SWEDEN
Fairer Schools: Fairer Society
For the special edition of the educational journal
Utbildung & Demokrati 2005
of Örebro University, Sweden.
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Unlimited
competition leads to a huge waste of labour and to a crippling of the
social conscience of individuals. Our whole education system suffers
from this evil. An exagerrated competitive attitude is inculcated into
the student, who is trained to worship acquisitive success as a preparation
for his future career. I am convinced there is only one way to eliminate
these grave evils, namely .. an education system ..orientated towards
social goals.i
When
mathematicians present their work, they are permitted to hide what may
have been years of painful struggle, all the errors and blind alleys
that they have had to fight through, to produce instead a seamless flow
of faultless logical precision. When I was young, and realised that
this is usual for mathematicians, I was indignant. All the time
these wicked people probably made as many mistakes as me! Later I learnt
that tidying away confusion is not such a bad habit. As a young engineer,
I was taught that my bench must be cleared of all dust, dirt, and tools
at the end of every day. The metal bin in the corner of the workshop
was there to receive failed attempts. All that should be left for final
inspection was finished work: edges sharp, curves smooth, all facets
polished and oiled. Perfection was expected - not trials !
Philosophers, too, like to give the impression
that their thoughts have emerged from a flow of effortless insights,
and that all they have to do is to explain their significance into a
finely organised text.
There is, of course, a good reason for omitting
mistakes. They get in the way of explanation. But in teaching a subject,
especially a subject like mathematics in which perfection is so highly
prized, mistakes are not just of interest to the students of its history,
it is essential that those studying the subject are at least allowed
to know that they are a part of the process - and that they will make
them too. How else will they learn to master disappointment, to benefit
from errors, and to hold to a course through uncharted thoughts? We
now know that science which is ever thought to be complete is wrongly
understood. We still look for perfection - but not for finished perfection.
Still the aim, it is no longer the end. Where generations of scientists
have sought, and have often claimed to have achieved it, we know that
science is a search which probably has no end. Some scientists are even
beginning to suspect that mankind seeks truth for a different reason.
And now I am older, I know that there is another
reason why mathematical ideas - and those of science - must be presented
in the most succinct form. It is not for those who find them to crow
they are masters of the universe, but rather to show younger minds the
standards of science and the challenge they must meet. But I have also
come to realise that presenting mathematical ideas like this, as finished
perfect statements, also creates a serious problem: a contradiction.
We want our pupils to see the elegance, economy, even the beauty, of
these ideas in their final, polished form. But we must wish that some
at least will want to continue; and then they must know that almost
always this will be a messy, often hazardous adventure. It will require
of them resilience, determination, self-confidence and courage. How
to warn them of this?
Mathematicians appear to have the hardest task:
for it is often said of them - the cruellest challenge - that most achieve
their best results whilst young. In physics, we may depend on Richard
Feynman's famous refusal of other responsibilities: that good physics
requires 'really solid periods of time'. And perhaps this is why philosophers
tend to be older. The aim of philosophy is not only to improve our knowledge
of the world around us - so that it can justly claim to intersect with
most other fields of thought - but to improve our knowledge of the world
within ourselves. When philosophers believe they have enough of this
knowledge, then they should be able to suggest ways to improve our understanding
of the causes and the pitfalls of society itself. This is nearly Marx's
definition: 'The philosophers have only interpreted the world in various
ways; the point is, to change it.'ii
The last, of course, is tricky. Good philosophy
may require so much 'solid time' that no individual may hope to achieve
it in a lifetime. It may need a period of evolution, with ideas being
passed on to generations. Saying what good philosophy is, or what emerge
from this metamorphosis, is rather like asking which tree in the forest
will outlast the rest. No-one can tell. We may try suggesting that good
philosophy should help a society last longer. But survival is also hard
to predict. Will our civilisation survive the next hundred years - or
ten? Recently I discovered in a class of eleven year olds brought up
on a diet of catastrophe stories, that the majority thought that it
might not last their lifetime.
What we may therefore do is to define the
attempt to create good philosophy. This definition may be much the
same as in other fields. The good attempt should not just echo, or summarise,
or plagiarise, or even - for actually this is very similar - only argue
against or about what has gone before. It must surely be an attempt
to improve on orthodoxy: not by overturning everything, but rather by
improving the obvious, making it easier to understand, more inclusive,
easier for more to use.
And this is most difficult of all. Many in the
past who have tried it are now known to have improved their society's
philosophy - but only eventually. Quite often, they have not lived to
see it - or, if they have been too forceful, have appeared too much
to interfere with established powers, the attempt may have cost their
life. Where it is accepted, it is perhaps the greatest benefit that
the philosophy of science has achieved that it does try to prevent such
heretics from being exiled, poisoned, crucified, hanged, stoned, or
tortured to death. The journals of record may still refuse their ideas;
but not being published is still better than being roasted alive.
I have never set out to be heretical. I have
invariably hoped to be thanked - just modestly - for my efforts better
to understand the world, even when, rather like Mister Andersen's little
boy, my contribution has been not to see what so many others see and
admire. In this present case I know I can annoy many people who contribute
to mathematics education. But I do not see what many seem to admire.
I am also bound to annoy a few who have been enjoying the effect on
others of their prestige as philosophers of sociology. I do not see
that they know their subject very well. In this respect I am on the
side of currently the most famous modern French sociologist Bernard
Latour, who I heard respond to the question: 'How can sociology be improved?'
- with the reply: "By making it more like a science." iii
Mathematics education is not a science: it is
about a science; and the most shocking discovery I have made in teaching
mathematics for nearly thirty years is that virtually no teachers of
mathematics know how bad their teaching methods really are, how seriously
they damage democracy, and how badly and irretrievably they change society's
structure.
I do not see it as an intellectual success when
only a fraction of our pupils end up actually competent in thinking
mathematically. I do not see that its teaching benefits societies morally
when it rewards obedience, ignores dishonesty, and ignores dissent -
or, rather, calls it laziness or stupidity. I do not see that education
benefits children spiritually, when first it robs most of their autonomy;
then removes their compassion or their self-respect.
I will stop here to save your nerves. Compared
with huge sums spent on modern education, the Emperor's new clothes
cost little and did little harm. But Andersen's fable should be told
to children daily. It is not entertainment. The essence of science is
to try to see reality as it really is, not just as the powerful wish
us to see it. Our education systems are infinitely more costly and harmful
than his Emperor's attempt to be admired. It is true that universal
education offers everyone access and instruction. But the benefits stop
there. The minority who achieve real competence in mathematics - or
almost anything - are highly valued. A larger number who appear to have
some competence - again in almost anything - even when this is accompanied
by little understanding, are valued too, for they can watch dials, follow
instructions, even manage large organisations, with an apparently equal
degree of ability. But for those who fail to learn a rôle to suit
any of the modern re-enactments of Hans Christian's moral fable, the
effects are life-long - but not satisfactory.
Why is mathematics so important? Because in
advanced societies competence in mathematics has become a universal
test of social value - and it cannot be escaped. If other routes are
available to social preferment, it does not matter. For the majority,
it does matter. I agree entirely with those (H.Köhler; P.Ernest;
O.Skovsmose) who insist that some mathematical ability is essential
for useful citizenship. I disagree with others who argue that all methods
of teaching are equally good, provided only that the teachers are equally
competent,
This is not true. When taught through instruction,
pupils are obliged to depend on understanding through a language not
their own, but their teacher's. A second source of failure appears.
By the middle years of secondary school three distinct groups of students
have already appearing. The lines of demarcation between them are hardening.
They are regarded by teachers as responsive; passable; and useless.
What is rarely noticed is that these groups do not share a common language.
The first, with a fair degree of competence, has begun to use the language
of instruction. It will soon meld perfectly with their thinking. The
second can use this language, but it will remain for them the language
of authority, never their own demotic. The third, increasingly frustrated
and resentful, is also beginning to define its identity by cultivating
a far cruder demotic, even less articulate, but gloriously ugly and
distinctive: Fuck the fucking fucked-up fuckers: they might say. Many
come from homes where there is never any thoughtful conversation. They
cannot even copy it. Disadvantaged from day one, they will be dropped
in the bin.
Ah! you may say: but all of this can
only be true of bad instruction. I will argue that all teaching
by instruction to children over ten years of age is likely to be bad
teaching, unfortunately more prevalent than you may believe. In most
schools it remains the main form of teaching from first to last.
Since this is also what we teachers find familiar, we are reluctant
to see its faults. '.. [T]he existence of certain political institutions,
and of a general climate of opinion that allowed and even promoted fundamental
criticism, does not mean, of course, that the entire gamut of popular
beliefs would be [so] scrutinised, or that every manifestation of the
irrational - including those in philosophy .. - would be exposed. ..
[T]he power of rational arguments to uproot deep-seated convictions
is only a limited one.'iv
That was the historian Geoffrey Lloyd, writing
of Plato's Athens. Many are eager to point out that these early democracies
were far from ideal. But in one respect their citizens were more fortunate
than us. Rational arguments had not destroyed their convictions that
lives can be purposeful - or that such concepts as honour, truth, trust,
loyalty, valour, and dignity have real and essential meaning. For many
youngsters, perfectly 'rational' arguments - as well as their own experience
- has made these concepts dependent on who uses them, and why, and when.
Accompanied by the twisting and fracture of moral structures within
and between the three groups - as well as between them and authority
- this relativism reduces most of these words to nothing that is common
to them all.
What is created is a kind of spiritual sickness.
Its symptom is a hatred of others; in many, hatred of self; overall,
a hatred of life. Democracy fails on the way: for it requires that others
treat us with the respect that we offer them. (The sense of this is
often reversed, but I think this is the original.) Nothing seems to
fit any more. 'The best lack all conviction, whilst the worst are full
of passionate intensity.'v
This sickness is created in the classroom. It
must be treated there. Political solutions are as band-aids to AIDS.
You may think it unfair to blame virtually all of the political
ills of society on the most non-political teachers: of mathematics.
There are many reasons why people can lose belief in themselves, can
lose belief in truth.. There was a time when anyone who caught a cold
could blame standing in the moonlight, crossing a fairy pathway. Now
we know that common symptoms usually point to a common cause. In the
case of a cold, a virus. There are many different viruses, and many
tens of thousands of mathematics teachers. Common to viruses is the
method of attack. Common to teachers, generally, is the method of instruction.
Most schools use this method all the time. Its
great benefit to both the school's managers and teachers - but also
to their inspectors - is that once teachers have completed their instruction,
no-one can blame them for their pupils' failure. It can only be the
pupils' fault. They must have been too lazy, or have missed too many
lessons, or too - let us not be harsh - challenged intellectually.
This is in every sense a bad method. Not just
a bad alternative. It is criminally, hopelessly stupid - and deeply
cruel. Simple bald instruction, each step carefully and uselessly recorded,
certainly protects all but the most appalling teachers and their schools
from being blamed for failure, but it is absolutely the worst way to
learn how to understand anything.
Instruction does not create understanding. Understanding
is achieved through discussion, through argument, and through patient
individual thought, best provoked by discussion. School managers, inspectors,
ministers, and their civil servants, should know this. If they do not,
they are incompetent. My belief is that precisely the people who could
insist on change, for mostly private reasons, do not care. Usually they
were in the middle category of student. They want selfishness to continue
to succeed; contempt for their weakness to continue to punish those
who 'have not tried'; honesty to remain what you can get away with.
They believe this is normal and that it has profited them. They think
it will profit their children. They will not change. If there is to
be a change, it must be because teachers become ashamed - like me -
of what they are doing to their pupils.
The Change
There
is a powerful connection between mathematics teaching and democracy.
It has always been there, ever since the Greeks began to make a system
of the mathematics. But the democracy was there long before anyone had
the notion to make mathematics a system. Without this beginning, in
a society which had already succeeded in making democracy systematic,
it is possible mathematics would have remained a hermetic science guarded
by a jealous elite. The Egyptians managed this for at least as long
as Western history. To make mathematics a common language for all the
sciences Islamic scholars took the next important step by importing
from India the modern form of number, with which all could calculate.
The effects on the secular and spiritual history of the West have been
immense.
I am very conscious of the honour of being invited
by your University to explain again, first, why I think these connections
exist and, second, why they can produce either a very healthy or a very
dangerous state of society. Recently a major survey of young people
was published in Britain. The majority are miserable in their lives,
doubtful of their future, distrustful of authority, prepared to be dishonest
when necessary, and not interested in politics. I shall argue that virtually
all of this dysfunction comes from education. Only ask young people
in which subject were they most disappointed at school. The majority
will say: mathematics.
The usual response to my first claim - it will
usually be more doubtfully rephrased: that mathematics has any connection
with democracy - is that it is obviously absurd. "Of course we
work democratically!" a seriously irritated mathematician snarled
at me many years ago. "How else do you think we could work without
murdering each other?"
I have forgotten his name, but this was a clear
acceptance of the principle and I was grateful. "But," he
went on, "we don't actually vote, y'know, to agree on our
results. That works rather differently."
He missed the point. Mathematics is not democratic
to keep mathematicians from each other's throats. This is incidental.
It doesn't always work. Sir Isaac Newton is remembered for exulting
at the death of his rival, Leibniz. (Leibniz published in 1684; Newton
in 1687.) Others no doubt exchanged congratulations at the death of
Georg Cantor. Neither his mental or physical health survived the malevolence
with which distinguished mathematicians had attacked his ideas .vi
Didier Nordon has pointed out that mathematicians
are just as different as everyone else.vii
Mathematics is democratic because the arguments
typical of mathematics are never intended to command obedience. No matter
how great the authority or reputation of those who create them, they
are placed before an audience to persuade it of their truth. They invite
questions, they provoke criticisms, they challenge to be disproved.
But this should be directed at the argument, not the arguer. The aim
is not to overpower resistance by bullying, or vehemence, but to achieve
an agreement through marshalling evidence, patiently connecting the
familiar with the less familiar, until either a wholly surprising result
appears, or it becomes evident that at least one of the original assumptions
is wrong. Any mathematics lesson different from this is not a mathematics
lesson at all. Plato imagined the attempt of his old tutor Socrates
to persuade a young slave to recall some mathematics that his mind had
forgotten but which his soul had not. Socrates succeeded. It is not
clear that this was good evidence of Plato's theory of knowledge, as
he believed. What is certain is that this is amongst the first recorded
demonstration of a good mathematics lesson.
I cannot claim to be the first to have noticed
that the arguments of mathematics are not given to us as instructions
but as requests for our assent,. Here is André Lichnérowicz,
one of France's most distinguished algebrist, writing, as I found it
later, in 1988, almost nearly ten years before my first efforts appeared
in print:
'Mathematics was created for us in Ancient Greece
by men who conceived a type of argument which would be without misunderstanding
or ambiguity, an argument capable persuading any kind of person, citizen
or slave, Greek, metic, or barbarian; an argument capable of achieving
agreement because its very form forbids disagreement.'viii
The italics are mine. Although he comes within
a hair's breadth, what is interesting is that Dr Lichnérowicz
still does not make the connection with democracy. Possibly the political
connection is far from his mind. Here is Dr Sune Bergstrom, a Swedish
Nobel Laureate in Medicine, once again describing the consequences without
once using the word 'democracy'.
'The traditional boundaries between various
fields of science are rapidly disappearing. The scientists of the world
are forming an invisible network with a very free flow of scientific
information - a freedom accepted by the countries of the world irrespective
of political systems or religions. The scientists have become close
to creating the 'open world' that the Danish Nobel Laureate Niels Bohr
as a prerequisite for peaceful development in his famous letter to the
UN in 1950.'ix
I am sure there is an excellent reason for this
timidity - especially real in the case of mathematics. For virtually
fifty years an extraordinary range of sciences: from aeronautics, through
biology to chemistry, geology, physics, of course, and on to oceanography
and zoology - all, sooner or later, were drawn into the web of secrecy
and secret funding which the Cold War created. A scientist might never
suspect, could never know, that his work was really paid for by a government.
Although there were always some fields of study
in mathematics which governments had to regard as 'assets', throughout
this period it was generally only the mathematicians who both sides
regarded as being ideologically and politically naive. They enjoyed
their own 'invisible network'. Their ideas, reports, and discoveries
could flow freely around the world. And they could also move, almost
as freely, themselves.
This is a highly valued asset of professional
mathematicians. Mathematicians can go, come, write, speak, correspond,
conference - without anyone giving two hoots about security. The possibility
that I might create the smallest cloud of suspicion that the way mathematics
is taught - in thousands and thousands of classrooms everywhere, completely
unobserved, beyond political control: that this might have social and
political consequences. No, no, no! The anger that I provoked was serious
emotion. It was intended to discourage such a dangerously silly idea.
But, of course, it never was a silly idea. Classical
historians have known this for a long time. The original purpose of
the kind of argument that we now call mathematics was not to do mathematics.
There was too little of it and it was much mixed up with magic. Its
original purpose was to rescue democracy from rhetoric. Here again is
Geoffrey Lloyd, who was the first to confirm my suspicion that this
knowledge had got lost in the great gulf between young sciences and
the older arts:
'The new professionalism in the art of speaking,
provoked hostile reactions from such writers as Aristophanes and Plato...The
citizens of Athens had ample opportunity to exercise their judgement
of skilful argument: but by the end of the fifth century [BC] they were
also being frequently warned, by different speakers, and in different
contexts, not just against those who set out to make the worst appear
the better cause, but also more generally against rhetoric itself...
Claims to particular wisdom in other fields beside the political were
similarly liable to scrutiny, and in the competition between the many
and varied new claimants to such knowledge those who deployed evidence
and argument were at an advantage compared with those who did not.'x
By now - as you may have gathered - I had begun
to think that we should start to treat this subject of mathematics teaching
seriously.
What is that you do to our children in your school!
Ideas
- for most of us - do not come up from nowhere one at a time, perfectly
spaced and sorted, like pearls on a string. Sometimes it is like having
a dumper truck dropped on one's head. Sometimes it is like hearing the
jump of a flea. To accept the first, one must be built like a bomb-proof
bunker, to detect the second, be even more sensitive than the flea.
And they do not occur in any special order.
I was a soldier in my first profession, defending
democracy in Europe from the Soviet threat. I discovered that shortly
after major hostilities began both sides intended to use nuclear weapons.
This would make much of Europe uninhabitable for centuries. Eventually
this seemed such a very silly idea, that I became a teacher;
a mathematics teacher. Although very privileged in this new career,
and although I found it most preoccupying, a lingering sense remained
that democracy does always need to be defended. There are so many who
would much rather be in charge.
For twenty-five fascinating years I was one
of the highest paid mathematics teachers in the world, working in one
of the twelve official European Schools. These twelve Schools form perhaps
the most important experiment in education in Europe since Charlemagne
ordered his officials to learn to read and write. I found this important.
But I had been teaching for about ten years when one day one of my pupils
hit me with the dumper truck. Despite my hard work: examples on the
board; clear instructions; tireless demonstrations; careful marking;
testing; correction; a new red ink pen every month - some of my pupils
always failed. Not all, of course: the majority usually did well. Why
did always about the same fraction fail every year? After five years
a class could only very rarely be taught together. Usually it would
be divided into an upper and a lower stream. This was still far better
than in some schools. Most people seemed to think it natural. 'Aptitude'
they called it. Some children have it; some less; some none.
I had just finished one of my demonstrations,
the examples filling the board behind me, had set the class to work,
and I was enjoying just a few moments peace.
"I've finished." said a voice. I examined
its source with an entirely insincere appearance of pleasure. An exercise
book was placed on my desk and its owner: a smartly dressed eleven year
old, placed a peremptory finger on his calculations: "There they
are."
And there they were: and all correct. Clearly
lots of aptitude. I ticked them off whilst I wondered what else to give
this little blighter to do. Then I had an interesting idea. "I
tell you what," I told him. "Why don't you go back to your
desk, and write down what you think your brain is doing when
you do these!"
I had not the least idea what the result might
be, but given that young Einstein would be working on it, it could be
revealing. What was more revealing was to see his happy smile disappear
like chalk under a wet cloth. He literally recoiled. "No fear!!"
he blurted: "If I try to do that, I'll start getting everything
wrong!"
Of course the suggestion was unfair. But this
reply was truly a revelation. I stared at him astonished, then with
delight. He stared back indignantly; snatched up his book and marched
back to his place.
My mystery was solved. It was in his reports:
first in maths all through primary school. But suddenly I knew what
it was that made him so clever: 'He's doing them automatically!'
This 'natural aptitude' was just the ability to copy, almost always
without fault, all of the actions that I showed the class. This
aptitude had nothing whatever to do with understanding. When I asked
a class, as I often did: "Do you understand?" what they believed
I meant was: 'Do you understand what to do?' and not what does the doing
of it mean!
For ten years I had been looking in the wrong
direction. He had just shown me what was wrong with my teaching. But,
by inference, I now knew what was wrong with almost everyone's teaching!
Being satisfied with my pupils' ability to copy my actions - which is
how most children learn nearly everything; by never properly investigating
what they understood of the reasons for their actions; by mainly
rewarding their accuracy in imitating ...
This was appalling. I was simply driving them
into a cul-de-sac. A few would begin naturally to detect some reason
in what they were doing. They would continue to succeed. For the rest,
however, the trap had almost closed. For them, everything depended on
their memory: in this case, you do that; in that case,
you do this; and so on. This was all they really knew.
If you listen to children showing each other
what they have learnt, this is what you will hear: just instructions
to do things. If you ask: why do you that - they will falter.
"But that's what you're supposed to do. Isn't it!"
They do not know any reasons. Reasons are strange and frighten them.
Like rats in a maze - I do not mean to be disrespectful,
this is what we create - by now they must memorise hundreds of responses
to simple stimuli. They do not look ahead. Even as they act, neither
stimuli nor responses have much meaning: except that pressing levers
like this produces rewards; following routes like this
avoids shocks. Eventually, however, but inevitably, around the age of
13 to 15 years, memory is not enough. The levers demand untested sequences.
The maze has too many turnings. The shocks become too painful. There
are no easy rewards.
So now they must choose. They begin to conceal
that they really do not understand what they are doing. Usually they
will find allies in this stratagem in their own teachers, anxious that
not too many will fail. Each year this can be avoided by making tests
as similar as possible to the last. As well as this dishonesty they
also begin to find that selfishness is an asset, for this helps make
clear the real inferiority of the weakest in the class, whom they will
now neither help, nor treat them kindly, nor allow into their circle.
And these last poor souls, confused and isolated, are usually the ones
we teachers say: 'did not try hard enough.'
The next shock, less violent but more painful,
came within a year. I was leaving a meeting with a primary colleague.
The meeting was to discuss the future of a young boy she had once taught.
She was a good friend of mine. His future was bleak. Once bright, eager,
confident and trusting, he had turned sullen, dishonest, disruptive;
he was about to crash and burn. Suddenly she stopped. So I stopped.
"What is it that you do in your school to our children!"
She asked, then she left me standing.
To explain more precisely what we do, I wrote
a book, which I took to a friend, Dr Philip Stewart, an Oxford don and
polymath (not all are both) for comment. He read only a part when he
stabbed a line with his finger and told me: "No-one has written
that before."
It was my turn to be surprised. I protested:
"But everyone must know that must be true." He shook his head
impatiently. "No. You know it because you teach it: But
-" and this was the scholar speaking: "no-one has written
it down!"
I cycled home through Oxford in a daze, repeating
to myself. "They don't know! They don't know! Everywhere in the
world they're teaching mathematics: and they don't know what they're
doing!" What I had written, obvious it seemed to me, was: "The
early Greeks developed democracy at the same time as mathematics, because
both use the same kind of argument.' And they don't know.
On the way home I decided, it was time they
did.
A year or so later, I did a brave thing. German
achievements in mathematics and other sciences are famous. At a conference
in Bavaria I was introduced by a smiling chairman to his cheerful colleagues,
to whom I said: "Gentlemen, I have come here to tell you that if
your predecessors had taught mathematics in Germany in the 19th
century correctly connected with mathematics, your country would
not have lost its democracy twice: first to the Kaiser; and then to
Adolf Hitler."
Their smiles disappeared. But they heard me
out. Then they published my paper: first in English; then German. It
was later published in France. Just three years after this, Dr Köhler,
who had come to my rescue in those first fearsome moments of silence,
directed the first Socrates conference in Stuttgart for the EU. He chose
its title: Mathematikunterricht und demokratische Erziehung.xi
I thank you for reading so far. It is a large
topic to tackle in one essay. I hope this has not been too muddled.
In the summary I offer a description of the method that I have used
with my classes for many years and always with satisfying and enjoyable
success. I could tell many anecdotes, but one I remember with special
pleasure is of another little boy whose class I had been teaching like
this for less than a year. Towards the end of a lesson he came to ask
if he might tell me something in private. I agreed, wondering what it
would be.
When the rest his class had disappeared, he
leant closer, and whispered in my ear: "You know this method you
are teaching us: reading the book aloud, and then listening to ourselves?"
I did. "Well," and he leant even closer, to whisper even more
fiercely: " What I've found out is: it works with other subjects
too! "
A Summary - and The Method
In
1999 New Scientist published an article of mine to which the editor
gave the title: Class Talk. In bold type he printed the sentence: 'Maths
teaching can hardly be said to be politically neutral.'
Mathematics teaching has inescapable moral,
social, and political effects on the widest scale. Depending on the
method of education, it will either preserve the stratification of society
in mutually uncomprehending classes, each with a different morality,
social structure, language, all inimical to each other; or it can promote
a kinder, more compassionate, cohesive society, whose young people have
learnt to work and to think together with common goals, to share a common
morality, structure, and language, and to accept each others' natural
differences with patience, understanding and compassion. The latter,
I suggest, is naturally conducive to a healthy diverse democracy, the
former cements in place social distinctions and structure which an older,
cruder emphasis on personal position, influence, and power has created,
and which they will work to sustain.
Mathematics education is so powerful in technically
advanced societies for several reasons. It has immense intellectual
prestige. But ability in mathematics has also become the common instrument
used to measure technical intelligence. Conversely, it is used to decide
which others lack this intelligence, who will not achieve it, and who
therefore - unless they have or find other means of social preferment
- may be treated as having lower social value.
The method selects the outcome. When taught
by instruction, the success of which being measured by testing individuals
privately, the outcome will be a small number who may learn to understand
its logic; a larger number who find that obedience, even without understanding,
is equally rewarded; and the rest who can neither understand, nor obey,
nor reproduce results sufficiently well to be allowed to continue. The
moral effect, generally, is to persuade the first group that their ability
is of the highest order; the second that obedience is also highly valued;
and, finally, there must be those whom the system has systematically
humiliated and terrorised, and who it now discards. The social consequences
of this destruction of innocence, dignity, and value are all around
us.
The alternative is learning through discussion.
This depends on understanding knowledge as a network of associations
built up over time in every individual mind through critical discussion
of ideas. The source of these ideas is not the teacher. It is the text-book
which they can take home every day. Everything the pupils need to know
is read by them one after the other, line by line, out of their own
textbook, always aloud. "And what do you think that means?"
the teacher asks of every line; and every answer offered must be in
a pupil's own words. Unpredictably but repeatedly, real mental
effort is required of everyone. In this way the meaning of the text
is discussed and defined by the class together, never monopolised by
the most forward pupils. Everyone has their chance. When their understanding
finally satisfies them - and their teacher - they choose the exercises
to test themselves; do them, and mark them or correct them.
Working together like this, connecting ideas
slowly together, children learn to enjoy the effort of co-operation,
to be patient with each other, to be respectful of the difficulty of
formulating and of combining their understanding. This is democracy
working!
There is also, I think, a glimpse here of a
cure for the spiritual sickness I mentioned earlier: for, although spiritual
reality has been long a matter of indifference to science, it is just
sharing understanding - which is what science is itself.
That must be another story.
i)
A.Einstein,
US Monthly Review, 1949.
ii) K.Marx, Theses on Feurbach, 1845.
iii) In colloquium at Oxford University's Said
Business School, Oxford, 2002.
iv) G.E.R. Lloyd, Magic, Reason and Experience,
CUP, 1979, p.263.
v) Irish poet W.B.Yeats, The Second Coming,
1921.
vi) Marcus Kline, Mathematics: The Loss of
Certainty, OUP, 1980.
vii) See, for example: Norden, Les Mathématiques
pures n'existent pas!; Actes Sud, 1981,1993.
viii) ibid
ix) Obituary, The Times, died Aug 15, 2004.
x) G.E.R. Lloyd, ibid.
xi) ed. H. Köhler, Landesinstitut für
Erziehung und Unterricht Stuttgart, 1998.
Colin
Hannaford,
Oxford, 20th September 2004