed as the improvement of teaching methods, but as creating a better learning situation.
In fact, the whole aim of education is not to keep children dependent on information communicated by a teacher,
but rather to teach them to read, to use libraries, to seek out the sources of in formation for themselves.
Mathematics and things
For many years it has beeen recognized in principle, though rarely applied in practice, that mathematics can be
made more intelligible if it is taught, rather like physics, as a laboratory subject . I have worked in this field, and have in the past mainly thought of it in connection with
the less academic students in secondary schools. A recent experience has made me think that this should also be
considered as a possible approach for the gifted. I was teaching selected groups of gifted children in Saffron
Walden, with ages from 8 to 10 years old. I had been doing some work on coincidences with numbers, which on other
occasions I had found led to great interest, but which was clearly falling flat. Accordingly, I gave out voting
papers on which the children could arrange possible topics in order of preference. At first the girls voted for
puzzles and problems; some boys voted for electricity and some for other topics. At this stage the general impression
was that I was going to talk about the mathematics of electricity. When they found that we were actually going
to use apparatus, with a few transistors and other components, they all left their former choices and came to electricity.
Much of the activity, of course, was simply playing about with the equipment and making it do interesting things.
This is the permanent desire of the gifted, and probably of all children, that something spectacular should happen.
But there were mathematical threads; observing that the current though an ordinary flashlamp bulb was 6000 times
too strong to be measured with a 50 microamp meter, or studying the relation between capacitors and resistors needed
to make a rather primitive electronic organ from a multivibrator.
The teacher.pupil ratio
In the last year I have been giving voluntary help to schools both with gifted groups and with remedial classes.
It has struck me that sometimes the greatest service I have done for a teacher is take a rather turbulent individual
into a separate room. Teachers sometimes describe these individuals as temperamental, sometimes as hyperactive
- which may mean little more than full of life. Channelling this exuberant energy so that it ceases to disrupt
the work of others is an extremely difficult task for a teacher of a large class. In a slightly different context,
Homerton College in Cambridge from time to time provide events for mathematically gifted boys and girls. Their
experience was that such boys and girls came up with a brilliant idea
every twenty seconds, and it was not practical to work with less than
one teacher to every two students.
It is clear that a great increase in the quality of life would occur if we could improve the teacher-pupil ratio.
In the present economic climate of the world, such an aspiration appears hopeless. However, most official economic
thinking relates to an age long dead. It is concerned with greater efficiency of production. But it was evident
in 1930, and is still more evident in the age of the micro-chip, that the problem is not to produce but to distribute.
The main social problem is to keep people occupied; the main economic problem is to spread incomes so that people
who need things can afford to buy them.
Of course this is not going to happen in the immediate future. With the present arrangements, firms and countries
have no option but to seek to be more competitive by 'de-manning' and by heavy capital investment in robots. But
by so doing they are destroying their own and their competitors' markets, and bringing the world closer to a situation
in which the robots produce masses of goods which the investors do not want to buy and others are unable to. Sooner
or later something must give and economic arrangements adapted to modern conditions be introduced. Then a very
much enlarged educational system will be one of the ways of providing an income and a meaningful activity for many
citizens. Until then we have to improvise as best we can.
Anyone who has ever been gripped by enthusiasm for a subject or a hobby knows
the infinite difference between the way the mind works in such a situation and the way it works when we are engaged
in something that does not appeal to us.
The first duty of a teacher is not to talk but to listen; to try to understand the direction the energies of each
pupil are taking and not to expect activity in places that Energy has yet to reach.
The commonest complaint from the abler pupils is that if they finish a set of exercises ahead of the rest of the
class they are simply given more of the same kind, which they find very boring.
1. Ostwald on Education. Mathematics in School, Volume 10, No. 2,
pp 28-33. March 1981.
2. Mathematics in Primary Schools. The Schools Council, Curriculum
Bulletin No. 1. H.M.S.O. 1965.
3. 'Two Avenues to Advance in Mathematical Education', Bulletin of
the Institute of Mathemdtics and its Applications, Volume 16, pp.
146-149, 1980: also published in Mathematics Teaching, 1980, pp.
4. For details and justification of this apparently wild remark, see my
article 'Oscillations in Systems of Mathematical Education', Bulletin
of the Institute for Mathematics and Its Applications, 1978, Volume
14, pp 259-262.
The following books have been written by W. W. Sawyer
Integrated Mathematics Scheme, Book C.
(Bell and Hyman, London, 1982.) 36 investigations for children in the top 3% ability range. For ages 10 to 14
A Path to Modern Mathematics
(Penguin Books, 1966 Translated into German, Japanese, Polish, Russian.) This shows how modern abstract theories
grew from classical mathematics.
A Concrete Approach to Abstract Algebra (1959, Freeman, U.S.A.
A First Look at Numerical Functional Analysis (Oxford University Press, 1978.) A detailed account of how modern theories can be applied to numerical
work with computers.
Mathematician's Delight (Penguin
Books, 1943) This deals with the central ideas and processes of calculus and trigonometry. Suitable for bright
thirteen-year olds; gifted younger than this.
Vision in Elementary Mathematics (Penguin
Books, 1964. Translations into Dutch, French, Italian, Japanese, Rumanian.) A pictorial approach to algebra, as
taught to a class of bright 9-year-olds in U.S.A. Suitable br gifted at an earlier age.
Prelude to Mathematics (Penguins,
1955. Dover reprint, 1983.) This deals with mathematics that is novel, surprising or apparently impossible! Secondary
The above article originally appeared in Gifted Education International 1983 Vol 1 pp 65-69 Copyright © 1983
A B Academic Publishers
Copyright © W. W. Sawyer & Mark Alder 2001
This version 20th February 2001