The following is a mathematical process best not repeated.
An adolescent boy discovers his intermediate teacher has left open the door to her resource room. He estimates her likely return to class; he then factors in the distance and time it would take to enter said room and safely remove a book containing answers to every maths homework assignment he might face in the oncoming year.
Though the calculation seems simple enough (risk versus reward), the 11 or so years the boy has walked the Earth provide an insufficient weight of wisdom. Had he worked out other variables (how would this move affect his maths abilities in, say, one, two or 30 years time?) he might have kept his bony bottom on that wooden chair.
The result of that pre-teen's less-than-standard deviation? A computational power more inert than innate. By sixth form, maths had been replaced with art. After all, life drawing seemed so much more ... sexy.
However, there are some for whom maths is both a career calling and a pleasurable pastime. A passion even. In Dunedin this week, the New Zealand Association of Mathematics Teachers annual conference, "Mathematical Journeys 2011", attracted more than 250 secondary school educators from the Far North to the deep South.
The influence of mathematics on the advancement of human civilisation is so well documented it almost defies repeating. From Euclid to Fermat to Gauss and onwards, maths has cross-pollinated many aspects of our existence, from science to art to politics to war.
(Without Einstein's famous E=mc2, published in 1906 as part of his Special Theory of Relativity, an equation that describes how a large amount of energy can be gained from a small amount of material, would there be an atomic bomb?)
As Iain Raeburn, an expert in pure mathematics at the University of Otago, points out, "it has taken thousands of years to reach our present state of mathematical knowledge, which is now so broad and deep that it requires the full-time attention of a professional mathematician to keep on top of even a small part of the subject".
"The ever-accelerating demands of technology mean that mathematics is now developing faster than ever, and it seems likely that it will continue to become even more important through the foreseeable future," Prof Raeburn notes.
"Maths is important because it arises every time we try to quantify something or to describe the relationship between two quantities, which is what science is all about."
Derek Holton, who retired as professor of pure mathematics at the University of Otago in early 2009, now lives in Melbourne and was due to speak at this week's Mathematical Journeys conference before a family issue prompted his withdrawal, highlights two examples of maths' wider influence.
"Although these things have impinged on everyone in some sense, they are beyond the realms of most people's personal experience.
"The first is a very particular case - the RSA encryption scheme. This is a way of coding data where you tell everyone how to encode messages but only you are able to decode it.You can probably see that this is extremely valuable and almost certainly has been used by the United States military as well as banks. This is a system that used a very old number theory result to produce something very useful.
"The second is a more general case - statistics.
"Now, some people feel that this isn't really maths and I think that can be debated, but there is no doubt that statistics has made an extremely important contribution to human life in the last 100 years. Its contribution in the medical area alone has essentially saved millions of lives," Mr Holton points out.
Yet maths has a function (pun intended) that goes beyond global implications; it requires both logical and creative thought.
"Mathematics has its own way of stripping a problem down to the bare essentials and I think in that regard it has something extra to give to people's enjoyment of life and to many people's working lives," Mr Holton says.
Having had extensive experience advising on changes to the New Zealand curriculum over the past two decades as well as co-founding the Ministry of Education website www.nzmaths.co.nz and being heavily involved in the International Mathematical Olympiad, an annual mathematics competition for secondary students, Mr Holton's work has provided a platform from which he has pondered, deeply, the fundamental aspects of mathematics teaching.
To him (and others), the answer lies in problem-solving. As a former maths teacher, he has witnessed just how motivating problem-solving can be (in the same way as solving murder mysteries is).
"You have me on my hobbyhorse now. I believe that we are not doing enough in this direction. For many defensible reasons, many of them to do with public examinations, we are spending more time on teaching students how to solve particular problems than how to solve any problem.
"You see, when we leave school, most of us are not engaged in solving quadratic equations or finding the turning points of functions. The problems that we meet, mathematical or otherwise, are not problems we encounter in school. So I feel that more time ought to be given to the approach to problem-solving in school. This is not the sole domain of mathematics; this ought to be a major goal of every subject."
However, as the Ministry of Education emphasises, there is no point in trying to teach basic facts before their meaning is understood.
For example, in the case of multiplication it is common for primary school children to know 6 x 5 = 30 but some struggle to make up a problem that is answered by this fact.
A more complex example: recalling 5 x 5 = 25 and 2 x 5 = 10, a child may deduce 7 x 5 has the same outcome as 25 + 10 = 35. This requires a "part-whole" thinking process, the emergence of which is preceded by basic counting (additive) stages where the size of numbers is severely restricted and there is typically only one way to solve problems. As such, counting represents a relatively low level of thinking, whereas part-whole thinking opens up the potential to grasp larger numbers and multiple strategies.
There is also the balance between strategy (or thinking) and knowledge, the latter being the ability to recall sequences (remember all those times tables learnt by rote?).
For example: Mikey works out 8 + 3 by counting on. The strategy Mikey uses is that he must start counting at 9, (not 8), he must say the next three words in the counting sequence, and must know that the answer to 8 + 3 is the final number he says. The knowledge component is in the process of counting forward without having to think what comes next.
"You have to learn with understanding first," says Pip Arnold, among several keynote speakers at the Dunedin conference.
"But there are some instances where you just have to learn by heart. For example, 7 x 8; I remember that as being one that'd drive me insane. In fact, there are only about 10 to 15 hard times tables; the others are really very simple."
At secondary school level, mathematics offers an opportunity to explore more deeply the use of patterns and relationships in quantities, space, and time. Statistics, meanwhile, accesses patterns and relationships in data.
The two disciplines are related, but use different means of solving problems. According to the New Zealand Curriculum, both equip students with "effective means for investigating, interpreting, explaining, and making sense of the world".
Ms Arnold, who has also helped develop the mathematics curriculum, worked as a consultant to teachers, held down jobs as head of mathematics at Sacred Heart Girls College and Auckland Girls Grammar School and conducted doctorate-level research into statistics education, is passionate about maths.
"I probably have been since the age of seven or eight. It was a subject I found I was good at and when I started teaching I decided to teach maths because it was a subject some didn't seem to enjoy or it wasn't taught well. Also, girls need maths role-models."
A stumbling block for some (though a door to a fascinating world for others) is the language of mathematics.
"You have a lot of symbols and words that have a particular meaning that is quite different to what they mean in everyday life. In statistics, that's even more so. We talk about samples and populations and means; what does the mean mean?
"I think if we, as teachers, aren't careful, kids will just think the language is too hard. We need to teach the right language early on. From the research we've been doing, we've found kids can actually handle the language a lot earlier; they are pretty switched on and are probably more confident than we are."
In regards the continuum of maths learning, there are several key points in a child's number-crunching ability.
The first is moving from counting numbers to adding them; next is multiplicative thinking, then proportional reasoning, which involves concepts such as understanding ratios, percentages as part of a whole or finding fractions of quantities.
"Then you start to get into algebraic thinking, although that can be at an additive and multiplicative stage as well," Ms Arnold explains, adding she believes the maths ability of most adults in New Zealand lies somewhere between multiplicative and proportional thinking.
The biggest jump of all perhaps looms around the middle of those high-school years, the leap from year 11 to year 12.
"In year 11, your first compulsory exam year, it is still quite a general curriculum, so you are still covering numbers, algebra, geometry, measurement and statistics," Ms Arnold says. "When you get into year 12, there is a much bigger focus on algebra and calculus; statistics becomes more prominent. You get more abstract."
Regardless of level, maths still needs to be enjoyable. A pastime even.
John Major, head of mathematics and assistant principal at Logan Park High School and president of the Otago Mathematics Association, says there are three or four reasons why maths is important at school: "Students are going to need maths to get further qualifications - and not just qualifications in maths, but in other things; you can't do a lot of science if you haven't done a lot of maths; you can't get into certain university courses without it.
"But you also do maths because it is fun. You can do maths just for the hell of it. You want to try and get that across as much as you can at a junior level," Mr Major says.
"But you have to live it yourself. If students don't see you getting a kick out of it, they aren't going to. It has to rub off on them a wee bit."
Mr Major admits maths might have a PR problem among a certain sector of the population.
"One of the biggest problems is people who can't do it are telling others that it's OK not to be any good at it. A kid goes home from school and Mum or Dad says, 'oh, well, I was never very good at maths ...'
"Survey primary school kids and maths is often one of their favourite subjects. But get to high school and it has been reversed. It becomes more formal, with assessments and so forth, but we can still get back to the fun side of things," Mr Major enthuses.
"I saw it this morning in my year 11 class when they suddenly thought, 'oh, I can do that', and were getting a buzz out of it.
"It is more about teaching a kid to give something a go. You're not going to get something right unless you attempt to put something down in the first place."
A final question: Why do humans do maths?
Because we can, Mr Major says.
"This power to think is what makes us different from other, lesser beings."
Wonder of numbers
Maths is a language, and a powerful one at that. In fact, it's the only language that can explain the way the world is made, says Tony Burnett, author of The Search for Simplicity: a fascinating journey through scientific discovery (2002).
The sentiments of the Cheviot-based author, who has also penned a series of educational resources for gifted children, are hardly unique.
Euclid, the Greek mathematician of 300BC who is often referred to as the father of geometry, once said, "The entire universe is written in the language of mathematics".
That language can be likened to a microdot, Mr Burnett says.
"A microdot looks just like a full stop typed at the end of a sentence but if you examine it closely - say, using a microscope - you can see that it contains whole pages of information.
"Pages of information are sometimes contained within the symbols of the equation and, if you know how to read them, you can often learn things that you never suspected."
Heard of Einstein's E=mc2 equation? What does it mean? Well, the symbols are easy enough to grasp:
"E" represents energy (in its various forms, including heat, sound, kinetic, elastic, nuclear), "m" is mass (the quantity of material in any given body), "c" the speed of light (300,000,000m per second) and the small "2" after "c" means it is squared, i.e. multiplied by itself.
Here's a simple calculation: take an object weighing just one gram (m=0.001kg), multiply it by 300,000,000 x 300,000,000 (c2). The answer: E=90 trillion joules (a joule being a standard metric unit for energy).
"Until Einstein discovered the E=mc2 equation, no-one had any idea of how the stars could possibly give out the energy that they do. E=mc2 is probably the most famous equation in the world. The most famous physics equation anyway," Mr Burnett says, adding he still feels a sense of wonder every time he thinks about it.
Game for the young
Maths expert Derek Holton on number-crunching ability and age:
"While it is certainly true that many mathematicians produce their best work early on, this is not the case for everyone.
"A few years ago Andrew Wiles proved Fermat's Last Theorem. It doesn't matter what this was. The importance of the event is that he was just over 40 when he finally completed the proof. So here is concrete evidence of someone producing new and, to mathematics, important work when he had passed his 30s.
"My second example is Paul Erdos. He was a Hungarian who remained active up to his dying day. In his lifetime, he managed to produce more research papers than anyone else has done. He died in his 80s.
"However, I think there is no doubt you have to keep mathematically fit, in the same way a rugby player needs to keep fit for a game.
"Almost certainly, one reason that a lot of good mathematical work is done early is because the young mind is somehow more supple. But you have to realise, too, that other things happen to people - even mathematicians.
"Good ones with promise get promoted up the ladder, get married and have children; they are involved in many of the things that other people are involved in. This means that they lack the time they had to work on mathematics when they were in their 20s and 30s."
• After nearly 24 years in the job, Derek Holton retired from his position as professor of pure mathematics at the University of Otago in early 2009.
