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The Importance Of Maths

rynner2 said:
The idea is that it is more important for a child to understand that 7.8 x 103 is roughly 800 than for him or her to be able to work out the answer precisely with paper and pencil (though we still need to know our times tables).
I seem to recall being taught the art of approximation at school, and the point above is absolutely correct (although, when I was a 9-year-old maths swot, I would have disapproved of such laziness!) - even when using a calculator, it's helpful to know in advance the kind of answer you'll be getting, so as to minimise the chances of typing errors ruining your maths homewok (or, possibly, your Mars landing...).

In the example above, knowing that the answer is roughly 800 will mean you'll know that something is amiss if your calculator tells you it's actually 8034...

So, in summary, while I have no idea what chunking, number bonds or the grid method are (even though I have a degree in the ruddy subject), I approve of these zany modern concepts, if they give kids some sort of grounding in the basics of maths.
 
Went to the library today, and on the New Books shelf was a book for parents, explaining the modern teaching of maths!

(I don't think it was the one mentioned at the end of the article:
Maths for Mums and Dads by Rob Eastaway and Mike Askew, Square Peg, £9.99 - I'll have to double-check next time.)
 
Magic numbers: A meeting of mathemagical tricksters
13:44 24 May 2010
by Alex Bellos
Magazine issue 2762.
(Martin Gardner, who inspired the gathering we report on in this article, died on Saturday. New Scientist consultant Jeff Hecht has written an assessment of his career on our CultureLab blog.)

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

The event was the Gathering for Gardner earlier this year, a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question.

"The first thing you think is 'What has Tuesday got to do with it?'" said Foshee, deadpan. "Well, it has everything to do with it." And then he stepped down from the stage.

The gathering is the world's premier celebration of recreational mathematics. Foshee's "boy born on a Tuesday" problem is a gem of the genre: easy to state, understandable to the layperson, yet with a completely counter-intuitive answer that can leave you with a smile on your face for days. If you have two children, and one is a boy, then the probability of having two boys is significantly different if you supply the extra information that the boy was born on a Tuesday. Don't believe me? We'll get to the answer later. 8)

As a melting pot of outside-the-box abstract thinking, this gathering is one of a kind. Attendees were invited to make the world's first snub dodecahedron out of balloons, shown how to solve the Rubik's cube while blindfolded and given tips on how to place a lemon under a handkerchief without anyone knowing. The 300 guests included magicians, origamists, artists, maze designers, puzzle writers, toy inventors and cognitive psychologists, as well as some of the world's most gifted mathematicians.

Origami letters
Erik Demaine, for example, is a former winner of the MacArthur fellowship, aka the "genius award", and the youngest person in recent years to be made a professor at the Massachusetts Institute of Technology. Now aged 29, he presented to the gathering some typefaces that he had invented as a result of his academic work in computational geometry. The "hinged dissection font" is a font in which each letter can be made from the same linked chain of 128 identical isosceles triangles. (A hinged dissection is a technique in which a large shape is divided up into smaller shapes, linked together with "hinges", and then refolded into another large shape.)

The letters looked distinctive, if a little clunky. Demaine reckoned the typeface had another more serious problem: "It's way too easy to read!" So he revealed a brilliantly incomprehensible font based on his work in origami: the "origami maze font", for which each letter of the alphabet is the origami fold-pattern that, once folded, would make that letter protrude from the page.

When he started to write in this font, and the screen at the front of the hall filled with an impenetrable grid of red and blue lines – red lines are the "mountain" folds and the blue ones the "valley" folds – the audience clapped and guffawed.

Recreational mathematics may be the "mathematics of fun" but it often inspires serious science. Hinged dissections were invented by the British puzzle creator Henry Dudeney a century ago. Though Demaine became interested in them and in origami for purely playful purposes, they have resulted in some of his most important academic research. Hinged dissections form the basis of his designs for reconfigurable robots, in which blocks hinged together in a chain can be made to fold into any three-dimensional shape.

The pop-up lettering technique behind his origami font could, he says, be used with paper that can pucker up to form tactile shapes: to make, for example, a map you can read in the dark. "There is no way to predict from the recreational side to the product side," Demaine says.

Games master
The four-day Gathering for Gardner, or G4G, owes its name to the journalist Martin Gardner, who died as this article was going to press, aged 95. Between 1957 and 1981 Gardner wrote the monthly "Mathematical games" column in Scientific American, which inspired a cult following. A decade after he put down his pen, Atlanta businessman and puzzle collector Tom Rodgers decided to pay tribute to Gardner by organising a gathering in his honour.

etc...

http://www.newscientist.com/article/dn1 ... ?full=true
 
I heard that discussed on R4 this week. Pfft, the chances are STILL 50:50, Tuesday or not. ;)
 
escargot1 said:
I heard that discussed on R4 this week. Pfft, the chances are STILL 50:50, Tuesday or not. ;)
Not according to Gary Foshee, it's not!

A surprising simple yet subtle puzzle. :D
 
Nah, the Tuesday business is a red herring. Your bullshit detector is on the blink again. :lol:
 
Maths crops up in unexpected places:

Plato's stave: academic cracks philosopher's musical code
Historian claims Plato's manuscripts are mathematically ordered according to 12-note scale
Julian Baggini guardian.co.uk, Tuesday 29 June 2010 14.21 BST

It may sound like the plot of a Dan Brown novel, but an academic at the University of Manchester claims to have cracked a mathematical and musical code in the works of Plato.

Jay Kennedy, a historian and philosopher of science, described his findings as "like opening a tomb and discovering new works by Plato."

Plato is revealed to be a Pythagorean who understood the basic structure of the universe to be mathematical, anticipating the scientific revolution of Galileo and Newton by 2,000 years.

Kennedy's breakthrough, published in the journal Apeiron this week, is based on stichometry: the measure of ancient texts by standard line lengths. Kennedy used a computer to restore the most accurate contemporary versions of Plato's manuscripts to their original form, which would consist of lines of 35 characters, with no spaces or punctuation. What he found was that within a margin of error of just one or two percent, many of Plato's dialogues had line lengths based on round multiples of twelve hundred.

The Apology has 1,200 lines; the Protagoras, Cratylus, Philebus and Symposium each have 2,400 lines; the Gorgias 3,600; the Republic 12,200; and the Laws 14,400.

Kennedy argues that this is no accident. "We know that scribes were paid by the number of lines, library catalogues had the total number of lines, so everyone was counting lines," he said. He believes that Plato was organising his texts according to a 12-note musical scale, attributed to Pythagoras, which he certainly knew about.

"My claim," says Kennedy, "is that Plato used that technology of line counting to keep track of where he was in his text and to embed symbolic passages at regular intervals." Knowing how he did so "unlocks the gate to the labyrinth of symbolic messages in Plato".

Believing that this pattern corresponds to the 12-note musical scale widely used by Pythagoreans, Kennedy divided the texts into equal 12ths and found that "significant concepts and narrative turns" within the dialogues are generally located at their junctures. Positive concepts are lodged at the harmonious third, fourth, sixth, eight and ninth "notes", which were considered to be most harmonious with the 12th; while negative concepts are found at the more dissonant fifth, seventh, 10th and 11th.

Kennedy has also found that the enigmatic "divided line" simile in the Republic, in which Plato describes a line divided by an unstated ratio, falls 61.7% of the way through the dialogue. It has been thought that the line refers to the golden mean, which expressed as a percentage is 61.8%.


Copies of the paper have been circulating among senior scholars, who believe Kennedy's argument should be taken seriously.

Professor Andrew Barker, a leading authority on ancient Greek music, said that "the results he's come up with look too neat to be accidental" and that if scholars confirm them, "he will have shown something quite startling about Plato's methods of composition".

Kennedy believes his findings restore what was the standard, mainstream view which held for 2,000 years "from the first generation of Plato's followers, up through the renaissance". This held that "he wrote symbolically and that if you worked hard and became wise you could understand the symbols and penetrate his text to his underlying philosophy." Only in the last few hundred years has an emphasis on the literal meanings of texts led to a neglect of their figurative meanings.

It also explains why it is that Aristotle, Plato's pupil, emphatically claimed that Plato was a follower of Pythagoras, to the bafflement of most contemporary scholars.

The secrecy was because Plato's was "a dangerous idea", claims Kennedy. "It meant that mathematical law governed the universe and not Zeus." Given that Plato's teacher, Socrates, had been executed for sowing impiety among the youth he would have been "very cautious abut revealing doctrines that threaten the gods of Olympus".

For once, Alfred North Whitehead's description of western philosophy as "a series of footnotes to Plato" looks like being an understatement. "We've got some 2,000 pages of Plato," says Kennedy. "We now know that underneath all of those genuine dialogues there's another layer of symbolic meaning. This is the beginning of a big debate. It will take years to make sense of all this."

http://www.guardian.co.uk/world/2010/ju ... sical-code
 
The Story of Maths - 1. The Language of the Universe

Series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.

After showing how fundamental mathematics is to our lives, du Sautoy explores the mathematics of ancient Egypt, Mesopotamia and Greece.

In Egypt, he uncovers use of a decimal system based on ten fingers of the hand, while in former Mesopotamia he discovers that the way we tell the time today is based on the Babylonian Base 60 number system.

In Greece, he looks at the contributions of some of the giants of mathematics including Plato, Euclid, Archimedes and Pythagoras, who is credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today.

http://www.bbc.co.uk/iplayer/episode/b0 ... _Universe/
 
Thank you for posting that. If I'm not too tired, I may watch it after work.
 
You won't find me posting about maths often - it is usually beyond me, but I found this fascinating. It's an extract of a much longer article:

Ye cannae change the laws of physics
Or can you?

RICHARD FEYNMAN, Nobel laureate and physicist extraordinaire, called it a “magic number” and its value “one of the greatest damn mysteries of physics”. The number he was referring to, which goes by the symbol alpha and the rather more long-winded name of the fine-structure constant, is magic indeed. If it were a mere 4% bigger or smaller than it is, stars would not be able to sustain the nuclear reactions that synthesise carbon and oxygen atoms. One consequence would be that squishy, carbon-based life would not exist.

Why alpha takes on the precise value it does, so delicately fine-tuned for life, is a deep scientific mystery. A new piece of astrophysical research may, however, have uncovered a crucial piece of the puzzle. In a paper just submitted to Physical Review Letters, a team led by John Webb and Julian King from the University of New South Wales in Australia presents evidence that the fine-structure constant may not actually be constant after all. Rather, it seems to vary from place to place within the universe. If their results hold up to scrutiny they will have profound implications—for they suggest that the universe stretches far beyond what telescopes can observe, and that the laws of physics vary within it. Instead of the whole universe being fine-tuned for life, then, humanity finds itself in a corner of space where, Goldilocks-like, the values of the fundamental constants happen to be just right for it.
...
Despite its convoluted origin, alpha has a real meaning. It characterises the strength of the force between electrically charged particles. As such, it governs—among other things—the energy levels of the electrons in an atom. When electrons jump between these energy levels, they absorb and emit light of particular frequencies. These frequencies show up as lines (dark for absorption; bright for emission) in a spectrum. When many different energy levels are involved, as they are in the spectrum of a chemically mixed star, the result is a fine, comb-like structure—hence the constant’s name. If it were to take on a different value, the wavelengths of these lines would change. And that is what Dr Webb and Mr King think they have found.

The light in question comes not from individual stars but from quasars. These are extremely luminous (and distant) galaxies whose energy output is powered by massive black holes at their centres. As light from a quasar travels through space, it passes through clouds of gas that imprint absorption lines onto its spectrum. By measuring the wavelengths of a large collection of these absorption lines and subtracting the effects of the expansion of the universe, the team led by Dr Webb and Mr King was able to measure the value of alpha in places billions of light-years away.

Dr Webb first conducted such a study almost a decade ago, using 76 quasars observed with the Keck telescope in Hawaii. He found that, the farther out he looked, the smaller alpha seemed to be. In astronomy, of course, looking farther away means looking further back in time. The data therefore indicated that alpha was around 0.0006% smaller 9 billion years ago than it is now. That may sound trivial. But any detectable deviation from zero would mean that the laws of physics were different there (and then) from those that pertain in the neighbourhood of the Earth.

What they found shocked them. The further back they looked with the VLT, the larger alpha seemed to be—in seeming contradiction to the result they had obtained with the Keck. They realised, however, that there was a crucial difference between the two telescopes: because they are in different hemispheres, they are pointing in opposite directions. Alpha, therefore, is not changing with time; it is varying through space. When they analysed the data from both telescopes in this way, they found a great arc across the sky. Along this arc, the value of alpha changes smoothly, being smaller in one direction and larger in the other. The researchers calculate that there is less than a 1% chance such an effect could arise at random. Furthermore, six of the quasars were observed with both telescopes, allowing them to get an additional handle on their errors.

If the fine-structure constant really does vary through space, it may provide a way of studying the elusive “higher dimensions” that many theories of reality predict, but which are beyond the reach of particle accelerators on Earth. In these theories, the constants observed in the three-dimensional world are reflections of what happens in higher dimensions. It is natural in these theories for such constants to change their values as the universe expands and evolves.

http://www.economist.com/node/16941123? ... highlights
 
rynner2 said:
Now I know I've lived too long... :(

‘Mum, Dad, that’s not how we do it at school’
When it comes to maths homework, parents do division and children do ‘chunking’. We should all learn to speak the same language
Rob Eastaway
http://www.timesonline.co.uk/tol/life_a ... 126927.ece
Another Eastaway article here, covering much the same ground, but including a graphic and a video to illustrate Gridding and Chunking:

http://www.bbc.co.uk/news/magazine-11258175

I'm happy to say it's not as alien as I'd feared! And the comments are interesting too. Essential stuff for parents with kids at primary school.
 
rynner2 said:
The Big Question: How close have we come to knowing the precise value of pi?
By Steve Connor, Science Editor
Friday, 8 January 2010

Why are we asking this now?

A French computer programmer has calculated pi to a world record of almost 2,700 billion decimal places. Fabrice Bellard said he did it for the programming challenge rather than any particular interest in this infinitely long number. All the more remarkable is that he did it on a desktop computer costing less than £2,000.
...
http://www.independent.co.uk/news/scien ... 61197.html
Pi record smashed as team finds two-quadrillionth digit
By Jason Palmer, Science and technology reporter, BBC News

A researcher has calculated the 2,000,000,000,000,000th digit of pi - and a few digits either side of it.

Nicholas Sze, of technology firm Yahoo, determined that the digit - when expressed in binary - is 0.

Mr Sze used Yahoo's Hadoop cloud computing technology to more than double the previous record.

The computation took 23 days on 1,000 of Yahoo's computers, racking up the equivalent of more than 500 years of a single computer's efforts.

The heart of the calculation made use of an approach called MapReduce originally developed by Google that divides up big problems into smaller sub-problems, combining the answers to solve otherwise intractable mathematical challenges.

At Yahoo, a cluster of 1,000 computers implemented this algorithm to solve an equation that plucks out specific digits of pi.

The pursuit of longer versions of pi is a long-standing pastime among mathematicians.

But this approach is very different from the full calculation of all of the digits of pi - the record for which was set in January at 2.7 trillion digits.

Instead, each of the Hadoop computers was working on a formula that turns a complicated equation for pi into a small set of mathematical steps, returning just one, specific piece of pi.

"Interestingly, by some algebraic manipulations, (our) formula can compute pi with some bits skipped; in other words, it allows computing specific bits of pi," Mr Sze explained to BBC News.

Fabrice Bellard, who undertook the full calculation announced in January, told BBC News that the single-digit and full pi calculation are vastly different in the degree to which they can be "parallelised" - that is, cut up into manageable pieces among different computers.

He said the current, single-digit record is "more a demonstration of the Hadoop parallelisation framework... it can demonstrate the power of new algorithms which could be useful in other fields".

The record-breaking MapReduce approach, he said, is useful in physics, cryptography and data mining.

Mr Sze added that the calculation was also a good test for the Hadoop hardware and approach.
"This kind of calculation is useful in benchmarking and testing," he said.
"We have used it to compare the [processor] performance among our clusters."

http://www.bbc.co.uk/news/technology-11313194

"the digit - when expressed in binary - is 0"...

I think zero is zero in any number base!
8)
 
rynner2 said:
"the digit - when expressed in binary - is 0"...

I think zero is zero in any number base!
8)

Maybe his calculations are a bit 'off' :D
 
Gah, we have different, just as important qualities. Name one famous chinese writer or artist now, off the top of your head. We need yin and yang in life, what does it matter if they're at the opposite sides of the world.
 
Mathematics, the makeover
Sums, it seems, are hot. Arithmatic rocks. A sexy band of number crunchers is taking maths away from the geeks and frumps
By Susie Mesure
Sunday, 3 October 2010

It has long vied with physics for the dubious honour of having the worst image on the curriculum. But that was before stars such as the former Wonder Years actress Danica McKellar stepped in to give maths a makeover.

The actress turned mathematician, who is best known for playing Winnie Cooper in the hit US television show, has made it her mission to sex up maths with a series of books aimed at convincing girls that the subject isn't just for geeks. As the raunchy cover of her latest how-to guide – Hot X: Algebra Exposed – makes clear, being a maths whiz doesn't make you a frump.

Penguin, which has just released McKellar's first book, Maths Doesn't Suck, in the UK, thinks her approach will revolutionise the way girls look at the subject. It plans to publish the author's two other books, which have both made The New York Times's bestseller list.

McKellar said she wanted to "break stereotypes... [that have] trained girls from a young age to believe that maths is too hard, too boring and just for boys, and that if they are smart, they can't be popular or beautiful," while making maths "more fun to learn". She said teaching maths in a "non-mathsy" context – her books are based on teen magazines and use examples intended to capture girls' interest, from crushes on boys to lipstick – attracted the "most mathsphobic girls and helped them to succeed".

The actress, who has had a paper published in the esteemed Journal of Physics which proved a theorem on magnetism, joins a host of big names who have helped to boost the subject's appeal. They include the actress Natalie Portman, who has guest-edited the teenage maths magazine Scholastic Math as well as gracing its cover. The Oscar-winner Rachel Weisz has also done her bit by starring as Hypatia, the fourth-century Greek mathematician and astronomer, in the 2009 film Agora. Simon Singh, who unpicked Fermat's last theorem to great acclaim in a bestselling book, and Marcus du Sautoy, the populist Oxford University mathematician, are also credited with inspiring more students to study maths.

But Mary Wimbury, director of the UK Mathematics Trust, said girls still needed encouraging by the likes of McKellar because they were "easily put off". She added: "We still need to get over the attitude that women can never do maths. There can be a perception that it's quite a geeky thing to do. Plus all the big names are still male, so it's good to have Danica to challenge the stereotypes."

etc...

http://www.independent.co.uk/news/educa ... 96268.html
 
Large cardinals: maths shaken by the 'unprovable'
A shocking discovery has unsettled the world of numbers, says Richard Elwes.
Published: 7:50AM GMT 09 Nov 2010

In the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities.

To understand them, we must understand what makes mathematics different from other sciences. The difference is proof.

Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction.

But proofs have to start somewhere. For all its sophistication, mathematics is not alchemy: we cannot conjure facts from thin air. Every proof must be based on some underlying assumptions, or axioms.

And there we reach a thorny question. Even today, we do not fully understand the ordinary whole numbers 1,2,3,4,5… or the age-old ways to combine them: addition and multiplication.

Over the centuries, mathematicians have arrived at basic axioms which numbers must obey. Mostly these are simple, such as "a+b=b+a for any two numbers a and b". But when the Austrian logician Kurt Gödel turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His "incompleteness theorems" showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms.

Gödel's theorems showed that maths meant that mathematicians could not hope to prove every true statement: there would always be "unprovable theorems", which cannot be deduced from the usual axioms. Most known examples, it's true, will not change how you add up your shopping bill. For practical purposes, the laws of arithmetic seemed good enough.

However, as revealed in his forthcoming book, Boolean Relation Theory and Concrete Incompleteness, Harvey Friedman has discovered facts about numbers which are far more unsettling. Like Gödel's unprovable statements, they fall through the gaps between axioms. The difference is that these are no longer artificial curiosities. Friedman's theorems are "concrete", meaning they contain genuinely interesting information concerning patterns among the numbers, which must always appear once certain conditions are met. Yet, Friedman has shown, the fact that such patterns always appear does not follow from the usual laws of arithmetic.

These patterns are not yet affecting physicists or engineers, but mathematicians are having to take unprovability seriously. In the past, they just had to show whether an idea is true or false. Now, results such as Friedman's raise the awkward possibility that the standard laws of mathematics may not provide an answer.

There is one easy way to make an unprovable theorem provable: adding more axioms. But which axioms do we need? The new axioms require a hard look at one of the most contentious issues in mathematics: infinity.

For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3… On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever.

This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not.

At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants.

Dr Richard Elwes is the author of 'Maths 1001: Absolutely Everything That Matters in Mathematics' (Quercus Publishing)

http://www.telegraph.co.uk/science/8118 ... vable.html
 
Crikey I had a school days flashback reading that article, always crap at maths, I used to have a Charlie Brown Teacher moment whenever more complex maths was explained (i.e it all sounded like blah blah blah blah). Got about half way down that article and experienced Charlie Brown Teacher.

:?
 
Heckler20 said:
Crikey I had a school days flashback reading that article, always crap at maths, I used to have a Charlie Brown Teacher moment whenever more complex maths was explained (i.e it all sounded like blah blah blah blah). Got about half way down that article and experienced Charlie Brown Teacher.

:?

My maths teachers were crap at teaching maths. ;)

I only really took to maths after I left school.
 
Learn maths to boost the economy, scientist advises
Britons should improve their understanding of mathematics in order to boost the economy and stimulate growth, a scientist has claimed.
By Laura Roberts 8:16AM GMT 18 Nov 2010 Comments

A report by the Organisation for Economic Co-operation (OECD) has found that a country's general competence in the subject is a major component in its economic success.

The level of mathematical knowledge among the general population was found to have a direct relation to Gross Domestic Product.

Prof Brian Butterworth, an Emeritus professor from the Centre of Educational Neuroscience at UCL, said that if the least able section of the population brushed up on their maths they would contribute significantly to the British economy.

He said: "It's not just raising the overall level that helps.
"If you just get the lowest 10 or 11 per cent – the percentage of our population that fails to reach the OECD minimum standard at 15 – to the minimum level this will increase GDP growth by 0.44 per cent per annum.
"It might not sound like much but actually over the years this creates an enormous improvement in GDP for the country."

He told BBC Radio 4 Today programme: "The UK is not very good at maths. We are about average looking at all OECD countries. So, we are significantly worse than Canada and Australia and much worse than China and Japan although we are a bit better than Germany and significantly better than the United States.

"We know from a recent OECD report that maths ability in the population is correlated with GDP growth. So the better at maths the country is the better their GDP growth."

Prof Butterworth said that maths is particularly important in terms of GDP but that science also had an impact.

http://www.telegraph.co.uk/education/ed ... vises.html
 
Diagrams that changed the world

A picture, the old adage goes, is worth 1,000 words. But in science a diagram can describe things that transcend the written word. A single image can convey the simple underlying pattern hidden by words or equations, says Marcus du Sautoy.

Draw the right picture and you can literally transform the way we see the world. But a diagram is more than just a physical representation of what we see with our eyes.

The power of a diagram is to crystallise a new way of seeing the world.

Often it requires throwing away information, focusing on what is essential. Other times it changes a scientific idea into a visual language providing a new map where the mathematics of geometry takes over and helps us to navigate the science at hand.

Copernicus certainly understood the power of a good picture. In his great opus De Revolutionibus Orbium Coelestium published shortly before his death in 1543, Copernicus takes 405 pages of words, numbers and equations to explain his heliocentric theory.

But it is the diagram that he draws at the beginning of the book that captures in a simple image his revolutionary new idea: it is the Sun that is at the centre of the Solar System, not the Earth.

His picture encapsulates some of the essential elements of the best diagrams. The concentric circles are not meant to describe the precise orbits of the planets.

Copernicus knew they weren't circles. The uniform distances between the circles aren't meant to tell you how far the planets are from the sun. Rather this picture conveys the simple yet shocking idea that we aren't at the centre of things. His diagram transformed our view of our place in the universe

But some diagrams do more than just crystallise the essential underlying structure of a complex system. A diagram has the power to create a whole new visual language to navigate a scientific idea.

Newton's optics diagrams for example transform light into geometry.

By representing light as lines Newton is able to use mathematics and geometry to predict the behaviour of light. It was a revolutionary idea. Look at the light that illuminates the world around you. There are no lines. Newton's diagram translates the slippery science of optics into the concrete world of geometry where mathematics becomes the eyes to see what is happening to light.

Sometimes a diagram is the crucial step in making people believe in the impossible.

Mathematicians had been struggling with the idea of the square root of minus one. There seemed to be no number on the number line whose square was negative. Yet experts knew that if such a number existed it would transform their subject.

But where was this number? It was a picture drawn independently by three mathematicians at the beginning of the 19th Century that brought these numbers to life.

They created a two dimensional map of numbers where the numbers we'd known about since the Ancient Greeks ran east-west along the horizontal axis while these new imaginary numbers like the square root of minus one extended vertically in the north-south direction.

Called the Argand diagram after one of its creators, this picture helped mathematicians to believe in these new numbers. Not only that, the diagram was a potent tool in manipulating these new numbers since the geometry of the diagram reflected the underlying algebra of the numbers they depicted.

One of the most powerful uses of diagrams though has been in visualising data. Given that we live in an age that generates huge reams of numerical information, finding ways to make sense of all these numbers is essential.

One of the first to use the visual world to navigate numbers was Florence Nightingale.

Although better known for her contributions to nursing, her greatest achievements were mathematical. She was the first to use the idea of a pie chart to represent data.

Nightingale had discovered that the majority of deaths in the Crimea were due to poor sanitation rather than casualties in battle. She wanted to persuade government of the need for better hygiene in hospitals.

She realised though that just looking at the numbers was unlikely to impress ministers. But once those numbers were translated into a picture - her Diagram of the Causes of Mortality in the Army in the East - the message could not be ignored. A good diagram, Nightingale discovered, is certainly worth 1,000 numbers.

One of the strengths of all these diagrams is that they transcend language. They can be read and understood by people across the globe.

Which is why when we launched our first space craft out of our Solar System in 1972, scientists recognised that a diagram was probably our best bet at communicating to any intelligent life out there in space.

Frank Drake and Carl Sagan created in some sense the ultimate diagram [see link], an engraving that was attached to the Pioneer space probe which would communicate in visual language who we are and where we come from.

It's unlikely that anyone or thing has received our first message to outer space but when they do, it is the clever use of diagrams that has the best chance of saying hello.

Marcus du Sautoy is the Simonyi Professor for the Public Understanding of Science at the University of Oxford. His new series, The Beauty of Diagrams, is on BBC Four at 2030 on Thursdays and also available on iPlayer.
http://www.bbc.co.uk/news/magazine-11798317
 
Why a head for figures is good for your wealth
By Sean O'Grady and Rula Awad
Wednesday, 24 November 2010

It's bad news for creative arty types, good news for accountants. Researchers have discovered that numeracy, rather than literacy or pure intellectual power, is the key to accumulating wealth and ensuring a comfortable retirement.

A team from the Institute for Fiscal Studies and University College London have found that in the years leading up to retirement "those who are more numerate accumulate financial assets at a faster rate than those who are less numerate", and detect a strong correlation between a facility with numbers and economic and financial literacy.

They argue this has become much more important when individuals have to make more choices on investments and retirement planning than in the past. Numeracy pays its biggest dividends in old age – almost 80 per cent of income for the least numerate group comes from the state, compared with approximately 30 per cent for the most numerate group. Similar research in the US suggests that the more numerate households are worth some $22,000 (£14,000) more.

The authors of the study, James Banks, Cormac O'Dea and Zoe Oldfield said: "As the UK has moved towards a system of individual provision for retirement income, the importance of an individual's or household's ability to make the right choices when it comes to providing for their retirement has increased."

One solution for those who are hopeless at maths is to marry someone who has discovered the magic of compound interest or unlocked the mysteries of pound-cost averaging.

"One may not need to be particularly numerate if one is married to someone who is so, and they are taking an active role in the management of household saving and consumption decisions," the report authors found.

Numeracy is the key factor, rather than intelligence or educational achievement. The authors say that literacy, memory function and executive ability are less important.

The researchers suggest a more "paternalistic" approach to making the less numerate save more, given the "strikingly low levels" of financial nous among large sections of the population. In one example, 30 per cent of low financial literacy households said they had a fixed rate mortgage when in reality it was variable rate.

Financial education was due to become a compulsory part of the curriculum in schools from September next year, but the plan was scrapped at the end of the last parliament.

Economic literacy in the UK is low by international standards. Britain lies in 30th place in a survey of 54 nations, slotting between Hungary and Slovakia, and behind the US, Germany and – somewhat surprisingly given its current economic woes – Ireland. Singapore is regarded as the most economically literate place on earth – it is also one of the wealthiest nations per head.

Virginia Wade
The tennis player, who is still the last British person to win a full Wimbledon singles title, gained a BSc in mathematics and physics at the University of Sussex, graduating in 1966.


Art Garfunkel
One half of the pop duo Simon & Garfunkel, he also studied an MSc in mathematics at Columbia University in New York. He also started working on a PhD, but quit early to pursue his music career.

Leon Trotsky
The revolutionary began a maths degree in Odessa in 1897. His studies were cut short by his exile to Siberia.

Michael Jordan
The basketball star started a maths degree at the University of North Carolina but changed to cultural geography during his first year. He went on to become a success in sponsorship terms for Nike, and is now a businessman with a controlling stake in the Charlotte Bobcats basketball team.

http://www.independent.co.uk/news/busin ... 42101.html
 
I am trying to improve my maths.

Im studying GCSE statistics and an independet Algebra and Graphs
 
Tesco gets a little help with its credit card interest calculations
Maths professor forces Tesco to backtrack after interest charged on his 12-month 0% interest card after 11 months and a week
Rupert Jones guardian.co.uk, Friday 10 December 2010 22.07 GMT

A maths professor has forced Tesco into a humiliating climbdown after exposing how the retailer got its sums wrong when calculating interest on its credit card.

Nigel Cutland, emeritus professor of mathematics at the University of York, spotted that Tesco charged him interest on his credit card – which promised 0% for 12 months – after 11 months and a week.

Tesco initially argued that "12 months means 12 monthly statements, not calendar months". But its response prompted the professor, more used to dealing with the precision of stochastic equations, to take his case further.

"My simple point was that they couldn't define 12 months to mean what they like – that only happens in Alice's Wonderland. While a relatively small amount, if this is the way they act for all such situations, they are making quite a bit of gain. I suspect it is a systemic fault."

After what he said was a "stern phone call to someone senior in complaints", Tesco looked at his case again and accepted there had been a "technical error". His interest bill, initially £76.90, was cut to £22.80.

A significant number of Tesco card holders may now be in line for refunds after the supermarket giant said it would run checks on other accounts to identify if there has been overcharging elsewhere.

etc...

http://www.guardian.co.uk/uk/2010/dec/1 ... lculations
 
Students taking maths post-16: Japan 85%, UK 14%
A new report shows the UK performing poorly in the numbers doing maths after 16
Warwick Mansell The Guardian, Tuesday 14 December 2010

The United Kingdom has a problem, it seems, when it comes to maths education. Results from an international testing study last week triggered a fresh round of soul-searching after they revealed that the UK is falling down the global rankings for reading, science and particularly maths.

And, today, new research revealed exclusively in Education Guardian shows England, Wales and Northern Ireland finishing bottom of another kind of league table, which compares countries according to the number of young people persisting with any kind of maths education beyond the age of 16.

The study, funded by the Nuffield Foundation, shows that among 24 states, these three "home" nations are the only ones where fewer than 20% of pupils take mathematics in any form during what the researchers classify as the "upper secondary" years.

Scotland is one of three [garbage snipped!] other countries where the figures stand in the 20-50% range. The remaining 18 countries all have post-16 participation rates for maths of more than 50%, the paper reports, with rates running at more than 95% in eight of them, including Sweden, Finland, Japan and Korea.

"The findings are stark," says the paper. "By any standards, we are out on a limb."

Not all students who take maths post-16 in other countries are studying advanced mathematics along the lines of A-levels. However, even when only pupils taking maths to advanced level are compared, the UK is still towards the bottom of the class, the research reveals.

In Japan, some 85% take advanced maths, equivalent to at least AS-level, in upper secondary; in Taiwan, the figure is 70%; in South Korea, it is 57%; and in New Zealand, 41%. By contrast, advanced maths take-up in the UK ranges from 11% in Wales through 13% in England to 15% in Northern Ireland and 23% in Scotland, with a UK average of between 13% and 14%.

Is this a problem? And if so, how should it be tackled? Among those calling for urgent action have been the CBI, the business lobby group, which published a paper in August recommending that all young people continue with some form of maths or numeracy education post-16, whether they were on an academic track or training in the workplace.

Young people leaving school with poor maths skills are costing the economy £2.4bn a year, the CBI said, and without an "uplift" in mathematics and numeracy capabilities, the UK would find itself falling behind in fields such as environmental technology, pharmaceuticals and the creative industries. :(

etc...

http://www.guardian.co.uk/education/201 ... erformance
 
More disappointing news on maths education:

England's teacher trainees 'do worse' in maths tests
By Hannah Richardson, BBC News education reporter

England's trainee teachers do worse in mathematical tests than their peers in some major economic competitors, a study suggests.
Teacher trainees in schools in Japan, China and Russia, easily outperformed those from England in the tests.

The study for charity CfBT Education Trust found a huge variation in the mathematical knowledge of England's trainees.
It suggests raising the maths entry requirement for primary teachers.

To teach at primary level in England teachers need a GCSE grade C in maths or above. But the report recommends increasing this, over time, to an AS level.

And it calls for secondary maths teachers, who are currently expected to have an A-level in maths, to take specialised mathematics enhancement courses which concentrate on the mathematical and teaching skills needed to be an effective teacher.

The two-year study for the CfBT charity, carried out by researchers at the Centre for Innovation in Mathematics Teaching at Plymouth University, subjected 1,400 teachers to a series of mathematical tests.
There was a group of between 100-200 volunteers taking part at both primary and secondary level in each country.

England's primary trainee teachers came second to last out of eight countries with a score of 32.2 out of 60.
Japan led the pack with 52.9 out of 60, followed by China on 43.1 and Russia on 41.7.
England was also more narrowly outperformed by Finland, Ireland, Hungary but finished above Czech Republic.

At secondary level, where teachers specialise in subjects, maths trainees in England came second to last out of seven countries, with only those in Hungary finishing lower.
Russia, China, Japan and Singapore were significantly ahead of England's score of 26 out of 40.

The greatest variation between individual trainees' results was found in the English group, pointing to a variable quality of new teachers.
Study author Professor David Burghes said England had a problem with maths teaching that did not seem to be replicated in other countries.

He added: "I don't think many of our trainee teachers have enough conceptual understanding of mathematics at the primary level.
"Countries that do well at mathematics tend to have a strong foundation at primary school."

The researchers also highlighted the high turnover of maths teachers in England's secondary schools.

Researchers gathered more information from the English group through an attitudinal survey, which suggested the trainees were highly motivated and the majority had intended to remain in maths teaching for their whole working life.
But the reality is that most stay for about three or four years. :(

The report said: "This does lead us to question what happens to these trainees over the next few years which results in such a poor retention rate in English schools?"

It also called for new university training schools to be set up in which trainee teachers could train.
They would combine theoretical and practical learning, and would maintain a link between trainees and university tutors.

Education director at CfBT Education Trust, Tony McAleavy, said: "The establishment of university practice schools otherwise known as university training schools, is the most important decision that could be made for taking the profession forward.
"This would ensure less variation in standards and would ensure that there would be peer support for new teachers in their first practice; something that has currently been lacking."

A spokesman for the Department for Education said an improvement in maths teaching was needed: "Currently we languish at 27th in the international league tables for maths, plummeting 19 places in under 10 years."
"In the schools White Paper we set out how we will attract the best maths graduates into the teaching profession, by expanding the Teach First programme and providing stronger incentives, whilst raising the bar for entry into PGCE teacher training by requiring at least a 2:2 degree to receive DfE funding," the spokesman said.

http://www.bbc.co.uk/news/education-12806182
 
When maths goes wrong:

AS-level maths error: students set impossible question
By Angela Harrison, Education correspondent, BBC News

An "unfortunate error" meant maths students were set a question that was impossible to answer in an AS-level exam.
Just under 6,800 teenagers took the paper - set by the OCR exam body - last Thursday.
OCR has apologised, saying it will make sure candidates are not disadvantaged by the mistake.

But some students writing on social networking sites have been calling for the test to be re-run.
The error was in an exam paper taken in 335 schools and other exam centres in England, Wales and Northern Ireland.
The question carried eight marks out of 72 being awarded for the paper.

Candidate Thomas Fay, who contacted the BBC News website, said he had been distressed to find a question that appeared "impossible".
"This threw me in the exam and many people found this to cause much added stress in the exam," he added.
"Many people are worried that the mistake made by the examining board will severely affect the mark and grade they achieve in the paper. For many this was a final exam and will most likely influence final grades and university admission."

Dozens of other students have messaged the BBC News website to voice their anger and fears about their grades.
Aron De Vos, 17, from St Albans, said: "I spent a good 15 minutes trying to answer that question. I was getting very frustrated about why I couldn't get the answer.
"I want to retake that exam. I can't believe how much time was wasted on a question where we were only able to get zero marks."

OCR has said it deeply regrets the "unfortunate error" and says it has a range of procedures in place to ensure candidates are not disadvantaged.
A spokeswoman said: "We very much regret that there was a mistake... and that our quality assurance procedures failed to identify this error.
"Because we have been alerted to this so early, we are able to take this error into account when marking the paper. We will also take it into account when setting the grade boundaries. We have sent a letter to all schools and colleges explaining in more detail what we shall do.
"We do apologise again that this has happened."

The exam body says it is not going to discount the question from the marking, because that might disadvantage candidates who spent a lot of time trying to answer it.

Students will be awarded points for their attempts to work out the question and measures are also in place which are designed to recognise that other candidates may have discovered the error quickly, OCR says.

...

http://www.bbc.co.uk/news/education-13627415

More student comments follow the article. It would be tragic for the students, and for the country (which deperately needs mathematicians), if some fail to get into Uni as a result of this cock-up. :evil:
 
Surely if the scores are normalised it won't have any effect?

Or perhaps they could grade it as normal, with "This is impossible" being the correct answer. Bonus marks for anyone who can actually show why it's impossible.
 
Anome_ said:
Surely if the scores are normalised it won't have any effect?
Exams are naturally stressful and a bum question only made this one more so. Psychologically, it wasn't a level playing field. A few students realised the question was wrong and left it, but many didn't and wasted too much time on it.

As a father said in one comment:
I honestly don't believe that a statistical approach to marking this question does anything than assure that the results distribution matches an average of previous results. It does nothing to reassure an individual they weren't disadvantaged over another student sitting a different board's exam or module who also needs the same grade to get to university.
 
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