MathsBank Blog

Happy 40th Birthday to the HP-35

2012-02-01T09:37:00.001Z

The HP-35 scientific calculator
Image via Wikipedia

1st February 1972 marked a birth that would revolutionisemathematics education across the world. This was not, this time, the birth of agreat mathematician, but of a machine. The first scientific hand-heldcalculator (HP-35) was introduced to the US market, and later to the UK. Costing$395, it was made by Hewlett-Packard and its name came from the fact that ithad 35 keys.

The HP-35 measured 79 x 147 x 34 mm, pretty chunky bytoday’s standards, but ultra-sleek in its day, when a computer still filled asmall room. It ran on rechargeable batteries, and its electronics used severalintegrated circuits. ‘Scientific’ meant the calculator was able to performlogarithmic and trigonometric functions with one keystroke. It featured a redLED display which could give scientific notation up to 10 digits, with 2 digitsfor an exponent (power of 10).

The price was reduced several times, eventually to $195 inthe US. But this was, of course, still too high for the HP-35 to become amainstream part of classroom teaching. Production of the HP-35 was stopped in Feb1975, 3 years after its launch. 300,000 units had been sold.

The numbers and functions for calculations were entered in"Reverse Polish Notation". This would seem very strange to today’s mathsstudents, since the operator always appears last: a calculation such as 3+5 wasperformed by typing “3 5 +”. It avoided the need for parentheses or an "="key.

Further models from HP followed. The introduction of theHP-35, its descendants and similar scientific calculators by Texas Instrumentssoon brought about the demise of the slide rule in the classroom.

The appearance of the calculator gave rise to the “should we,shouldn’t we?” debate in the media and among educators, with some feelingstrongly that the introduction of a calculating device would bring about adecline in students’ own calculating abilities. These arguments had little realrelevance in the early days, since the cost, fragility, and short battery life ofthese early machines meant the calculator had limited use in the classroom, andwas certainly not available to every pupil.

By the early 1980s, those deterrents began to decline. Solar-poweredscientific calculators began to appear, with hard cases, costing around £20. TheUK exam boards bowed to the inevitable and made them a standard piece ofequipment in O-Level and A-Level mathematics. The calculator was here to stay.

Later, the first hand-held calculators appeared that couldgraph functions. Like their simpler counterparts, these calculators were tooexpensive to be widely adopted when they first appeared, but today many schoolsencourage their use, although they are still not all tolerated withinexaminations.

Nowadays, the line between a calculator and a handheldcomputer is very blurred. Some calculators such as the TI-89, the Voyage 200and HP-49G are able to differentiate and integrate functions, solvedifferential equations and run word processing software. Other handheld devices can, of course, connectto the Internet. Whether such devices will ever be used in our exams, only timewill tell.

An emulation of the HP-35 is available for the Apple iPad.

David Hilbert (Jan 23 1862 - Feb 14 1943)

2012-01-23T14:37:00.000Z

David Hilbert
Image by aldoaldoz via Flickr

Today marks the 150th anniversary of the birth of David Hilbert. Hilbert was a German mathematician, recognized as one of the most influential mathematicians of the 19th and 20th centuries. Much of his work was not of direct relevance to A-Level mathematics, but his greatness demands a mention in this blog.

Hilbert worked in many areas of mathematics and developed many fundamental theorems. He also formulated the theory of Hilbert spaces, one of the cornerstones of the emerging "functional analysis". But he is perhaps most famous for his collection of problems that set the course for much of the mathematical research of the 20th century. He presented 10 of these unsolved problems at the Paris conference of the International Congress of Mathematicians, with the complete set of 23 published in 1902 in the Bulletin of the American Mathematical Society.

Many of Hilbert's problems remain unsolved, including the famous Riemann hypothesis. Many mathematicians believe the Riemann hypothesis to be unsolvable, at least given the current confines of mathematical knowledge and methodology.

But all of the problems generated a vast amount of research and investigation throughout the 20th century. They have led to new mathematical techniques and ideas, which may never have come about were it not for the problems posed.

Hilbert's 23 problems at the turn of the 20th century inspired another list of unsolved problems one hundred years later, at the turn of the millennium. The 21st century list of seven Millennium Prize Problems was chosen in 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, which offered no rewards, the solution of each problem included a million dollar prize. One of the Millennium Prize Problems, the Poincaré conjecture, was solved relatively quickly, with the Russian Grigori Perelman presenting the solution in 2002 and 2003. He famously turned down the prize money and the prestigious Fields Medal that was offered.

Hilbert strived to establish rigor in all mathematics and developed important tools used in modern mathematical physics and he is known as one of the founders of proof theory and mathematical logic.

Modern mathematics would be in a different, poorer place if it were not for the work of David Hilbert, who would have been 150 today.

Sir Francis Galton

2012-01-17T02:30:00.000Z

Sir Francis Galton lived from 16 February 1822 to 17 January 1911.

Sir Francis Galton
Image from Wikipedia

He was a cousin of Charles Darwin, and like many educated Victorian gentlemen, dabbled in many different areas of learning. Among Galton's interests were anthropology, genetics including eugenics (the idea of improving the human race by genetic selection), exploration, geography, invention, meteorology and statistics. He was knighted in 1909. He actually invented the term eugenics, and is considered the first person to use the expression "nature versus nurture".

Galton's wide range of interests led him to publication of over 340 papers and books during his lifetime. But it is his mathematical contributions, particularly in the field of statistics, that we are interested in. He created the statistical concept of correlation and widely promoted regression towards the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, and introduced the use of questionnaires and surveys for collecting data on human communities, which he used in his genealogical studies and his analyses of the behaviour of people, anthropometrics.

So Galton's contributions were considerable and, in some cases, revolutionary. But some of his ideas were also highly controversial, even for the time in which he lived. He attempted to draw up a 'Beauty Map' of the British Isles, and for this he classified passing girls into three categories: attractive, indifferent and repulsive, surreptitiously making pin-pricks in paper stored in his pocket as a means to build a database.

Eugenics itself is today considered a brutal concept. The practices involved in favouring certain families deemed genetically superior was practised by some governments during the early years of the 20th century, resulting in deprivation and a loss of human rights for millions.

Galton's book, Hereditary Genius (1869), was one of the first scientific attempts to study genius and greatness. It demonstrates some of the statistical techniques that Galton would develop through his life, but it is steeped in language that is far from acceptable in today's scientific literature, such as "idiots" and "imbeciles".

He founded psychometrics (the science of measuring mental faculties) and devised a method for classifying fingerprints that was later used in forensic science. He also conducted research on the power of prayer, concluding it had none, because of the fact that those prayed for lived no longer than those not prayed for.

Finally Galton was a pioneer of scientific meteorology. He devised an early weather map, proposed a theory for the formation of anticyclones, and was the first to establish a complete record of short-term climatic phenomena on a European scale.

Francis Galton was truly a great British eccentric, whose ideas were often controversial, but whose contributions to statistics and science in general cannot be ignored.

A Brief History of Hawking, 70 Today

2012-01-08T08:00:00.000Z

Stephen Hawking

Happy birthday to Stephen Hawking, who is 70 today.

Hawking will be known to most readers of this blog, an example of an individual whose extremely challenging disabilities have not prevented a life of incredible achievement. He has motor neurone disease, a condition that has progressed over the years and has now left him completely paralysed.

Born on 8th January 1942, Hawking is a theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity. Among the many awards he has been given throughout his lifetime, he was, in 2009, awarded the Presidential Medal of Freedom, the highest civilian award in the United States.

The diagnosis of motor neurone disease came when Hawking was 21, shortly before his first marriage. Forty-nine years later, he has been almost completely paralysed, but is still producing some of his finest work.

To speak, Hawking uses a computer fitted into his wheelchair. In latter years, he has operated it using a muscle in his cheek, selecting words, and those words are spoken by a synthesised voice. The voice, familiar to all of us, is no longer available as a speech synthesiser, but Hawking continues to use it because he considers it now to be his own.

Hawking was the Lucasian Professor of Mathematics at the University of Cambridge for 30 years. The post has previously been held by Isaac Newton and Charles Babbage, to name just two of the former eminent incumbents. Hawking retired from the post in 2009.

In 2007, to celebrate his 65th birthday, Hawking took a zero-gravity flight, during which he experienced weightlessness eight times. He became the first quadriplegic to float in zero-gravity. This was the first time in forty years that he moved freely, without his wheelchair. His plan is to take a sub-orbital space flight in 2013 on Virgin Galactic's space service.

Stephen Hawking is now Director of Research at the Centre for Theoretical Cosmology in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He is best known for his contributions in the fields of cosmology and quantum gravity, especially in the context of black holes. Hawking's key scientific works have included theorems regarding gravitational singularities in the framework of general relativity, and the theoretical prediction that black holes should emit radiation, today known as Hawking radiation.

He has also written popular science books, in which he discusses cosmology in general, going a long way to making the subject accessible to the general public; A Brief History of Time was by far his best-selling title, staying on the Sunday Times best-sellers list for a record-breaking 237 weeks.

At the time of the diagnosis of motor neurone disease in 1963, Hawking's doctors gave his life expectancy as a further two or three years. Everybody who meets the man says he is a highly entertaining person, with a great sense of humour and an ability to inspire. Although he has continued to deteriorate slowly, gradually losing the use of his arms, legs, and voice, and now completely paralysed, Stephen Hawking has lived to celebrate his 70th birthday today. The world of science, indeed the world in general, is a better place because of it.

Make a Great Resource and Win a Great Prize

2011-11-27T08:17:00.001Z

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.

Caleb Gattegno (1911 - 1988)

2011-11-11T10:30:00.000Z

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 linguistics, Gattegno pioneered the learning of reading and foreign languages with ‘infused reading’ and ‘the silent way’. He was the first English translator of Piaget, he was influential in spreading awareness of developmental psychology.

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.

A-Level Exams Earlier, University Applications Later?

2011-10-31T09:35:00.001Z

Image by Getty Images via @daylife

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

Currently, pupils in their final year of school must maketheir UCAS applications by mid-January. Universities judge each applicationbased on predicted A-Level grades, references from teachers, personalstatements and possibly an interview. The universities then award conditionaloffers, dependent on certain A-Level grades being gained.

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

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

An application later in the year would also give pupils moretime 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-Levelexaminations, the marking, awarding of grades, university applications anddecision-making would all need to take place in the summer term. In NorthernIreland and Scotland, this problem would be exacerbated because schools breakup for their summer holiday earlier than in England and Wales.

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

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

Evariste Galois (25 Oct 1811 – 31 May 1832)

2011-10-25T14:23:00.000+01:00

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.

The Forefather of Computing

2011-10-18T10:02:00.003+01:00

Part of the Difference Engine. Image via WikipediaOne 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".

Alan Turing Competition

2011-09-08T11:29:00.001+01:00

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

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!

Paul Meier 1924-2011

2011-08-26T10:08:00.003+01:00

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.

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

2011-08-25T10:29:00.001+01:00

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.

The Code

2011-08-19T10:29:00.002+01:00

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.

Results Up, Places Down

2011-08-18T13:55:00.005+01:00

The A-Level results are out.

Students sitting a maths exam
Image from Wikipedia

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

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.

Pierre de Fermat

2011-08-17T09:34:00.003+01:00

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

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.

International Mathematical Olympiad

2011-08-15T09:49:00.005+01:00

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.

Why Our Schools Still Fail At Maths

2011-08-09T09:17:00.002+01:00

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.

Firstly the problem: the education system in the UK is producing vast numbers of 16 year olds without a basic grasp of mathematics. The report says that 300,000 pupils complete their education at the age of 16 every year without a sufficient understanding of maths and that 24% of adults are "functionally innumerate".

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.

The report makes a number of recommendations, the one which has caught the media's attention being to extend compulsory maths education up to the age of 18. The report makes it clear that this extra study would not necessarily be in the form of an AS or A2 qualification, but should take the form of very practical courses.

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.

Happy Tau Day

2011-06-28T09:38:00.000+01:00

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?

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

2011-06-09T08:41:00.003+01:00

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:

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.

Women In Maths Day

2011-05-05T23:11:00.000+01:00

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.

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

2011-04-15T16:45:00.005+01:00

The UK team at the 2010 China Girls' Mathematical Olympiad

One year from now, the first European Girls’ Mathematical Olympiad will take place at Murray Edwards College (formerly New Hall) of the University of Cambridge. In April 2012, the competition will be similar in style to the International Mathematical Olympiad (IMO).

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.

This venture is inspired by the very successful China Girls’ Mathematical Olympiad (CGMO) which has encouraged many more girls to develop and pursue their interest in mathematics. The Chinese Olympiad is now an international event. The UK sent a team to compete in the CGMO for the first time in August 2010.

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.

More Universities to Charge Maximum Fees

2011-03-31T10:31:00.002+01:00

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?

A New Era in Super-Computing?

2011-03-23T12:08:00.001Z

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.

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

2011-03-22T11:30:00.000Z

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.

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

2011-03-02T08:44:00.002Z

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.

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

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?