Tuesday, 21 October 2014

Martin Gardner, born 21 October 1914

Martin Gardner, an inspirational problem-setter and fun mathematician, was born this day 100 years ago. He died only 5 years ago and this blog celebrated his life then. Re-read the article on the life of a true 20th century great.

Thursday, 25 September 2014


Leonhard Euler is widely considered one of the...
Leonhard Euler is widely considered one of the greatest mathematicians. (Photo credit: Wikipedia)
In Our Time on Radio 4 introduces us to e, the irrational number that underpins so much of our modern mathematics.

e is an infinite decimal, like pi, and is approximately 2.718. It is irrational, transcendental and a part of what is often described as the most beautiful equation ever written.

Pi has been known about since the times of the ancient Greeks, but e was not discovered until the 17th century, because the mathematics required simply did not exist. The ancient Greeks' fear of the infinite was a part of the reason they did not stumble across it.

Jakob Bernoulli was the first mathematician to discover e cropping up in his studies of compound interest. A little later, John Napier invented logarithms, using a base approximately equal to 1/e. These huge tables of numbers took Napier 20 years of his life to devise and were designed to make multiplication easier and more accurate. Logarithms were later refined by Henry Briggs.

e also appears a lot in modern calculus. The function e to the power x has a gradient which always has the same value as the function itself. When Leibniz and Newton discovered calculus they used infinite sums to form the derivative of the exponential function.

Later the natural logarithm was found to be the area under the curve y=1/x.

e was named by Euler, the great mathematician of the 18th century, possibly the greatest mathematician of all time and probably the most prolific in terms of publications. But he did not, we are told, name e after himself. Instead, he thought e to be the first letter of the alphabet not widely in use in mathematics. Euler showed e was the sum of infinite series and introduced his famous Euler identity, which many mathematicians consider the most beautiful equation ever written.

To give just a few applications, e crops up in radioactive decay, in the normal distribution in statistics, and in the prime number theorem, which tells us roughly how many primes there are below any given integer.

Thursday, 21 August 2014

GCSE Results Day: Maths Results Rise

GCSE students in England, Wales and Northern Ireland have been receiving their results today. The results show that 68.8% of entries scored A*-C, up from 68.1% last summer, although there was a marked fall in English GCSE grades.

Students sitting their GCSE Maths exam.
Photo: Wikipedia.
There have been warnings of volatility in this set of results following an overhaul of the exam system. The most significant impact on this year's results has been the big fall in students taking their GCSEs a year early. Schools have been discouraged from such multiple entries following changes in the way school league tables are compiled.

Fewer fourth years taking maths GCSE meant there was a sharp improvement in maths results: the percentage achieving A* to C grades rose by 4.8 percentage points to 62.4%.

The overall pass rate was 98.5%, down 0.3 percentage points. 6.7% of entries were awarded an A* grade.

Girls are still doing better than boys at GCSE, with 73.1% of girls' entries achieving A* to C compared with 64.3% for boys.

In England, but not in Wales or Northern Ireland, this is the first year of results following moves towards exams at the end of two years, rather than including coursework and modular units. The results for GCSE English seem to have been most affected by this change, with the number of A*-C grades down 1.9% to 61.7%.

In Wales and Northern Ireland, these changes were not introduced and the three regional sets of GCSE exams are now beginning to diverge in various ways, including the subjects being taken by students.

While the government are defending the changes being made, Chris Keates of the NASUWT teaching union said this year's students had to "cope with a raft of rushed through and ill-conceived changes to the qualifications system and so today's results are especially commendable".

The National Union of Teachers' leader Christine Blower said that the headline figures "mask underlying issues which will only become clear over time".

Have you had your GCSE results today, or are you teaching GCSEs? How did your school fare following changes to the structure of GCSEs this year? Let us know at info@mathsbank.co.uk .

Thursday, 14 August 2014

A-level maths now most popular subject

Pupils in England, Wales and Northern Ireland receive their A-level results today and they appear to have fallen slightly this year.

The Joint Council for Qualifications, issuing the results, said there has been a slight fall in A* and A grades and the overall pass rate is down for the first time in over 30 years. The percentage gaining the very highest A* grade has risen from 7.6% to 8.2%. 8.5% of boys' grades were A*, with girls' grades at 7.9%.

A-Level results this year are "broadly stable".

For the third successive year overall A* and A grades have fallen slightly (this year down from 26.3% to 26%), but exam officials are saying A-level results are broadly "stable".

For school leavers planning to go to university, there are suggestions this could be an unusually good year to apply. There are a record number of university places on offer this year - over 500,000 for the first time, which is a rise of over 30,000. Students may still get places even if they have missed their grades. The Ucas admissions service says initial figures show a 2% increase in students getting their first choice place.

Education Secretary Nicky Morgan says the government is "lifting the cap on aspiration". Universities Minister Greg Clark says the increase in the number of places is an "important source of social mobility".

There is a trend for more students to take so-called "facilitating subjects" at A-level, such as maths and physics, which can help university applications. Maths is now the most popular subject, overtaking English this year for the first time.

It is the first set of results following the Government's scrapping of January A-Level sittings. However, the fewer opportunities to take modules does not seem to have affected students' overall performance too badly.

Regarding other proposed changes, Labour's shadow education secretary Tristram Hunt said he would reverse the government's plan to remove the link between AS and A-levels. This de-coupling of the two exams would limit young people's "opportunity to realise their full potential", said Mr Hunt.

Would you like to share your results story with us? Comment on this article, or email info@mathsbank.co.uk.

Thursday, 17 July 2014

Michael Gove Out

So Michael Gove is no longer our Secretary of State for Education. He presided over a time of headlong change in our education system. Some would describe him as a visionary; others, such as the teaching unions, would probably say he attempted too much too quickly, and that would be the polite version.

Nicky Morgan is the new Secretary of State
at the Department for Education.
Photo credit: Wikipedia.

As education secretary, Michael Gove was a deeply controversial figure. He brought in free schools, rewrote the National Curriculum and rapidly increased the number of academies in England, so that now around 50% of schools in England have academy status. For academies and free schools he brought in legislation allowing these institutions to employ unqualified teaching staff. He also presided over the dramatic rise in the maximum level a university can charge in tuition fees from £3000 to £9000.

His supporters would say Michael Gove took on an education system unwilling to change. He brought forward difficult but necessary changes despite fierce opposition. His critics would say that he is a deeply divisive figure, a zealot bent on his own view of what an education system should look like, stuck in the past and out of touch with the modern realities of teaching.

It is true he got himself into many arguments: he fell out with Deputy Prime Minister Nick Clegg, with Ofsted Chief Inspector Michael Wilshaw, with his own Conservative colleague, Home Secretary Theresa May over the alleged plot by Islamic extremists to seize control of certain schools in Birmingham.

Did Prime Minister David Cameron think his personal friend Gove was a liability as the general election approaches next year? He has been replaced by Nicky Morgan and time will tell whether her tenure makes for more harmonious relationships with teaching bodies, Ofsted and other interested parties. She takes over at the Department for Education at a time of change, but will not be blamed for the series of reforms that she will have to preside over. We wish her well.

Friday, 17 January 2014

Happy 40th Birthday HP-65

First pocket programmable calculator
First pocket programmable calculator (Photo credit: Wikipedia)
Two years ago we celebrated the fortieth birthday of the HP-35, the first mass-produced, pocket-sized, scientific calculator.

Today marks the fortieth birthday of its younger sibling, the HP-65, released on 17 January 1974. This was the world's first programmable calculator.

The first programmable calculators were introduced in the mid-1960s by Mathatronics and Casio, but these machines were very heavy and expensive.

So the miniaturisation involved in the HP-65 was a breakthrough. Bill Hewlett is supposed to have insisted that the calculator should fit in his shirt pocket and this was partly achieved with the tapered body.

The HP-65 had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader. The magnetic program cards were fed in at the thick end of the calculator under the LED display.

Examples for programs provided with the calculator included algorithms for hundreds of applications, including the solutions of differential equations, stock price estimation and statistical functions.

Tuesday, 3 December 2013

Pisa Results: UK Could Do Better

test documents for the Programme for Internati...
The Programme for International Student Assessment (PISA) test on a school table in Germany (Photo credit: Wikipedia)
The UK is ranked 26th out of 65 countries in maths, in the results of the 2012 Pisa tests (Programme for International Student Assessment), which were published today.

The tests in maths, English and science were taken by 500,000 15-year old students around the world. The top 7 places are filled by far eastern countries and cities. Shanghai came top of the league table (Chinese cities are entered individually, rather than the country as a whole). The highest place gained by a European country is 8th place, for Lichtenstein. The US came even further down the league table than the UK, coming 36th in maths. Wales was the poorest performing region of the UK.

Many educationalists have criticised the league table. Prof Alan Smithers from the University of Buckingham pointed out that in many Asian countries, pupils are coached specifically for these tests.

However, Christine Blower, leader of the National Union of Teachers, said the results showed that countries with successful education systems "pay teachers well, respect the profession and encourage collaboration between teachers and schools".

Naturally, Education Secretary Michael Gove blamed the last Labour government for the UK's poor performance. For Labour, Tristram Hunt said that the coalition government's policies were taking the UK towards a poorer education system, not a better one.

What do you think about the results of the Pisa tests? Is the UK lagging behind in these key subject areas? Are the Pisa tests an accurate measure of performance?

You can take the Pisa test here.

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