A SUBMISSION TO THE ROYAL SOCIETY

by Colin Hannaford
Foreign Correspondent EdNews.org

Our British Foreign Correspondent, Colin Hannaford, with long experience in the mathematics classroom, was recently invited by the Advisory Committee on Mathematics Education (ACME) of the Royal Society in London to submit his proposals for improving mathematics teaching in Great Britain. He submitted the following suggestions in the same week that the Guardian newspaper published a very different proposal by an Oxford academic: that schools 'should consider abandoning it as a compulsory subject … All it does is reinforce [children's] sense that maths is boring and difficult.' The following is not recommended for those wishing to agree that the best way to win battles is to retreat.

Area 1: Pedagogy

Q1. What are the effective pedagogies for maths teaching? Is there a single, most effective pedagogy? What would you like the Review team to recommend in this area?

I was a head of mathematics for twenty-five years at Britain's official EU European School. Being able to work in this State supported international comprehensive school - outside the National Curriculum and in practice virtually autonomously - I was able to discover that my success in teaching my pupils was actually extremely superficial. Instead of helping them explore mathematics as it really is: a collection of the most fascinating and powerful arguments ever constructed - I was considered a satisfactory teacher, as well as being one of the most highly paid in the world, if the majority of my mixed ability, multi-national, and even multi-lingual, pupils simply learnt to apply selected techniques by rote, in order, primarily, to 'complete the syllabus 'and pass their tests and exams.

The fact that they understood very little of what they were doing was never seriously examined, either by me or by anyone else. This appeared to be officially irrelevant.

Being substantially free to find a better way to teach mathematics, I found that I could greatly increase my pupils' enjoyment and understanding of what they were doing as well as learning what mathematics should achieve, in a most elementary and easily transferable fashion.

I simply learnt to teach them how to read, discuss, and understand the explanations in their expertly written text-books!

As I became more confident in this new approach, I gradually broadened its application until I was using it from the first day of the First Year to remedial Baccalaureate classes in the Sixth and Seventh Years preparing for university entry. (I may add that in twenty-five years none of the latter pupils failed. Although they had all chosen to avoid the strong maths exam, the final class Baccalaureate examination average of my pupils was always around 70 %.)

Recently I wrote to describe the general results to Professor Lord Martin Rees, President of the Royal Society, and to the distinguished moral philosopher, Professor Lord Raymond Plant:

'When a class is encouraged to use its mathematics lesson as a forum for the discussion of its arguments, a forum in which their understanding is collectively developed and collectively shared, in which they learn automatically to listen to, respect, and value others' interventions, opinions, and perspectives - then the basis of a true understanding of the enormous depth and potential of mathematics together with a proper respect for democracy are both being formed. Then the majority begin to see that attempts to impose any one belief by ridicule, abuse, or force - even if later it is found to be correct - are generally unhelpful and immature. We can start helping children to achieve

this level of maturity from a very early age. We very rarely do this. The consequences are all around us. The consequence of correcting this failure - so simply - would be an enormous step away from wars and towards peace; and not only in the future:the experience of their children can actually help and encourage the parents in the present.'

Although I have lectured on this theme in the past fifteen years to receptive audiences in universities from Prague to Istanbul, I will only detail the most salient of these occasions here.

First was an address I was invited to give to the German national working party on mathematics education (die Arbeitskreis für mathematische Bildung) in Bavaria, in 1992. There I boldly told my audience of distinguished academic educationalists and working teachers: "I have come here today to tell you that if your predecessors in the 19^th century had taught mathematics correctly connected with democracy, your country would not have lost its democracy twice: first to the Kaiser; then to Adolf Hitler."

It was intended to shock: and it did. But my further explanation impressed the delegates so much that my paper was published that year in its journal, first in English. Later it also appeared in German. The French national association of mathematics teachers has also published it in French.

Visiting Budapest in 1993 I met Professor Dr Eva Vásárhelyi of the Department for Mathematical Didactics of Eötvös Lorand University. Her colleagues and their post-graduate students had carried out research in a number of Hungarian schools to show that learning mathematics primarily from discussion is more effective in schools at all levels than when taught primarily through instruction. (I have asked Professor Vásárhelyi for a copy of this research - even if in Hungarian. I have assured her that the Committee will certainly be able to find a reader of it!)

In 1996 I was instrumental in initiating a two-year European Union Comenius study, directed by the Stuttgart Landesinstitut für Erziehung und Unterricht (LEU) entitled: 'Mathematikunterricht und demokratische Erziehung' (Mathematics teaching and democratic education). This involved invited teachers from Austria, Germany, England, and Spain. My thesis for this study was published by the premier European English-language education journal Zentralblatt für Didaktik der Mathematik (ISSN 0044-4103). I was also twice invited to lecture in Stuttgart by the Baden-Württembergische Landeszentrum für politische Bildung.

The success of these lectures no doubt contributed to the decision of the Baden-Wurttemberg government to back the EU-LEU study and its subsequent development. It charged Dr Hartmut Köhler of LEU, the director of the EU project, to create an entirely new series of text-books for the German secondary mathematics syllabus. The emphasis in these books is developing solutions through discussion, rather than learning techniques through instruction without understanding. Fifty schools have been involved. Four volumes of exercises have so far been published under the aegis of Stuttgart LEU with the general and evocative title: 'Kreative Ideenbörse' ('Exchange for Creative Ideas': ISBN 3-7892-1624-0)

In 2004 I was informed by Professor Hani Khoury of Mercer University, Georgia (one of America's largest private foundations) that he had been teaching my pedagogical philosophy to his students for the previous six years. He invited me to visit Mercer and to lecture to his colleagues and their students. During my visit I was also asked to address two evening classes of mature student teachers on the subject of Kant's philosophy of moral absolutes.

I believe my audience was intrigued by my presentation of Kant's understanding of moral absolutes in the context of a long neglected chicken-house on a remote island in the Inner Hebrides, and the response of its chickens to my attempt to clean it - but on the whole it was well received. I, in turn, was then rewarded in the conversation with my audience that followed on the subject of New Orleans, the hurricane Katrina, and the apparent invisibility of a major fraction of the population of a major American city to their own government. (The full account appears in my book '473959' [Hannaford, 2006, ISBN 1-4251-0942-x]

Subsequently Professor Khoury proposed that our joint approach would be better called a Socratic Methodology: since it may in fact be applied to any discursive subject, and since it is especially useful in classes with very different levels of ability in reading and comprehension, or in

which a national language is not natural to all the children present. Incidentally, it is an important and valuable fact that children will listen far more attentively - and eventually will listen and comment far more sympathetically to one another - in the course of one of these lessons, than they will to most teachers!

Also in 2004 I was invited to attend a conference in Oxford of the British society SAPERE (Society for the Advancement of Philosophical Enquiry and Reflection in Education) on social and moral problems faced by religious and ethics teachers from over a dozen countries. At the plenary session I offered to show delegates the degree to which their efforts are systematically contradicted by their mathematics teaching colleagues, who have the dual advantage of far more time and the powerful compulsion of repeated and competitive examinations. This paper: 'The Social, Moral, and Political Implications of Mathematics Teaching' has been translated into German and French, and has been frequently accessed on my website. It also shows why the Socratic Methodology avoids doing the same social, moral and intellectual damage.

In 2005 I was invited by Her Highness Sheikha Mozah of Qatar, consort of the Emir, President of the Qatar Foundation, and UNESCO Special Envoy for Basic and Higher Education to attend the Foundation's second Symposium on Innovations in Education. On the second day I spoke for three minutes on the proper sense of the themes of 'Technology, Empowerment, and Education' - giving, incidentally, the shortest speech of the conference, for which I was later congratulated by Dr Mir Asghar Husain, Director of UNESCO's Educational Strategies and Policies Division, who claimed that this made it the most memorable speech of the conference. The most essential part for me was to explain to the audience that techne logos - in the original Greek - had nothing to do with machines. It actually refers to 'reasoned debate': that is, debate between people. It was originally about improving interactions between human: not between humans and machines. The ability to work with machines cannot replace the ability to work with people.

After this Symposium I was invited by Dr Sheikha Al Misnad, President of Qatar University, to address her graduating class of student teachers. I presented to them the paper 'Evaluating Change' which I had been asked to prepare for the Symposium.

This paper explains how the Socratic Methodology can be introduced into schools without noticeable disruption over a five year period, the final aim being to prepare students still at school to work alone from their text-books, when necessary, as will definitely be expected of them at university or in professional applications.

In 2007 I made a formal proposal to the Qatar Foundation for a conference in 2008 to describe to an invited audience of education ministers the advantages and the consequences of the Socratic Methodology. Its formal title is: Adopting the Socrates Method for teaching mathematics: encouraging a culture of democratic behaviour to foster inter-cultural and inter-faith understanding and tolerance.

The Foundation has been a major force for progressive social reform in the Arab region, and Doha, the capital of Qatar, is famous as the venue of a series of important international meetings and accords. In August 2007 I receive a letter from the private office of Her Highness expressing her support and, later, her decision to send two Qatari professors to participate. It may be noted here that Her Highness has been awarded the 2007 Chatham House Prize for her commitment to progressive education and community welfare in Qatar and because of her strong advocacy at home and internationally in favour of closer relations between Islam and the West.

The Warden of St George's House at Windsor Castle has offered accommodation for the conference in December 2008. The emphasis will be on the need to learn critical, constructive and receptive argument everywhere, for no country is immune to the effects of its young people's alienation, frustration, anger; from individual suicide to mass violence. Your Committee will be aware that our own educational establishment either does not have or cannot mobilise any new ideas to deal with this disturbing new phenomenon. Not inapposite is the remark by Dr Martin Stephen, the High Master of St Paul's School, one of Britain's largest and most successful private schools, in an interview with Professor Michael Shaughnessy, of Eastern New Mexico University, published

recently in Education News, America's largest on-line education journal: "I find it interesting that as a working Headmaster there is no educational theorist in the UK at the moment who is really speaking to me."

I find this rather more than 'interesting'. It is most worrying. It is also the reason why I place my hope in your Committee and the Royal Society to persuade our government - or any future government - that, again in Dr Stephen's words: "education is actually very simple; after all, we've been doing it one way or another since we started to walk on two legs."

Simplicity must begin with a single simple aim. It needs first to be understood that mathematics is not intended to train children to be obedient. This is a very dangerous corruption of its powers. It is the science of receptive observation, critical understanding, constructive argument and persuasion. To learn these skills most fruitfully, children must be allowed freely to observe, to attempt to understand, to argue their own point of view, to be persuaded, or to persuade others, that one form of understanding is most useful - as far ahead as they can see.

But the reason for reforming mathematics teaching in British schools must not depend on any single teacher's experience, nor on the support of any others, even if, asin Germany, they are now far ahead of others in reforming mathematics teaching.

The weakness of such a position is obvious. For any group of teachers with some success in developing a new approach to teaching mathematics, governments can produce a thousand teachers who have never felt the least insufficiency in teaching mathematics through orthodox instruction.

Our argument is much more general. The observable facts are that supposedly successful schools are producing youngsters who have passed an exam in mathematics - and in other subjects - but without the majority having the least experience of independent, critical thinking or independent study.

The Socratic Methodology provides these elements automatically. Children's experience of them can begin as soon as they can read. In mixed-ability, multi-national, multi-lingual classes it will assist their comprehension in reading and listening, in both literacy and numeracy together. It is highly enjoyable. It employs all children's natural skills. It teaches them the value of working together as a cheerful team, rather than trying to survive in a tankful of cannibalistic predators.

But the most general argument must address the most fundamental question: What exactly is mathematics education for? Why are so many socially successful adults unembarrassed to declare: "Of course, I was never good at maths!" - as if inviting agreement that even basic mathematical competence is socially unimportant?

My colleague Dr Köhler in Germany has pointed repeatedly to their error. Modern societies literally cannot function without mathematics. Industries cannot function without a steady supply of competent, creative mathematicians. Anyone unable to participate in even a simple mathematical conversation is therefore like a visitor from a very foreign country: unprepared; unrepresented; uninformed.

A popular belief appears to be that mathematics teaching is intended to discover a few gifted mathematicians; that to another minority of youngsters it will be of value if they are aiming for careers in science and some industries; but for the majority it will have mainly served as a measure of their 'innate intelligence'.

Not only is this untrue, but providing this very unreliable information will have cost the majority several years of humiliation, leaving most of them ever mistrustful of mathematics. I have argued elsewhere that they will also have been divided, socially and morally, into mutually distrustful groups with entirely different moral codes. My friend Professor Didier Nordon at the University of Bordeaux, a celebrated author and a regular contributor to 'Pour la Science', the French edition of Scientific American, has also written angrily that calibrating children's and students' intelligence is not what mathematics is for. (See, for example, his 'Les Mathématiques pures n'existent pas!' [Actes Sud, 1993])

We both argue that maths education will always have these disastrous accidental consequences (detailed and explained in my SAPERE paper), unless it has more general - and far more deliberate - social and moral aims.

In the first instance, it should be made generally known that the earliest teaching of the forms of arguments we now called mathematical did not take place in classrooms. They were developed in the open fora of the early democratic Greek states. They were developed more or less spontaneously in reaction to the increasingly dangerous success of rhetoric, learnt by the rich and their lawyers. They were learnt by ordinary people because of the obvious success of these far more efficient forms of argument. Their political success preceded their later adoption by mathematicians!

It is a costly mistake for modern societies to neglect this potential in their schools.

It may also be called a spiritual mistake. Mathematics has had a highly visible success in engendering a spirit of techne logos between mathematicians and other scientists. Since every child, even in the most elementary education system, must learn some mathematics, much social and cultural strife might be prevented by using mathematics education to engender the same spirit in and between young people. To any who believe that God is universally active and benevolent towards humankind, it must be obvious that this will be in accordance with His will and purpose!

The Cambridge physicist John Polkinghorne abandoned a distinguished career in nuclear physics to become a Christian priest, and later won the 2002 Templeton Prize for his attempts to reconcile science and religions. He has argued that evolution is guided by what he calls 'not meaningless randomness, but contingent particularity'.

To attempt to translate this from the realm of particle physics to the human level is not easy, but I think what he may mean - and what then appeals to me - is that either we can certainly act as if there is no moral guidance - 'anything goes' is surely the moral equivalent of random meaninglessness - or we can trust in our deepest human instinct to distinguish love from hate, and try always to choose the first rather than the second: to act according the constructive rather than the destructive emotions.

It is because of this that I believe the social and moral aims of mathematics education should be both empirical and spiritual. This will sound rather strange to many. Let me try to explain what I mean.

Their empirical mathematics education should provide young people with at least five years of practice in perceptive, critical, constructive, receptive dialogue. As a consequence of this, I believe that they, in their turn, will tend to prefer an innovative, diverse, self-critical, open society.

As a consequence of their empirical training, they will also ultimately know how to continue learning without tutorial support. This must be a deliberate pedagogical aim of any school. It should be achieved at least by their penultimate year. In their final year - the last before further independent study at university, or other careers - the students must be able to study, research, and produce appropriate finished work alone.

But then a crucial spiritual advance can also be made. The Socratic Methodology allows children to discover, naturally and progressively, that accepting the criticism of their neighbours without getting indignant or angry is more useful than any violent response.

Two thousand years have passed since the highest level of human moral attainment was set at not merely loving your neighbour, but loving your enemy too. Generations of spiritual leaders have sought since then a practical means to teach this remarkable achievement.

It is entirely intrinsic to the Socratic Methodology. Whenever a child learns to return the correction of another with neither a frown, nor a kick, nor with a promise of a punch in the mouth, but with a truly surprised and grateful: "Hey, thanks!" another instance of 'contingent particularity' may be thought to have occurred - and this at a very human level.

In short, we learn best to return God's faith in us when we accept that His plan may not be ours. Or, to express the same in an entirely secular form, we learn best to be human when see that others are just as human as we are.

(For further comments, see earlier publications in the Times Educational Supplement, the New Scientist, the journal 'Literacy Today' of the National Literacy Trust, and articles already published over my name in EdNews.org.)

Area 2: Provision<

Q1.What range of provision best supports children across the full ability range? How about the most gifted pupils? And what about those who are not progressing fast enough to reach national expectations? And what about pupils with special educational needs? What would you like the Review team to recommend in this area?

Every child must be able to read aloud from a textbook: or, at the very least, must be able to follow the correct line of words. In very poor schools, a text-book may be shared between two or more pupils. In the poorest, explanations may be transcribed, sentence after sentence, from a textbook onto a blackboard. Throughout these lessons three essential facts must be communicated continually.

First:children must understand that if only they can read, books contain knowledge which can lift anyone from a life of poverty and social and economic subordination to the possession of valued abilities and social and economic independence.

Second: reading is not just a matter of their brains recognising shapes of words and making appropriate sounds. It involves a harder task of trying to understand the meaning that the words and the sentences are intended convey.

Third: full understanding is proven by translating into their own words what they believe this meaning to be. This must not only please them. If they are working with others, it must be an honest attempt to satisfy others too. The final translation that they all achieve must be at least as satisfactory as the original. Sometimes it may be even better!

The full ability range may extend from the barely literate to the precociously able. Whatever their age, the teacher must direct individual pupils to read successive sentences aloud to the rest of the class. Without causing any deliberate embarrassment, every pupil should be given an opportunity within their capacity in this way.

The precociously literate must not be allowed to dominate. They can be allowed to read on silently as far as they wish - even to start to work independently. When, in the course of the lesson, the pupils are asked to explain what the sentences mean which they have just heard or which they may even have just read themselves, the teacher's astute direction will ensure that everyone is involved in achieving a new formulation which satisfies then all (naturally, including the teacher!) Finally the whole section of the text the lesson is intended to elucidate must be tested in this way.

During attempts at new formulations, the teacher can also ensure that support is given by the more to the less able - even if this means showing that the more able are not always perfect in responding; also by ensuring that the least able have occasional opportunities to succeed.

It is most of all important for the teacher to communicate to the pupils that there is nothing so advantageous for them all as belonging to a class which is cheerfully confident, co-operative with their teachers and tolerantly supportive of each other: in belonging, in other words, to a class which can succeed as a whole!

Area 3: Intervention

Q1. Are you aware of any numeracy intervention programmes for Year 2 pupils? What should the key features of a successful intervention programme look like? How do you identify if a child needs additional support? How do you measure a child's starting points, the progress the child is making and whether the child has reached the desired level of attainment? How long should the intervention last? And how intensive / frequent should it be? What would you like the Review team to recommend in this area?

My opinion is that the practice of'intervention' must be exercised with extreme caution, especially in already unhappy classes. The reason is that children in an unhappy class, or in an already divided class, will likely to exploit any opportunity to be treated as needing special care.

This will include exaggerating any apparent deficiency that has been noticed - even to a degree that makes it incorrigible. Except in cases of gross disability (and even here sometimes the other children can help most), the best therapeutic environment is a happy class that is working together as a class. This should be the main aim.

At this point I want to refer you to the work of the Haberman Foundation in the United States, and to an opinion of its founder Professor Martin Haberman. It relates specifically to a frighteningly pointless exchange which was broadcast by the BBC this morning (Nov 2^nd) between Professor Peter Tymms of Durham University and Lord Adonis, a government education minister.

The former described the research he had completed for the government after it has spent five hundred million pounds (approximately one billion US dollars) on improving reading in primary schools over the past ten years. He and other independent researchers had concluded that any apparent improvement was 'basically illusory'. Lord Adonis, speaking for the government, claimed, to the contrary, and continued to claim at length, that these results were 'the best ever'. Professor Tymms dryly replied: 'One has to look at the actual evidence as well as official figures'. Which I suppose is a polite way of saying that there may be a reason why they differ.

I admire Lord Adonis. He was plucked from a seriously deprived childhood by the Oxfordshire social services; was placed in a select public (private) boarding school; went from there to Oxford University to read history; later gained a PhD, also at Oxford, on the history of the 19^th century British aristocracy, briefly was a journalist, was finally made a member of the 21^st century British aristocracy, being appointed to a ministerial post in the present government.

My understanding is that he has never taught in a school, or anywhere else, in his life. He appears, however, to have an entirely understandable belief in the benefits of 'intervention'.

My own opinion - identical with the result of studies by Professor Haberman in the United States - is that education research on learning in schools has been misdirected for decades.

Professor Haberman points out that virtually all mainstream studies have concentrated on learning by individuals. As a consequence of this, teachers are trained to address a class as if the pupils remain discrete individuals. When it becomes apparent that some of them are not reaching the 'desired level of attainment', 'intervention' is then said to be justified. The individuals can then benefit from the results of all the research studies which have concentrated on individual learning. Lord Adonis also recommends that this should even be 'one-to-one'.

I hope my repetition has made the fallacy apparent. A class of pupils is not a collection of discrete individuals who remain continuously and magically isolated from others. The class is a social group. It is to a group to which pupils consciously 'belong'.

"And it is powerful!" Baroness Estelle Morris, also a former school-teacher, commented to me recently at an All-Party Parliamentary Committee meeting to examine why British children are the unhappiest out of 21 developed societies.

This is exactly the point repeatedly ignored by inexperienced academics.

The class, as a group, has a powerful social dynamic. It can either be engaged by the teacher in useful and enjoyable co-operative learning - that is what the Socratic Methodology achieves - or it will divide in a remarkably short time into the 'goodies', the 'stupids', and, in modern schools, those needing to be isolated from the rest to be treated by 'intervention': one-to-one, of course.

The cost of this mistake, stubbornly persisted in by the major part of the British academic educational establishment, is, I suggest, a major reason why British children are reported to be the most unhappy of 21 developed nations. One group of children, the All-Party Committee was told, explained the reason why they were unhappy. It was: "Because all our teachers are so depressed!"

Why do we not listen to children: the real experts on growing up?

In 2003 an article of mine for the National Literacy Trust called "Read Aloud and Learn" received special editorial comment and praise because in it I pointed out that for the majority of children numeracy and literacy - aspects of learning which appear very different to adults - are to them very similar.

It is clear that learning to write is inextricably linked with learning to read. Recently Dr Robert Rose in Georgia, United States, sent me a report of a group of independently minded kindergarten teachers, and the extraordinary success that they have had in teaching children, first to write capitals, then to announce the sound of single letters, then of linked letters, and finally to write cursive script.

I quote from his paper: 'Historically, many authorities on the subject of literacy instruction have stressed the importance of adequate practice in printing alphabet letters. The first-century Roman writer and rhetorician, Marcus Fabius Quintilianus (ca A.D. 70) wrote that with regard to becoming literate, "Too slow a hand impedes the mind."

In 1912, Maria Montessori wrote, in effect, that teaching young children to print letters is easy, that it is easy to teach children to read after they have practiced printing alphabet letters, but that it is difficult to teach children to read if they have not practiced writing them.'

Rose describes the rejection of his report of their success by education journals in the United States, including the Harvard Educational Review. One assistant editor told him: "That couldn't possibly be true!"He may be comforted to know that I received a very similar response in Britain. I was recently told by a representative of the British Ministry of Children, Schools, and Families that: "Early writing is not taught through the use if capitals!" No explanation for this was offered.

It seems that in both the United States and Britain there is a very similar reluctance to return to methods of education of the whole class which are simple, enjoyable and effective. There is instead a very similar enthusiasm for making teachers' tasks more and more similar to industrial management. There are ever more complex and jargon-heavy plans; ever more 'interventions' to be justified, made, and supervised; ever more stress imposed on their pupils through ever more frequent tests of their supposed 'progress'. Perhaps it is driven by a fear that teachers can too easily escape the control of the state.

In any event, this is regress: not progress. It is also deeply uninformed. The most important fact to be firmly re-established - I hope by your Committee - is that when teachers attempt to teach a class of individuals as if they can still be addressed separately as individuals, they are attempting to ignore the social dynamic of the class.

If the teacher is sufficiently powerful and charismatic, this can work. But if the class decides to do so, and this decision may be more or less at random, it can break the heart of any teacher.

Instructing individuals only works with individuals. To succeed - and survive - teachers must learn to engage the willing attention and co-operation of the whole class.

Area 4: Subject Knowledge, CPD, ITT

Q1.What conceptual and subject knowledge of mathematics should we expect of primary school teachers and early years practitioners? What do you see as the key issues for teachers / practitioners CPD and why is it so important? What do you see as the key issues for teachers / practitioners ITT and why is it so important? What would you like the Review team to recommend in this area?

In my experience, primary teachers generally do their work very well. However, the transition from primary into secondary is notoriously difficult for very many children.

My observation is that the main reason for this is simply the sudden change in the way that pupils are treated.From being members of a group - of their class! - suddenly they find that they are

being treated as discrete individuals! Suddenly they find they are solely responsible for their achievements; that they are individually rewarded or, as they see it, individually punished for their success or failure - even when, most distressingly, there is no obvious difference to them; suddenly they are made to view all their previous companions as competitors.

None of these problems would arise if children began to be made conversant with the Socratic Methodology from the age of six or seven - nor would it be beyond a creative teacher to begin to follow its philosophy much earlier; nor a proactive CPD course to teach it - so that entering the second tier of education would not involve any abrupt change in their treatment, or in their status as individuals. It would be a continuation of their previous experience of contributing to the work of their class individually - but this without the fear of isolation, of ridicule, or - the most damaging label of all - of 'intervention'.

In other words the erroneous concept of mathematics as a covert or declared measure of intelligence must be replaced by the far healthier and true concept of mathematics as training in careful observation and in creative argument: an activity to be enjoyed, not hated, and not feared.

Area 5: Curriculum

Q1. What is the most effective design and sequencing of the mathematics curriculum? Are there any gaps /

Wednesday

November 14th, 2007

Colin Hannaford

British and Foreign correspondent EducationNews.org