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 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?

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.

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

• String Theory Finally Does Something Useful (wired.com)
• The mystery of mass: What makes one particle light and another heavy? | Ian Sample (guardian.co.uk)

The Forefather of Computing

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

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

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.

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.