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Posted: 04-06-2011 02:15 Post subject:

But then you could argue, as many do, that you need to have a standard entrance exam, rather than allowing different schools to set their own papers. Many places do use standard tests, but these can be fraught with other problems.

Exams are never going to be a "level playing field" psychologically or otherwise. Some people cope with the stress of an exam better than others. If people are sitting different exams, there is always the possibility that the students at one school are getting easier questions than at another. Who's to say that resitting the exam won't disadvantage the students who did well the first time? If it's a voluntary resit, then the people taking the second paper have more time to study.

The father's comment applies equally to any exam, whether it had an impossible question in it or not. I just don't think this situation requires any particular special remedy.

Posted: 08-06-2011 16:45 Post subject:

After years of falling standards, it seems we can't even get people clever enough to set exams any more: Evil or Very Mad

Multiple-choice AS-level exam answers 'all wrong'
By Sean Coughlan, BBC News education correspondent

Another incorrect question has appeared in an AS-level exam - the third such mistake pupils have faced this summer.
The multiple-choice biology question, set by Edexcel, offered a selection of wrong answers but not the correct one.

The exam board said the question was worth one mark out of a possible 425 and marking would be adjusted to ensure "no candidate is disadvantaged".

The two other "unanswerable questions" were in maths and business studies AS-level papers set by other exam boards.

Students have complained that even if the questions are discounted, it is difficult to know how much the overall grade could have been affected by the time wasted trying to interpret a wrong question.

etc...

http://www.bbc.co.uk/news/education-13697116

Posted: 10-06-2011 10:33 Post subject:

...and counting

Three new blunders discovered in school exam papers fiasco
Six mistakes have now emerged, mostly in AS-level questions which proved impossible to answer
By Richard Garner, Education Editor
Friday, 10 June 2011

Three more exam paper blunders are being investigated by the regulator, it emerged yesterday. Ofqual said it was now trying to establish how at least six mistakes in total appeared in questions on this year's papers – most of them at AS-level.

The latest to emerge concerned a geography paper set by the Assessment and Qualifications Alliance (AQA), taken by 88,000 teenagers.

etc...

http://www.independent.co.uk/news/education/education-news/three-new-blunders-discovered-in-school-exam-papers-fiasco-2295507.html

Posted: 14-06-2011 08:20 Post subject:

Universities 'dumbing down on maths' to fill places
By Hannah Richardson, BBC News education reporter

Universities are having to dumb down the maths requirements on some of their courses in order to fill places, a report says.
It maintains that nearly two-thirds of the students accepted on courses needing post-GCSE maths do not have those skills.
The Advisory Committee on Mathematics Education report argues that this causes problems for students.

Ministers want students who fail GCSE maths to take it up to the age of 18.
The Acme report says 180,000 UK students entering university every year will encounter a significant amount of maths.
A further 150,000 students in social sciences face some maths.
But only about 125,000 students continue studying maths beyond GCSE level.

The report said that for more selective universities, such as those in the Russell Group who tend to ask for higher A-level grades, mathematics requirements can be a useful filtering tool.
For less prestigious universities, "the inclusion of mathematical requirements can reduce the number of applicants to unsustainably low levels", it says.

The report quotes a tutor on a foundations-of-computing course, which includes mathematics that students need to complete their computing degrees.
The tutor says 70% of pupils do not have maths beyond GCSE and some of these are not the higher level of the qualification.
"The tutor of this course estimates that if a stronger mathematics background could be assumed for the cohort then this module could be shortened," it adds.
This would make room for an extra computing topic, the tutor said.

And the report refers to two chemistry tutors who said although their course did not require A-level mathematics it should do so.
It added: "There are clearly significant mathematical demands within chemistry degrees, and chemistry departments address this by aiming to ensure that their students study mathematics within their course.
"However, if their mathematics is weak, students will struggle."

Dame Julia Higgins, who chairs the Advisory Committee on Mathematics Education, said all students should study maths until the age of 18.
She said: "In the last 30 years, many university subjects have become more mathematical but the number of students with the appropriate level of mathematical skills has not risen far enough to match this.
"In order to do this, additional courses need to be developed for study at the post-16 level."

The report also warned that some schools were discouraged by the system of league tables in England from entering many students for high-level maths courses, either at GCSE or A-level.
It also suggested that some students were being drilled to pass exams rather than understand mathematical concepts.

A Department for Education spokesman said: "All young people must be able to demonstrate their understanding of maths, whether they are going onto further study or into employment.
"A good qualification in maths is demanded by employers.
"That is why we have announced plans for all young people who fail to get a C or better in GCSE maths to study the subject up to 18, until they get a good qualification."

http://www.bbc.co.uk/news/education-13751233

Posted: 16-06-2011 20:35 Post subject:

Higher maths can get very abstract, especially when more than three dimensions are involved. But everyday problems involving just two dimensions (on a map) can still be interesting:

Computer helps solve Bloom's riddle
DAVID MOLLOY

“Good puzzle would be cross Dublin without passing a pub,” according to Leopold Bloom, the main character of James Joyce’s Ulysses.
Just in time for Bloomsday, an Irish computer programmer thinks he’s cracked one of Dublin’s oldest mazes.

Rory McCann plotted a course through the city from the North Circular Road at Blackhorse Avenue to the canal at Baggot Street, which he says avoids every pub, even if it does require a detour through a museum and a park.

In a city famous for its public houses, Mr McCann set himself a few rules: “cross Dublin” required moving from north to south, and from east to west, roughly within the boundaries of the city’s canals.
“It was something I’d heard about, like most Dubliners, and it was an interesting idea,” Mr McCann said.

The UCD graduate used pub locations from OpenStreetMap, an online editable map, and designed a piece of computer code that ignored any routes across the city within 35 metres away from anywhere marked as a “pub”.
The result is a winding route through the city, around the outside (or through) Arbour Hill cemetery and cuts across the National Museum at Collins Barracks. At more than one point a trip around a block or down some narrow lanes is essential.

The Irish Times tested the route yesterday and found it accurate, despite a few close shaves.
The route takes the liberty of passing a few hotels. On the quays, just after passing James Joyce House on Usher’s Island, the route passes the now-closed Noel Leonard’s pub, and on Harcourt Street, Club Conradh na Gaeilge, a private club for Irish speakers with a bar in the basement.
Cutting across Iveagh Gardens, a park only open until 6pm, is essential to avoid the busy nightspots on Harcourt Street.

The route will take Dubliners down many streets they may not be familiar with, from the terraced housing on Murtagh Road on the northside to the abandoned Iveagh Market on Lamb’s Alley -an equally pub-less alternative to nearby Back Lane.

Mr McCann’s first version of the route passed two pubs, which readers on his blog quickly pointed out and he corrected them in time for Bloomsday. Some commentators argued that bars inside hotels should be counted, but Mr McCann disagreed.

At the very least, his efforts have offered a possible and inventive solution to an 89-year-old brainteaser.
“It’s something that a lot of people have been wondering about in Dublin, and a lot of people have been cynically pointing out I’ve ruined a great pub conversation,” Mr McCann said.

http://www.irishtimes.com/newspaper/breaking/2011/0616/breaking13.html

Here, distance was no object, but avoiding certain sites was. In a way this is the opposite of the Travelling Salesman problem:

http://en.wikipedia.org/wiki/Travelling_salesman_problem

Posted: 28-06-2011 23:38 Post subject:

Just to show that mathematicians can be as bonkers as anyone else:

'Tau day' marked by opponents of maths constant pi
By Jason Palmer, Science and technology reporter, BBC News

The mathematical constant pi is under threat from a group of detractors who will be marking "Tau Day" on Tuesday.
Tau Day revellers suggest a constant called tau should take its place: twice as large as pi, or about 6.28 - hence the 28 June celebration. Rolling Eyes

Tau proponents say that for many problems in maths, tau makes more sense and makes calculations easier.

Not all fans of maths agree, however, and pi's rich history means it will be a difficult number to unseat.

"I like to describe myself as the world's leading anti-pi propagandist," said Michael Hartl, an educator and former theoretical physicist.
"When I say pi is wrong, it doesn't have any flaws in its definition - it is what you think it is, a ratio of circumference to diameter. But circles are not about diameters, they're about radii; circles are the set of all the points a given distance - a radius - from the centre," Dr Hartl explained to BBC News.

By defining pi in terms of diameter, he said, "what you're really doing is defining it as the ratio of the circumference to twice the radius, and that factor of two haunts you throughout mathematics."

The discrepancy is most noticeable when circles are defined not as a number of degrees, but as what are known as radians - of which there are two times pi in a full circle. With tau, half a circle is one-half tau.
Dr Hartl reckons people still use degrees as a measure of angle because pi's involvement in radians makes them too unwieldy.

He credits Bob Palais of the University of Utah with first pointing out that "pi is wrong", in a 2001 article in the Mathematical Intelligencer.
But it is Dr Hartl who is responsible for the Tau Manifesto - calling tau the more convenient formulation and instituting Tau Day to celebrate it.

Kevin Houston, a mathematician from the University of Leeds, counts himself as a convert.
"It was one of the weirdest things I'd come across, but it makes sense," he told BBC News.
"It's surprising people haven't changed before. Almost anything you can do in maths with pi you can do with tau anyway, but when it comes to using pi versus tau, tau wins - it's much more natural."

Dr Hartl is passionate about the effort, but even he is surprised by the fervent nature of some tau adherents.
"What's amazing is the 'conversion experience': people find themselves almost violently angry at pi. They feel like they've been lied to their whole lives, so it's amazing how many people express their displeasure with pi in the strongest possible terms - often involving profanity.
"I don't condone any actual violence - that would be really bizarre, wouldn't it?"

http://www.bbc.co.uk/news/science-environment-13906169

Posted: 29-06-2011 01:23 Post subject:

Pi is actually the ratio between the square of the radius and the area of a circle.

I'm starting a movement to get Upsilon recognised as 4/3rds Pi, being the ratio between the cube of the radius to the volume of a sphere.

Or maybe we should use 4 Pi, as that's the ratio of the square of the radius to the surface area of a sphere, which approximates the shape of the surface of the Earth...

Or we could...

Basically, which ratio you use is largely arbitrary. 2 Pi turns up a lot in some calculations, but ultimately Pi is entrenched, and changing the public perception it is going to take a lot of work. Better not to worry about it too much.

Posted: 29-06-2011 21:44 Post subject:

Gove says 'vast majority' should study maths to 18

Education Secretary Michael Gove says he would like to see the "vast majority" of pupils in England studying maths to the age of 18 within a decade.
He said there were "strong arguments" for "making certain subjects compulsory for longer".
Mr Gove said he did not want to prejudge a national curriculum review, but also suggested that pupils study calculus and algebra at younger ages.
He said the UK was lagging behind its global competitors in the subject.

Speaking at the Royal Society, the education secretary said he was concerned about a post-16 "maths gap", where even pupils who did well at GCSE have forgotten what they knew by the time they start university or work.
The "maths gap" that most pupils now experience after the age of 16 means that even those who did well at GCSE have forgotten much of the maths they learnt by the time they start their degree or a job.

He said not enough students had sufficient maths to be able to study subjects such as physics, engineering and social sciences at university.
"I think we should set a new goal for the education system so that within a decade the vast majority of pupils are studying maths right through to the age of 18," he said.
"Of course, I am not prejudging the [curriculum] review. But there are strong arguments for introducing concepts earlier, for covering some topics more thoroughly, and for making certain subjects compulsory for longer."

Mr Gove said it was possible that England could follow East Asian countries in bringing in "much greater focus on fundamental number concepts, fractions and the building blocks of algebra in primary school".
He said he thought it "genuinely bizarre" that so many children left school "essentially unaware of the development of calculus".

Mr Gove said the review of the national curriculum that the government had commissioned would "set out the essential knowledge that children need to advance in core subjects", but not be "an attempt to prescribe every moment of the school day".

He said it would not spell out how issues and controversies, such as embryo experimentation and energy conservation, should be tackled.
Filling the national curriculum with topical subjects "only encourages a constant tinkering and rewriting which we should stop," Mr Gove said.

According to the Guardian newspaper, the head of the curriculum review panel, Tim Oates, said earlier this month that climate change should not be included in the national curriculum.

Maths is currently compulsory between the ages of 5 and 16.

Mr Gove has already said that teenagers who fail to achieve C grades in English and maths GCSEs by the age of 16 should continue to study the subjects.

Dame Athene Donald, Chair of the Royal Society Education Committee, welcomed Mr Gove's comments.
"The Royal Society's most recent State of the Nation report identified a significant shortage in the numbers of students studying mathematics past GCSE, given the needs of higher education and the economy, and advocates that all students should study some form of mathematics to 18," she said.

http://www.bbc.co.uk/news/education-13958422

Posted: 27-07-2011 22:21 Post subject:

One of my favourite topics:

Nature's hidden prime number code

Prime numbers are found hidden in nature, but humans have made spectacular use of them, writes mathematician Marcus du Sautoy.
Ever since humans evolved on this planet we have been trying to make sense of the world around us.
We have attempted to explain why the world looks and behaves the way it does, to predict what the future holds. And in our search for answers we have uncovered a code that makes sense of the huge complexity that confronts us - mathematics.

By translating nature into the code of numbers we have revealed hidden structures and patterns that control our environment.
But not only that. By tapping into nature's code we have been able to change our surroundings, have built extraordinary cities, and developed amazing technology that has resulted in the modern world.

Buzzing quietly beneath the planet we inhabit is an unseen world of numbers, patterns and geometry. Mathematics is the code that makes sense of our universe.

In the forests of Tennessee this summer, part of this code literally bursts from the ground. Nashville is usually home to the sound of blue grass and honky tonk.
But every 13 years, the banjos and basses get drowned out for six weeks by the chorus of an insect that has fascinated me ever since I became a mathematician. Only found in the eastern areas of North America, this cicadas survival depends on exploiting the strange properties of some of the most fundamental numbers in mathematics - the primes, numbers that are only divisible by themselves and one.

The cicadas appear periodically but only emerge after a prime number of years. In the case of the brood appearing around Nashville this year, 13 years. The forests have been quiet for 12 years since the last invasion of these mathematical bugs in 1998 and the locals won't be disturbed by them again until 2024.

This choice of a 13-year cycle doesn't seem too arbitrary. There are another two broods across north America that also have this 13-year life cycle, appearing in different regions and different years. In addition there are another 12 broods that appear every 17 years.

You could just dismiss these numbers as random. But it's very curious that there are no cicadas with 12, 14, 15, 16 or 18-year life cycles. However look at these cicadas through the mathematician's eyes and a pattern begins to emerge.

Because 13 and 17 are both indivisible this gives the cicadas an evolutionary advantage as primes are helpful in avoiding other animals with periodic behaviour. Suppose for example that a predator appears every six years in the forest. Then a cicada with an eight or nine-year life cycle will coincide with the predator much more often than a cicada with a seven-year prime life cycle.

These insects are tapping into the code of mathematics for their survival. The cicadas unwittingly discovered the primes using evolutionary tactics but humans have understood that these numbers not just the key to survival but are the very building blocks of the code of mathematics.

Every number is built by multiplying primes together and from numbers you get mathematics and from mathematics you get the whole of science.

But humans haven't been content simply with observing the importance of these numbers to nature. By understanding the fundamental character of these numbers and exploring their properties humans have literally put them at the heart of the codes that currently protect the world's cyber-secrets.

The cryptography that keeps our credit cards secure when we shop online exploits the same numbers that protect the cicadas in North America - the primes.
Every time you send your credit card number to a website your are depending on primes to keep your details secret. To encode your credit card number your computer receives a public number N from the website, which it uses to perform a calculation with your credit card number.

This scrambles your details so that the encoded message can be sent across the internet. But to decode the message the website uses the primes which divide N to undo the calculation. Although N is public, the primes which divide N are the secret keys which unlock the secret.

The reason this is so secure is that although it is easy to multiply two prime numbers together it is almost impossible to pull them apart. For example no one has been able to find the two primes which divide the following 617-digit number:

25,195,908,475,657,893,494,027,183,240,048,398,571,429,282,126,204,

032,027,777,137,836,043,662,020,707,595,556,264,018,525,880,784,406,

918,290,641,249,515,082,189,298,559,149,176,184,502,808,489,120,072,

844,992,687,392,807,287,776,735,971,418,347,270,261,896,375,014,971,

824,691,165,077,613,379,859,095,700,097,330,459,748,808,428,401,797,

429,100,642,458,691,817,195,118,746,121,515,172,654,632,282,216,869,

987,549,182,422,433,637,259,085,141,865,462,043,576,798,423,387,184,

774,447,920,739,934,236,584,823,824,281,198,163,815,010,674,810,451,

660,377,306,056,201,619,676,256,133,844,143,603,833,904,414,952,634,

432,190,114,657,544,454,178,424,020,924,616,515,723,350,778,707,749,

817,125,772,467,962,926,386,356,373,289,912,154,831,438,167,899,885,

040,445,364,023,527,381,951,378,636,564,391,212,010,397,122,822,120,

720,357

The primes are the atoms of the arithmetic. The hydrogen and oxygen of the world of numbers.

But despite their fundamental character they also represent one of the greatest enigmas in mathematics. Because as you count through the universe of numbers it is almost impossible to spot a pattern that will help you to predict where the next prime will be found.

We know primes go on for ever but finding a pattern in the primes is one of the biggest mysteries in mathematics. A million-dollar prize has been offered to anyone who can reveal the secret of these numbers.

Despite having cracked so much of nature's code the primes are as much an enigma today as when the cicadas in the forests of Tennessee first tapped into them for their evolutionary survival.

http://www.bbc.co.uk/news/magazine-14305667

Posted: 28-07-2011 10:37 Post subject:

The prog can be viewed here:
http://www.bbc.co.uk/programmes/b012xppj

And here's a review:

The Code, BBC Two, review
Marcus du Sautoy's new series set out to prove that numbers are the "truth" of the universe.
By Chris Harvey
10:05PM BST 27 Jul 2011

For many years now, I’ve had a recurring dream. It’s the day before my Maths A-Level exam and I haven’t been to school for a whole year. I’m sitting at a desk with a towering stack of textbooks, looking at page after page of formulae, as panic rises. The imagery is fairly consistent.

After seeing the first episode of mathematician Marcus du Sautoy’s The Code on BBC Two, however, I’m confidently expecting this to change. In future maths dreams, I will be sitting forlornly on Brighton beach holding up first a big fish, then a little fish, or standing beneath a medieval siege engine in the dark as flaming projectiles are fired over my head. If I have to draw a graph, I’ll get out a sparkler and sketch it D’Artagnan-style with the fiery light trail.

Because The Code, in which du Sautoy set out to reveal the hidden ways in which our universe is governed by numbers, shapes and patterns, threw its all into changing how we see maths, employing all of the stylistic touches that have already proved successful in the television subgenre that should perhaps be described as Programmes About Physics with Brian Cox.

There was the constant gadding about in exotic locations (classrooms, boo); an eclectic soundtrack (eerie synths, cool) and some fabulously arty photography. One shot was of complete darkness except for a small fuzzy oval that turned out to be du Sautoy’s face. There was another sequence in which the professor travelled by train as mathematical conclusions echoed around him. And there was a trip to Chartres cathedral that might have been lifted straight from The Da Vinci Code. Cool

At 11 minutes in, though, I spotted a blackboard. There were numbers chalked on it. I nearly turned over right then until du Sautoy began carefully placing small, coloured silk cicadas on it. The colours represented different species of the winged insect, and du Sautoy was explaining how their breeding cycles appeared to be governed by prime numbers. Reassured that this wasn’t going to be some sort of “maths lesson”, I decided to carry on.

This gave du Sautoy a chance to develop his theory. If I’m not mistaken, it was that numbers had been used by God to construct the universe. Or, at least, that pretty much everything we perceive as reality could be seen as part of an elaborate mathematical system he called The Code (du Sautoy is an atheist).

The du Sautoy Code, however, hinted at eternal significance. Chartres cathedral was physical proof of how “12th-century scholars had stumbled across elements of The Code”. The mysterious number pi was “the essence of circleness distilled into The Code”. The “truth” of the universe was, perhaps, The Code.

The Code was everywhere. We knew this because du Sautoy kept repeating it. The actual nature of it – equations and stuff – was mostly conspicuous by its absence. And what there was, was fairly basic. There was probably too much assume-the-viewers-know-nothing for the maths itself to be consistently interesting. But many of the segments were fascinating. The way numbers govern what sounds uncomfortable to our ears, for instance. The Code was engaging, sometimes wondrous and at all times visually stimulating. It clearly knew how to make use of a formula.

http://www.telegraph.co.uk/culture/tvandradio/8666486/The-Code-BBC-Two-review.html

Posted: 17-09-2011 12:26 Post subject:

This is interesting:

The special trick that helps identify dodgy stats
Using Benford's law, forensic statisticians can spot suspicious patterns in the raw numbers, and estimate the chances figures have been tampered with
Ben Goldacre guardian.co.uk, Friday 16 September 2011 21.15 BST

This week we might bust an entire nation for handing over dodgy economic statistics. But why would they bother? Countries have an interest in distorting their accounts, just like companies and individuals. If you're a eurozone member such as Greece, for example, you have to comply with various economic criteria, and there's the risk of sanctions if you miss them.

Government figures are subjected to various audits already, of course, but alongside checking that things marry up with one another, forensic statisticians also have ways of spotting suspicious patterns in the raw numbers, and thus estimating the chances that figures from a set of accounts have been tampered with. One of the cleverest tools is something called Benford's law.

Imagine you have data on, say, the population of every world nation. Now, take only the "leading digit" from each number: the first number in the number, if you like. For the UK population, which was 61,838,154 in 2009, that leading digit would be "six". Andorra's was 85,168, so that's "eight". And so on.

If you take all those leading digits, from all the countries, then overall, you might naively expect to see the same number of ones, fours, nines, and so on. But in fact, for naturally occurring data, you get more ones than twos, more twos than threes, and so on, all the way down to nine. This is Benford's law: the distribution of leading digits follows a logarithmic distribution, so you get a "one" most commonly, appearing as first digit around 30% of the time, and a nine as first digit only 5% of the time.

Next time you're waiting for a bus, you can think about why this happens (bear in mind what leading digits do when quantities repeatedly double, perhaps) but reality agrees with this theory pretty neatly, and if you go to the website testingbenfordslaw.com you'll see the proportions of each leading digit from lots of real-world datasets, graphed alongside what Benford's law predicts they should be, with data from Twitter users' follower counts to the number of books in different libraries across the US.

It doesn't work perfectly: it only works when you're examining groups of numbers that span several orders of magnitude, for example. So, for age, in years, of the graduate working population, which goes from around 20 to 70, it wouldn't be much good, but for personal savings, from nothing to millions, it should be fine. And of course, Benford's law works in other counting systems, so if three-fingered sloths ever develop numeracy, and count in base-6, or maybe base-12, the law would still hold.

This property of naturally occuring data has been used to check for dubious behaviour in figures for four decades now: it was first used on socioeconomic data submitted to support planning applications, and then on company accounts: it's even admissible in US courts. But in 2009, an economist from Bundesbank suggested using Benford's law on countries' economic data, and last month the results were published (hat-tip to Tim Harford for the paper).

Researchers took macroeconomic data on all 27 EU nations, looking specifically at the accounting data countries have to hand over for monitoring, which is all posted for free at the online repository Eurostat: things such as government deficit, debt, revenue, expenditure, etc. Then they took the first digits from all the numbers, and checked for deviations from what you would predict, using Benford's law.

The results were fun. Greece – whose economy has tanked – showed the largest and most suspicious deviation from Benford's law of any country in the euro.

This isn't a massive surprise: the EU has run several investigations into Greece's numbers already, and the ones from 2005 to 2008 were repeatedly revised upwards after the fact. But it's neat, and if you wanted to wile away a very nerdy afternoon, I reckon you could even download the data, for free from Eurostat, and repeat the analysis for yourself. Joy!

http://www.guardian.co.uk/commentisfree/2011/sep/16/bad-science-dodgy-stats

More here:
http://en.wikipedia.org/wiki/Benford%27s_law

Posted: 03-10-2011 13:46 Post subject:

A formula for justice
Bayes' theorem is a mathematical equation used in court cases to analyse statistical evidence. But a judge has ruled it can no longer be used. Will it result in more miscarriages of justice?
Angela Saini guardian.co.uk, Sunday 2 October 2011 21.30 BST

It's not often that the quiet world of mathematics is rocked by a murder case. But last summer saw a trial that sent academics into a tailspin, and has since swollen into a fevered clash between science and the law.

At its heart, this is a story about chance. And it begins with a convicted killer, "T", who took his case to the court of appeal in 2010. Among the evidence against him was a shoeprint from a pair of Nike trainers, which seemed to match a pair found at his home. While appeals often unmask shaky evidence, this was different. This time, a mathematical formula was thrown out of court. The footwear expert made what the judge believed were poor calculations about the likelihood of the match, compounded by a bad explanation of how he reached his opinion. The conviction was quashed.

But more importantly, as far as mathematicians are concerned, the judge also ruled against using similar statistical analysis in the courts in future. It's not the first time that judges have shown hostility to using formulae. But the real worry, say forensic experts, is that the ruling could lead to miscarriages of justice.

"The impact will be quite shattering," says Professor Norman Fenton, a mathematician at Queen Mary, University of London. In the last four years he has been an expert witness in six cases, including the 2007 trial of Levi Bellfield for the murders of Marsha McDonnell and Amelie Delagrange. He claims that the decision in the shoeprint case threatens to damage trials now coming to court because experts like him can no longer use the maths they need.

Specifically, he means a statistical tool called Bayes' theorem. Invented by an 18th-century English mathematician, Thomas Bayes, this calculates the odds of one event happening given the odds of other related events. Some mathematicians refer to it simply as logical thinking, because Bayesian reasoning is something we do naturally. If a husband tells his wife he didn't eat the leftover cake in the fridge, but she spots chocolate on his face, her estimate of his guilt goes up. But when lots of factors are involved, a Bayesian calculation is a more precise way for forensic scientists to measure the shift in guilt or innocence.

In the shoeprint murder case, for example, it meant figuring out the chance that the print at the crime scene came from the same pair of Nike trainers as those found at the suspect's house, given how common those kinds of shoes are, the size of the shoe, how the sole had been worn down and any damage to it. Between 1996 and 2006, for example, Nike distributed 786,000 pairs of trainers. This might suggest a match doesn't mean very much. But if you take into account that there are 1,200 different sole patterns of Nike trainers and around 42 million pairs of sports shoes sold every year, a matching pair becomes more significant.

The data needed to run these kinds of calculations, though, isn't always available. And this is where the expert in this case came under fire. The judge complained that he couldn't say exactly how many of one particular type of Nike trainer there are in the country. National sales figures for sports shoes are just rough estimates.

And so he decided that Bayes' theorem shouldn't again be used unless the underlying statistics are "firm". The decision could affect drug traces and fibre-matching from clothes, as well as footwear evidence, although not DNA.

"We hope the court of appeal will reconsider this ruling," says Colin Aitken, professor of forensic statistics at the University of Edinburgh, and the chairman of the Royal Statistical Society's working group on statistics and the law. It's usual, he explains, for forensic experts to use Bayes' theorem even when data is limited, by making assumptions and then drawing up reasonable estimates of what the numbers might be. Being unable to do this, he says, could risk miscarriages of justice.

"From being quite precise and being able to quantify your uncertainty, you've got to give a completely bland statement as an expert, which says 'maybe' or 'maybe not'. No numbers," explains Fenton.

"It's potentially very damaging," agrees University College London psychologist, Dr David Lagnado. Research has shown that people frequently make mistakes when crunching probabilities in their heads. "We like a good story to explain the evidence and this makes us use statistics inappropriately," he says. When Sally Clark was convicted in 1999 of smothering her two children, jurors and judges bought into the claim that the odds of siblings dying by cot death was too unlikely for her to be innocent. In fact, it was statistically more rare for a mother to kill both her children. Clark was finally freed in 2003.

Lawyers call this type of mistake the prosecutor's fallacy, when people confuse the odds associated with a piece of evidence with the odds of guilt. Recognising this is also what eventually quashed the 1991 conviction for rape of Andrew Deen in Manchester. The courts realised at appeal that a one-in-three-million chance of a random DNA match for a semen stain from the crime scene did not mean there was only a one-in-three-million chance that anyone other than Deen could have been a match – those odds actually depend on the pool of potential suspects. In a population of 20 million adult men, for example, there could be as many as six other matches.

Now, Fenton and his colleague Amber Marks, a barrister and lecturer in evidence at Queen Mary, University of London, have begun assembling a group of statisticians, forensic scientists and lawyers to research a solution to bad statistics. "We want to do what people failed to do in the past, which is really get the legal profession and statisticians and probability guys understanding each other's language," says Fenton.

Their first job is to find out how often trials depend on Bayesian calculations, and the impact that the shoeprint-murder ruling might have on future trials. "This could affect thousands of cases," says Marks.

They have 37 members on their list so far, including John Wagstaff, legal adviser to the Criminal Cases Review Commission, and David Spiegelhalter, the Winton professor of the public understanding of risk at the University of Cambridge. Added to these are senior statisticians and legal scholars from the Netherlands, US and New Zealand.

Fenton believes that the potential for mathematics to improve the justice system is huge. "You could argue that virtually every case with circumstantial evidence is ripe for being improved by Bayesian arguments," he says.

But the real dilemma is finding a way to help people make sense of the calculations. The Royal Statistical Society already offers guidance for forensic scientists, to stop them making mistakes. Lagnado says that flowcharts in the style of family trees also help jurors visualise changing odds more clearly. But neither approach has been entirely successful. And until this complex bit of maths can be simply explained, chances are judges will keep rejecting it.

http://www.guardian.co.uk/law/2011/oct/02/formula-justice-bayes-theorem-miscarriage

http://en.wikipedia.org/wiki/Bayes%27_theorem

Posted: 04-10-2011 08:31 Post subject:

Statistics and justice came together in this case:

Colin Norris case: Murder convictions 'unsafe'

A Scottish nurse may have been wrongly convicted of the murders of four women, a BBC investigation suggests.
Colin Norris was found guilty of poisoning five patients with insulin, one of whom survived, at two hospitals in Leeds in 2002.
Norris was jailed for life in 2008 but new evidence is to be referred to the Criminal Cases Review Commission.
New studies suggest naturally-occurring hypoglycaemia is much more common than the jury were led to believe.

....

The prosecution argued that naturally-occurring hypoglycemia was so rare that a cluster of four or five cases must mean foul play.

....

But Prof Vincent Marks, one of the world's foremost experts on insulin poisoning, has carried out a forensic international review of all the new medical research.
He believes there was insufficient evidence for insulin in four of the cases.
Prof Marks said: "I was surprised at how very common it is in this particular group of elderly, sick people.
"In one very detailed survey, of thousands of patients, it was up to 10%. It's not that rare after all."

....

The BBC has uncovered evidence of other similar cases of hypoglycaemia which occurred in the hospital where Norris worked but while he was off duty.

His lawyer, Jeremy Moore, believes there were serious flaws in the investigation and the convictions need to be quashed.
He said: "It seems that they trawled through hospital records looking for evidence of patients that might have died suspiciously but it seems they only cherry-picked those cases when Colin was on duty and ignored any others that might have occurred in the hospital."

....

Hospital Serial Killer: A Jury In The Dark will be broadcast on BBC1 Scotland on Tuesday 4 October, at 22:35 and on the BBC News channel at 00:30. It will also be available on the BBC iPlayer

http://www.bbc.co.uk/news/uk-scotland-15127072

Posted: 04-10-2011 08:35 Post subject:

Rynner, you don't need to tamper with the raw data to produce dodgy stats. That's a very unprofessional way to go about it Smile

Posted: 02-12-2011 09:57 Post subject:

Subtracting calculators adds to children's maths abilities, says minister
National curriculum review to look at use of calculators in primary schools amid concerns students are too reliant on them
Jeevan Vasagar and Jessica Shepherd guardian.co.uk, Thursday 1 December 2011 15.13 GMT

Calculators may be restricted in primary schools until children have mastered basic arithmetic including knowing times tables by heart, a minister has said.

The use of calculators will be looked at as part of a national curriculum review, after the schools minister, Nick Gibb, expressed concern that children's mental and written arithmetic was suffering because of reliance on the devices.
Gibb said: "Children can become too dependent on calculators if they use them at too young an age. They shouldn't be reaching for a gadget every time they need to do a simple sum.
"They need to master addition, subtraction, times tables and division, using quick, reliable written methods. This rigour provides the groundwork for the more difficult maths they will come across later in their education."

In a parliamentary debate on Wednesday led by the Conservative MP Elizabeth Truss, she warned of the dangers of producing "a satnav generation of students overly reliant on technology".

The review is under way, and ministers are due to announce new programmes of study for English, maths, science and PE early next year. These will be introduced in schools next September.

Students are encouraged to use calculators in the national curriculum, which has a section on "calculator methods" for children aged between eight and 11.
The use of calculators is common in Year 5, when children are aged between nine and 11. A 2007 study found that only 2% of Year 5 pupils in England were banned from using calculators, compared with the international average of 54%.

A recent Ofsted report on maths, which looked at 20 successful primary schools, found most only introduced calculators in the later stages of primary schooling, and then only for checking answers for calculations carried out without them.

Gibb compared England to three high-performing regions and countries around the world: Massachusetts, Singapore and Hong Kong.
In all three, curriculums state that calculators should not be used as a replacement for basic understanding and skills, the government said. Fourth and sixth grade state assessments in Massachusetts – equivalent to Years 5 and 7 in England – do not permit the use of a calculator. Massachusetts is the highest performing US state for maths.

In Singapore, which is ranked second for maths in the Organisation for Economic Co-operation and Development (OECD) league tables, almost no 10-year-olds use calculators in the classroom. The UK is currently rated 28th for maths in the same league table. One in five 11-year-olds in England failed to reach the expected level in maths this year, according to provisional figures.

Gibb added: "You can't expect children to cope with complicated quadratic equations if they don't know their times tables by heart.
"Without a solid grounding in arithmetic and early maths in primary school, children go on to struggle with basic maths skills throughout their school careers. It also means they leave school without the knowledge they need to complete everyday tasks in their adult lives.
"The use of calculators in primary schools must be appropriate."

http://www.guardian.co.uk/education/2011/dec/01/subtracting-calculators-adds-children-maths