Prologue
Given the critical dependence of our civilisation on mathematics, it is paradoxical that it lacks any general acclaim or popularity, and usually invokes fear and mysticism. Perhaps this is because for most of us, the process of acquiring mathematical skills is somewhat protracted and painful. It is often perceived as a nightmare topic, which engenders a defence mechanism of open ignorance that is often lauded. In truth, most people can become reasonably competent. However, the time and effort needed to acquire the formal rigour for even a modest capability is often seen as excessive by modern standards.

'Mathematics is often defined as the science of space and number ... It was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns'
Lynn Arthur Steen

The purpose of mathematics, its application, relevance, and teaching is a source of constant debate. In reality it is a subject driven by the need to understand. Whether it is for pure science, logic, philosophy, engineering or economics, the quest is the same: a search for clarity, truth, understanding and precision. Despite being driven by the need to find solutions to real problems, mathematics has, in most respects, been ahead of practical applications. Today, we can see evidence of this historically advantageous situation gradually being eroded and reversed.

'The art of asking the right questions in mathematics is more important than the act of solving them'
George Cantor

Many of the problems we now address are non-linear and chaotic, and defy our tried and tested mathematics. Curiously then, our education system still concentrates on feeding students a diet of linear problems with convenient closed form solutions. Not surprisingly students form a picture of a universe as depicted in Fig 1(a). Only later, and mostly in industry, do they discover that their universe a sea of non-linearity Fig 1(b). So perhaps we should not be surprised at the general confusion and misunderstanding in the mass of society on matters mathematical. Even in well-educated groups the level of misunderstanding and expectation rests firmly on the misguided assumption that prediction with precision is always a reasonable possibility. The reality is that modelling our world is inherently complex as the simplistic realm of well behaved linear convenience has long been overtaken!

'The power of dealing with numbers is a kind of "detached lever" arrangement, which may be put into a mighty poor watch. I suppose it is about as common as the power of moving the ears voluntarily, which is a moderately rare endowment'

Olive Wendell Holmes

This is compounded by the small percentage of the population able to grasp even the most fundamental aspects of non-linear dynamics. The decision makers who impact on all our lives - those in positions of power and control - often lack any understanding. But at a more fundamental level, to have a work force and influential people generally ignorant in mathematics severely limits the progress of society. Without such skills, decision making becomes increasingly flawed, leading to under-achievement, and reduced wealth creation. How then are we to teach and educate people to use the power of mathematics as we move to the 21st century?

Society Speed Up
We are living through a technological and social revolution driven by IT and global competition. In much of industry hierarchical structures and vertical integration are giving way to virtual organisations that operate in a continually changing amorphous manner. Whole sectors are growing new dependencies that make inter-working fundamental to survival. The increase in the contract labour force, fewer full time employees, working from home, car, hotel, and office are gradually becoming the dominant mode. As a result there is a growing requirement for continual, on-line, education and training. The old world of going from school to university, and an education for life, has largely gone. It is now necessary to attain a series of higher degree 'top ups' throughout a career that is increasingly subject to radical changes in direction and discipline. Such is the pace that many science and engineering degrees have a half life of < 5 years, and for more vocationally based courses, it is < 2.

Not surprisingly then, education is under pressure to radically change. Publicly we lament the decline in standards whilst continually changing the curriculum to cram more into a fixed internship, and fudge the figures to make it all look more egalitarian. Just 30 years ago 1 in 15 of our young people entered university, today it is 1 in 3. However, they have not grown any smarter in the intervening period, so the net result has been a watering down of standards. In an ideal world we would have streamed them into ability groups, injected more individual tuition, and expanded the teaching time. In truth we have gone in the opposite direction. Even if technology offers new solutions it seems clear that there is an increasing need for a foundation layer of basic understanding across a broad front. We need late, rather than early, specialism to realise medics, lawyers, managers, and politicians who understand systems and mathematics.

Good-Bye Stability
The process of education has been relatively stable for millennia. Since ancient times we have only progressed from crude pictures in the sand, to the blackboard, white board, overheads, and more recently pictures on the screen. Education is now about to face a more radical paradigm of on-line, just-in-time, access to information, experience, tuition, coaching, understanding and training. Teleworking, information networks, CD books and instructive, interactive games already give a clear indication of the way IT can change old ways and thinking.

'Computers are useless, they can only give you answers.'
Pablo Picasso

Give a child the option of a book or a multi-media encyclopaedia, and watch them choose the electronic version! Watch the child at home working on an interactive CD based learning package, and see a 50% speed increase in their learning process, and an 80% improvement in retention. Watch the same child at school trying to get individual attention in a crowded class where the IT is 5 years old, feared by the teacher or non-existent. See the university student with a lap top in their room and watch their progress. See a student without this facility struggle to get enough time on university machines. Observe those without IT skills and you are watching - a dying species!

Today's distinct and separate education sector will become increasingly tenuous as community and customer requirements begin to undermine current frameworks. Organisations designed to cope with stable environments, and limited flows of information will find it increasingly difficult to survive. This new turbulent phase will trigger the absorption of education into a new Edutainment industry.

Technology
During the past 10,000 years the power of the human brain has changed little and we expect very little change in the next 10,000. In contrast, a mere 100 years have seen computers evolve from mechanical to electromechanical, thermionic valve, transistor, and integrated circuit based machines. An electronic wrist watch now has more processing power than a computer the size of a domestic washing machine 20 years ago (Fig 2). Today's machines have a staggering ability that will continue to double every 12 - 18 months for at least another two decades.

'For those, like me, who are not mathematicians, the computer can be a powerful friend to the imagination. Like mathematics it doesn't only stretch the imagination. It also disciplines and controls it'
Richard Dawkins

So, we now face something of a dichotomy as the human mind is fundamentally incapable of formulating many complex problems, processing them and providing answers using paper and pen. Whilst computing power can analyse, synthesise, model, and solve problems of staggering complexity, in so doing they very often mystify even the well educated. Understanding is often lost in sophistication, and in short, we are often data rich and information poor!

'Mathematics is often defined as the science of space and number ... It was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns'
Lynn Arthur Steen

Education - Past
Looking back to my formative years, I see a poor education in mathematics founded on the teaching methods unchanged since Archimedes! I was expected to tackle problems requiring multitude sequential steps to get an answer. The right numerical answer was the primary goal, with little emphasis on conceptual understanding. To my mind it was largely a 'handle turning' and unenlightening exercise full of mystery, the very antithesis of education!

In my professional life things became more interesting and rewarding. I found that "handle turning" had at least equipped me to expect to perform steps '1' to 'n' before an answer would appear. At this stage I focused on the inner mechanisms to gain an understanding of the processes. As a result I found mathematics to be far more enjoyable - and beautiful!
'Mathematics, rightly viewed, possess not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture'

Bertrand Russell
Since that time education has become something of a political football with everyone seeming to be experts on the topic. Changes in teaching methods and the curriculum have seen more emphasis on discovering new things, by neglecting basic processes, and putting less emphasis on getting the right numerical answer. We now have generations of children indoctrinated to expect that if an answer doesn't emerge within steps 1, 2, 3,... then the problem is too tough, and there must be an easier way of doing it! In many respects this change in philosophy seems to have engendered a worsening of our situation. There is now a gross inability to appreciate the necessity and power of mathematics. As a result, we see degree courses extended to cover the inadequacy of an early education system up to A Level. We also see professionals emerging from courses where mathematics has been wholly removed because it is unpopular! Medics, accountants, lawyers, engineers, scientists, and teachers who have a lamentably poor appreciation of mathematics are now common - and probably the norm!

'Helmholtz - the physiologist who learned physics for the sake of his physiology, and mathematics for the sake of his physics, and is now in the first rank of all three'
William Kingdon Clifford

Education - Future
Visualisation
What hope then for the future? Probably the single most powerful tool available for teaching mathematics is visualisation. Even as recently as the 13th Century the general use of pictures in mathematics was decried by many. A parallel situation exists today with those who decry the use of IT. And yet without computers the world of complexity would remain hidden from view - chaos and Fractals cannot be explored without the screen!

Turning the complex into animated, 3D, colour, pictographic forms gives a new means of understanding often invisible physical and abstract situations. It almost certain that visualisation technology holds the vital key to creating a generalised understanding of the non - linear (Fig 3).

'Who knows what secrets of nature lay buried in the terabytes of data being generated each day by physicists studying the results of numerical simulations or the image of a distant galaxy. Given the volume and complexity of scientific data, visualisation in the physical sciences has become a necessity in the modern scientific world'
Robert Wolff

In a recent R&D programme the thorny problem of understanding a large software programme of 1.6M lines of code was tackled. By mapping areas of activity on a plane (Fig 4) it became obvious that vast tracts of code were seldom if ever used.

The suspicion was that successive teams of software writers had invoked the "if it works, don't touch it"; maxim beloved of the industry. In a second mapping (Fig 5), the amount of activity was reflected as rotating coloured 3D spheres, connected by pipes of interaction. This revealed for the first time, and has subsequently been confirmed, that large software suites strongly resemble neural networks. So the problem seems to be that software and neural networks are strongly related. In fact, they may even be the same - both are inherently non-linear, and lack any general theory. A powerful result through simple visualisation!

'Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different'
Goethe

On a simpler note I vividly remember my introduction to Maxwell's equations as a student where my entire understanding was dictated by the limited artistry of a mathematician. This involved the use of coloured chalks and sketches representing 3D and 4D on a 2D blackboard! Today that same problem can be represented on an animated computer screen the difference is stunning - but would be no more than planar on this passive paper! With such tools understanding is achieved far faster, and more effectively. The same is true of turbulent flow, transformations, chaos, catastrophe and much more that is conceptually difficult to grasp. A static picture in black and white, or indeed an equation, are insufficient for the human mind to visualise the subtleties of interaction encountered (See: Interesting Sites to Visit). We can no more understand the underlying mathematical and physical processes than we could have predicted the big bang without a telescope!

Virtual Reality
While the popular view of VR is games orientated and full of hype, it does have serious applications that far exceed standard visualisation techniques. It is principally a medium for direct experience - flight simulators, military exercises, and stepping inside the atom or molecule. For the first time we can now fly a proton and experience fission, rather than just gaze on a set of complex equations, or watch the resulting trajectories at the output of an accelerator. We can also see and feel the binding energies in the alignment process of long chain molecules, thereby adding to our fundamental understanding.

VR also offers significant potential as a mathematical tool, and for many, it augments the left-right brain connect so vital to understanding. This may in turn signal the early demise of the well established mathematical paradigm. Perhaps it will soon no longer be the preserve of a knowledgeable elite!

For many situations, we can now view and handle functions in a new way as they are no longer frozen in time and space, but animated by real time interactivity. This raises the possibility of the poorly apt or educated being able to gain direct understanding through 'hands on' experience. This could be both advantageous and dangerous. Alone and unaided it is easy to draw the wrong inferences and conclusions, but with educated help, the full power can be realised.

'There is an astonishing imagination, even in the science of mathematics...We repeat, there was far more imagination in the head of Archimedes than in that of Homer'
Voltaire

New Dangers
During the 1960's, before the arrival of the pocket calculator, when professionals avidly used the slide rule, mathematicians often rebuked students for this practice as logarithm tables were the 'true way'. At A-Level the rot had already set in! Students given a problem with an integer solution would use a slide rule to arrive at the answer 3.99, when a moments thought would have revealed it to be 4. This was followed by the pocket calculator able to return an answer to 9 decimal places. Unthinking students lost the ability to ask the most fundamental of questions: what might be a reasonable answer, and where should the decimal point be? Unfortunately computers have compounded this problem. I was recently confronted with a solution that suggested a man should weigh 700 kg. When asked if this was a reasonable answer the student's response was - "well that's what the computer said!" The assumption that the computer arrives at an answer and it must be right is stunning, but it happens!

Approximations
'Truth is much too complicated to allow anything but approximations'
John Von Neuman

During my engineering education I had the benefit of a number of Professors who imparted numerous wisdoms including the notion of the good approximation. In fact, getting an approximate answer was essential during the early days of computers due to the cost of processor time (can anyone remember paying for computer time?). Without a focused effort, vast amounts of money could be expended to no avail. A good approximation allowed a focus on the area of interest and gave a faster return. Today, few people bother because computing power is so cheap that it does not matter. In my own case, I rarely use a pocket calculator, but resort to a spreadsheet for even the most mundane and simple of calculations. I have not lost the ability to perform mental arithmetic, but I have become extremely adept at using computer power, even though it may be a back hoe digger to crack a walnut!

How then are we create future generations knowledgeable in mathematics, and able to use the available tools and interpret the results correctly? I think the problem now starts with teachers and parents who wince at the very thought or use of the word mathematics. Mathematics can be fun, and indeed, for many of us it is fun. Unfortunately for most, the time taken to get to the point where they might discover this is far too precious. They either lose interest through a blindness to the need, or perhaps worst of all, sheer boredom imparted by the people teaching the topic.

'Blindness to the aesthetic element in mathematics is widespread and can account for a feeling that mathematics is dry as dust, as exciting as a telephone book ... On the contrary, appreciation of this element makes the subject live in a wonderful manner and burn as no other creation of the human mind seems to do'
P. J Davis and R. Hersh

Software
Recent developments have seen multi-media products that are very reinforcing for most minds. Individuals can learn at their own pace, there is no reproachful eye or an indignant voice, only encouragement. It seems to me that we should be combining such tools with enlightened teachers to explore new ways to teach mathematics. Whilst plotting graphs on paper might be good for the soul, it is also extremely limited in scope. Children now come from a culture of computer games, working on the screen, and being creative in an environment that gives almost instant gratification. Putting mathematics into that same framework could have a devastating effect on the speed of the learning process.

For the professionals too there are now an abundance of mathematical and visualisation tools that give instant skills (See: Interesting Sites to Visit). The real challenge is getting the right advice and guidance - just in time. Interestingly, a number of institutions now run MSc courses straight from the lecture theatre to the desk. By using multi-media technology and telecommunications, the expert of your choice can be on your screen. In my company our internal MSc and MBA courses use a mix of technology and traditional teaching methods - it works. Perhaps the most revolutionary step here is the ability to access the best of the best at reasonable cost. Why have a second rate professor when you can get the best?

This all offers a solution to a long standing problem. Can I get access to the mathematical expertise to solve my current problem in real time? With the right computer terminal, and network connection it is now possible - it has been tried and it works!

Finally, we should expect an increasingly pictographic, interactive and virtual world to predicate radical changes in mathematical representation. The ultimate challenge is to achieve faster and higher levels of understanding for an audience with inherently less time availabile and shorter attention spans. Electronic visualisation is currently the only available tool.

'Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite'
Paul Erdos.

About The Author
Peter Cochrane joined BT Laboratories in 1973 and has worked on a wide range of technologies and systems. In 1993 he was appointed as the Head of Advanced Research. A graduate of Trent Polytechnic and Essex University he is also a Visiting Professor to UCL, Essex, and Kent Universities. As a consultant to numerous international companies and organisations he has travelled and lectured widely and has published extensively on technology and the implications of IT on society.

He received the Queen's Award for Innovation & Export in 1990; the Martlesham Medal for contributions to fibre optic technology in 1994; the IEE Electronics Division Premium in 1986, Computing and Control Premium in 1994, the IERE Benefactors Prize in 1994, and the James Clerk Maxwell Medal in 1995.

Interesting Mathematical Sites To Visit
http://bambi.ccs.fau.edu/ccs.html (Complexity home page)
http://blanche.polytechnique.fr/lactamme/Mosaic/descripteurs/demo_14.html (Fractal world)
http://math.furman.edu/~mwoodard/mquot.html (Quotations)

Bibliography
Emmot, S: Information Super Highways - Multimedia Users & Futures, Academic Press, London, 95
MacDonald L & Vince J: Interacting with Virtual Environments, Wiley, Chichester, 1994
Earnshaw R A & Vince J A: Computer Graphics - Developments in Virtual Environments, Academic Press, London, 1995
Cochrane P: Desperate Race to Keep Up With Children, The Times Educational Supplement, 23rd June 1995, p 25
Milne R & Montgomery A: Proceedings of Expert Systems 94, British Computer Society, Information Press, Oxford, 1994
Gell M & Cochrane P: Turbulence Signals a Lucrative Experience. The Times Higher Education Supplement, 10 March 1995, p 11.
Hague, D: Beyond Universities - A new Republic of the Intellect, Hobart Paper, Institute of Public Affairs, London, 1991
Cochrane P: The Virtual University. Business of Education, Issue 5, March 1995
Ravitch D: When School comes to you. The coming transformation of education, and its underside, The Economist, September 11th, pp 53-55.
Pagels H R: The Dreams of Reason - The Computer and the Rise of the Sciences of Complexity, Bantam New Age Books, London, 1988
Davis P J & Hersh R: The Mathematical Experience, Houghton Mifflin Books, Boston, 1981
Rogers E: Wonders & Delights - Essays in Science Education, Institute of Physics, Bristol 1994