Thursday 10 December 2015

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.

Tuesday 17 November 2015

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.

Tuesday 3 November 2015

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.


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?

Monday 2 November 2015

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.

Thursday 20 August 2015

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.

Monday 17 August 2015

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.

Friday 5 June 2015

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:

Monday 25 May 2015

John Nash 1928 - 2015

English: John Forbes Nash, American mathematic...
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.

Monday 4 May 2015

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?

Friday 16 January 2015

UCAS Expresses Doubts Over A-Level Changes

English: Entrance to UCAS The organisation whi...
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.