Joseph Fourier and the Vuvuzela

Joseph Fourier (1768 – 1830), more than anything else, wanted his mathematics to bring usefulness to society. During his lifetime, he formulated a theory of heat, which was later applied to light and, especially recently, sound.

He postulated that there is fundamentally one form of wave, to which the same mathematical principles can be applied. And he possessed a remarkable ability to generalise results and to find some unexpected applications for his work.

He worked hard from a young age; at his boarding school, he would collect the ends of old candles and studied into the night.

He became an intellectual figurehead for the French Revolution and when the Republicans won the day, he flourished in the new society. New centres of learning emerged and nobody was denied a place in the education system because of a lack of funds.

Fourier was appointed head of mathematics at the École Polytechnique. He had been a great orator during the revolution and now practised these skills in the lecture theatre. Debate was expected between teacher and pupil and Fourier, the great performer, thrived, filling his lectures with interesting historical references and practical applications.

In May 1798, Fourier received a summons to set sail for Egypt with Napoleon Bonaparte and 30,000 soldiers and academics. There he helped with the discovery of the Rosetta Stone. He also began to develop his obsession with heat, posing himself the question “How does heat vary over time?”

Eventually he would set forth his mathematical laws of heat, and this would become one of most important branches of mathematics, through the startling revelation that almost any natural phenomenon could be described as the sum of simple sine waves. In sound, for example, the idea was that every noise could be made up from the sum of the pure sounds created by a tuning fork.

There was vigorous opposition, notably from Laplace, who described considerable difficulties in Fourier’s work and a “lack of rigour”. But the final version of Théorie Analytique De La Chaleur was published in 1822 and remains a seminal work.

Lord Kelvin described Fourier’s work as one of the most beautiful theories of modern mathematics.

One example of modern day applications of Fourier’s work is the way in which computers can break music down into its component waves. Then, these waves can be manipulated using Fast Fourier Transforms to enhance, warp or distort the overall effect, or to remove the noise on a particular frequency. And hence the application of Fourier to the very topical, sometimes maligned vuvuzela. Could Fourier analysis really help to block the sound of the monotonous football trumpets? The BBC seem to think it would detract from the erudite commentary being offered. But at least one other source seems to think it could work (if you've got a Linux machine and lots of leads). Let us know your experience.

info@mathsbank.co.uk

Martin Gardner

Photograph: Konrad Jacobs, Erlangen/Creative Commons

Martin Gardner died on May 22 aged 95. His writings on mathematics and science inspired the concept of mathematics as a fun pursuit. He achieved further fame with his shrewd analyses of Lewis Carroll's Victorian fantasy stories about Alice.

In 1956 he wrote his first article for Scientific American magazine. This he followed up with an essay about hexaflexagons – hexagons made from strips of paper that show different faces when flexed in different ways. This clear and entertaining exposition impressed the publisher so much that Gardner was given a regular column in the journal, to be written on similar topics. At this point, he had not studied any mathematics since leaving secondary school, so Gardner became a self-taught expert, gaining second-hand books in order to find enough material to keep his "Mathematical Games" column running. He did, and it ran for 25 years until 1981. It earned Gardner the American Mathematical Society's prize for mathematical exposition.

During this time, he was also a prolific writer for other mathematical publications; he published articles in hundreds of magazines, newspapers and various journals.

His lack of a formal mathematical education meant that Gardner rarely relied on academic jargon, but instead he gave the column a broad appeal by packing it with anecdotes, cultural references, jokes, tricks and many games. He introduced concepts such as fractals and Chinese tangrams, which have all become favourites with today's mathematics teachers as they seek mathematically rich and challenging, but fun activities.

Gardner was also a writer on the subject of debunking the paranormal and pseudo-science. In his 1952 book "Fads and Fallacies in the Name of Science", Gardner argued against many bogus concepts, such as alien abductions and a belief in UFOs. Later he became an antagonist of the spoon-bender Uri Geller.

His followers have created a regular convention known as "Gatherings for Gardner" (G4G), at which mathematicians, magicians and fans of all sorts of puzzle congregate from around the world.

There can be few mathematicians over 40 who would not say they have been influenced in some way by Gardner's blend of mathematics and mathematical fun.

Many congratulations to Fabrice Bellard, the French computer scientist who has calculated the value of pi to a jaw-dropping 2.7 trillion decimal places. This is a new record, which took 131 days of computation time and checking. And all of this without supercomputers - he performed his calculations on a trusty desktop PC. The previous record (a mere 2.8 trillion places) was set last August by a team from the University of Tsukuba in Japan.

This area of work, the ability to calculate irrational numbers to ever-increasing accuracy, has become known as 'arbitrary precision arithmetic'. In itself, it seems unlikely that such an accurate value for pi will prove useful in any practical context. But the constant is often used in testing computers and software algorithms. And Monsieur Bellard's techniques, which he claims were 20 times faster than those used by the Japanese team, will contribute to further advances, and may even be used in other areas of mathematics and computer science.

We love this story because it highlights the fact that ordinary mathematicians can still make significant, ground-breaking contributions to mathematics.

Now, I wonder whether Monsieur Bellard can recite the value of pi he has calculated. Then I would be really impressed.

November newsletter is out today.

Battle lines drawn over transfer tests

You may not have heard about it on the mainland, but here in Northern Ireland, the excellent education system has reached a crisis point.

The grammar schools were told that last year, 2008, would be their last chance to use academic selection to decide who would make up the new intake of pupils. The transfer tests, or 11+, were abolished.

But the grammar schools have devised their own tests, one being used by the predominantly Catholic schools, another by the schools with a Protestant tradition. Pupils in their final year of primary school (they call it P7 here) began the new tests, based on mathematics and English, this weekend, with the first of three papers to take place over the coming weeks. They must visit the grammar school of their choice on a Saturday morning, unlike the old official transfer tests, which took place in the primary schools. And, unlike the old tests, they are not free.

Education minister Caitriona Ruane of Sinn Fein believes the old system was unfair, and placed too much pressure on children at a young age. She has so far refused to countenance any form of compromise. Ruane seems to believe so strongly in her cause that she has now threatened the schools with legal action if their tests continue.

Ironically, the unintended consequences of Ruane's actions may be to allow a private education system to develop, which until now has been largely unnecessary because of the high standards of the grammar schools. Fees for the tests is perhaps a first step. School fees and opting out of the state system may be the next, if the grammar schools feel their statuses and reputations are being threatened.

Here at mathsbank, we would like to see a resolution to the increasingly hostile dispute one way or the other, before the education of the children of Northern Ireland begins to suffer as a result of it.

Over the past five years there has been a very encouraging increase in the number of students choosing to study both maths and further maths at A-Level.

This year the increases in both subjects were particularly large, far greater than for other science and technology subjects.

Mathematics in Education and Industry (MEI) would like to understand this trend a little better and are collecting some data from teachers, through an online survey.

The aim of the survey is to find out what our A-Level maths teachers believe to be the main causes of the rise in maths student numbers. Teachers of all specifications in England, Wales and Northern Ireland may take part.

Your feedback is valuable and the closing date is 11 October.

http://www.mei.org.uk/index.php

What next for post-16s?

Many A-Level students have just had their results and are now forming a good idea of how they will be spending their next year, or two or three.

But what for this age group in general? How is the future of post-16 education shaping up? Will A-Level continue to be the Gold Standard of school-leaving exams and university requirements?

There are several factors to consider here. There is no doubt, with the continued improvement in the overall results our students are achieving, that there is pressure for change. The universities are finding it increasingly difficult to use A-Level results as a benchmark on which to judge an applicant. And many schools are now hoping to prove the ability of their pupils in other ways.

It is important to note, at this point, that the improvement in grades is a good thing. Improving teaching methods, and more diligent, better-prepared pupils are the primary factors giving us these results. But, if there is a downside, it is how to then distinguish between the increasing numbers who are getting the best grades.

The new A* grade, to be awarded for the first time this year, was introduced specifically to address this problem. There will be no change for the vast majority, but students who are gaining an A grade easily, with a truly exceptional score, will be awarded the new grade.

Other initiatives have been introduced. In mathematics, Edexcel are now offering the Advanced Extension Award, which, although examining the same content as the A2 exam, demonstrates a higher understanding of the material.

The International Baccalaureate has been adopted by many schools, and one of its aims is to give the student a more rounded qualification, proving ability across a wider field of study.

Cambridge University has devised the Pre-U and at least one private school, Harrow, has said it will consider abandoning A-Levels entirely in favour of this new exam, which is now in its second year of teaching.

Finally, there are bound to be changes to the structure of the GCSE within the next few years. What evolved as a school-leavers' exam is now looking less and less useful in that role, since almost all pupils now go on to some kind of post-16 education. GCSE could be brought forward a couple of years, to fill the 'SATS void' and already a large number of schools are giving their pupils the chance to sit GCSE up to two years early. How all of this impacts on post-16 education remains to be seen, but any decisions made about GCSE will clearly have to be made in conjunction with decisions made about the structure of our ever more critical post-16 assessments.

Whichever party forms our next government, we hope to see some very careful consideration of these issues.