MEI (Mathematics in Education and Industry) and Tarquin Books are joining forces to offer an annual prize to post-16 maths students.

Entrants must design an electronic resource - e.g. Geogebra file, Excel spreadsheet, a video, etc. - that communicates the mathematical ideas needed to solve a practical problem of some kind. These mathematical ideas used should be relevant to A-Level Mathematics or Further Mathematics A-Level, or to some equivalent qualification. Your entry could be designed by you individually, or it could be a collaboration between a group of students.

Entries to the competition could also form the basis for an Extended Project Qualification.

The winner will receive books to the value of £100 and Tarquin Books will also donate £400 worth of books or materials to the winners’ school or college.

To register your interest in the competition, follow the Prizes link on the Tarquin website. The closing date for entries is 30 April 2012 and the winner will be announced by 31 May 2012. For further information email Andrew Griffin at Tarquin.

## Sunday, 27 November 2011

### Make a Great Resource and Win a Great Prize

Labels:
Extended Project Qualification,
Math,
Mathematics,
MEI,
Tarquin

## Friday, 11 November 2011

### Caleb Gattegno (1911 - 1988)

Image via Wikipedia |

Today marks the 100th anniversary of the birth of Caleb Gattegno. He was an educator, particularly in mathematics, but also in linguistics, publishing many seminal works on the theory of education. But none of this theory was detached from the real world. His ideas were based on practical, real-life situations, and what, according to his observations, helped his students the most. He was also an inventor, creating tools for the classroom that are still used and found invaluable by today's teachers.

Gattegno grew up in Alexandria, then lived in Cairo, London and New York. He worked all over the world and devoted his life to a study of learning, not only creating a number of important innovative techniques for the teaching of languages and of mathematics, but also made a remarkable, seminal contribution to the understanding of the learning process at all ages. In 1952 he founded the organisation that would later become the Association of Teachers of Mathematics, and its magazine Maths Teaching. This year, the ATM is making Gattegno and his work the theme of their annual conference.

Caleb Gattegno made a significant impact on teaching and thinking about education not only in the UK but also in many countries around the world. Within mathematics, apart from the creation of ATM, his work included the promotion and use of Cuisenaire Rods, the creation of geoboards, developments of the animated geometry films of Nicolet, and the Gattegno ‘tens’ chart for number. Cuisenaire Rods were so effective, he thought, that he founded the Cuisenaire company in the UK.

In addition he challenged many in the educational world to consider what is involved in learning, encouraging them to allow an explorative approach. The teacher's actions should be subordinated to the way in which the child learns through exploration and investigation. His idea that only awareness is educable breaks the learning process down into four stages. The first, the most important, is a single act of awareness, for example that something is there to be learned or can be explored. Without this realisation, or any notion that there is a challenge, problem or issue to be explored, the pupil will not be able to proceed to the next three stages. These stages can be described as exploration, transition (which begins with the new skill being something that can be achieved with a lot of concentration, and ends with it being automatic) and transfer, in which the new skill can be used and applied to the learning of yet other skills and ideas.

Use of his teaching aids was developed in classrooms, and to demonstrate their power he was prepared to teach children of any age or ability, at any time, in front of other teachers. He ran many seminars through his lifetime and his personal influence was felt profoundly by those who saw him in action.

## Monday, 31 October 2011

### A-Level Exams Earlier, University Applications Later?

Image by Getty Images via @daylife

UCAS, the body that administers UK university admissions,
has put forward proposals for changes to the admissions system. These
recommendations include bringing the A-Level examinations forward and
completing most of the university application process when the exam results
have been released. This system, UCAS argues, would be fairer and less complex.

Currently, pupils in their final year of school must make
their UCAS applications by mid-January. Universities judge each application
based on predicted A-Level grades, references from teachers, personal
statements and possibly an interview. The universities then award conditional
offers, dependent on certain A-Level grades being gained.

Many schools, particularly private schools, give university
admissions advice that can maximise the chances of successful entry. UCAS
argues that this system makes it unfair for pupils who do not have such a system
of support available to them. In short, as the Guardian puts it, the current
system favours the rich.

An overhaul would lead to a fairer and more transparent applications
process, with the actual grades gained being central to a university’s decision.

An application later in the year would also give pupils more
time to discover their real interests, which subjects they are excelling in,
and would like to spend further time studying. The downside to such a plan would be the timing. A-Level
examinations, the marking, awarding of grades, university applications and
decision-making would all need to take place in the summer term. In Northern
Ireland and Scotland, this problem would be exacerbated because schools break
up for their summer holiday earlier than in England and Wales.

The last Labour government attempted to bring in similar
reforms of the universities applications process, without success, largely
because of opposition from teaching unions. Although many teachers see some benefits
to such a scheme, the amount of teaching time for the A-Level examinations
would be shortened. There is also a feeling that there would be simply too much
to achieve during the summer term.

What do you think? Would you be happy to delay your university
application until after you have received your A-Level results? How do other
countries manage university admissions? Would a compulsory gap year be one radical
solution to the problem (which they once called National Service), giving
pupils further time to think about their futures and time to do something
useful in the workplace, while ensuring our students are more mature when entering university? Let us know your thoughts.

## Tuesday, 25 October 2011

### Evariste Galois (25 Oct 1811 – 31 May 1832)

**A sketch of Galois, image via Wikipedia**

Of all the mathematicians who ever lived, Évariste Galois, born 200 years ago today, is the one who has given me the most inspiration for the subject.

He was a hot-headed genius, living in a period of political instability in post-revolutionary France. The ideas he formulated, during his incredibly brief life, were not only ground-breaking, they were so ahead of his time they could be described as alien to the 18th century world he lived in. Only now are Galois's ideas becoming of dramatic importance. If you are interested in quantum computing, or particle physics, to name just two of today's hottest scientific topics, you should know about Galois.

At the age of 16, with a head full of novel mathematical ideas, he sat the entrance exam for the École Polytechnique, failing to get in. The following year, he tried again. This time, the application ended in frustration and acrimony. The examiner reported the student ‘Knows nothing’ and was of ‘Little intelligence’. More likely, Galois’s approach took too many logical leaps, confusing his supposedly superior examiner. He entered the inferior École Normale, despite being described by the authorities there as ‘Obscure in expressing his ideas’.

Aged 19, Galois sent two papers to the Academy of Sciences as an entry to the Academy’s prestigious Grand Prix. This first attempt at recognition failed for mysterious reasons, although it seems that Cauchy, the eminent mathematician refereeing the works, simply wanted the two papers combined. Galois’s second submission, however, failed in farcical circumstances. The revised paper was sent directly to Fourier, who was secretary to the Academy and appeared sympathetic to Galois’s ideas. But Fourier died shortly afterwards and the paper was lost.

One year later, political tension in France was running high. Galois had friends in revolutionary circles and attended a banquet with them. He was overheard, perhaps by spies, uttering the name of the king, Louis Phillipe, while brandishing a dagger. This being construed as a threat on the king’s life, Galois was arrested. This time, he was released, but he was later re-arrested and this time sentenced to 9 months in prison.

He was not well treated in prison, being forced into drinking matches with his fellow inmates. When Galois complained about his ill-treatment, he was put into solitary confinement.

When out of prison, Galois returned to his mathematics, but his political interests continued to divert his attention. He finally received a response from the Academy of Sciences. His submission was declared “neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”.

The exact circumstances surrounding the final days of Galois’s life and the fateful early morning duel, on 30 May 1832, are unclear. It has been rumoured that he was involved with a young lady, and perhaps the duel was instigated by Galois for her honour. The night before the duel, he set out to document all he could of his life’s mathematical work. He finished his rushed document with “… j'espere, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.” (I hope some people will find it to their advantage to decipher all this mess.)

Galois lost the duel and his short life was brought to a dramatic and violent end, the lack of sleep perhaps contributing to his slower reactions. His final words were “Do not cry, it takes all my courage to die at the age of 20.”

The work of Galois that survives amounts to no more than 60 pages. In its modern form, it bears little relation to the scrawl of his last document, but the ideas remain. He analysed the symmetry in the solutions of equations in ways never previously imagined. These new approaches, combined with his chaotic presentation, were what had confused his superiors at the various academic institutions.

Today, Galois Theory is of fundamental importance in the field of Quantum Theory. It is helping in answering questions about the structure and origin of the universe. It has been applied not only to describing the sub-atomic particles that have already been discovered, but to predicting the existence of new ones. The workings of the Large Hadron Collider, CERN’s giant particle accelerator, are dependent on the strength of Galois Theory.

Who knows how differently mathematics - even human civilisation - may have evolved, had Galois's life not been brought to such a sudden and tragic end.

###### Related articles

## Tuesday, 18 October 2011

### The Forefather of Computing

**Part of the Difference Engine. Image via Wikipedia**One hundred and forty years ago today, on 18 October 1871, Charles Babbage died.

Born in 1792, this English mathematician and inventor is known as a pioneer of modern-day computing, and is credited with conceiving the concept of a programmable computer. He developed an obsession for mechanising computation, in order to eliminate inaccuracies in mathematical tables. By 1822, he had developed a small calculating machine able to compute squares. He then produced prototypes of a larger Difference Engine.

In the 18th century, numerical tables were calculated by humans who were known as 'computers'. While at Cambridge, Babbage saw the high error rate involved in this human process and realised the potential for the work to be mechanised. In 1822 he began to develop the Difference Engine, firstly to compute values of polynomial functions. By using the method of finite differences, it was possible to avoid the need for more complicated multiplication and division operations, hence the machine’s name.

However, the Difference Engine was never finished, despite no shortage of funding for the project, neither was the successor he designed “Difference Engine Number 2”. The first difference engine would have been composed of around 25,000 parts, weighed 13,600 kg, and been 2.4 m tall.

During Babbage’s lifetime, the Swedish engineer Per Georg Scheutz and his son Edvard constructed the first working models of the Difference Engine. They used Babbage’s design and it was successfully deployed in certain applications, primarily printing tables of logarithms.

While the Difference Engine would have been capable of making calculations, had he completed it, Babbage’s most ambitious work began in 1833 when he started to develop his programmable Analytical Machine. If the Difference Engine is to be compared with today’s pocket calculator, the Analytical Engine can be considered the first mechanical computer, a forerunner of all programmable computers.

In fact, the Analytical Engine was not a single machine but a succession of designs that Babbage modified until his death in 1871. The Analytical Engine could be fed punched cards, whose patterns of holes contained programming instructions, a system that survived until the 1970s in modern computers.

The programs stored on the cards were created initially by a person, and then fed into the machine for processing. The analytical engine would have used loops of punched cards to control a mechanical calculator, which could formulate results based on the results of previous calculations. The engine would also use several features used in modern computers, such as sequential control, branching, and looping.

The mathematician Ada Lovelace, was one of the few people who appreciated Babbage's work, and formulated a program for the Analytical Engine, which would have been able to calculate a sequence of Bernoulli numbers, had the machine ever been built. Because of her work, Lovelace is now thought of as the world’s first computer programmer and the modern programming language Ada was named after her.

Charles Babbage made many other inventions, some of which now seem utterly bizarre. Some of the more practical ideas, however, include the standard railroad gauge, uniform postal rates, occulting lights for lighthouses and Greenwich time signals.

Later in life, Babbage became a bitter and bad-tempered old man. He especially hated street performers, who he blamed for depriving him of a quarter of his working potential.

Parts of Babbage’s uncompleted mechanisms are on display in London's Science Museum. There, in 1991, a perfectly functioning difference engine, modelled on "Difference Engine Number 2" was constructed from Babbage's original plans. Built only using tolerances achievable in the 19th century, the performance of the completed engine indicated that Babbage's machine would have worked. It performed its first calculation to 31 digits, far more than the average modern pocket calculator.

Nine years later, the Science Museum finished building a working model of the printer Babbage had designed for the difference engine, an remarkably complex device for the 19th century.

The modern world and its age of computers owes a huge amount to Charles Babbage, the Victorian engineer and mathematician. As Isaac Newton once said (in a different context) we are "standing on the shoulders of giants".

## Thursday, 8 September 2011

### Alan Turing Competition

Next June marks the centenary of the birth of Alan Turing.

To commemorate the centenary on 23rd June 2012, the IMA have organised a competition. A £100 Amazon voucher is up for grabs for the best article, written in a maximum of 500 words, about the mathematician and his work.

If you’re 17 to 19 why not write a brief article on the work of this great mathematician. The winning article will feature on the Maths Careers website.

Send your entry by email to mathscareers@ima.org.uk or by using the entry form on the competition page. Include your name, age, name and address of school and a contact email address. Or if you are home-schooled, send your name, age, name and address of parent/guardian and a contact email address.

Good luck!

Alan Turing(Photo from Wikipedia) |

To commemorate the centenary on 23rd June 2012, the IMA have organised a competition. A £100 Amazon voucher is up for grabs for the best article, written in a maximum of 500 words, about the mathematician and his work.

If you’re 17 to 19 why not write a brief article on the work of this great mathematician. The winning article will feature on the Maths Careers website.

Send your entry by email to mathscareers@ima.org.uk or by using the entry form on the competition page. Include your name, age, name and address of school and a contact email address. Or if you are home-schooled, send your name, age, name and address of parent/guardian and a contact email address.

Good luck!

Labels:
Alan Turing,
competition,
IMA,
MathsCareers

## Friday, 26 August 2011

### Paul Meier 1924-2011

What have mathematicians ever done for us? Well, here's one who has saved millions of lives, quite possibly you or somebody you know well.

Paul Meier (July 24, 1924 – August 7, 2011) |

The American statistician Paul Meier died earlier this month, age 87. It is thought that the techniques introduced through his work in medical statistics have saved millions of lives.

In the field of medicine, hundreds of new drugs and medical practices come into use each year. Today it is standard that such treatments are tested by a randomised clinical trial. Some patients are given the new treatment, some are given the old; but the key point is that the decision who gets which treatment is made randomly. The randomisation ensures that the statistics coming out of the trial - provided the trial was large enough - give reliable results.

Sixty years ago, this approach was unheard of. In the 1950s it was common practice to offer a new treatment to patients it was thought would benefit from

it. Often, those patients had the best chances of recovery anyway. Any analysis of the survival or improvement rates would show that the new treatments were better than they really were.

Randomisation seems obvious to us now, but it was not at that time. Paul Meier said in 2004 ‘When I said “randomize” in breast cancer trials I was looked at with amazement by my clinical colleagues.'

In general, the differences are not very big. A new drug may improve the chances of survival rates from, for example, 57 to 59%. But with large numbers of patients involved, statistical techniques can show whether these small improvements are a result of the new treatment. And 2% of a large group of people is still a large number.

The randomisation process also introduced a new rigour to medical trials. Only after a randomised trial is a drug or treatment released to the wider medical community. This rigour, and the unbiased evidence resulting from it, is in very large part due to Meier.

The polio vaccine trial of 1954 was an early test of the randomised control trial. The trial involved more than a million children and was the largest medical experiment in history. The results showed the benefit's of Jonas Salk's vaccine, which went on to be used widely, and successfully, against the disease.

Breast cancer treatment has also benefitted enormously from randomised control trials, with the positive effects of both chemotherapy and hormonal treatments being shown. Survival rates have increased dramatically as a result. Think also of AIDS treatment, drugs for TB and many others.

The Kaplan-Meier estimator was a second legacy of this revolutionary statistician. The estimator has many other uses, but in the field of medicine it gives statisticians a simple way of comparing patients’ survival rates after different treatments. The basic idea is that the probability of a patient surviving up to the start of a certain time interval is the product of the probabilities of his not dying during each of many previous intervals. The Kaplan-Meier estimator is now universally used in medical research. The journal article introducing the method in 1958 remains one of the most cited research papers in any field of science, with about 34,000 citations so far.

## Thursday, 25 August 2011

### GCSE results. Well done girls. Well done boys in maths.

GCSE results are out today, with the results showing maths as one of the few remaining areas in which boys are outperforming girls.

About 26% of papers taken by girls were given an A or A*, while just under 20% of those taken by boys were. Overall, those awarded between an A* and a C grade has risen for the 23rd year in a row, up 0.8% to 69.8%. But the overall pass rate (grades A* to E) dropped slightly to 92.7%.

Mathematics is compulsory at GCSE. As with A-Levels, more pupils are taking individual sciences, and fewer are taking modern languages, geography and history, although RE numbers are up.

The gap between girls and boys is now at its widest ever. 26.5% of grades awarded to girls were A or A*, with only 19.8% of boys. The gap has also widened when considering A*-C grades. Maths is one subject that bucks this trend: boys have beaten girls at GCSE maths for the third year in a row. It is widely believed that the decision to drop coursework in GCSE maths has given boys better chances. The proportion of boys getting grades A* to C in maths has risen again this year from 57.6% to 58.6%. The proportion of girls passing has also risen, from 56.8% to 58.3%.

Northern Ireland again gained the best results with 75% of papers being awarded A*-C grades.

The government has introduced the English Baccalaureate to demonstrate whether a pupil has gained a good grade in 5 key subject areas. But the fall in the uptake of the humanities, which are one of the 5 key areas, will not be welcome news for ministers.

Critics say that the inexorable rise in GCSE grades is a sign of increasingly easy examinations. The NCETM have welcomed the rise in maths grades, saying that the improvement is a result of better teaching methods.

About 26% of papers taken by girls were given an A or A*, while just under 20% of those taken by boys were. Overall, those awarded between an A* and a C grade has risen for the 23rd year in a row, up 0.8% to 69.8%. But the overall pass rate (grades A* to E) dropped slightly to 92.7%.

Mathematics is compulsory at GCSE. As with A-Levels, more pupils are taking individual sciences, and fewer are taking modern languages, geography and history, although RE numbers are up.

The gap between girls and boys is now at its widest ever. 26.5% of grades awarded to girls were A or A*, with only 19.8% of boys. The gap has also widened when considering A*-C grades. Maths is one subject that bucks this trend: boys have beaten girls at GCSE maths for the third year in a row. It is widely believed that the decision to drop coursework in GCSE maths has given boys better chances. The proportion of boys getting grades A* to C in maths has risen again this year from 57.6% to 58.6%. The proportion of girls passing has also risen, from 56.8% to 58.3%.

Northern Ireland again gained the best results with 75% of papers being awarded A*-C grades.

The government has introduced the English Baccalaureate to demonstrate whether a pupil has gained a good grade in 5 key subject areas. But the fall in the uptake of the humanities, which are one of the 5 key areas, will not be welcome news for ministers.

Critics say that the inexorable rise in GCSE grades is a sign of increasingly easy examinations. The NCETM have welcomed the rise in maths grades, saying that the improvement is a result of better teaching methods.

Labels:
English Baccalaureate,
GCSE,
grades,
results

## Friday, 19 August 2011

### The Code

Have you been watching the BBC's The Code?

This three part series was a fascinating glimpse into the world of mathematics, presented by the ever-inspiring Marcus du Sautoy. It finished last week and unfortunately the BBC have already removed it from the iPlayer. But there are several clips you can still find there.

There is also a code-breaking challenge on The Code website. The cryptic questions all have answers provided by way of clues throughout the series. As far as I can tell, the clips that are still on the iPlayer contain some or all of these clues, so it is not too late to get involved.

Marcus du Sautoy, with the help of some whizzy BBC graphics and a good budget, has a great way of teaching the subject. He gives an insight into a wide variety of mathematics, even complicated areas, making them accessible to a wide audience. It's a children's programme really, as the treasure hunt on the website shows, but the depth of the presentation ensures that there is something to be learned whatever your level of maths.

All of this is really impressive stuff. It's just a shame there were only 3 episodes of The Code. Keep up the good work BBC and please, re-run the programme some time.

Labels:
BBC,
Code,
codebreaking,
Marcus du Sautoy

## Thursday, 18 August 2011

### Results Up, Places Down

The A-Level results are out.

First the statistics.

Once again the results have improved; this is the 29th year in a row that the overall number of passes has increased.

The percentage of A-level grades A*-E awarded has gone up very slightly, from 97.6% to 97.8%

But for the first time in 15 years there has been no increase in the total proportion getting A or A* grades. Just over 27% of entries scored these grades, with a small rise in the proportion awarded A*.

The gap seems to be closing between boys and girls. The number of A* grades for boys has gone up from 7.9% to 8.2%. For girls, the number of A* grades has fallen slightly from 8.3% to 8.2%.

More people took A-levels this year - the number of A-level grades issued is up 1.6% to 867,317.

There is good news for mathematics. Maths and the sciences have all seen significant increases in the number of entries. Maths (including Further Maths) has gone up by 7.4%. There has been a 40% increase in students taking maths over the past 5 years.

And in these subjects, the rate of improvement for boys is bigger than that for girls. The gap between boys and girls at grade A in these subjects has fallen from 0.9% to 0.3%.

Sadly, although maths and the sciences are faring well, modern foreign languages continue to decline. French and German continue their downward trend, with the number of entrants down 4.7% and 6.9% respectively

A student beginning university this year will pay a maximum of £3000 per year for the duration of their course. A student beginning next September will pay up to £9000 per year.

The fact that there are few jobs available is another factor driving people towards university.

The UCAS tracking website crashed this morning due to the sheer number of visits

Michael Gove, the education secretary, has promised a through review of the A-Level system. He is reportedly interested in moving away from the current modular structure, and towards a system whereby more emphasis is placed on a single final examination.

So congratulations if you have achieved the grades you wanted. And good luck if you are still looking for a uni place.

Students sitting a maths examImage from Wikipedia |

Once again the results have improved; this is the 29th year in a row that the overall number of passes has increased.

The percentage of A-level grades A*-E awarded has gone up very slightly, from 97.6% to 97.8%

The gap seems to be closing between boys and girls. The number of A* grades for boys has gone up from 7.9% to 8.2%. For girls, the number of A* grades has fallen slightly from 8.3% to 8.2%.

More people took A-levels this year - the number of A-level grades issued is up 1.6% to 867,317.

There is good news for mathematics. Maths and the sciences have all seen significant increases in the number of entries. Maths (including Further Maths) has gone up by 7.4%. There has been a 40% increase in students taking maths over the past 5 years.

And in these subjects, the rate of improvement for boys is bigger than that for girls. The gap between boys and girls at grade A in these subjects has fallen from 0.9% to 0.3%.

Sadly, although maths and the sciences are faring well, modern foreign languages continue to decline. French and German continue their downward trend, with the number of entrants down 4.7% and 6.9% respectively

All these facts and figures do not help those who are now facing the very real scramble for places in the increasingly competitive race for university places. The increase in tuition fees, scheduled for September 2012, has been the biggest factor pushing up the number of students applying this year. Whereas many students would have previously opted for a year out while they ponder their futures, this does not make financial sense for those taking on a student loan.

A student beginning university this year will pay a maximum of £3000 per year for the duration of their course. A student beginning next September will pay up to £9000 per year.

The fact that there are few jobs available is another factor driving people towards university.

The UCAS tracking website crashed this morning due to the sheer number of visits

Michael Gove, the education secretary, has promised a through review of the A-Level system. He is reportedly interested in moving away from the current modular structure, and towards a system whereby more emphasis is placed on a single final examination.

So congratulations if you have achieved the grades you wanted. And good luck if you are still looking for a uni place.

Labels:
A-Level,
Mathematics,
Michael Gove,
results,
UCAS,
university

## Wednesday, 17 August 2011

### Pierre de Fermat

Today, were he still alive, Pierre de Fermat would be 410 years old.

Fermat was born on 17 August 1601 in Beaumont-de-Lomagne in southern France and lived until 12 January 1665.

He was an amateur mathematician (he was actually a lawyer), but he is now often known as the founder of modern number theory.

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. His work anticipated differential calculus, which was not formally laid out by Newton and Leibniz for another 50 years.

As ground-breaking as this work on calculus was, Fermat is most commonly remembered for the famous theorem that bears his name: Fermat's Last Theorem, which he studied after analysing the work of the ancient Greek mathematician Diophantus. The theorem, which is beautifully simple, states that if n > 2, there are no integer solutions to the equation:

In his notebook, Fermat wrote in pencil in the margin, "I have discovered a truly remarkable proof which this margin is too small to contain".

Did Fermat really have a remarkable proof? This will remain one of the mysteries of mathematics. But what we do know is that the world had to wait over 450 years until a proof was finally delivered by Andrew Wiles in 1995.

Another thing we know with certainty is that Wiles's proof is not the same as Fermat's proof, if indeed Fermat's existed. Wiles drew on a vast field of 20th century mathematics that simply didn't exist in Fermat's day. It is over 100 pages long and took 7 years of Wiles's research time.

Today's google logo also commemorates the birthday of the remarkable Frenchman Fermat.

Pierre de Fermat (1601-1665)Picture from Wikipedia |

Fermat was born on 17 August 1601 in Beaumont-de-Lomagne in southern France and lived until 12 January 1665.

He was an amateur mathematician (he was actually a lawyer), but he is now often known as the founder of modern number theory.

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. His work anticipated differential calculus, which was not formally laid out by Newton and Leibniz for another 50 years.

As ground-breaking as this work on calculus was, Fermat is most commonly remembered for the famous theorem that bears his name: Fermat's Last Theorem, which he studied after analysing the work of the ancient Greek mathematician Diophantus. The theorem, which is beautifully simple, states that if n > 2, there are no integer solutions to the equation:

When n = 2, you will probably recognise the equation as Pythagoras' Theorem, which, as any GCSE student will tell you, has many solutions, e.g.

The idea that there were no solutions for n > 2 seemed compelling, and the simplicity of an idea can sometimes lure us into a belief that the proof must be just as simple. Not in this case.

In his notebook, Fermat wrote in pencil in the margin, "I have discovered a truly remarkable proof which this margin is too small to contain".

Did Fermat really have a remarkable proof? This will remain one of the mysteries of mathematics. But what we do know is that the world had to wait over 450 years until a proof was finally delivered by Andrew Wiles in 1995.

Another thing we know with certainty is that Wiles's proof is not the same as Fermat's proof, if indeed Fermat's existed. Wiles drew on a vast field of 20th century mathematics that simply didn't exist in Fermat's day. It is over 100 pages long and took 7 years of Wiles's research time.

Today's google logo also commemorates the birthday of the remarkable Frenchman Fermat.

## Monday, 15 August 2011

### International Mathematical Olympiad

Congratulations to the team who represented the UK in the 2011 International Mathematical Olympiad. The team was put through 10 days of gruelling competition in Amsterdam from 13th – 24th July. Between them, they secured two gold, one silver and two bronze medals and also received one honourable mention.

The team and the medals awarded are as follows:

James Aaronson of St Paul's School (gold medal)

Andrew Carlotti of Sir Roger Manwood’s School (gold medal)

Ben Elliott of Godalming College (silver medal)

Adam Goucher of Netherthorpe School (bronze medal)

Jordan Millar of Regent House School (bronze medal)

Joshua Lam of The Leys School (honourable mention)

It was the first time in 15 years that the UK team was awarded two gold medals. The team finished 17th out of 101 countries. The UK entry was organised by the UK Mathematics Trust.

The problems the team faced can be downloaded from the IMO website.

More information about future olympiads can be found at the British Mathematical Olympiad website and the European Girls' Mathematical Olympiad website.

## Tuesday, 9 August 2011

### Why Our Schools Still Fail At Maths

There are some good points in the Maths Task Force's report to the government on the future of maths teaching in our schools. The task force, headed by Carol Vorderman, was established by the Conservative Party while they were in opposition.

Carol Vorderman Image from Wikipedia. |

This is not good for those individuals, whose options will be limited, and not good for employers, who increasingly need workers with at least a basic mathematical ability.

Why is this happening? The report blames a shortage of maths teachers, although it must be made clear, this is not the situation in all parts of the UK. As a result, the report concludes, a quarter of maths classes are currently being taken by non-specialists, i.e. teachers whose degree was not in mathematics. It also points out that there are shortcomings in primary education. With non-specialists being used to teach (primary teachers generally teach all subjects), they are not adequately prepared for the rigour required in mathematics.

Secondly, the report blames the way in which the curriculum has been devised. It has been formulated, it says, not by educators, but by administrators, whose understanding of what really needs to be taught in the maths classroom, is lacking.

Another key point of the report is to scrap the final vestiges of the SATs tests, those taken by 11 year olds. By "teaching to the test", teachers are narrowing the learning of their pupils, thereby preventing a broader mathematical understanding, which would give them a far better grounding for secondary school.

Michael Gove, the education secretary, backs the suggestions in the report. Even before it was released, Gove was talking about compulsory maths teaching up to the age of 18.

So, what should we make of the Task Force report?

The shortage of maths specialists in our schools is fundamental and must be addressed. There is no substitute for clear, authoritative, imaginative and well-thought out teaching. Addressing this problem alone would go a long way to solving the problem. This raises questions of teachers' pay, attracting academic excellence into the teaching profession, and why it is impossible to hire good quality teachers at certain schools, but these are issues that must wait for another time.

It is easy to blame SATs, which have been condemned by teachers ever since their introduction. Key Stage 2 SATs are a waste of time and they force teachers to focus on teaching to the test alone, which narrows the field of the pupils' learning. These things are true, and SATs should certainly be abolished. But it is hard to believe they can seriously impact on a child's mathematical progression through secondary school and beyond.

Thirdly, teaching of maths to all pupils to the age of 18 would be very difficult to implement. Trying to teach a class full of pupils who had failed their GCSE, who thought they were going to be free of mathematics, only to find they have another 2 compulsory years of it, does not sound like a good idea, especially if these compulsory lessons were not going to lead to any proper qualification.

So the report's findings are mixed. Compulsory maths teaching to the age of 18, on its own, is not going to fix this chronic, very real problem. Unless the pupils who emerge without a C at GCSE are given some serious incentives to study their loathed subject for another two years, this idea is doomed to failure. What would? Teaching better, by good quality teachers, not necessarily for longer.

The CBI, some universities and FE colleges have welcomed the report and praised the suggestion of compulsory maths education to the age of 18. However, the National Union of Teachers has said it does not understand the need for such a report, given that a full review of the National Curriculum is underway. We say: care and serious consideration is required. We have discussed the curious decisions of the education secretary before on this blog. Be careful what you wish for Mr Gove.

## Tuesday, 28 June 2011

### Happy Tau Day

Happy Tau Day!

Happy what day? Tau is a mathematical constant, whose value is 6.28... . Is that ringing any bells? Correct:

OK, so tau being 6.28 explains why 28th June is Tau Day (blame the Americans - they write their dates backwards). But why do we need a new mathematical constant, especially one that is simply double another one?

Some maths teachers and academics have been in favour of using tau instead of pi in maths teaching, particularly in early years. It's because 2 pi seems to crop up a lot, probably more often than a single pi, particularly in geometry and trigonometry. For example, the circumference of a circle is given by:

If tau were widely adopted, this would be replaced by:

Personally, I'm not convinced. Tau would certainly be useful in a number of formulae and mathematical solutions. But I think students would end up using a half of tau just as often as they currently use 2 pi.

And if we went to a system where both constants were in use, would this not just add to the confusion, rather than alleviate it?

Finally there are thousands of years of pi tradition. The ancient Greeks obsessed over pi, just as much as modern mathematicians do.

What do you think of tau?

Happy what day? Tau is a mathematical constant, whose value is 6.28... . Is that ringing any bells? Correct:

OK, so tau being 6.28 explains why 28th June is Tau Day (blame the Americans - they write their dates backwards). But why do we need a new mathematical constant, especially one that is simply double another one?

Some maths teachers and academics have been in favour of using tau instead of pi in maths teaching, particularly in early years. It's because 2 pi seems to crop up a lot, probably more often than a single pi, particularly in geometry and trigonometry. For example, the circumference of a circle is given by:

And if we went to a system where both constants were in use, would this not just add to the confusion, rather than alleviate it?

Finally there are thousands of years of pi tradition. The ancient Greeks obsessed over pi, just as much as modern mathematicians do.

What do you think of tau?

## Thursday, 9 June 2011

### Exam Standards Are Slipping (That's The Papers, Not the Candidates)

You may be used to the idea of going into exams to find questions you find impossible to answer.

If you sat the OCR Decision Maths 1 exam on 26th May you will certainly have found yourself in this situation; one of the questions had no solution.

This embarrassing slip-up was the first of five errors on AS exam papers this summer. The other mistakes came to light on:

Reports are also circulating of mistakes on a CCEA business studies GCSE and one more, undisclosed paper.

- an AQA Business Studies paper, in which there was not enough information to answer the question;
- an Edexcel Biology multiple choice exam, in which none of the answers given were correct;
- an AQA Geography paper, in which the flow of a river was incorrectly labelled at one point;
- an AQA Computing AS Level paper, in which an arrow was shorter than it should have been.

Reports are also circulating of mistakes on a CCEA business studies GCSE and one more, undisclosed paper.

Jim Sinclair, Director of the Joint Council for Qualifications, said that procedures were in place to cope with these errors.

In the case of the Decision Maths, the question was worth 8 marks out of 72, or 11% of the total marks awarded for the paper. The exam board OCR have decided not to discount the question from the paper, since it was worth such a large percentage of the marks. Instead, they will award marks for correct working, which will reward pupils who spent some of their time attempting to answer the question.

But many students are unhappy with this proposal. Facebook groups have been formed to demand a complete rerun of the exam. Some students think the time they wasted will jeopardise their chances of them getting the grades required for their university applications.

More details about the impossible D1 question are here.

Our advice to examination candidates has always been: if you can't do a question on the paper, leave it and come back to it if you have time at the end. It may be time to modify this advice: leave it; it may not be possible anyway.

## Thursday, 5 May 2011

### Women In Maths Day

Following on from our European Girls’ Mathematical Olympiad, here's some news about the Women In Maths Day, taking place tomorrow, 6 May 2011.

The event is being staged by the London Mathematical Society, at De Morgan House, 57-58 Russell Square, London, WC1B (Nearest tube: Russell Square).

The event is particularly aimed at women, but men can also attend. There is also an invitation to the post-conference dinner.

For more information about Women in Maths Day, see this PDF on the LMS website.

The event is being staged by the London Mathematical Society, at De Morgan House, 57-58 Russell Square, London, WC1B (Nearest tube: Russell Square).

The event is particularly aimed at women, but men can also attend. There is also an invitation to the post-conference dinner.

For more information about Women in Maths Day, see this PDF on the LMS website.

## Friday, 15 April 2011

### Female? At school? Good at maths? You should be at the EGMO next year!

The UK team at the 2010 China Girls' Mathematical Olympiad |

The aim of the competition is to give more girls an exciting opportunity to show their mathematical ability on an international stage. Through the competition and the associated talent search and preparation, the organisers hope it will increase the participation of girls in mathematics competitions and in the UK IMO squad.

It is hoped that EGMO 2012 will be the first of many European Girls’ Olympiads. The organisers are currently seeking a host country for the 2013 competition. They intend that the competition will become an annual event, moving around Europe every year with host countries providing the competition funding. Preliminary enquiries indicate that about 20 nations will send teams to the inaugural competition.

The first EGMO will be held from 10 to 16 April 2012. The event is being jointly organised by Murray Edwards College and the United Kingdom Mathematics Trust, the educational charity that organises mathematics competitions in secondary schools in the UK.

Participating countries will send teams consisting of their strongest four female mathematicians of school age. The competitors will try to solve eight difficult problems over a two day period: four in each of two 4½-hour examinations.

There will be a nationwide talent search to find the girls to form the UK team at EGMO 2012. A Competition Challenge (the UK Mathematical Olympiad for Girls) will be held in June 2011 to select those students who will engage in training for the competition.

The exams will be held on 12 and 13 April 2012. Approximately half the contestants will be awarded medals, with gold, silver and bronze medals being awarded. Each contestant not awarded a medal but with a perfect score on at least one problem will receive an Honourable Mention.

Teams consist of up to four female contestants, accompanied by a Leader and Deputy Leader (who may be male or female).

If you are interested in the EGMO, take a look at the competition website, http://www.egmo2012.org.uk .

To give an idea of the standard, some of the challenges set at the CGMO can be found here. Be warned: they are tough! But the training and preparation is aimed at bringing competitors up to the right level.

To find out more about the competition or about how you can get involved, contact the EGMO from their website.

Labels:
CGMO,
EGMO,
European Girls’ Mathematical Olympiad,
IMO

## Thursday, 31 March 2011

### More Universities to Charge Maximum Fees

Last month we reported that Exeter University had announced it will charge the maximum of £9000 tuition fees when new legislation allows them to do so in 2012.

Since then, a growing number of universities have declared similar intentions. Indeed, with 21 universities now having declared their fees for the 2012-13 year, it is obvious that fees at a maximum level will be the norm rather than an exception. This is in complete contrast to the comments of David Willets, the government's universities minister, who said only last month that maximum fees would only be charged in exceptional cases.

The list so far can be found on the BBC news website. It includes eight out of the 20 members of the elite Russell Group, with the remaining 12 members still to declare.

All 8 of the second tier "1994 Group" universities to have declared will be charging maximum fees.

Only four out of the 21 who have declared will be charging less than the maximum, all of these being former polytechnics or higher education colleges.

All universities charging maximum fees will have to provide special provision for applicant students from poorer families and a system of bursaries.

We have covered the protests against the rise in tuition fees here. And if you are an A-Level student thinking about university, but worried about the increasing costs, we offer some words of advice.

If, as now seems possible, nearly all universities charge the maximum £9000 fees, there could be serious consequences for our entire education system. Firstly, the number of young people who can afford university will fall. With a limited number of bursaries on offer, there is a danger that the universities system will become accessible only to a privileged minority of the population, with students from low and average income families simply unable to afford it.

Secondly, the universities may suffer themselves if the number of applicants falls. With vastly reduced grants from central government, universities will have to take cold decisions about which departments are attracting the required numbers of students, with the remainder facing severe cuts or even closure.

Finally, these government plans may begin to look like a step too far for the country as a whole. Keeping students in education beyond sixth form is vital to the success of British companies and the wider economy. It also provides our young people with a valuable experience that they cannot get in the workplace.

Are these prospects a part of the fairer Britain the coalition government wanted to achieve when it took office?

## Wednesday, 23 March 2011

### A New Era in Super-Computing?

If you're looking for areas in which maths can be applied to the real world, look no further than computing and in particular the latest exciting advances coming from universities in the US.

At a meeting of the American Physical Society in Dallas, scientists from University of California, Santa Barbara have been demonstrating the latest steps on the road to a quantum computer.

A quantum computer, something that, as far as we know, has yet to be built, would be able to perform calculations on a scale that would vastly out-perform today's super-computers.

The UCSB device is one step along the road towards such a computer. It houses a chip containing 9 quantum devices, four of which are "quantum bits" or Qubits, which do the calculations. Later this year, the team hopes to increase the number of Qubits to 10. When scientists are able to increase the number of Qubits to about 100, they think the chip will be the basis of a viable, usable computer.

All of this opens the possibility that in the near future we'll have the power of today's super-computers on our desks, on our laps, even in our mobile phones.

For these developments, we owe a lot to Erwin Schroedinger, whose work on quantum physics and wave equation paved the way for the weird world of quantum mechanics.

At its heart, quantum computing depends on "super-position", which is the seemingly unnatural ability for a particle to be in two states at the same time. A particle spinning in one direction could be given a weak pulse of energy, which might be enough to set it spinning in the opposite direction, but maybe not. As long as the particle is not being observed or interacted with in any way, quantum physics says that the particle is in both states at the same time.

Now, we could use a whole line of these particles to represent the binary digits of a number. If a calculation is performed using a traditional computer, we would need to feed each number into the computer separately. But because a quantum computer can operate on particles in super-position, it can perform the calculation on all the possible combinations simultaneously. A number whose binary representation is 7 digits long is between 0 and 127. A traditional computer would need to do a calculation on each of these 127 numbers. A quantum computer could do them all at once.

But the power of quantum computing brings huge challenges for society. In fact, as things stand, a fully functioning quantum computer would jeopardise the stability of the world. This is because world commerce depends on the use of secure ciphers to protect and verify financial transactions. Additionally, many secure conversations between governments and government institutions are carried out using the same sets of ciphers. With the unimaginable computing power that quantum computing would bring, these ciphers, which we have previously considered unbreakable, would be rendered useless.

So the race is on: will the quantum computer arrive first, posing threats to international security and commerce? Or will a new quantum cryptography be developed first, securing transactions in a new unbreakable way? (It's another story, but such a form of encryption has already been shown possible over short communication distances. And it is absolutely unbreakable.)

Yesterday's science fiction really is tomorrow's reality.

## Tuesday, 22 March 2011

### Special Offer - Extend Your Subscription at MathsBank.co.uk

Exam season is drawing near and, as ever, we'd like to help you in your teaching or revision.

If you have previously taken out a subscription on the MathsBank site, there is now the chance to renew your membership at a special rate.

If you are interested in getting 25% off a new subscription, please email and we will forward instructions on how to proceed.

## Wednesday, 2 March 2011

### Exeter's Fees will be Maximum - Others to Follow?

Image via WikipediaExeter University has this morning announced it will charge the maximum of £9000 tuition fees when new plans allow them to do so in 2012.

Exeter joins a growing list of universities including Oxford, Cambridge and Imperial College intending to charge the highest rate, but it is the first university outside the elite Russell Group. If others in the second tier "1994 Group" declare similar intentions, it could be a sign that the government is going to struggle to keep fees to an average level of £7500, as planned. The government stated, when the plans were drawn up, that a level of £9000 would only apply in exceptional circumstances.

David Willetts, the government's Universities Minister, has put himself on a collision course with the universities by stating that there is no reason arts courses should require fees of more than £6000 and scientific and medical courses £7500.

With the Russell Group universities, there is no doubt that setting tuition fees at the highest level is intended to signal exceptional quality. But the Exeter announcement appears to be a decision based purely on the university's finances. A spokesperson for Exeter University said that students would be expecting increases in the number of academic staff and better facilities. Since government cuts mean that the university is losing funding, it will need to increase its income in some other way. Students appear to be those who will pay.

Students at universities and FE Colleges, along with sixth form pupils, have all joined protests against the increases in tuition fees over the last few months. We have covered the student protests here. And if you are an A-Level student thinking about university, but worried about the increasing costs, we offer some words of advice.

Our feeling is that, with a growing number of universities announcing maximum fees, there will be serious consequences for the number of young people who can afford university. With a limited number of bursaries on offer, there is a danger that the universities system will become accessible only to a privileged minority of the population, with students from low and average income families simply unable to afford it. Universities may suffer themselves if the number of applicants falls. Are these prospects a part of the fairer Britain the coalition government wanted to achieve when it took office?

Exeter joins a growing list of universities including Oxford, Cambridge and Imperial College intending to charge the highest rate, but it is the first university outside the elite Russell Group. If others in the second tier "1994 Group" declare similar intentions, it could be a sign that the government is going to struggle to keep fees to an average level of £7500, as planned. The government stated, when the plans were drawn up, that a level of £9000 would only apply in exceptional circumstances.

The announcement will mean Exeter will have to provide special provision for applicant students from poorer families and a system of bursaries.

With the Russell Group universities, there is no doubt that setting tuition fees at the highest level is intended to signal exceptional quality. But the Exeter announcement appears to be a decision based purely on the university's finances. A spokesperson for Exeter University said that students would be expecting increases in the number of academic staff and better facilities. Since government cuts mean that the university is losing funding, it will need to increase its income in some other way. Students appear to be those who will pay.

Students at universities and FE Colleges, along with sixth form pupils, have all joined protests against the increases in tuition fees over the last few months. We have covered the student protests here. And if you are an A-Level student thinking about university, but worried about the increasing costs, we offer some words of advice.

Our feeling is that, with a growing number of universities announcing maximum fees, there will be serious consequences for the number of young people who can afford university. With a limited number of bursaries on offer, there is a danger that the universities system will become accessible only to a privileged minority of the population, with students from low and average income families simply unable to afford it. Universities may suffer themselves if the number of applicants falls. Are these prospects a part of the fairer Britain the coalition government wanted to achieve when it took office?

Labels:
Exeter University,
tuition fees

## Tuesday, 15 February 2011

### Want a decent education? Don't do A-Levels.

Image via Wikipedia

The Royal Society, one of Britain's leading scientific institutions, has declared A-Levels "not fit for purpose".

This conclusion stems from the fact that relatively few pupils deciding on the usual 3 A-Level subjects will choose to do a science or maths. Across the UK in 2009, 17% of 16 to 18-year-olds took one or more science A-level, or equivalent qualification, the report says. This compares unfavourably with Scotland, where pupils generally take 5 Highers and often spread their choice between the sciences and the arts.

The report says this difference indicated problems with the structure of post-16 qualifications provision and demonstrates an "underlying cultural indifference" towards the sciences and maths.

Other findings included a need for more specialist maths and science teachers and a broadening of the diversity of A-Level subjects.

The Royal Society is calling for the rapid introduction of an English Baccalaureate at A-Level, which would require students to take a spread of subjects, including maths and a science subject. A similar qualification has already been announced for GCSE level by education secretary Michael Gove. The government thinks this new qualification will drive up the take-up of maths and science at both pre- and post-16 levels.

A review of the A-Level system has already been promised, although so far there are few details of the shake-up. Gove previously announced a desire to return to more emphasis on a final examination, moving away from a modular system.

The Chair of the Royal Society's Education Committee, Professor Dame Athene Donald, said in the report that, at a time of economic uncertainty, it was deeply worrying to find maths and science numbers so low, when these specialities have such potential to revitalise the country's financial outlook.

The Royal Society, one of Britain's leading scientific institutions, has declared A-Levels "not fit for purpose".

This conclusion stems from the fact that relatively few pupils deciding on the usual 3 A-Level subjects will choose to do a science or maths. Across the UK in 2009, 17% of 16 to 18-year-olds took one or more science A-level, or equivalent qualification, the report says. This compares unfavourably with Scotland, where pupils generally take 5 Highers and often spread their choice between the sciences and the arts.

The report says this difference indicated problems with the structure of post-16 qualifications provision and demonstrates an "underlying cultural indifference" towards the sciences and maths.

Other findings included a need for more specialist maths and science teachers and a broadening of the diversity of A-Level subjects.

The Royal Society is calling for the rapid introduction of an English Baccalaureate at A-Level, which would require students to take a spread of subjects, including maths and a science subject. A similar qualification has already been announced for GCSE level by education secretary Michael Gove. The government thinks this new qualification will drive up the take-up of maths and science at both pre- and post-16 levels.

A review of the A-Level system has already been promised, although so far there are few details of the shake-up. Gove previously announced a desire to return to more emphasis on a final examination, moving away from a modular system.

The Chair of the Royal Society's Education Committee, Professor Dame Athene Donald, said in the report that, at a time of economic uncertainty, it was deeply worrying to find maths and science numbers so low, when these specialities have such potential to revitalise the country's financial outlook.

Labels:
GCE Advanced Level,
Highers,
Mathematics,
Michael Gove,
Royal Society

## Monday, 14 February 2011

### Gove acted unlawfully

Two weeks ago we reported that the education secretary, Michael Gove, was facing a legal challenge to his scrapping of the Building Schools for the Future (BSF) scheme.

Yesterday, the High Court in London ruled in favour of the Education Authorities bringing the case to court. Michael Gove was told he must reconsider the decision to scrap the scheme because he had unlawfully failed to consult local councils. Seven hundred school building projects were shelved because of the decision.

In the House of Commons, Gove continued to defend his actions, telling MPs that his decision was clear and rational. He said that the scale of the deficit meant cuts were inevitable. He also pointed out that the judge had not ordered a reinstatement of the scheme, nor any compensation payment to the schools involved. The ruling gives schools the chance to "make representations".

For Labour, shadow education secretary Andy Burnham said that the ruling was damning. He demanded full details of conversations between Mr Gove and his civil servants and called on David Cameron to remove him from any future involvement in these matters.

Unfortunately, it seems unlikely that any of this will ever lead to the school rebuilds taking place.

Yesterday, the High Court in London ruled in favour of the Education Authorities bringing the case to court. Michael Gove was told he must reconsider the decision to scrap the scheme because he had unlawfully failed to consult local councils. Seven hundred school building projects were shelved because of the decision.

In the House of Commons, Gove continued to defend his actions, telling MPs that his decision was clear and rational. He said that the scale of the deficit meant cuts were inevitable. He also pointed out that the judge had not ordered a reinstatement of the scheme, nor any compensation payment to the schools involved. The ruling gives schools the chance to "make representations".

For Labour, shadow education secretary Andy Burnham said that the ruling was damning. He demanded full details of conversations between Mr Gove and his civil servants and called on David Cameron to remove him from any future involvement in these matters.

Unfortunately, it seems unlikely that any of this will ever lead to the school rebuilds taking place.

## Thursday, 27 January 2011

### Carl Friedrich Gauss (30 April 1777 – 23 February 1855)

Carl Friedrich Gauss Oil painting by G. Biermann |

Remarkable as it is, it is not so much this particular piece of work we wish to celebrate in this article. In mathematics Gauss is remembered for so many things, but perhaps most often for the Gaussian or "Normal" statistical distribution.

Gauss was a child prodigy. There is a story, which may or may not be true, about his remarkable mathematical ability at primary school. His teacher, realising Gauss had completed a task well ahead of the rest the class, asked Gauss to calculate the sum of the numbers from 1 to 100. The child thought for a few seconds, then told her the answer: 5050. He had added 1 and 99, 2 and 98, etc, to turn the sum into 49 x 100, after which he added the only unpaired numbers, 50 and 100.

When Gauss was only 14, Charles William Ferdinand, the Duke of Brunswick, who was always on the lookout for bright students, paid for the boy to go to the university in Brunswick.

Subsequently, he studied further in Göttingen. 1796 was Gauss’s most prolific year, in which he documented at least 5 major discoveries and proofs in number theory and geometry. He demonstrated the construction of a 17-sided polygon using only straight-edge and compass, something that had eluded all mathematicians since the ancient Greeks.

Gauss sorely wanted to be accepted by the elite group of Parisian mathematicians. But he presented his ideas in cryptic ways and didn’t believe in showing much working and this ultimately went against him. He was to spend all his life in what is now Germany.

In 1801, there was great excitement in the world of astronomy as an 8th planet was discovered between Mars and Jupiter. It was named Ceres, and today we know it as the largest asteroid. This discovery was considered a great omen for a new dawn of science at the beginning of the 19th century.

Shortly afterwards, however, the astronomers lost sight of the newly discovered object. Gauss announced he knew where to find it, using mathematical techniques now used to analyse data of many different types. To solve the problem of measurements of Ceres, Gauss had invented the Gaussian (or Normal) Distribution, now used widely in statistics, enabling patterns to be seen in seemingly random data.

The breakthrough was his realisation that the measurements, accurate and inaccurate, would, when plotted on a graph, be distributed around the true value in the shape of a bell-shaped curve. With this and other tools, statistical analysis has become a powerful weapon to analyse data, to test hypotheses, to separate statistical fact from fiction.

Gauss had an aversion to teaching and said that a professor “loses his precious time” lecturing students. This led to his taking the non-teaching post as Professor of Astronomy in the University of Göttingen in 1807 and he spent the rest of his career tracking the paths of planets.

The Gaussian distribution crops up everywhere; it is one of the statistician’s tools for understanding the real world, in the fields of chemistry, medicine, engineering, finance and many others.

Although arguably the most famous of Gauss’s mathematical innovations, the Gaussian distribution is probably not the greatest of his many mathematical achievements.

Despite his enormous influence in the fields of statistics and astronomy, the properties of numbers was Gauss’s true mathematical penchant. He once said “Maths is the Queen of the Sciences and Number Theory the Queen of Maths.”

## Wednesday, 26 January 2011

### Legal Challenge to Gove's Cuts

Image via WikipediaThe coalition government looks set to face yet another problem over education.

After the fiasco with the increases in tuition fees and scrapping of the Education Maintenance Allowance (EMA), the Education Secretary Michael Gove now faces a legal challenge. This revolves around his decision to abolish the Building Schools for the Future (BSF) scheme. The scheme, devised by Labour, was scrapped last July after the coalition government came to power.

Under the scheme, every secondary school would have been given funding for a rebuild. Now, 700 schools will have this money withdrawn. Seven councils (Waltham Forest, Luton Borough Council, Nottingham City Council, Sandwell, Kent County Council and Newham) have brought a legal challenge to the High Court in London over this volte face, saying that stopping the programme in their areas was arbitrary and legally flawed.

Gove is accused of not consulting properly and failing to give adequate reasons for his decision. His lawyers said in reply that the coalition had inherited the largest ever peacetime deficit and that spending cuts had to be made quickly.

Have you suffered because of the cuts to the BSF programme? Are you teaching or learning in shabby classrooms? Let us know why your school should qualify for a rebuild.

The Government is less than a year old, but is already under considerable pressure, not only in the field of education funding, but in defence, social care, employment, health and just about any area you care to choose. The state of the economy continues to look parlous.

How long can things go on? Will the Coalition Government continue with its controversial cuts? Or will it come up with a Plan B, as shadow chancellor Ed Balls advised yesterday? Will the Conservatives be "out of office for a generation", as Bank of England governor Mervyn King predicted, even before the election.

We live in interesting, if difficult, times.

After the fiasco with the increases in tuition fees and scrapping of the Education Maintenance Allowance (EMA), the Education Secretary Michael Gove now faces a legal challenge. This revolves around his decision to abolish the Building Schools for the Future (BSF) scheme. The scheme, devised by Labour, was scrapped last July after the coalition government came to power.

Under the scheme, every secondary school would have been given funding for a rebuild. Now, 700 schools will have this money withdrawn. Seven councils (Waltham Forest, Luton Borough Council, Nottingham City Council, Sandwell, Kent County Council and Newham) have brought a legal challenge to the High Court in London over this volte face, saying that stopping the programme in their areas was arbitrary and legally flawed.

Gove is accused of not consulting properly and failing to give adequate reasons for his decision. His lawyers said in reply that the coalition had inherited the largest ever peacetime deficit and that spending cuts had to be made quickly.

Have you suffered because of the cuts to the BSF programme? Are you teaching or learning in shabby classrooms? Let us know why your school should qualify for a rebuild.

The Government is less than a year old, but is already under considerable pressure, not only in the field of education funding, but in defence, social care, employment, health and just about any area you care to choose. The state of the economy continues to look parlous.

How long can things go on? Will the Coalition Government continue with its controversial cuts? Or will it come up with a Plan B, as shadow chancellor Ed Balls advised yesterday? Will the Conservatives be "out of office for a generation", as Bank of England governor Mervyn King predicted, even before the election.

We live in interesting, if difficult, times.

## Thursday, 13 January 2011

### What is Randomness?

The BBC Radio 4 programme 'In Our Time' looked at the topic of randomness today. The In Our Time website has a link to the programme on the iPlayer, if you missed it first time round.

What is meant by randomness? Well, a truly random event is not deterministic, i.e. it is not possible to determine the next outcome, based upon the previous outcomes, or on anything else.

In actual fact, random processes are very important in lots of areas of maths, science and life in general, but truly random processes are remarkably difficult to achieve. Why should this be the case? Because in theory, many processes that we consider to be random, such as rolling a dice, are in fact deterministic. You could, theoretically, determine the outcome of the dice roll if you knew its exact position, size, etc.

The ancient Greek philosopher and mathematician Democritus (ca. 460 BC – ca. 370 BC) was a member of the group known as Atomists. This group of ancients were the pioneers of the concept that all matter can be subdivided into its fundamental building blocks, atoms. Democritus decreed there was no such thing as true randomness. He gave the example of two men meeting at a well, both of whom consider their meeting to have been pure chance. What they did not know is that the meeting was probably pre-arranged by their families. This can be considered an analogy for the deterministic dice roll: there are factors determining the outcome, even if we cannot measure or control them precisely.

Epicurus (341 BC – 270 BC), a later Greek mathematician, disagreed. Although he had no idea how small atoms really were, he suggested they swerve randomly in their paths. No matter how well we understand the laws of motion, there will always be randomness introduced by this underlying property of atoms.

Aristotle worked further on probability, but it remained a non-mathematical pursuit. He divided all things into certain, probable and unknowable, for example writing about the outcome of throwing knuckle bones, early dice, as unknowable.

As with many other areas of mathematics, the topic of randomness and probability did not resurface in Europe until the Renaissance. The mathematician and gambler Gerolamo Cardano (24 September 1501 – 21 September 1576) correctly wrote down the probabilities of throwing a six with one dice, a double six with 2 dice, and a triple with three. He was the first person to notice, or at least to record, the fact that you're more likely to throw 7 with 2 dice than any other number. These revelations formed part of his handbook for gamblers. Cardano had suffered terribly because of his penchant for gambling (at times he pawned all his family's belongings, ended up in a poor house, and in fights). This book was his way of telling fellow gamblers how much they should bet and how to stay out of trouble.

In the 17th century, Fermat and Pascal collaborated and developed a more formalised theory of probability and numbers were assigned to probabilities. Pascal developed the idea of an expected value and famously used a probabilistic argument, Pascal's Wager, to justify his belief in God and his virtuous life.

Today there are sophisticated tests that can be performed on a sequence of numbers to determine whether or not the sequence is truly random, or if it has been determined by formula, human being, or some other means. For example does the number 7 occur one tenth of the time (plus or minus some allowable error)? Is the digit 1 followed by another 1 one tenth of the time?

An increasingly sophisticated series of tests can be fired into action. We have the "poker test", which analyses numbers in groups of 5, to see whether there are two pairs, three of a kind, etc, and compares the frequency of these patterns with those expected in a truly random distribution. The Chi Squared test is another statistician's favourite. Given that a particular pattern that has occurred, it will give a probability, and a confidence level, that it was generated by a random process.

Another naturally occurring, seemingly random distribution is that of the prime numbers. The Riemann Hypothesis provides a way to calculate the distribution of the primes, but it remains unsolved and nobody knows whether the hypothesis remains valid for very large values. However, like the digits in the irrational number π, the distribution of the primes does pass all the tests of randomness. It remains deterministic, but unpredictable.

Another useful measure of randomness is a statistic called the Kolmogorov Complexity, named after the 20th century Russian mathematician. The Kolmogorov Complexity is the shortest possible description of a sequence of numbers, for example the sequence 01010101.... could be described simply as "Repeat 01". This is a very short description, indicating the sequence is certainly not random.

However, for a truly random sequence, it would be impossible to describe the sequence of digits in any simplified form. The description would be just as long as the sequence itself, which indicates that the sequence would appear to be random.

During the last two centuries, scientists, mathematicians, economists and many others have begun to realise that sequences of random numbers are very important to their work. And so in the 19th century, methods were devised to generate random numbers. Dice, but can be biased. Walter Welden and his wife spent months at their kitchen table rolling a set of 12 dice over 26000 times, but these data were found to be flawed because of biases in the dice, which seems a terrible shame.

The first published collection of random numbers appears in a book of 1927 by Leonard HC Tippet. After that, there were many attempts, many flawed. One of the most successful methods was that used by John von Neumann, who pioneered the middle-square method, in which a 100-digit number is squared, the middle 100 digits are extracted from the result, and squared again, and so on. Very quickly, this process yields a set of digits that pass all the tests of randomness.

In the 1936 US presidential election, all the opinion polls pointed to a close result, with a possible win for the Republican Party's candidate Alf Landon. In the event, the outcome was a landslide to the Democratic Party's Franklin D Roosevelt. The opinion pollsters had chosen bad sampling techniques. In their attempts to be high-tech, they had telephoned people up to ask them about their voting intentions. In the 1930s, it was far more likely for wealthier people - largely Republican voters - to have a telephone, and so the results of the surveys were deeply biased. In surveys, truly randomising the sample population is of prime importance.

Likewise, it is also very important in medical tests. Choosing a biased sample set (e.g. too many women, too many young people, etc.) can make a drug appear more or less likely to work, biasing the experiment, with possibly dangerous consequences.

Well, so far, atomic and quantum physics have come closest to providing us with truly unpredictable events. It is, to date, impossible to determine precisely when a radioactive material will decay. It seems random, but maybe we simply don't understand. At the moment, it remains probably the only way to generate truly random sequences.

Ernie, the UK Government's premium bond number generator, is now on its fourth reincarnation. It must be random, in order to give all the country's premium bond holders an equal chance of a prize. It contains a chip that exploits the thermal noise within itself, i.e. the amount of movement in the electrons. Government statisticians perform tests of the number sequences that this generates, and they do indeed pass the tests for randomness.

Other applications are: the random prime numbers used in internet transactions, encrypting your credit card number. The National Lottery machines use a set of very light balls and currents of air to mix them up, but like the dice, this could, in theory, be predicted.

Finally, the Met Office uses sets of random numbers for its ensemble forecasts. Sometimes it is difficult to predict the weather because of the well-known "chaos theory" - that the final state of the atmosphere is highly dependent on the precise initial conditions. It is impossible to measure the initial conditions to anything like the precision required, so atmospheric scientists feed their computer models various different scenarios, with the initial conditions varying slightly in each. This results in a set of different forecasts and a weather presenter who talks in percentage chances, rather than in certainties.

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