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: