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(A kinder way to learn mathematics)
I
know exactly when I first began to realize that my career as a mathematics
teacher must end soon. For over a decade I had been developing a new
way to teach children to learn mathematics directly from their textbook.
I was so pleased by one of my classes - they were 12 to 13 year olds;
but I teach all my classes like this, right up to their Baccalaureate)
that I invited one of my school's inspectors to a lesson. He was not
pleased. He was incensed. "If you carry on teaching these children
like this," he told me angrily - although he did at least wait
until they had all left the room, "in a few years time they won't
need a teacher at all! You must stop it at once!"
He had much more to say. Whilst he was saying
much more, I could only stare at him in silence. If the sides of his
head had pealed away revealing a monstrous kind of stick-insect drooling
green slime, I might have been more surprised, but not very much.
He was comparatively young for a senior post,
far younger than I, and as far as I knew his subject was geography.
He had probably taught this for less than ten years. I had been teaching
mathematics for 26 years: 24 of them at this school. I had begun teaching
at Magdalen College School in Oxford. My specialism is remedial mathematics,
so that in my section of our school I generally took the youngest classes,
then the weaker pupils in the final two years before their Baccalaureate.
For at least a decade our School had never fallen below the highest
average Baccalaureate results in our entire group of twelve international
schools. What ever did he imagine I had done to contribute to this success?
What ever did he suppose is the purpose of school? And, finally, but
not to emphasize a sordid point too much: I was one of the highest paid
mathematics teachers in Europe, possibly in the world. If I was not
expected, and even required, to do better than the average in teaching,
who would?
In the moments in which he drew breath, I attempted
to suggest that if indeed I could produce classes of 15 year olds capable
of learning mathematics by themselves - and I can; and if, at this age
or younger, such pupils are able to explaining mathematics unselfishly
to each other - and they are, both achievements might just possibly
be regarded as generally good educationally - and valuable socially
too. Children are usually taught to be ruthlessly selfish with their
own understanding in schools. They are conditioned to this route to
success; and they are, as a result, also conditioned, at all levels,
to hate those who are more able than themselves and to be contemptuous
of those who are less. Then we wonder why our societies are full of
selfish, angry, frustrated, disappointed people! If children learn instead
the pleasures of learning independence with their teachers (if they
have not already been conditioned irretrievably to hate them too) together
with the pleasure of helping - and then receiving thanks from - the
less able than themselves, they may as adults be less inclined to ignore
injustice and to treat the less fortunate with derision. Both of these
aims seemed to me valuable additions to any school's curriculum. With
me my pupils were learning both.
But none of this appeared to enter into his
notion of good teaching. He was still explaining this: how a good teacher
should write everything pupils need to know on the blackboard; then
they should copy all this into their notebooks, and then they should
take their own notes home, and there they must attempt, now of course
alone, to do the exercises which I should set to test that they had
learnt from the lesson. What lesson?
"But if," I was finally forced to
reply, "the lesson is mainly spent copying notes from the blackboard
of what is already printed in their textbook, they will not have time
to discuss what it means; some of them will make mistakes in copying;
they will make more mistakes at home, and when they find out that they
do not understand, they have no other source of information, and no
way to develop their understanding - because they have never learnt
to use their textbook!"
He waved this aside. He had another explanation
ready. If all of this was done, their failure then could only be the
result of their laziness or their stupidity. It could never be blamed
on the inadequacy of their teacher, or on their instruction. Most of
all, it could never be blamed on the school!
And he was apparently worried that my teaching
my pupils to learn by themselves might finally remove employment from
other teachers. But - I wanted to say: he gave me no time - even when
schools can get enough teachers, and even when they do teach like this,
all the evidence is: it isn't working. Universities do not run extra
classes for students - especially in mathematics - because their lecturers
like to give them. They give them because the students have not learnt
enough at school. They do not know how to learn.
But my heart was sinking. What I was doing should
never have surprised him. Published articles explaining this method
- and praising it - had appeared twice in The Times Educational Supplement;
once in the New Scientist. An entire two year European Union project
had been devoted to it in Germany. I had written its fundamental thesis
in which I had revealed - to a somewhat startled audience - the historic
link between mathematics arguments and democracy. It is delightful simple.
Mathematical arguments are not intended to command
obedience. They are intended to persuade that certain statements or
true - or, at least, that they are useful. Lessons in which mathematics
is taught mainly through authoritarian instruction: instruction which
has only to be uncritically obeyed to be rewarded; in which understanding
is an obstacle to progress - such teaching will certainly produce a
fine crop of young fascists, and eventually adult too; but very little
respect for democracy will result. This is created in lessons in which
mathematics is mainly taught through patient, reasoned, thoughtful discussion
- eventually through reaching agreement. Not only has careful university
research prove lessons like these to be far more effective in teaching
mathematical understanding, but they are also lessons in democratic
good manners. You cannot learn patience for the careful shaping of an
argument in a shooting gallery.
Many schools in Germany are now using this approach
under the direction of my German colleagues. It was also for the reason
that I had first pointed out the connection - forgotten for some millennia
- that an earlier school director, earlier a Danish government minister,
asked me to teach ethics as well as mathematics. Most recently the British
National Literacy Trust has published an article, with an editorial
praising it, in its journal Literacy Today. It also published statistics
showing that the kind of teaching he was demanding - "I am very
disappointed. I shall have to come back, and see you again!" were
his final words - is up to ten times less effective than when children
are encouraged to explain their ideas to others.
What do children complain of most at school?
Boredom; and of course not understanding their teacher's explanations.
But what is more boring than endlessly copying notes? Who is more likely
to succeed: pupils who depend on understanding the teacher, who has
tried to explain something once or thrice in the classroom whilst they
were either daydreaming or distracted; or the pupils who have been taught
to read and understand their textbook, which they can consult again
and again, alone and undistracted, anywhere, at any time?
Later our director, a lady from Portugal, previously
a highly placed state inspector of schools, let me know that I was not
to be obliged to change my methods. But I had had occasional spasms
of angina for a year, and the unpleasantness of this encounter triggered
an attack which woke me the next night, and the next. My doctor stopped
me working at once, and further investigations kept me off work for
months. At the John Radcliffe Hospital the first of the specialists
I saw, one of the best cardiologists in Britain, gave an immediate explanation
of the cause. "It's stress." he told me. "Twenty-five
years of teaching has an effect. You are feeling in the heart."
He wrote his report: that my workload should be reduced. The DfES, the
British Ministry of Education insisted that I should see its own medical
adviser in London for independent report His advice was exactly the
same: reduce this teacher's workload.
At the beginning of the next year a hastily
arranged meeting brought a senior civil servant down from London to
discuss my timetable. Before I could speak, however, he had already
told my director: "I am entirely satisfied that the School has
done everything necessary to respond to this problem." Perfectly
bewildered, I pointed out that actually nothing had been done! My timetable
hours were exactly the same as the year before.
But this meeting, it was made clear to me,
had ended.
In fact, I discovered, all my ethics classes
had also been taken from me, and replaced with more maths. The increase
of workload would be considerable. I had always enjoyed having a few
ethics lessons each week. As in my maths lessons, I was most successful
in getting my pupils to talk. One of my old pupils came back to school
one day to tell me how much she had valued this. "We never really
get asked to talk about our ideas to someone who listens seriously."
This is probably true in many homes today. The most effective way to
help children to mature is to treat them as if they are mature. To develop
mature opinions, they need to talk. Democracy needs mature opinions
too.
But I was becoming convinced - actually sadly,
and for the first time - that nothing that I was doing was seen to be
valuable by my employers. I was not regarded, as in Germany, France,
Spain and Hungary, and other countries, as a valuable innovator in mathematics
and democratic education (which I had been able to show are linked),
but rather more as a nuisance and a malcontent. I was also causing my
heart condition by working too hard! "That's obvious." the
civil servant told me. "You are doing too much beside teach!"
To show her sympathy, my director added: "And you must know that
according to the regulations, you may not do anything more without my
permission!" No effective notice was taken of the opinion of either
medical adviser, but the DfES then offered to renew my contract only
for a further year instead of the usual four: "because your medical
condition makes your future uncertain." This was the final absurd,
insulting, careless nail that made my decision for me. I was clearly
no longer wanted.
But lazy, I decided, I could certainly be. I
estimated that my work load had been increased by 20 percent. I responded
by reducing my effort by 25 percent. From then on I wrote very little
except the day's date on the blackboard. Everything was read out of
the textbook by my pupils. One after the other, line by line, aloud.
The meaning of the text, the diagram, or the demonstration was then
discussed and defined, and when their understanding satisfied them -
and me - they chose the exercises to test themselves; and they, finally,
marked their results. I explained very little myself. To any seriously
perplexed individual who came to me later for individual help, I simply
pointed out where the explanation should be found in the text. "Go
back three pages," I would suggest, "and start reading from
there." As always, I gave regular tests, and marked them accurately,
but I marked almost nothing in their exercise books. In most cases they
would already have found their errors - and corrected them.
The results of this total abuse of my visitor's
idea of my responsibilities as a teacher were astonishing. My classes
got even better. The day-dreamers woke up, and began to work. The most
irritating stopped - or at least reduced - being irritating, and began
to work. The most distracting lost some of their friends, and began
to work. My angina became sharper for some months, but drugs - and possibly
my new routine - got the better of it, and by the end of the year I
was told by my first cardiologist that I had now no problems with my
heart at all. By this time, however, my mind was fully made up. When
a further year of the same treatment at school was offered, I refused.
Towards the end of the year I was working with
a class when the door at the end of the classroom opened and two tall
young men stood there. I had last seen them both aged about 15. They
were now nearer 20. The darker of the two wore black, with a gold chain
at the neck. The taller, dressed less dramatically, loomed over his
friend like a blond colossus. Both beamed at me shyly.
'Come back in twenty minutes, lads', I told
them, and they left. Twenty minutes later they were back, and I invited
them to address the class, suggesting they start with their names. "Well:"
began the dark one. "My name is William Anastasiadis. I'll spell
that for you!" Which he did by counting off the letters on his
fingers: "A- N-A-S-T-A-S-I-A-D-I-S!" The class sat as if petrified.
"And this," he turned and pointed at me, "is the best
teacher we ever had." The blond colossus, now known to be Mr. Scott
Walker, nodded solemnly.
On the very last day one of the nicest rewards
of my career was left behind on my blackboard last week by a girl of
one my senior classes as she and her friends left me for the summer.
"Good-bye, Mar H," she wrote, "We love you." I had
not had the courage to tell them that they would not be seeing me again.
I was afraid there might be tears. And some of them could be mine. Similarly
I did not warn most of my colleagues: for which I hope I may be forgiven.
I am not really sorry not to be teaching. I
write this on the second day of their new term, and I know I shall miss
many of my pupils. I should like them to know that I did not willingly
desert them. I also want to make clear that I do not believe that I
have been treated badly: or that anyone at any time has been discourteous
or unkind. In this last year the new deputy director, a clever and courteous
Dane, went out of his way to tell me how often parents had told him
of their thanks for my work with the children.
But the fact is that I had begun to feel suffocated.
It is not me that they have treated badly, but my work. Again: this
is not because anyone involved has been a bad person. They are simply
working in a system which, despite its astronomical costs, neither looks
for nor wants new ideas. It does not discuss them, notice them, test
them, or promote them. It simply does not want them. They disturb it.
It is better to get rid of anyone who is unorthodox rather than to find
out whether what he does is actually worthwhile.
For years, therefore, I had been living with
an illusion - perhaps that should be a delusion. Around twice a year
a glossy journal is produced for the schools. Two years before it had
contained my article explaining in detail how this method works. It
was called "Programmed to fail - or taught to succeed?" For
this is the real, basic, unavoidable choice. Virtually all teaching
in mathematics today will ensure that a major fraction of pupils become
increasingly puzzled by what they are doing. Except for the most talented
or the most slavishly obedient, anxiety over their actual often secret
incomprehension of what it is they are actually doing! eventually becomes
so serious that they start to lose heart and to fail. This careless
destruction of young people's confidence, this humiliation of hundreds
of thousands by their failure to copy meaningless routines, is as disgusting
and senseless as any carnage in war. I had found an elementary way to
stop this carnage, easy to use with children as soon as they can read.
In my classroom I had proven repeatedly - as has been proven in Hungary
in careful university studies - that this method not only works, but
produces better mathematical understanding enjoyably - and yet in this
school, supposedly the most important in the history of Europe (in the
words of my first British director), it was not to be allowed out of
my classroom.
In the end, this was unacceptable. No matter
what the salary may be, any job ceases to be interesting when there
is no possibility of progress in it. When my bosses dismissed my efforts,
they dismissed me. I am happy to report that my visitor has since been
promoted. I am unlikely to harm him; and I bear him no ill-will, for
he at least showed me my delusion plain. As he told to me, regretfully,
at the time: 'I must report what I see.'
So this, then, is my report.
The method of teaching that he wanted me to
use is used in many schools. Its most important benefit, as he pointed
out, to both the managers of the schools and to teachers who use it,
is that no-one can blame them for their pupils' failure. And far, far
too many of their pupils - without question - will fail.
This is, in every other sense, a bad method.
Not just a bad alternative. It is criminally, irredeemably, hopelessly
stupid. Simple bald instruction, each step carefully but 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 creates no association.
It does not create understanding. Understanding is achieved through
creating associations with other knowledge through discussion, argument,
most of all through the patient individual thought which only discussion
provokes.
The truth is that most teachers just do what
they are told to do. This is not reprehensible in itself: they are,
after all, the lowest tier in the profession's food chain. But school
managers, inspectors, ministers - and their civil servants: all ought
to know this. If they do not, or if they are afraid to change, they
are incompetent. My own belief is that for mostly private reasons, many
of those who have the power to make the change simply do not care. Or
not enough. They want selfishness and ruthlessness to continue to succeed;
for contempt for the weak to continue to punish those who 'have not
tried'. There is no real benefit for them in improving on the present
situation. They will make noises about improvement; spend vast sums
of our money on their new initiatives: but without changing the strategy
of teaching in the classroom to give children more autonomy in learning,
they will improve, essentially, nothing.
What I can therefore offer to anyone who wants
their children to avoid the horrors which most schools will blithely
inflict on them is a free explanation showing youngsters how to learn
maths alone, or with friends, from any reasonably good textbook. In
a few years they really should not need a teacher. It is called the
Socrates Workbook for Nine to Nineteen Year Olds - and there is also
a Teacher's Guide. Both can be downloaded free - and in five languages
- from my website: www.gardenofdemocracy.org. A more formal explanation
of this method and its historical connection with democracy was published
in 1997 by the important European English language journal International
Reviews on Mathematics Education, and this article, entitled 'Mathematics
Teaching is Democratic Education', may be downloaded from www.zblmath.fiz-karlsruhe.de/.
The proceedings of the two-year European Union project with much the
same title may be obtained from LEU, the Landesinstitut für Erziehung
und Unterricht, Stuttgart. The NLT journal Literacy Today published
the article Read Aloud - and Learn! in Summer 2003. The TES and New
Scientist have of course their own archives.
If, in addition, if any group of parents - almost
anywhere in the UK - would like to organize a meeting of around fifty
or more people, I should be happy to come to explain how, by using this
method, they can themselves help their children to become happier and
more successful in learning than ever before. It's actually very simple,
even obvious, but I doubt very much that they will ever learn it in
their school.
Colin
Hannaford,
Oxford, 6-11th September 2004
