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.

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.

Thursday, 18 August 2011

Results Up, Places Down

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.

Wednesday, 17 August 2011

Pierre de Fermat

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.

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.
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.

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