Sir Francis Galton (16 Feb 1822 - 17 Jan 1911)

Today marks the 200th anniversary of the birth of Sir Francis Galton. He 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.

Happy 50th Birthday to the HP35

The HP-35 scientific calculator
Image via Wikipedia

1st February 1972 marked a birth that would revolutionise mathematics education across the world. This was not, this time, the birth of a great mathematician, but of a machine. The first scientific hand-held calculator (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 it had 35 keys.

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

The price was reduced several times, eventually to $195 in the US. But this was, of course, still too high for the HP-35 to become a mainstream part of classroom teaching. Production of the HP-35 was stopped in Feb 1975, 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 maths students, since the operator always appears last: a calculation such as 3+5 was performed by typing “3 5 +”. It avoided the need for parentheses or an "=" key.

Further models from HP followed. The introduction of the HP-35, its descendants and similar scientific calculators by Texas Instruments soon 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 feeling strongly that the introduction of a calculating device would bring about a decline in students’ own calculating abilities. These arguments had little real relevance in the early days, since the cost, fragility, and short battery life of these early machines meant the calculator had limited use in the classroom, and was certainly not available to every pupil.

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

Later, the first hand-held calculators appeared that could graph functions. Like their simpler counterparts, these calculators were too expensive to be widely adopted when they first appeared, but today many schools encourage their use, although they are still not all tolerated within examinations.

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

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

GCSE Results 2017

GCSE results are out today, with a fragmented set of exams now being sat across the UK. In England, new GCSE courses have been designed to be more challenging. Overall across the UK, the pass rate has dropped across the full range of subjects, from 66.9% to 66.3% of entries. A pass under the old system was a C grade; under the new system of grades from 1 to 9 (with 9 the best), a 4 is considered a standard pass.

In England the pass rate for maths, one of the new tougher exams, it dropped slightly from 71.4% to 70.7%. The proportion of entries receiving the top grades (A/7 or above) has also fallen, to 20%, down 0.5 percentage points on last year, the lowest since 2007.

However, exam boards appear to have worked the grade boundaries to ensure the distribution of grades is broadly consistent with last year's. For example, on the higher tier maths paper, a grade 4 pass would have been achieved with just 18% of the overall marks. For a grade 9, 79% mark was required. For a grade 7, the equivalent of an A, candidates needed just over half marks.

Students and teachers have complained about a lack of textbooks and practice papers in the run-up to the summer exams.

In Northern Ireland and Wales, exams are still graded A* to G. In Northern Ireland, female students continue to outperform their male counterparts at this level, with an 8.1% difference in the pass rate (A*-C). One in ten entries are now awarded the top A* grade here.

A-Level Results Day 2017

A-Level results are out today in England, Wales and Northern Ireland. This year's results have shown an increase in the number of A and A* awards given, the first rise in 6 years.

26.6% of entries from boys were awarded these top grades, with 26.1% of girls' entries (overall 26.3%, up 0.5%). This is first time in 17 years that the boys have been ahead at this level of education.

In England, 13 A-Level subjects were reformed this year (with mathematics and the rest to follow). Changes include assessment based entirely on a final exam, eliminating coursework entirely. There was a fall in the top grades awarded in these subjects. Another change means that pupils now have to decide from the outset whether to take AS or a full A-Level because the two qualifications have been decoupled. This has resulted in a number of pupils continuing to their second year, who would previously have stopped after the first year, resulting in a larger number of less able pupils taking the full A-Level. A lack of text books for the new syllabuses and a lack of past papers may be other contributing factors.

For those who didn't make the grades they required for university, there is a very good chance they will be able to take a place through clearing this year. University applications from the UK and European Union countries have fallen compared with last year and there is a demographic dip in the number of 18 year olds.

In Northern Ireland, 30.9% of entries were awarded the top two grades, but girls are still ahead. Indeed they increased the gap to 6.5% with 33.3% of girls and only 26.8% of boys getting the top two grades.

John Napier (1550–1617)

Napier's "Bones", a set of wooden rods used for muliplication

If you have studied logarithms, you have John Napier to thank for their discovery. He died 400 years ago today.

John Napier was a Scottish landowner, mathematician, physicist, and astronomer, but is probably best known as the discoverer of logarithms.

Before calculators became widespread in the second half of the 20th century, logarithms were widely used to perform multiplication of numbers, particularly large numbers. Logarithms were looked up in "log tables".

Logarithms have many uses and come into many areas of modern maths, and if you have studied beyond GCSE level, you will be very familiar with them. The natural logarithm (log base e) is sometimes called the Napierian logarithm, although Napier didn't explicitly work with base e.

Napier also invented "Napier's bones", which were a set of wooden rods inscribed with digits, and were another tool for performing multiplication. Napier was also one of the earlier mathematicians to make frequent use of the decimal point in his work and helped to popularise this idea.

Napier's birthplace, Merchiston Tower in Edinburgh, is now a part of Edinburgh Napier University, which is named after him.

Napier died in 1617 at his home at Merchiston Castle from the effects of gout, but his revolutionary work on what became a key part of modern mathematics will ensure he is remembered for a long time to come.

GCSE Results 2016

GCSE results across England, Wales and Northern Ireland this year have suffered a significant fall. This is in marked contrast to last week's A-Level results, which showed a great deal of stability when compared with recent years. GCSE results peaked in 2011 after a long period of "grade inflation", and have been falling year on year ever since.

Of the five million results being released today, the proportion of entries being given A* to C grades has fallen from 69% to 66.9%, an unprecedented fall of 2.1%. The proportion of top A* and A grades was down overall by 0.9% to 13%. Awards of the very top A* grade have also fallen slightly, down from 6.6% to 6.5%.

One reason for the fall in the overall pass rate is that more pupils in England are re-taking GCSE English and maths, following measures by the government to attempt to ensure that all pupils reach a grade C in these subjects. As a result there were re-sits for tens of thousands who did not reach a C grade last year. Among over 16s, GCSE entries were up by 380,000, up roughly 25% on last year.

By subject, there were falls in the pass (A*-C) rate in maths (down 2.3%), English (down a worrying large 5.2%) and in other subjects.

But even without the re-sit figures, there was a fall in the results of 16 year olds, with the proportion getting A* to C falling by 1.3%. It is thought that a move to an alternative qualification, the iGCSE, for many able students, may account for some of the drop.

While the overall results were downwards, in Northern Ireland the proportion of passes increased slightly to 79.1% and top A* grades rose to 9.3%. In Wales, the level of A* to C passes remained at 66.6%, with A* grades rising slightly to 6.1%.

Girls continue to do better than boys at GCSE. The A*-C rate for girls was 71.3% compared with 62.4% for boys, a gap some would describe as worrying, or even unacceptable. The percentage of entries gaining A* grades for girls was 7.9% and 5.0% for boys.

Next year will start to see the phasing in of a radical change to the way that GCSEs are graded. The new GCSE exams will be graded from 1 to 9 (with 9 the highest), rather than the current A* to G.

A-Level Results Day 2016

Hundreds of thousands of teenagers in England, Wales and Northern Ireland are receiving their A-level and AS results today. The A-level results show a stable pattern compared with last year.

Maths remains the nation's most popular choice of subject, as it was last year, and increasing numbers are also opting to take Further Maths at A-Level.

The proportion of top grades (A and A*) is 25.8%, down by 0.1% on last year. The overall pass (grades A*-C) rate of 98.1% remained the same. Northern Ireland remains the region with the biggest proportion of top grades, 29.5%.

Girls fared better than boys, once again, with 79.7% of girls getting a pass grade, compared with 75% for boys. Boys are getting more A* grades (8.5% compared with 7.7% for girls), although this gap between the very top-performing girls and boys has narrowed for the first time in five years. The overall level of A* grades (8.1%) has been falling now for 2 years.

According to Ucas, the universities' admissions service, 424,000 university places have been offered to hopeful students, which is up by 3% on the same time last year. But many places are reportedly still available through clearing, including at leading universities and for highly sought-after courses, such as medicine.

The increase in the number of places still available is mainly due to two factors: a roughly 2% fall in the number of school-leavers and the removal of the cap on the number of places universities in England can offer.

AS levels are being "decoupled" from being part of A-levels - and this year's figures show a 13.7% drop in entries for the AS course. Previously the AS has carried with it the option of continuing on to a full A-Level, but this will now cease to be the case: if you register for an AS, this is the qualification you will get.

Teachers and head teachers' leaders have warned that, although the overall results appear to show stability when compared with recent years, individual schools and pupils are facing some unpredictable outcomes.

The current cap of £9000 for university tuition fees is being removed, so students starting at England's universities in the autumn could face higher fees. Exeter University has been the first to announce that it will increase fees to £9,250 for all current and new students.

With the prospect of increasing fees, school leavers may opt not to go to university at all. Financial services firm PwC says that it has had a 20% increase over two years for new recruits of those leaving school with A-levels.

Government considers compulsory maths lessons up to 18

George Osborne wants to make
maths compulsory to the age of 18.

We already discussed plans outlined in the budget to force all schools to become academies by 2020. Another budget announcement was that the government is considering making all pupils study maths to the age of 18.

The move was welcomed by prominent mathematician Marcus du Sautoy, but critics claim it is “undeliverable” because of a critical shortage of maths teachers.

It was one of several education measures outlined by George Osborne in his budget speech, but teachers point out that there are barely enough qualified teachers to deliver GCSE maths in schools, without the extra burden of a course for all 16 to 18 year olds.

The chancellor said, in his budget speech: “We are going to look at teaching maths to 18 for all pupils. Providing great schooling is the single most important thing we can do to help any child from a disadvantaged background succeed."

Currently schools are facing increasing difficulties trying to fill maths posts. Because of this pressure, 20% of maths lessons are taught by teachers without a maths degree. Teacher leaving rates in this subject area are also above average.

The government has called on Prof Sir Adrian Smith, vice chancellor of London University and former president of the Royal Statistical Society, to assess the feasibility of the idea. Presumably he will take into account chronic teacher shortages as one of the factors that may render the scheme impossible.

At the last General Election, the Labour Party called for English and maths to be studied until the age of 18.

The Labour Party's education spokeswoman, Lucy Powell, supported the new proposal but questioned its current viability and criticised the government for failing to achieve its target for recruiting maths teachers four years in a row.

“There is nothing more important to our global competitiveness than mathematics, which will drive success in digital skills, automation and other important jobs of the future,” she said.

There has long been concern about the maths skills of UK children, who fare poorly in international tests. The latest Pisa tests, from 2013, put England in 26th place for maths, behind countries such as Singapore, Taiwan, South Korea and Japan.

Academisation Phase 3 is on its way

Education Secretary Nicky Morgan will preside
over the third phase of academisation,
releasing all schools from  local authority control.

All schools are to be converted to academies by 2020 and then will be asked to join an "Academy Chain". That is the plan of the existing government, announced as a part of the budget on 16 March.

This move will effectively end the involvement of local authorities in school administration. Each academy takes a certain amount of its funding directly from central government. It is also free from the constraints of the National Curriculum and can set its own pay scales for its staff.

The first phase of "academisation" began before 2010. Failing and struggling schools were offered cash incentives to convert to academies. Outside sponsors provided some of the funding and because the schools were no longer constrained by external pay structures, could offer larger salaries to good staff, including to head teachers, to turn the school around.

After 2010, all schools were given the option to convert to academy status. Incentives were still available and a large number of state schools took up the offer. This was then education secretary Michael Gove's idea and they were known as "converter academies".

Phase 3 will see all schools being forced to take up academy status. Many have concerns about these plans, including the Labour Party, teaching staff and the unions. There will undoubtedly be extra demands upon central government and some suggest that the Department for Education is already unable to cope with its workload.

What do you think?
Do you think that academies have more freedom and flexibility to offer a better education to their students?
Do you think that effectively ending the National Curriculum in this way is a good thing?

Ada Lovelace, born 10 December 1815

Portrait of Ada Lovelace, 1840.
(Picture credit: Wikipedia)

Ada Lovelace was born 200 years ago today, on 10 December 1815. She became a highly respected British mathematician and writer, known mainly for her work on Charles Babbage's early mechanical computer, the Analytical Engine. Her notes on this machine include what is now thought of as the first set of instructions intended to be carried out by a machine, or the first computer program, and she as the first computer programmer.

She was born Augusta Ada Byron and inherited the title Countess of Lovelace later in life. Her father was the poet Lord Byron, but she never knew him, as he left his wife a month after Ada was born. She had her mother Anne Milbanke to thank for setting her up with an interest in mathematics and science, but Ada always remained interested in her father, and loyal to him, despite not knowing him.

Lovelace was still a teenager when her mathematical ability led her into a working relationship and friendship with fellow British mathematician Charles Babbage. In particular she helped in Babbage's work on his Analytical Engine. In 1842 and 1843 she translated an article about the engine from Italian into English. She also supplemented the translation with a very detailed set of notes of her own.

These notes contained what many now consider to be the first computer program, an algorithm designed to be carried out by a machine. Lovelace's notes are important in the early history of computers, although the algorithm was never used because the Analytical Engine was never built. She also had a realisation about the potential of computers to go beyond mere calculating. Others, including Babbage himself, focused only on their numerical capabilities.

Upon her death, Ada was buried next to her father Lord Byron, in Hucknall, Nottinghamshire, at her own request.

Ada Lovelace Day is now an annual event in mid-October that aims to "... raise the profile of women in science, technology, engineering and maths," and to "create new role models for girls and women" in these fields.

The Riemann Hypothesis Remains Unproven

Dr Opeyemi Enoch claims to have proved
the Riemann Hypothesis.

There were stories going around today that one of the longest-standing problems in mathematics had been solved, namely the Riemann Hypothesis.

The Riemann Hypothesis relates to a mathematical function called the Zeta function, and in particular where the value of this function is zero. Proving the hypothesis would give mathematicians new insight into the distribution of the prime numbers.

A Nigerian academic named Dr Opeyemi Enoch was reported to have finally proved the 156 year old problem, which is one of the Clay Institute's outstanding millennium problems.

As this article on the Aperiodical website makes clear, Dr Enoch has a very varied academic background, including designing a prototype silo for peasant farmers and detecting people on an evil mission.

His "proof" was presented at a poorly-attended conference, which doesn't seem to have attracted the audiences or world attention that such a ground-breaking piece of work would warrant. But the BBC ran an interview with Dr Enoch, in which he was asked, among other things, what he would do with the one million dollar prize for solving the problem.

In an article on Radio 4's Today programme this morning (scroll to 2:50:44), however, Professor Marcus du Sautoy of Oxford University refuted the claims of the proof and managed to plug his book at the same time, of course.

The Clay Institute is clearly unconvinced about the proof, stating that the problem remains unsolved.

Another Maths Question Goes Viral (this time from down under)

Another maths question has been popping up all over Twitter and other social media. This time the question comes from the Victorian Certificate of Education, an Australian maths exam.

Here it is:

While many were left puzzled, the question may appear easier if a simple straight line is drawn downwards dividing X into two equal parts, as shown below:

Now the angle marked is simply the external angle of a regular 12-sided shape. The formula for the external angle of any regular shape is:

where n is the number of sides, i.e. 12.

So

and

.

It was fairly easy when you know how! But like Hannah's sweets from the English Edexcel board, the trick is spotting how! Expect more of this type of question in years to come, as the exam boards try to separate out pupils who have just learnt the formulas from those who can spot when to use them!

And, of course, social media means we'll get to hear about all of them!

Nicky Morgan's Plans for Education

Nicky Morgan will announce her plans
for English schools today.

It seems that every education secretary wants to unveil their own big revolution for our education system, before schools, pupils and teachers have had time to adapt to the last seismic shifts.

Today it was Nicky Morgan's turn to announce a raft of measures that will bring profound changes to our schools.

The eye-catching suggestions are:

• Primary school pupils in England could face formal tests at the age of seven;
• A pool of "elite teachers" will be recruited to work in struggling schools in coastal towns;
• A target will require 90% of pupils to take core academic subjects at GCSE.

Mrs Morgan maintains her changes will help "every young person get the best start in life". At present, she says, there are 20 local authorities where most pupils do not achieve five good GCSEs, including English and maths. For Labour, shadow education secretary Lucy Powell said that rather than raising standards, the government has caused a "chronic shortage" of teachers.

In her speech today, Mrs Morgan will announce details of the pre-election pledge to create a National Teaching Service. It will recruit a pool of 1,500 high-achieving teachers over five years who would be deployed to struggling schools. It will also give local "commissioners" the power to intervene in under-performing state schools in a variety of ways. These commissioners are already in post, created to oversee academies and free schools.

The plans also include the re-introduction of testing all pupils at the age of 7, which was previously scrapped. Critics point out that the UK's pupils are already the most examined in Europe and our schools are in danger of becoming "exam factories".

In secondary school, there is clarification of the last education secretary Michael Gove's target that all pupils will have to take traditional GCSE subjects, in what he named the English Baccalaureate. This requires pupils to take GCSEs in English, maths, history or geography, two sciences and a language.
There will now be a target of 90% of pupils, which will allow for just a small number of exemptions, such as for pupils with special needs. At present, only about 39% of pupils take these subjects.

Brian Lightman, leader of the ASCL head teachers' union, said it would be "immensely challenging" for schools to get enough staff for subjects such as modern languages to meet these new targets.

Are you a teacher, head teacher, parent or pupil? What are your thoughts on the big changes being announced today?

George Boole born 2/11/1815

Today Google is celebrating the 200th anniversary of the birth of George Boole, with an animated logo.

So who was George Boole and why is his work relevant today?

George Boole -
picture credit Wikipedia

George Boole was a British mathematician and logician and was ahead of his time. Until his time, logic was a subject mainly applied to philosophy and natural language. Boole was the first to apply logic to mathematics in a systematic way. By doing this, his work paved the way for the digital revolution and "Boolean logic" was a precursor for the way computers perform their calculations.

Boolean logic treats variables as either on or off (true or false, or 1 or 0). In the 1930s, the American Claude Shannon applied Boolean logic to build the first electrical circuits.

Today Boolean logic underpins almost every computer program in some way. For example, if you search the web for a two-word term, such as "cheesy chips", the search engine's algorithms will apply an AND operation, and bring you web pages where the two words BOTH appear. You would not find it very useful if the results shown were pages where either "cheesy" OR "chips" were on the page.

Different parts of the Google logo light up when the x and y in the second g are shown. See if you can work out the rules.

As well as inventing this field of logic, George Boole published work on differential equations and probability theory. He died aged just 49 in 1864 and will be remembered as an important pioneer of the information age.

GCSE Results Day

Hundreds of thousands of teenagers in England, Wales and Northern Ireland have received their GCSE results today.

The results for more than five million GCSE entries show A* to C grades have risen slightly this year, but slightly fewer top A* and A grades have been awarded. The proportion of A* to C grades rose to 69%, up from 68.8% last year, but A* grades fell by 0.1 percentage points. In line with last week's A-Level results, the national GCSE results are stable compared with last year.

In maths, those achieving A*-C grades increased from 62.4% to 63.3%. There were also improvements in A*-C grades for English, physics, chemistry and biology. But fewer entries for the double science GCSE were awarded good grades.

The best results came in Northern Ireland, as last year, where the proportion achieving A*-C grades rose from 78% to 78.7%. In Wales, there was no change, at 66%.

There have been changes in the age groups of pupils taking GCSEs this year and this is thought to have influenced results. After changes made to the league tables, schools are entering fewer younger pupils (third and fourth year pupils) for GCSEs. In addition, more 17-year-olds are taking GCSEs, because of a government policy that requires pupils to re-sit maths and English if they failed to gain at least a C grade.

As with A-Levels, this year's results come just before a major overhaul to the system. Although the qualification will still be called a GCSE, grades 1 to 9 will be awarded and the content, especially in maths, will become more demanding. The new exams will be phased in over the next 3 or 4 years.

A-Level Results 2015

Hundreds of thousands of teenagers in England, Wales and Northern Ireland received their A-level results on Thursday.

The proportion of A-level entries being awarded top A* and A grades has fallen slightly this year to 25.9% of entries, down from 26% last year.

In a year of "stable" results, overall passes (A*-C grades) rose marginally by 0.1 percentage point to 98.1%. The proportion getting the very top A* grade remained the same at 8.2%, with A grades down by 0.1%.

Schools minister Nick Gibb said the results showed the impact of the government's drive for "core academic subjects" with a 20% increase in maths entries since 2010. Traditional subjects such as geography and history have also seen strong growth in numbers, but computer science has seen the biggest increase.

With caps on the number of students at each university being removed, record numbers have been accepted on university courses. The Ucas university admissions service said that 409,000 places had already been confirmed, up 3% on last year.

Michael Turner, director of the Joint Council for Qualifications, pointed out "The over-riding message from this year's figures is one of stability. There have been no significant changes to the system."

"As a result thousands more pupils, from all backgrounds, are studying subjects that will secure them a place at a top university or an apprenticeship and that will help to secure well paid employment," said Mr Gibb.

Chris Keates, leader of the NASUWT teaching union, said that the results showed that the "gold standard" A-level system had been maintained, despite the pressure on schools to prepare for forthcoming changes to exams.

But this year's lifting of the cap on university places in England has seen more students than ever accepted on to courses.

Northern Ireland A-Level students achieved slightly fewer A and A* grades compared with last year, but still outperform England and Wales. The Joint Council for Qualifications said that 29.3% of Northern Irish entries achieved A or A* grades, a drop from 29.9% last year. Mathematics is also becoming a very popular subject for A-Level students in the province, with a 10.6% rise in the number of girls taking maths at A-Level in Northern Ireland.

But maths saw a fall in the number of students being awarded the top A* and A grades as did the science subjects and English.

This year's relatively stable results come before a period of major transition for the "gold standard" A-Level. First teaching for the new A-Levels in some subjects begins this year. For other subjects it will be 2016, and for Maths and Further Maths the new syllabus will be taught in September 2017 for the first time.

Hannah's Sweets

Quite a few GCSE students were stumped by a question about Hannah's sweets on yesterday's higher tier Edexcel paper.

The question was:

Hannah has 6 orange sweets and some yellow sweets.

Overall, she has n sweets. She takes one sweet from the bag and then another.

The probability of her taking 2 orange sweets is 1/3.

Prove that: 

The question is not actually that difficult if you remember how tree diagrams work.

When Hannah first takes a sweet there are 6 orange sweets out of n, so the probability of her choosing orange is 6/n.

When she chooses her second sweet there are now only 5 orange (if she chose orange the first time) out of a total of n-1 sweets.

You multiply the probabilities along the branches of a tree diagram, so

 Multiplying the 2 fractions on the left gives:

Cross-multiplying gives:

John Nash 1928 - 2015

English: John Forbes Nash, American mathematician and winner of the Nobel Prize in Economics 1994, at a symposium of game theory at the university of Cologne, Germany (Photo credit: Wikipedia)

The American mathematician John Nash has died in a car crash with his wife, police have said. Nash is most famous for winning a Nobel prize in 1994 and for being played by Russell Crowe in the Oscar-winning film A Beautiful Mind.

John Nash was 86. He and his 82-year-old wife Alicia were killed when their taxi crashed in New Jersey. Police said they were thrown from their vehicle and media reports suggested the couple may not have been wearing seatbelts.

Nash is renowned for his work in game theory, the mathematical study of decision-making, which won him the Nobel Prize for Economics in 1994.

Nash married Alicia Larde in 1957. Alicia committed her husband for psychiatric care several times after the onset of severe schizophrenia. The couple divorced in 1962, but they remained close, and with Nash's condition improving by the 1980s, they remarried in 2001. Alicia Nash helped care for her husband, and the two later became prominent mental health advocates.

John Nash's work in the field of game theory, and his struggles with schizophrenia, were the focus of the 2001 film A Beautiful Mind.

Nash Equilibrium, named after the mathematician's work in game theory, has become a concept used in a wide variety of disciplines including chiefly economic analysis, but also computing, evolutionary biology and artificial intelligence. This is the basis of the work that led to the Nobel prize.

Should Google be allowed in public exams?

Mark Dawe of OCR
believes it is "inevitable"
that search engines
will be allowed in exams.

The head of exam board OCR sparked a controversy this week when he said it was "inevitable" that search engines, such as google, will be allowed in public exams, such as GCSEs and A-Levels.

Mark Dawe told BBC Radio 4's Today programme that allowing internet use in exam rooms would reflect the way pupils learned and how they would work in future.

He said that students would still need a basis of knowledge and that they would have limited time to conduct searches.

Regarding when these changes might be introduced, Mr Dawe said: "It's very unlikely to happen in the next few weeks or next few months, but it's certainly inevitable, I would suggest."

If you are a pupil reading this, you might be thinking that such a move would make your exams - and your revision programme - a lot easier. But would it? Clearly the exams themselves would adjust to the changes. There would be fewer questions where the answers were easily "googled". The questions would become more about how to apply the knowledge you have, rather than about how much you remember. A part of the skill set required for these new examination would be on your ability to find the relevant material on the web, how to collate this information in a sensible way and into a usable form, being able to discern between good reliable information and nonsense (of which, as you know, there is a lot on the web).

In A-Level maths, for example, it is possible to find the solution to any integral on the web. So the questions may become more about applications of integration: real world problems. Questions may become more wordy in mathematics. Problem solving skills will be required, since the candidate will need to know what form of integration is required, or even whether a question requires integration or differentiation, rather than the exam paper presenting an integral and asking the candidate for the solution. So examination writers would also have to take on an entirely new set of skills.

The Campaign for Real Education condemned the idea as "dumbing down". Their spokesman Chris McGovern said: "We have a crisis in standards in this country." He added: "You can have an exam in how to use Google - that's not the same thing as having a history exam or a geography exam.

It is important to note that when calculators were first introduced in public exams in the 1970s there was a similar furore from some parties.

What do you think? Should Google be allowed in public exams?

UCAS Expresses Doubts Over A-Level Changes

UCAS - the organisation which is responsible for managing more than 2 million applications to higher education courses in the UK.  (Photo credit: Wikipedia)

UCAS, the universities admissions authority, today expressed serious reservations about the changes to the structure of A-Levels, proposed to begin this year.

UCAS, citing the results of a survey it conducted, said that schools were still undecided about which courses to offer from September, when the changes begin. In addition, it stated that some pupils may be put at a disadvantage by the changes, in particular the proposed decoupling of the A-Level from the AS.

Some universities have already expressed grave concerns about this aspect to the change. Last month we reported that Cambridge University had written to schools to ask them to continue teaching the AS, which will become optional. Amid the confusion, the changes to maths and further maths have already been delayed until 2017.

The Department for Education say that the decoupling is intended to allow students to study a subject more deeply for two years, without being distracted by exams halfway through.

Universities make offers before final grades are published. One of their objections is that, without an AS result to look at, it will be more difficult to target offers at the right candidates.

In addition, the universities will be faced with candidates presenting complex combinations of results, with the introduction of the new system being staggered over three years in the various subjects.

Many pupils benefit from the AS-Level as a staging post to the A-Level, because it provides them with a confidence boost, especially those who had not gained top results at GCSE.

UCAS warned that the confusion caused by these changes will not be over until 2020.

The chief executive of UCAS, Mary Curnock Cook, said that the picture will be complicated further by the fact that pupils from different parts of the UK will be sitting exams with the same names (A-Level, AS-Level and GCSE), but with different structures and grading procedures.

Labour has committed itself to reversing the decoupling plans, but as we reported in our previous article, any reversal would be fraught with difficulty.