Benoît Mandelbrot (1924 - 2010)

Image via WikipediaBenoît Mandelbrot, who discovered the mathematical shapes known as fractals, died of cancer, aged 85, on 14 October 2010.

The visionary mathematician was born in Poland into a Jewish family that moved to Paris when he was 11 to escape the Nazis. He subsequently moved to the US and spent most of his life working for IBM. He eventually became a professor of mathematical science at Yale University.

During the 1970s, Mandelbrot developed a new mathematical way of understanding the infinite complexity of nature. In 1975, he introduced a new word to the world: fractal. Mandelbrot's seminal works, Fractals: Form, Chance and Dimension and The Fractal Geometry of Nature, were published in 1977 and 1982. In these, he showed that seemingly random mathematical shapes in fact often followed a pattern, made of ever smaller, repeating shapes.

So what are fractals? And why are they important? There is no simple definition of fractals that can be considered truly accurate, but The McGraw-Hill Concise Encyclopedia of Science and Technology defines them thus:

Geometrical objects that are self-similar when the distance at which they are viewed is changed ... The concept is helpful in allowing order to be perceived in apparent disorder, e.g. in the case of a river and its tributaries, every tributary has its own tributaries ... so that it has the same structure and organisation as the entire river except that it covers a smaller area ...

Fractal mathematics is closely related to chaos theory. The best way to get a feeling for fractals is to consider some examples: mountains, coastlines, cauliflowers and ferns are all natural fractals. The concept of the small piece resembling the whole object has allowed scientists to measure the areas and volumes of complicated, irregular objects, including the coastline of Britain, the size of a lung or a cauliflower. They are the sort of shape that mathematicians once shied away from in favour of regular spheres, cones and the like.

Clouds are another classically fractal natural phenomenon. In fact, fractal theory has myriad applications in physics, biology and astronomy. "If you cut one of the florets of a cauliflower, you see the whole cauliflower but smaller," Mandelbrot explained at the Technology Entertainment and Design (TED) conference in California earlier this year. "Then you cut again, again, again, and you still get small cauliflowers. So there are some shapes which have this peculiar property, where each part is like the whole, but smaller."

He also famously wrote: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." He used the word "roughness" to describe this chaos and irregularity inherent in the natural world.

In fact, although the cauliflower and clouds are famous examples of natural fractals, most things in nature are fractal. A sand dune features ripples that resemble the entire dune; a river system has tributaries that resemble the main river, and each of these has its own tributaries. The closer you look, the more you see the same patterns repeating. The whole universe is fractal, with "self-similarity" at all different scales, the defining characteristic of fractals.

The infinitely complex Mandelbrot Set
Created by Wolfgang Beyer with the program Ultra Fractal 3

Like many others, I first became aware of fractals in the 1980s. At school I saw a video on the subject produced by Sheffield University, taking a tour of the lurid, infinitely complex Mandelbrot Set, named in Mandelbrot's honour. At home, I generated my own version of it on my home computer, the single still image taking 20 minutes to appear pixel-by-pixel on the monochrome screen. Now, advances in computing power have allowed us to see the Mandelbrot Set in immense depth. Take a look at this awe-inspiring video of the Mandelbrot Set, generated by the fractal software Fractice. The familiar shapes of the Mandelbrot Set appear again and again, ghostlike, uninvited.

Few people realise how much fractals have revolutionised our understanding of the world, and how many fractal-based systems we depend upon. The fractal mathematics Mandelbrot pioneered, together with the related field of chaos theory and the advent of computers, unveil these hidden beauties of the natural world. It inspired scientists in many fields - including cosmology, medicine, engineering and genetics - and artists, including those producing remarkably realistic computer graphics. Most well-known are the fractal images, which have inspired countless intricate pieces of art.

Fractal geometry has many practical uses. It can provide a way to understand complexity in "systems" as well as in shapes. in the architecture of the networks comprising the internet, the timing and sizes of earthquakes and the variation in a person's heartbeat and the prevalence and diagnosis of certain diseases. Fractal mathematics has also been applied to such technological developments as digital music and in computer file compression systems.

Another application of fractal theory is in the financial markets, a field Mandelbrot first worked in while studying the mathematics of complexity, working as a researcher for IBM during the 1960s. Mandelbrot applied fractal mathematics to market behaviour, in particular the profits and losses made by traders over time. Mandelbrot returned to this idea in 2005. His book "The (Mis)Behaviour of Markets" sets out a highly critical attack on the world of banking and financial systems, he stressed that the economic models in use were unable to cope with their own complexity, and he warned against the huge risks being taken by traders, who, he claimed, tend to act as if the market is inherently predictable, immune to sudden shocks or changes.

Fractal mathematics cannot be used to predict individual big events in chaotic systems, but it tells us that such events will happen. As such, it reminds us that the world is complex at every scale, and inherently unpredictable.

Related articles

• Benoît B Mandelbrot: the man who made geometry an art (guardian.co.uk)

Help! - I Can't Afford to go to Uni!

The problem
Lord Browne's report on university funding is to be published on Tuesday. It's expected to come up with a list of proposals that fit in with the theme of the government's other budget adjustments: cuts and austerity.
In short, the universities need more money. There is nothing to spare in the Education Department's budget, so, one way or another, the student will have to pay more.

Clare College and King's Chapel, Cambridge
Photograph Christian Richardt

Some kind of Graduate Tax now seems to have been ruled out. This was the Liberal Democrats' preferred option. David Cameron has spoken out against the Graduate Tax, saying that it would be unfair and wouldn't actually raise any money until 2040. Lib Dem Vince Cable, business secretary, appears now to agree.

So, we are probably left with higher tuition fees. There are already expectations that the report will recommend allowing fees to rise from the current £3,290 per year to £7,000 or more. Adding on the cost of living, many students will be leaving university £40,000 in debt. There will be changes to the threshold at which loans are to be repaid and big increases in interest rates. Student loans will become proper, market-driven financial loans, with the interest rates that are typical for personal loans.

There are problems here. Accusations are flying that the Lib Dems are abandoning their core election promises, since they made their opposition to higher fees a flagship election issue.

At the general election, every Lib Dem MP, including party leader Nick Clegg, signed a personal pledge that they would vote against any increase in tuition fees.

The party are pre-empting Lord Browne's report by highlighting "progressive" measures within it, for example that repayments on student loans will be more expensive for graduates in higher-paid jobs.
But the National Union of Students (NUS) says it is an "insult to the intelligence" to try to "re-brand" an increase in fees as "progressive".

Tuition fees appears to divide the two coalition parties so sharply, it could be the first issue to create a serious rift between them. Many Lib Dem MPs are already openly opposing the moves and threatening to vote against any legislation. Some have even spoken about the government falling. Like or hate the current government, political instability and another general election are probably the last things we need.


Some calm words of advice

If you are going to have to make the decision about your post-school future soon, whether or not to apply for university, in these straitened times, what should you do? Do you need an alternative to university? Are there alternatives?

Take comfort in the fact that the A-Level is still a gold standard in the world of education. A-Level maths is held in particularly good esteem, especially with a good grade.

Here are some ideas to consider:


Get advice. Speak to your careers adviser, your parents, your friends. Don't feel you have to take anybody else's advice, but at the same time listen carefully. They may well have some valuable insight.


Work your way through uni. There are plenty of evening and weekend jobs available for students. Make yourself the guy selling the pints, not the one drinking them. There are downsides of course: any job is a serious commitment; the hours spent at work instead of studying could damage your exam results. Don't bite off more than you can chew. Wind down the hours as exam season approaches.


Don't go to uni, get a job. For example, the banks are still recruiting, despite their problems. The entry level may not be as high as for  those with degrees, but in time you can work your way up the ladder. You'll be earning money, not accruing debt. Temping agencies are often a good place to start, especially for office work. Prove yourself a good worker and any company will be tempted to take you on full-time - it's cheaper for them.


Take a year out. Travel or get work for a year, while you think about what to do next. There doesn't have to be an instant transition from school to university.


Study abroad. There are plenty of good universities in Europe, although the fees will probably work out as much as in the UK. You get to see some of the world at the same time as getting a degree. It's worth thinking about.

Whatever you choose to do after your A-Levels, don't take this decision lightly. Remember that an education from a good British university can and will be life-changing for you. If you can afford it.

Related articles

• Graduates to repay from £21,000 (bbc.co.uk)
• Browne review: Rich graduates to fare better in tuition fees shake-up (guardian.co.uk)

Free Schools have Fifteen Minutes of Fame

Michael Gove (Photo: Paul Clarke)

Education Secretary Michael Gove is putting a brave face on the numbers of schools applying for "Free School" status. He told Andrew Marr this weekend that of 700 initially expressing an interest, 16 schools have formally applied to adopt this status from September 2011 and that this has "exceeded his expectations". Free Schools will be state-funded, but will remain outside of local authority control. In Gove's vision, it will be parents, teachers and voluntary groups that will be interested in creating Free Schools. He cites similar systems in Sweden and the US; and legislation was passed just two months into the lifetime of this Parliament to allow for their creation.

Supporters of the scheme claim it will create more local competition and drive-up standards.

But there are many critics, even from Michael Gove's own side of the political divide. Some have argued that the scheme will only benefit middle class parents with the time to dedicate to setting up a school and that it will divert money away from existing schools. Moreover, if having Academy or Free School status becomes a symbol of success, schools remaining under local education authority control will be seen as inferior.

Peter Wilby, writing in The Guardian, thinks that such schools will end up being run by private companies rather than parents, teachers and voluntary groups.

Paul Carter, the Conservative leader of Kent County Council, stated that under present funding arrangements: "The more Academies and Free Schools you operate ... the less maintained schools would get."

The National Union of Teachers warns that the policy could "fuel social segregation and undermine local democracy". The union is already talking about strikes to oppose the Free Schools scheme and the expansion of the Academies programme.

Rushed legislation is so often bad legislation. Rather than inventing new types of school to fit in with the Prime Minister's "Big Society", Gove's attentions should instead focus on the buildings programme, whose scaling back he so badly botched after taking office. It has been shown that a good working environment is one of the most important factors in a child's educational development.

Besides, do we not have enough different types of school in this country already?

Ed Balls, the shadow education secretary, described it as "laughable" that Mr Gove considers his expectations to have been exceeded with the numbers declaring an interest.

Our prediction is that the distinction between the Academies and the Free Schools will blur and then disappear. With scant interest from the original audience, the parents and teachers themselves, and private business apparently the only sector of society interested, and in the face of much criticism and the prospect of strikes, the Free Schools programme will be slowly forgotten.

Just as with his plans for reforming A-Levels, it sometimes appears that Gove is shooting from the hip, saying whatever comes into his head without thinking.

Yes, there is room for improvement in our schools, and yes, there may be scope for greater involvement from the world of business, whether it be in the form of sponsorship or in the day-to-day tasks of school management. But there appears to be little appetite among our busy parents and teachers for taking on the responsibilities of running schools, nor so far from the voluntary sector. The Free Schools notion will be brief, will become subsumed into the Academies programme and will die, not with a bang, but with a wimper.

Related articles

• Michael Gove dealt fresh blow as only 20 'free schools' approved (guardian.co.uk)
• First 16 'free schools' revealed (bbc.co.uk)

The Battle for Britain’s University Places

Hundreds of thousands of students in England, Wales and Northern Ireland received their A-levels and AS-levels results on Thursday. More university places are on offer this year, but the numbers are capped and universities face fines if they breach these limits.

Yesterday, we discussed the new A* grade, introduced this year to help universities differentiate between the very best candidates.

To get the new grade, a student has to score an A overall, plus at least 90% in each paper in the second year of the course. Yesterday, we reported that about 8% of all students’ papers have been given an A*. A total of 8.3% of A-levels taken by girls were awarded the new A* grade, compared with 7.9% of those from boys.

Students from independent schools were proportionately more likely to get an A*. They provided 30% of all A* grades awarded, despite making up only 14% of entries. Candidates from comprehensive schools, which are responsible for 43% of A-level entries, gained 30% of the A* grades awarded. Further education and sixth-form colleges, which enter 30% of candidates, saw their students given 20% of all A*s awarded.

We are pleased to say that the highest percentage of A*s was awarded in further maths at 29.9%. 17.2% of maths candidates achieved the grade. The numbers taking maths continued to rise with an extra 4,526 entries, and an extra 1,209 for further maths. It is pleasing to see this trend after a marked fall back earlier in the decade.

In Scotland, the overall pass rate for Highers was up slightly, to 74.6%, creating a new record. Northern Ireland fared the best of the regions, with 35.7% of papers being given an A or A*, 9.3% the A*.

So why, despite these record successes, are many candidates facing disappointment over university places?

Applications this year are up by 12% on last year's record level, with the numbers increased by those re-applying after not finding a place last year.

Many universities are warning that the number of places available through the ‘Clearing System’ will be severely restricted. Last year almost 48,000 students found places through this system, which matches available university places to students who did not get the grades they needed for their first choices. Half of Scotland's universities have already said their courses are full.

Universities Minister David Willetts congratulated students on their results and said that those who did not get the offer of a university place had other good options.

"There are more university places than ever before and already 380,000 applicants have got confirmed places at university. For those who have sadly not done as well they hoped, there are places available in clearing.

"Of course, university is not the only route into well-paid and fulfilling work. That is why we are also investing so much in Further Education and 50,000 extra high-quality apprenticeships."

Some may accuse Willetts of having his head in the sand. We have heard stories of students with three A* grades without any offers of a university place. The president of the National Union of Students (NUS) Aaron Porter said: "With youth unemployment pushing one million, savage education funding cuts and arbitrary limits on places, the government is at risk of imposing poverty of opportunity on a generation of young people facing a very uncertain future."

The biggest problem this year is clearly the recession. With jobs scarcer, the competition for university places automatically heats up. But there are other factors at work too.

The system has become skewed by awarding points for A-Level grades, implying that an A-Level in media studies is as valuable or useful as the same grade in maths or a science. The fact that these softer subjects are attracting more teachers only exacerbates the trend.

The private sector and the best state schools have shown that it is possible to maintain good standards, while keeping the demand for trendy or less demanding subjects to a low level. Securing a place for your child at one of these schools and the rules for the child’s education are clear: traditional subject choices, respected by the top universities, rigorously pursued. It is a recipe that has sadly disappeared in too many places.

Our school-leavers deserve better than the mess they have found themselves in this year. What will become of our goals of greater social mobility when subjects such as German and physics thrive in the independent sector but are dwindling elsewhere? The knock-on effect is the closure of the university departments in these subject areas, and this has already begun.

If you are still unsure about where or whether to go to university, it’s essential to get some high-quality, targeted information, advice and guidance. Speak to your teachers and to the universities themselves. Don’t rush into a course that isn’t suitable for you. You may end up leaving university early without a degree, but with the debt.

Good luck.

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• A-levels out amid degree pressure (bbc.co.uk)

The long wait is over

So the long wait is over. A-Level results are out. Congratulations if you have achieved your goals. And good luck if you are still waiting to hear about your post-A-Level destiny.

Two over-riding themes emerge from this year’s bunch of results: the ever-improving grades achieved (including the awarding of the new A* grade this year) and the shortage of university places available for this year’s cohort, both of which are cited as evidence of at best a sign of desperate times, at worst a failed education system.

What of the first, the notion commonly referred to as ‘Grade Inflation’? (I’ll look at university places in my next post).

The statistics

First, the statistics. Well, it was another record-breaking year of results. One in 12 A-level exams (8%) has been awarded the new A* grade (which, we believe is given to those who gain roughly 90% in their exams).

27% of entries have gained an A or A* grade and this rate is currently increasing at about 1% each year (26% achieved grade A last year, 25% the year before). The overall pass rate rose for the 28th year in a row, with 97.6% of entries gaining an E or above, up from 97.5% in 2009.

Why?

In my mind, a number of issues are at play here.

Firstly, let’s give some credit to the students. They are working harder, becoming more organised and better mentally equipped. They support each other, get help from parents and sometimes personal tutors. The web and great textbooks also help.

Secondly, don’t forget the teachers. Teaching methods are improving and this is assisted by new developments in the classroom, such as interactive whiteboards, a whole host of educational websites, and considerable expertise in these resources, as well as in the subject matter. Well done teachers.

There has also been some talk, particularly in these competitive times, of pupils choosing ‘softer’ subjects, in which they are more likely to rack up the number of A-Level points to move on to their institution of choice.

But I don’t think this is the end of the story.

As I’ve hinted already, there’s fierce competition out there. Pupils are realising that it’s a battle out there, and the best university places, or the best jobs, come from good grades. They see the incentives and they are prepared to work to get the rewards.

But I’m going to highlight one more factor, which I haven’t heard mentioned in the debate so far. I have done a lot of work for the exam boards over the years. Edexcel’s mathematics A-Level provides good, challenging questions, in about 12 different maths modules. The questions are quality-controlled, the examiners are tested, their work is scrutinised. In turn, the examiners’ supervisors are put through a rigorous qualification and testing procedure. The planning and the chain of command are almost militaristic.

But – and here’s the point – the exam papers are not necessarily too easy, they are too samey. Year after year, the question papers feature questions from the same topics, with only the numbers changing, the ordering of the parts or the words changed to alter the subject of the question. There is no imagination required to solve some of this stuff. Learn the technique and you know how to pass the exam. The teachers know this and a thorough analysis of the last 5 years’ past papers give them all the teaching material they need to ensure a good bunch of grades from their class.

It’s only exams like Cambridge University's STEP papers and the Advanced Extension Award that really test the imagination, the inventiveness of our pupils.

In my mind, this is the way forward. Forget A* grades, A** grades, ad infinitum. Let’s change the exams. Edexcel and the other exam boards should put as much effort into devising a truly challenging set of exam papers each year as they do into ensuring a fair and unbiased system (which is, of course, also very important). Perhaps a little adjusting of the syllabus would be necessary.

The truly outstanding pupils would shine. An A grade at A-Level would be a true indication of excellence, one that would make us proud of the UK’s education system again.

Related articles

• Record A-level passes spark intense battle for uni places (guardian.co.uk)
• 1 in 12 A-levels have new A* grade (independent.co.uk)

Another Government, Another Shake-Up

The Conservative-led coalition government came to power in May and set about reforming every area of public life with what some would describe as indecent haste.

Michael Gove is our new education secretary and instantly went into negative territory on the credibility meter with his bungling of the announcements about schools that will undergo refurbishment work. This programme, of course, is being cut back, like everything else, but Gove's handling of the lists of schools made him look likely to be the second casualty of the already embattled new cabinet.

As I write, however, he is still in office. His latest pronouncements relate to the A-Level and, almost parodying his own party, he wants to take things back to some Golden Age he half-remembers, when the A-Level exam was indeed a single exam at the end of two years of study. He says he wants to "revive the art of deep thought". If his plans come to fruition, it would mark the end of the AS exams, which students take after their first year of A-Level study.

In a response to his comments this week, the Advisory Committee on Mathematics Education (ACME) said harder exams could put students off studying A-Level mathematics and that this could trigger a "collapse" in A-Level maths student numbers and even lead to university department closures. Professor Dame Julia Higgins, chair of ACME, warned making A-level maths harder could result in a drop in the number of students taking the subject.

She said: "We feel it is very important that we warn you that implementing such a policy runs a genuine risk of repeating the collapse in the numbers studying A-level mathematics". She went on to say it was very likely that university mathematics departments would close as a result.

A-Level maths has been in a modular format since the introduction of Curriculum 2000 at the turn of the millennium. Student numbers fell by about 19% over the following 3 years. There was an overhaul in 2004 to make the A-Level more accessible, and the old "Pure" modules were replaced by the current "Core" modules. After 6 more years, student numbers are now more or less back to where they were before modularisation.

Michael Gove said universities had complained A-levels were not preparing students sufficiently well and that he wanted them to be more academically rigorous. Modular A-levels have been criticised as being easier than the traditional variety, because the learning is broken up into many small units. Critics argue that the process of modularisation has contributed to grade inflation, where the percentage of pupils getting a grade A has continued to rise year-on-year. This year, the A* grade will be used for the first time to reward those achieving the very best results. Cambridge University's STEP papers and the Advanced Extension Award have also been used to differentiate pupils with top grades.

But Cambridge University have joined with ACME in criticising Gove's remarks. Cambridge admissions manager Geoff Parks said AS-levels are an "invaluable indicator of progress".

He added: "We agree with the secretary of state in some aspects of the reform. We would agree that at the moment A-levels are too modular and there is too much examination. We are not sure all the opportunities for re-takes are advantageous. But we think there is a middle way between where we are now and the proposals he has set out."

The Department for Education said there was a need to restore confidence in public exams. It said it would work with universities for a "robust and rigorous" A-Level and that reform plans would be set out later in the year.

mathsbank

How to win at World Cup predictions

Now the World Cup is over, the sound of the Vuvuzelas is fading almost to a melodic memory, the England shirts given away to the local charity shop, it's time to round up the success or otherwise of the various World Cup predictions, based on mathematical techniques.

Firstly, let's look at the banks. I have no idea why our esteemed financial institutions have employed people to make football predictions, when they should really be working rather hard at recouping the money we've given them. But here goes:

UBS failed dismally with a prediction that Brazil would win, and another that Italy would get quite a long way.

http://www.ubs.com/1/e/bank_for_banks/news/topical_stories/edition_10.html

Goldman Sachs also backed Brazil, predicting the other three semi-final teams to be England, Argentina and Spain. One out of four there, and no winner.

Danske Bank, not wishing to disagree, backed Brazil too, but JP Morgan must be feeling really silly. They tipped England for the title.

And a quick look at some of the academics. Metin Tolan, a German physics professor, thought that Germany would end up victorious:

http://uk.eurosport.yahoo.com/22042010/58/world-cup-2010-professor-proves-germany-win-world-cup.html

The most successful analysis we've seen so far has been that of the mathematics department at Queen Mary, University of London. They analysed each game's pattern of passing, giving each player a score called their 'centrality'. This measures how important each player is to the team's performance. Similar techniques are used in modelling computer networks.

http://www.maths.qmul.ac.uk/~ht/footballgraphs/index.html

From this analysis, the researchers say they predicted a Spanish win in the final against the Netherlands, although it is not clear from the website how exactly they reached this conclusion. Also, it seems impossible to predict the winner of a tournament before knowing which teams would play against each other in the latter stages.

What does all this tell us? The world is a very unpredictable place and you'd probably be better off becoming a bookmaker than visiting one. That or buy a psychic octopus.