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Category Archive • Maths
January 19, 2005
Alice does maths

I am very busy today, but Alice has a post up about arithmetic, and about maths, one of the points being that the teaching of the forner can often screw up the teaching of the latter. For her, the big breakthrough in her teaching came when she made an abacus with paper clips, thereby answering the question: why?


Proper abacus picture, here, and this:

The abacus was the first known machine developed to help perform mathematical computations. It is thought to have originated between 600 and 500 BC, either in China or Egypt. Round beads, usually made of wood, were slid back and forth on rods or wire to perform addition and subtraction. As an indication of its practicality, the abacus is still used in many eastern cultures today. The abacus, an ancient product of the middle east, is really a full blown hand-held decimal calculator!

Ten out of ten.

Posted by Brian Micklethwait at 01:12 PM
Category: Maths
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November 21, 2004
Darts education

DartsBoard.jpgSolid evidence of whether or not educational standards are actually declining is hard to come by. Here is some though:

The John Barras chain of public houses is installing calculators beside its darts boards. Declining standards in mathematics have left younger players unable to do the sums, the chain claims.

This is quoted by Giles Smith, at the top of an article about "darts education". (Gratuitous darts board picture there, drawn on maths type paper!) Smith then cracks a lot of jokes which I quickly got bored with. I preferred the other quote he stuck at the top, from darts champion Phil "The Power" Taylor:

Darts is fantastic for honing your maths skills. They should introduce darts calculations into the GCSE maths syllabus.

… although, like so many, Phil "The Power" Taylor jumps from "X might be a good idea", to "X should therefore be compulsory". But this habit is an educational defect shared by many more persons than Phil "The Power" Taylor.

Posted by Brian Micklethwait at 08:45 PM
Category: Falling standardsMaths
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September 17, 2004
Maths Tsarina

CeliaHoyles.jpgMore Tsardom:

The Government today appointed a new maths "Tsar" tasked with turning around years of decline in the subject.

Celia Hoyles, who starts her job as chief mathematics adviser next month, will "champion" the subject at all levels, from primary schools to university and beyond.

Education Secretary Charles Clarke said the appointment of Prof Hoyles was "critical" to revitalising maths education.

England is short of about 3,500 maths teachers, equivalent to more than one for every comprehensive in the country, a major inquiry found earlier this year.

Students, teachers and employers were all being let down by the current system, according to Professor Adrian Smith’s Government-backed inquiry into post-14 maths.

Mr Clarke said he was "delighted" with the appointment of Prof Hoyles, who is currently working at the University of London's Institute of Education.

"I believe this appointment is critical to the success of the mathematics strategy we outlined earlier this year," he said.

"The road ahead will be filled with opportunities to revitalise the study of mathematics and raise the profile of mathematics for everyone, not just pupils in schools and their teachers."

Prof Hoyles said: "I am thrilled to accept the role of chief adviser for mathematics, and look forward to the challenges ahead."

You get the strong feeling, don't you, what with all the "sneer quotes", that the writer of this report detected an air of false optimism about the show that was laid on in front of him. "Delighted". "Filled with opportunities". "Thrilled". Above all there is that gruesome word "challenge", which means insoluble difficulties of all kinds.

The original idea of a Tsar was that there was only one of him, and his word was law. But what happens when two Tsars bump into each other in a government corridor, both chasing the same money, or demanding the same slice of the school day or of the National Curriculum? Who wins? And what happens when the Tsar comes up against, you know, the Minister of Education?

Posted by Brian Micklethwait at 11:35 PM
Category: Maths
[0] [0]
July 25, 2004

My thanks to Arts & Letters Daily for picking up on this article from my local evening paper about dyspraxia, which is partly a reason but academic failure, but mostly an excuse for bad teaching.

Here are the key paragraphs of this particular story, about a boy thus branded, and about the expensive tutoring his parents subsequently set about rescuing him with:

So began 18 months of after-school sessions with puzzles and videos, complemented by special teaching from two other psychologists to teach reading. In addition, most evenings a tutor came to the house to help my son with his homework - the cost was phenomenal. By the end of the second year, the situation was probably worse. He was in the bottom set at school and scored miserable marks in exams. He was below the border line to pass the common entrance. Then came enlightenment.

"Your son," announced one educational psychologist suddenly, "is not dyspraxic." "What?" I exclaimed. "He just hasn't been taught maths," she continued. "It has undermined his self-confidence to learn everything else at school." The revelation was astounding. She recommended a maths tutor.
"Most of my work," the maths tutor told me "is with pupils from your son's school. They can't teach maths." Neither could he.

Desperate, I was told about a maths tutor who it was said could perform miracles, at £90 per hour. To save my son, there was no choice.

"No one has taught him maths," announced the miracle worker, "and he's got no self-confidence." Teachers at the school, he discovered, regularly humiliated my son because of his poor results. "Can you do anything?" I pleaded. "Oh, yes," he said. It was October. The exams were in June.

Over the following eight months I witnessed the most astonishing transformation. A cowed child became a confident student. Understanding maths transformed his mastery of every other subject. His common entrance mark in maths was 83 per cent and he achieved five A grades (over 75 per cent) with the rest Bs (over 65 per cent).

When I cautiously raised with one or two other parents the rather sensitive subject of poor teaching in the school, I was amazed by the response. Oh didn't you know, 75 per cent of the boys doing Common Entrance have private tuition at home? Nobody had declared their hand until after the exams. And when I told my story to an old friend, Anne Alvarez, a well known child psychologist, she told me: "Dyspraxia and other labels put on children are often too loosely used. Many diagnostic labels are used as wastebaskets."

Our son's headmaster recently announced the appointment of a new maths teacher. We later learned that this new teacher had not even passed A-level maths.

The writer of this article implies that the misuse of the word "dyspraxia" is more common in the private sector. But my guess would be that this is merely because in the private sector they at least have to provide some kind of reason for academic failure, or failing that, they have to contrive a plausible excuse. In many a state school, I should guess, failure of this sort could simply be allowed to run its course unchallenged.

Posted by Brian Micklethwait at 01:48 PM
Category: Maths
[1] [0]
June 21, 2004
How to teach arithmetic to boys

I've spent most of my blogging time today writing a ridiculously long piece about the complexities of qualifying out of the group matches at the European Soccer Championships, and a link from here to there is all I can offer today.

Here, gratuitously, is the picture I used to illustrate the kind of stuff I was writing about.


The educational relevance? Well, simply that sporting arithmetic is a great way to teach arithmetic to small boys. I still remember with pleasure the day I explained about fractions to a small boy, by talking about a soccer game.

And I dare say there's even the occasional girl who might be persuaded to take maths a bit more seriously with talk about sport.

Posted by Brian Micklethwait at 07:35 PM
Category: Boys will be boysMathsSport
[7] [1]
May 28, 2004
"A strategy is in place …"

More doom and gloom, to echo what those Cambridge professors (see previous posting) were saying:

The education system is "in danger of implosion" because of falling standards, North-East business leaders have warned.

And proposals to revamp schooling between the ages of 14 and 19 will do nothing to address the North's serious skills shortage, according to the CBI.

It discussed a plan to replace GCSEs and A-levels with a four-tier assessment at a regional council meeting this week.

The proposals, unveiled in February by a working group headed by former chief inspector of schools Mike Tomlinson, were designed to ensure everyone leaves school with basic skills.

But CBI North-East director Steve Rankin said: "Falling standards will not be addressed. There's a real need to concentrate on three things: basic numeracy, basic literacy and attitude."

This educrat reply does not inspire confidence.

A spokeswoman for Newcastle City Council said: "Pupils deserve to be congratulated on their success, which we are sure they will take with them into working life. Newcastle Local Education Authority already has a number of successful strategies in place to improve levels of literacy and numeracy."

"Successful strategies are in place." Not: "You are wrong, our kids can read and count." So, the problem is as it is said to be by the complainer, in this case the CBI man. And a "strategy" being "in place" means that so far no improvement in the situation has actually occurred. Right?

Plus, note that the spokeswoman doesn't even say that there is a "strategy in place" to deal with "attitude", so God knows what is happening to that.

Incidentally, Patrick Crozier has been looking over my shoulder and has been saying: "I can't believe it throws out numeracy". What did be mean? It turned out he meant my spellchecker. It puts a squiggly red line under "numeracy". Great. My spellchecker is illiterate about numeracy.

Posted by Brian Micklethwait at 10:58 PM
Category: LiteracyMaths
[1] [1]
May 19, 2004
Afghan girls doing maths

Joanne Jacobs, who is indifferent to pictorial content on account of having sufficient content of her own, links to this picture, which unlike her I here reproduce:


Here's how the accompanying Christian Science Monitor story starts:

A second-grade math class in Kabul, Afghanistan, met in the school breezeway with a blackboard on wheels. The young scholar was shy about speaking in front of her class. A proud teacher watched. A classmate reached out her hand to offer support.

This scene, so natural, so universal, was nonexistent in Afghanistan for many years when the Taliban were in power. Laws prohibited women and girls from attending school or even leaving their homes.

A breezeway sounds like something out of doors.

Posted by Brian Micklethwait at 03:42 PM
Category: IslamMaths
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April 20, 2004
Teachers – the best shall win prizes

I think that giving prizes to great teachers is a great idea.

Who was the best maths teacher in Britain, last year? Who was the best science teacher in Britain, last year? Is there any award ceremony which tries to find out? I seem to recall some kind of televised (in Britain) event at which teachers were given prizes and celebs took it in turns to recall their favourite teachers, but alas I missed it for some reason. Can anyone fill us (me) in on that?

We'll know when this process has worked. The great teachers will be celebs.

Oddly enough it was this prize, which has had an amazing effect (on space flight), which got me googling for teaching prizes.

A recent Glenn Reynolds TCS article about this X-Prize, and about prizes generally, ends thus:

NASA wonders too, and is establishing its own prize system called Centennial Challenges. At the moment the program is new and relatively small, but I hope that we'll see other government agencies – and private philanthropists – consider the prize approach. It's not a panacea, of course, but it's a way of bringing many minds to bear on a problem, and trying out many different approaches in parallel. I suspect that many of the 21st Century's problems will benefit from this sort of approach, and I hope that the X-Prize example will break new ground, not only in terms of spaceflight, but in terms of all sorts of other problems.

Why shouldn't that sort of thinking apply to teaching?

teachaward.jpgThis picture here is captioned as follows:

20 November 2001
Mrs Susan Burr from the Kyle Academy in Scotland wins the 'Most Inspiring TEACH SPACE 2001 Award'.

Well done TeachSPACE. I picked this picture simply because it looked nice, and illustrated the principle, of turning little known good teachers into slightly better known good teachers. It was pure coincidence that once again the space exploration angle asserted itself.

Posted by Brian Micklethwait at 03:49 PM
Category: MathsScience
[1] [0]
March 30, 2004
How India thinks and what India learns

There is a fascinating article by Cherryl Barron in the latest Prospect (April 2004 – paper only so far as I can work out) about the reasons for the Indian computer software miracle.

The emergence of India as a software superpower is still generally attributed to the cheapness of its programmers and software engineers. But the underlying reasons are more complex and interesting, lying in the subcontinent's intellectual and pedagogical traditions.

Software is ubiquitous. It is at the core of processes in every strategic industry, from banking to defence. And the depth of India's advantage in software suggests that it poses a bigger challenge to the western economies than even China. China, strong in manufacturing and computer hardware, has been almost as unimpressive in software as Japan. Indeed, no developing country has ever taken on the developed world in a craft as sophisticated and important as software.

Indian software aptitude rests on both the emphasis on learning by rote in Indian schools, and a facility and reverence for abstract thought. These biases of Indian education are usually considered mutually exclusive in the west, where a capacity for abstraction is associated with creativity. In India, learning by rote is seen by most conventional teachers as essential grounding for speculation.

An educational tradition that spans learning by heart and exalting excellence in higher mathematics is just right for software. It fits the mentality of computers. These are, after all, machines so fastidious as to refuse to send email with a missing hyphen or full stop in an address. Yet no product on earth is as abstract, boundlessly complex and flexible as software. It cannot be seen, heard, smelled, tasted or touched and is, to borrow Nabokov's description of chess – a game invented in India – a "spectral art."

India's software accomplishments reflect those extremes. Indian firms dominate a world elite of over 120 companies recognised for producing outstandingly accurate software, those which have earned a CMM Level-6 tag, software's equivalent of the Michelin 3-star rating. These establishments – of which America has less than half the Indian total—are certified to be following an exacting, detail-ridden methodology developed at Carnegie-Mellon University in Pittsburgh for producing reliable code.

At the other pole of cyber-sophistication, most of the reigning US technology giants – Microsoft, General Electric, Texas Instruments, Intel, Oracle and Sun Microsystems – have established software design and development facilities and even R&D laboratories in India to take advantage of the world-class brains produced by the Indian institutes of technology, willing to work for an eighth of the starting salary of their US counterparts.

This next bit also alludes, perhaps without intending to, to what used to be wrong with people educated in India.

Western programmers' view of their craft tends to stress its more rarefied dimensions, such as this description by the US computer scientist Frederick Brooks: "The programmer, like the poet, works only slightly removed from pure thought-stuff. He builds his castles in the air, from air, creating by exertion of the imagination. Few media of creation are so flexible ... so readily capable of realising grand conceptual structures."

Yet "pure thought-stuff" is also an encapsulation of ancient India's contributions to the world's scientific heritage. In about 600 BC, before the Greeks, some schools of physics in India developed atomic theories, based not on experiment but purely on intuition and logic. Some western physicists marvel at how much closer the imaginative speculations of Brahmin atomic theory have come to current ideas in theoretical physics than those of any other pre-modern civilisation.

"The Indians advanced astronomy by mathematics rather than by deductions elicited from nature," the science writer Dick Teresi has noted in Lost Discoveries. Indian mathematics was also distinctively airy-fairy. Whereas Greek mathematics was largely extrapolated from mensuration and geometry, the ancient Indians most distinguished themselves in abstract number theory. Zero, infinity, negative and irrational numbers – all concepts that the Greeks dismissed as ludicrous – were Indian concepts.

Airy-fairy. "Pure-thought-stuff." Yes, that sums up the cliché stereotype Indian university graduate of my (older) generation. Very big on abstraction, can talk the hind leg off a donkey, but no bloody use for anything except becoming a bureaucrat and driving the Indian economy – what little there used to be of it – ever deeper into the dust.

Spatial extension and quantities of objects were far less interesting to pioneering Indian mathematical minds. In fact, the Indian leaning towards abstraction – so deep-seated that theoretical physicists and mathematicians still outrank every other sort of egghead in status – explains India's relatively poor showing, historically, in more practical sciences. The sinologist Joseph Needham observed that more practical study would have entailed defying Indian caste rules about contact between Brahmins and artisans. Similarly, the progress of ancient Indian knowledge of physiology, biology and anatomy was held back by the taboo on contact with dead bodies.

All of this brings to mind a remark by Peter Drucker from long ago to the effect that computers have provided something never before seen in the world, namely: paying jobs for mathematicians.

Could it be that the way that computers have enticed all these airy-fairies and pure-thought-stuffers away from being government bureaucrats will turn out to be their most important beneficial contribution to the Indian economy? Yes, these people are doing splendid things with their computers, but think of all the abysmal things they used to do and might still be doing instead, were it not for computers.

I can confirm the excellence of Indians at maths with one extremely anecdotal anecdote. By far the cleverest attender (way ahead of me) of those Kumon maths sessions I occasionally mention here was an Indian boy of about eleven or twelve. (One of the "slumbering giant" glories of Kumon is that it enables Kumon instructors to accept and help to educate pupils who are cleverer than they are. I think this is the single most impressive thing about Kumon. Think about that. But I digress.)

Barron ends as she began, by contrasting India with China:

It was the supreme pragmatists, the Chinese – whose intellectual traditions favoured practicality and action over airy speculation – who were the technological geniuses of antiquity. They invented paper, seismographs, the magnetic compass, the wheelbarrow, irrigation, ink and porcelain. But reasoning for its own sake was of so little interest to them that, unlike the Greeks and Indians, they never developed any system of formal logic. It hardly seems accidental that it is through the manufacture of physical objects that China is making its mark today, while India floats on the ethereal plane of software.

As regulars here will know, I have been trying recently to liven up this blog with pictures. And I think it says something about the priorities of Indian civilisation just now that when I typed "India" and "Mathematics" into Google, the pictures were all either terrible or irrelevant. How do you illustrate an ethereal plane? Just an Indian guy in front of a blackboard covered in mathematical symbols would have done nicely, but I could find nothing like that.

Lots of stuff about Ramanujan, though.

Posted by Brian Micklethwait at 08:40 PM
Category: ChinaIndiaMathsTechnology
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March 09, 2004
Help with maths championship

This comment materialised just now, on this:

hello... i'm from Poland, and I am preparing myself to the math championship... This championship is in English. could someone please send me a www page, on which I could find same excericises? THaks! maciekkowalski@yahoo.co.uk

Any offers? We'll overlook that he can't spell "maths".

Posted by Brian Micklethwait at 10:08 PM
Category: Maths
[2] [1]
February 24, 2004
Is this maths – or is it actually history?

I do not yet have any idea what to think about this, other than to suspect that whatever the government does, it won't make that much difference:

Maths education is failing on every account and needs a fundamental multi-million pound overhaul, a government-backed review of the subject reported today.

The current system of GCSEs and A-levels is not meeting the needs of students, teachers, employers or universities, the report's author, Professor Adrian Smith, said today as he published the damning 186-page document, the result of a 15-month inquiry into the future of maths in schools.

Less than 10% of GCSE students go on to take A-level maths, and less than 10% of A-level students go on to a maths degree, the report says. Incentives should be considered to halt the "disastrous" decline in pupils taking maths at A-level - examples mooted include waiving university tuition fees for maths students.

Further incentives are necessary to recruit and retain more maths teachers. The report documents a shortfall of 3,400 qualified maths teachers - 40% of maths graduates would have to become teachers to account for the shortfall.

GCSE maths should be split into a two-tier structure covering "maths for life" and maths for further academic study to ensure pupils at both ends of the ability range are properly stretched.

The report calls on the government to set up a "maths tsar" to help revamp the structure and content of the maths curriculum and also to advise ministers.

Ah, a tsar. That's the giveaway. What they appoint a tsar, it means they don't know what the hell to do, and are praying for a miracle.

I suspect that this may be more than a mere error of British education policy, and more like a fundamental historical shift, away from making things, and towards supplying those aesthetic services that Virginia Postrel goes on about. After all, how much maths do you need to be a beautician? Or a lawyer?

If all that physical stuff that the West used to churn out is now going to be made in China, it makes sense for young people to shift their focus away from hard science and towards soft philosophising and grooming and chit-chatting. counselling, marketing, packaging, advertising, showbiz, coming first in reality TV contests. That does certainly seem to be the direction of the culture (and I am certainly in no position to complain about it). And against all that, as I say, I suspect that the government may be powerless.

Posted by Brian Micklethwait at 08:47 PM
Category: Maths
[2] [1]
August 21, 2003
Karl Popper and the defeat of boredom

Keeping up with Alice, who is now back from her camping trip, took me here, and to this article by Sarah Fitz-Claridge, entitled The Education of Karl Popper.

In about 1917, Popper came to a clear realisation about school: "... we were wasting our time shockingly, even though our teachers were well-educated and tried hard to make the schools the best in the world. That much of their teaching was boring in the extreme – hours and hours of hopeless torture – was not new to me. (They immunised me: never since have I suffered from boredom. In school one was liable to be found out if one thought of something unconnected with the lesson: one was compelled to attend. Later on, when a lecturer was boring, one could entertain oneself with one's own thoughts.)" On returning to school after an illness of over two months Popper was shocked to find that his class had hardly made any progress, so, at the age of sixteen, he decided to leave school. He enrolled at the University of Vienna, where the cost of enrolling was nominal and every student could attend any lecture course. "Few of us thought seriously of careers – there were none ... We studied not for a career but for the sake of studying. We studied; and we discussed politics."

At university Popper initially attended lectures in many different subjects, but he soon dropped all subjects other than maths and theoretical physics. He thought that in mathematics he would learn something about standards of truth. He had no ambition to become a mathematician, and says: "If I thought of a future, I dreamt of one day founding a school in which young people could learn without boredom, and would be stimulated to pose problems and discuss them; a school in which no unwanted answers to unasked questions would have to be listened to; in which one did not study for the sake of passing examinations."

I think that one of the best ways to write about education is to write about the educational experiences and opinions of people who are deservedly famous, or for that matter deservedly infamous.

I've had a pre-occupying day, so I've let Sarah Fitz-Claridge do most of my thinking and writing along these lines today. It's a formula I expect to use again many times in the future, and not necessarily with writings already available on the internet in their entirety. Linking to aready internetted stuff is useful, but it is also faintly parasitical. All I've really said here is: have a read of this. But that is something.

Posted by Brian Micklethwait at 08:35 PM
Category: Education theoryHigher educationMaths
[1] [2]
July 02, 2003
Numbers games

I've just done a piece for Ubersportingpundit about the way that statistics loom so large in sport generally, and in cricket in particular. I gave it the same title there as I've used for this posting here. At the end I digressed into mentioning how sport encourages boys (especially) to get better at arithmetic.

That's it really. That's my point.

Take cricket. An enormous amount in cricket depends on, to put it bluntly, sums. Sums like: at what rate (runs per over) must the batting side score to get to their target total. If Steve Waugh makes a century, what will that do to his test match average? If England make 550, and Zimbabwe then make 250, and then followed on and make 200, England win by an innings and … what? (The Zimababwe cricket team, like much else in Zimbabwe these days, has been much weakened lately.)

I remember once explaining fractions to a twelve year old boy by talking about a soccer match the previous night. Man United had beaten some hapless rivals by 8 goals to 2. One Man U player scored 4 goals, so he scored half of the Man U goals. Another Man U guy scored 2 goals, so he scored a quarter of the Man U goals. And so on. The big insight was that this poor kid had never connected those damned "fractions" they tormented him with at school with regular and much used English words like "half" and "quarter". Yes, those are fractions. Four divided by eight, four over eight, is a half. Talking about football brought it all alive. I should imagine that there's many a maths teacher who has used sport in this kind of way.

Posted by Brian Micklethwait at 01:52 PM
Category: Maths
[3] [0]
June 14, 2003
The maths isn't adding up

More news of the shortage of people willing to be teachers from the Independent. Maths teacher Stephen McCormack (his title is "Can you find the teachers to sack?" - lovely) writes:

Pupils need to build on relationships with their teachers. If these are absent, the effect on learning and behaviour is marked. This, for me, is where successive governments have failed in the long-term development of adolescents.

It would be bad enough if confined to poor and crumbling inner city housing estates, but it is not. Take Surrey: an affluent county with stable schools and academic tranquillity? Not so. Surrey's turnover rate is the highest in the country.

The current drive to do something in London might have its good points but it will undoubtedly make things worse in the surrounding counties. I am not saying that nothing is being done. One area where there has been a clear improvement is in the numbers attracted to teacher training. Applicants for places on PGCE courses are up, due in part to the bursaries on offer. But it's no good recruiting and training teachers if they don't stay to do the job.

Three years ago I was one of 13 idealistic people starting a maths PGCE course at a London college. Only seven of us will still be teaching in or near London this September. Of the rest, three have dropped out, two have left the UK and one has gone to teach in Devon. Not all schools face these problems. I could also point to numerous schools where turnover is low, and where most vacancies attract enough candidates. Fine. But that doesn't alter the fact that thousands of children are getting a raw deal because of our inability to get staffing right.

McCormack still thinks in terms of failure by "successive governments", rather than by the very idea of government, and says that in France things are done much better, and maybe that's so. But the story I hear is that things there aren't getting any better either.

Posted by Brian Micklethwait at 12:15 AM
Category: MathsSovietisation
[0] [0]
April 25, 2003
Prime bear

Dave Barry (archiving mess – scroll down to Wed April 23) calls this his "educational site of the year". Yes it's Alkulukuja Paskova Karhu, the Prime Number Shitting Bear.

I actually learned quite a lot about Prime Numbers from this eccentric animal, like how around 1,000 they are a lot more frequent than I had supposed, nearly as close together as they start out being.

I don't have to have it explained why mathematicians find Prime Numbers fascinating, because they find anything mathematical fascinating by definition, but do Prime Numbers have any uses other than as something for joke bears on joke websites to emit from their recta? I've heard they're used for encoding things, or maybe for making it impossible to decode things. How does that work?

And do primes have other uses? Surely they must. Commenters who know maths? Here's your chance to broaden the minds of all the maths-phobic humanities snobs who flock here by the thousand.

And linguists! What language is "Alkulukuja Paskova Karhu", and what does it mean? I'm guessing it's Russian, and it's the bear's name, but what do I know?

Posted by Brian Micklethwait at 02:11 AM
Category: Maths
[11] [0]
February 13, 2003
Maths – even more applicable than applied maths

Just in case you don't regularly read Rational Parenting (a mistake on your part, I think, but let that pass), do go there today for an interesting piece by Alice about the maths bit of Britain's current National Curriculum. This key paragraph comes right at the end:

P.S. I've just worked out what this complicatedness is all about: the maths syllabus is apeing real life. It's trying to make itself relevant, by being applied-maths instead of just simple maths-maths. This is nuts. The whole reason maths matters at all is that you can and do apply it; the real-life situations are the applying part. Sigh.

By all means teach maths by referring to particular applications of it in real life – points totals in the Premier League, lengths of bits of wallpaper, areas of wall to determine how much paint you need, speeds of trains to determine times of journeys, etc. etc. By all means, illustrate all those universal statements. But maths itself is of such universal applicability that to embed such illustrative embellishment in the very structure of what must be learned is to limit its universality, and to miss the point of the thing completely.

Posted by Brian Micklethwait at 10:34 PM
Category: Maths
[4] [0]
November 27, 2002
More about Kumon

I mentioned my brief involvement as a Kumon helper, in this post last week. Anyone wanting an outsider's view of the Kumon maths system that doesn't take too long to read, and which emphasises both its methods and effectiveness, and the wider implications of that achievement for education as a whole, may find this helpful. It's part of the Adam Smith Institute "Around the World in 80 Ideas" project, which looks very good, both literally in the sense of looking good on the computer screen and (it seems to me) being well organised and easy and intuitive to use, and in the sense of covering a wide variety of subjects briefly and interestingly.

Sample paragraph:

Kumon's individualist approach overcomes the problems of the collectivist grade system. It allows pupils to move at their own speed: slower pupils are able to move at a pace which does not intimidate and discourage them, and faster pupils are able to move at a pace which does not frustrate and bore them. The method thus allows people to acquire a skill to the maximum level, which their own abilities allow, which will be of enormous utility for the rest of their lives.

Oh, I didn't spot this until now, but I get a mention! Much more important fact: Kumon are now getting seriously stuck in to the teaching of basic literacy. This is a more complex task than maths, but they believe that the same basic methodology applies. I wouldn't dare to differ.

Yet more evidence of the continuing importance, global influence and general vitality of Japan and its culture. There's more to that place than electronic toys.

Posted by Brian Micklethwait at 01:07 PM
Category: Free market reformsMaths
[5] [0]
November 26, 2002
Ethnic mathematics?

Last Wednesday Giants & Dwarfs, who describe themselves as "A Blog on Academia and "Culture" (thanks for the email introducing yourselves gentlemen), had this tantalising report, tantalising because the link embedded in it went to something entirely different:

THE NEXT STEP into madness. The University of Hawaii at Hilo has been awarded a $2.5 million grant by the National Science Foundation, which will be used in part to teach "ethnomathematics" (everyone knows 1+1 doesn't equal 2 if you're of Hawaiian-American extraction).

Anyone know anything about this? It may not be as mad as G&D make it sound, but it does indeed sound decidedly unmathematical. I have never forgotten being told by the Professor of Maths at Essex University several decades ago that mathematics is the study of what everyone is compelled to agree about, regardless of race, colour or creed. And it's surely true that maths is that, even if that definition might be said to include some other things besides. Maths is a huge and expanding clutch of statements of the form "if this is true and this is true then it must follow that this also is true". If you're a Martian, never mind a Hawaiian, you may not get this or that bit of maths, but if the mathematicians have done their stuff right, you can't deny it. It says something very revealing about maths that when humans are trying to strike up relationships with aliens, by including messages in those rockets they fire off into the wild black yonder, they always include mathematical messages.

If all that "ethnomathematics" says is that the language in which the universally true statements of mathematics are expressed may be somewhat culture bound, then fine. But I suspect it of saying something more, something untrue. Comments are always welcome (and thanks very much for all the comments on postings here so far) but on this matter especially so.

Posted by Brian Micklethwait at 02:53 AM
Category: Maths
[4] [3]
November 22, 2002
The un-mis-informing of Ali

I promised the other day that by way of a change I would soon be mentioning (and I promised it rather sooner than now – apologies) some actual teaching that I'd done. Well, the most recent teaching I did was when I was helping my friend Mariana run something called a Kumon Centre, which is a franchised after-school maths club, managed locally but in accordance with a centrally imposed set of guidelines first developed and still presided over in Japan. I wrote a piece for the Libertarian Alliance about this experience, but the events I am about to write about happened after that piece had been published.

Regular school maths usually seems to involve the children working through only a few rather hard problems. Kumon makes them do many more much easier ones. Instead of hoping that they get, say, about half to two thirds of their stuff right, Kumon says they must get nearly everything right. At the heart of the Kumon method is the difference between a child painfully working out that seven plus six equals, er, thirteen? (anxious glance at face of teacher), and knowing with real certainty that seven plus six equals thirteen, with no doubts or hesitations. The usual educational emphasis is on "understanding". The Kumon literature talks of "mastery".

Each child does a clutch of sums selected for him or her personally (there is no everyone-in-the-class-does-the-same-stuff rule) each day, which are supposed to take about twenty minutes to complete. In schools, teachers do the teaching. With Kumon, the system does the teaching. All we did was mark the work the children had done, and then Mariana would follow the rules of the system to set them their next lot of sums.

It worked. Almost all children made steady progress, and in some cases - and in more than just maths by the way – progress was truly astonishing. Kumon sometimes seemed to administer nothing less than a psychological transformation.

But there was one boy for whom Kumon did not seem to be working its magic. Ali was the boy's name, and he seemed to be in such serious trouble that Kumon seemed beside the point. When he did sums they were all over the place. Answers were totally wrong, and figures written the wrong way round. He could hold a pencil and write, but what he wrote was crazy. We seriously doubted if there was anything we could do, and we were ready to give up right there. He would make repeated mistakes, both of calculation and in the way he wrote numbers, and we even started to believe that he might be "dyslexic", or even brain damaged. Also, Ali seemed to be an extremely arrogant little boy. He had a way of lowering his eyelids and raising his head that made him look as if he thought the world to be populated entirely by fools.

At which point I got very, very lucky. I said, let me have a try with him. I decided to do some teaching.

As I say, with Kumon, you're not supposed to teach. You simply shove the stuff in front of them and they do it with the minimum of guidance, and at the end you tell them how they've done, and they learn. The system teaches them, not us. But that wasn't going to work with Ali.

I sat Ali down in front of a clutch of Kumon sums and sat myself down right next to him. I got him to do each one exactly right, telling him exactly what to do and getting him to correct all errors immediately, as soon as he made them, and telling him exactly what to do whenever he didn't know.

I separated the task he faced into a succession of tiny steps and got him to do each step right before proceeding to the next. You start by writing your name there. No, there. What's your name? Ali. Good. Can you spell that? Good. Please write Ali there. Good. Now: what does this say? I point at a two. Two. Good. And what does that say? I point at a one. One. Good. What about that? I point at the plus in between the two and the one. No? That says plus. That means you are adding two to one. What does this say? Don't know? That says equals. That means what does two and one come to. What's it the same as? What is two plus one, two and one, two added to one? So. What's two and one? Don't know? It's three. Do you know how to write three? You do. Good. Please write three there, which is where the answer is supposed to go. Excellent.

And so on. I never made him guess more than once, and I was unfailingly polite. I always said please before asking him to do anything, and I never raised my voice. I never, that is to say, confused Ali being ignorant with Ali being stupid. I did nothing that would be unfamiliar to an averagely capable aerobics instructor working with a arthritic old-age pensioner, but for some reason this sort of thing, when needed by a child, is not always supplied, even in something as widely known as simple arithmetic.

Aside from not knowing the answers, Ali's biggest problem was writing the numbers the correct way around. He would routinely write mirror reflections of them instead. Not all the time, just rather a lot. (This was what had prompted the dyslexia diagnosis.)

When Ali did this - getting, say, the answer right but writing it mirrored - I would say well done, you got the answer right. The answer is five, and that's what you wrote. Well done. However, you wrote the five the wrong way round. Please rub out the five you did, and rewrite it the correct way round. Good.

As I say, you aren't supposed to do this in Kumon. If all the children were to get twenty minutes of solid attention, the way I was attending to Ali, the place would have stopped being the learning factory for everyboy and everygirl that it's supposed to be and would have reverted to being a few tutors helping a few rich kids. But I didn't care.

And the reason that I didn't care was that it worked. After about three sessions along these lines, Ali reached his personal plateau of arithmetical excellence (a few sums wrong but almost all of them right), just like any other Kumon kid.

There was nothing wrong with Ali's brain. Nothing whatsoever. He wasn't stupid, far from it. He had merely been misinformed.

Nor, in my opinion and in my brief experience, and despite my initial prejudice along exactly such lines, was he arrogant. My guess was there was something a bit wrong with his eyesight, and he did the lowered eyelids and raised head thing to correct it.

I don't know for sure how or why Ali had been misinformed and anyway, the cause of the problem wasn't important; what mattered was that the problem was being dealt with. My guess is that (a) his mother may not have been that sure about doing or writing out arithmetic and had consequently not been helping him with it, the way most mothers help most kids with easy sums. Confusions created at school were not cleared up at home. (The more I contemplate state education, the more important remedial home teaching seems to me to be.)

So how had Ali's school created these confusions? I surmise that at his school Ali had not had all his errors corrected, only some of them. Maybe, what with all the other kids to be worrying about, they just couldn't or didn't bother to find the time.

Maybe Ali's teachers had become gripped by the fallacy that ignorance is stupidity and that therefore to correct someone's mistakes - all of someone's mistakes, all of someone's numerous mistakes - is to launch an all-engulfing personal attack on them. (Better to boost the little kiddy's confidence and self-esteem by telling him he's doing better than he is.)

Maybe, what with Ali looking down his nose at everybody, they judged him to be difficult, and feared that if they told him the full story of how badly he was doing he'd make a scene. (That Ali and his family were Muslims might also have made them fear some kind of cross-cultural battle.)

Whatever the exact reasons, Ali had been getting wrong information. Here, two plus one was three, but here, where he'd put two plus one was two, no-one had objected. So presumably that was correct also. Numbers written both the right way round and the wrong way round were left unmolested, so presumably both answers were okay. Except that sometimes the wrong answers weren't okay.

Actually, I believe that Ali was very intelligent, and that had he been less intelligent he would have been less confused. He was getting a mass of bad information and, poor fellow, he was taking it all in, the way a less clever boy might not have.

As I say, Ali wasn't arrogant at all, but if he had thought arithmetic too silly and arbitrary and irrational to be worthy of his sustained attention, who could have blamed him?

I believe I did Ali some big favours, and in a very short time. With luck I convinced him that arithmetic could, if explained properly, make perfect sense and that he could make perfect sense of it. I told his parents that there was nothing – nothing whatsoever - wrong with his brain, only the information that had previously been fed into it, and with any luck they believed me. If Ali's learning environment later reverted to confusion then presumably Ali went back to being confused, but with more luck, both Ali and his parents will then blame society rather than Ali, and seek out non-confusion.

I could go on at even greater length, and in an earlier draft containing this (for another Libertarian Alliance Educational Note) I did. There are plenty of "lessons" here, I think. But this is blogging, and I'll leave this story to speak for itself.

Posted by Brian Micklethwait at 12:51 PM
Category: Brian's brilliant teaching careerMaths
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November 03, 2002
Maths - the traditional way

I saw this link, I kid you not, on a van parked in Twickenham. It is for a company called Accelerated Education Publications whose slogan is "Teaching the traditional way."

Which doesn't sound right. Traditional and accelerated don't normally go together. So, I decided to have a look at the website. It turns out that maths teaching has been so badly undermined by trendy teaching methods and the National Curriculum that no one knows how to teach it anymore and that old is the new new.

Sounds like a good market to be in.

Posted by Patrick Crozier at 05:09 PM
Category: Maths
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