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Felix Grant believes that software can open up the big ideas in mathematics to the wider public, democratising the subject, and making it part of popular culture

Charlotte, a 17 year-old student on a vocational childcare course, volunteered out of ecological and humanitarian idealism for an environmental study. With only a very hazy and biased idea of the issues involved, and having three times failed to get a 'C' grade in GCSE maths, she wasn't a promising candidate for anything more than manual contributions. Yet she cracked one of the three main scientific problems encountered.

Aaron, a self-taught independent film-maker with a blissfully maths-free past, made a documentary within a production plant belonging to a large multinational corporation. Fascinated by an intractable problem deriving from the industrial process that had been explained on camera, he spent some of his off-duty time on it - and provided the insight which led to its resolution.

Parveaz, a hairdresser turned temporary secretary in a medical school, was afraid of appearing incompetent. Unable to decipher some passages in hand-written mathematical script that she had been asked to type up, she tried to puzzle it out on her own, using her one year of college maths. She gave up and took the problem to one of the school's epidemiologists who, checking her work against that in the manuscript, realised that an early and fundamental mistake in an important study had never been corrected.

I could go on, but these anecdotes will suffice. The common factor in each case was the availability, to someone who would not normally access it, of mathematical software.

Before you make the obvious retort; no, I am not suggesting that powerful software does away with the need for mathematical knowledge, understanding, or experience. There will always have to be expertise to make appropriate use of intuition. However, from an initial position of scepticism, I now believe powerful mathematical software can tap deep wells of unused human potential. Not on the off-chance that employees can be exploited above their pay grade; rather, to bring mathematics into line with what is happening to other cultural tools.

I was started on this track by a chance conversation with the editor, and with fellow contributor Ray Girvan, about the strategic directions taken by Wolfram Research and, in particular, developments in CalculationCenter. At that time, Wolfram was trailing new releases of both CalculationCenter and its big sister, Mathematica. I said, rashly, that developments in CalculationCenter would 'put powerful mathematics in the hands of anyone with GCSE maths'. Having researched this piece, after a summer of more than 100 conversations and interviews, I no longer believe this was rash.

Most areas of life in technologically enabled societies have been democratised by progress. Just 150 years ago, most people in those societies couldn't read or write; today, anyone can publish their ideas worldwide. My four-year-old grandson may not be the next Rembrandt, but with a cheap camera he can keep visual records that Dürer would have died for.

In mathematics, though... what? There have been huge leaps in power for the experienced and confident mathematical or scientific user, but for the general population little has happened beyond the advent of calculators to take the drudgery out of arithmetic. Part of the problem is inherent. Mathematics is more hierarchic than many other areas of expertise: new learning builds on what went before to a greater extent. That's not the whole story, though.

Of course, there is more in the mix than just enquiring spirits and powerful black boxes. One crucial factor is attitude. On the aforementioned environmental project was a geologist who, recognising intelligence and intuitive insight, took Charlotte under her wing and had the patience to teach the uses of Mathcad. Aaron happened on a process engineer who treated him as an equal, explained how to use Matlab, and listened to what he had to say. Parveaz was prepared to stretch the knowledge she had into a new context and into Maple, and her epidemiologist didn't let professional pride stand in his way. But, sadly, mathematics remains by and large what it has always been: an esoteric area of activity, used by initiates and closed to the population at large.

Talking to 16- to 21-year-olds in the British, Italian, and US education systems, the people who will or might have been tomorrow's scientists and engineers, I find a clear common perception. Arts and humanities subjects are not perceived as inherently easier or more interesting than mathematics or the physical sciences. Rather, they permit a more immediate coming to grips with ideas that young people see as relevant to their lives. In literature, sociology, or history, for example, it is possible to take an intellectually exciting idea by the horns straight away. Learning the underpinning knowledge and processes can be done 'on the job', whilst using them to wrestle with the idea. The big turn-off in mathematics, for most, is the sequentially hierarchic learning process, which allows full engagement with the big idea only at its end.

That is where mathematical software could make a big difference - and it did for Charlotte, Aaron, and Parveaz. But the trend towards usability and accessibility has not moved outside the existing market for such tools. All of the leading mathematical software publishers progressively beef up their products' capabilities. All of them also work at value-added ways to make those capabilities more readily available, for less investment of effort, to their user community. Other value-added comes in the form of applications to encourage use within the training of future members of that user community. There is considerable effort to widen that user community; deepening it tends to get less attention.

Until that barrier is breached, making it possible for anyone at all to pick up mathematics and play with it, Parveaz, Charlotte and Aaron will remain oddities. Once it is breached, though, there is the potential for cultural sea change on a par with the growth of mass literacy.

The most promising signs, at the moment, come from Wolfram. For some time, I've had both Mathematical Explorer and Calculation-Center on the laptop that habitually accompanies me; both have been invaluable in tackling mathphobia in friends, acquaintances, students, and clients alike. Mathematical Explorer directly, and impressively, addresses the issue of access to ideas, but within a specific framework. CalculationCenter is less far along that road, but is a tool for general application.

Wolfram is a law unto itself, a reviewer's nightmare. As these words go to press, there is no way of knowing whether or not, by the time you read them, the new versions of CalculationCenter and Mathematica will have come to market. However, that doesn't matter here: the important issues are in conception rather than implementation. Earlier this year, Calculation-Center version 2 became network-enabled, making it much more widely available. Crucially, from my present point of view, it can be available to anyone in a networked environment: companies, schools or hospitals - by anyone from the CEO to the work-experience teenager.

This is the first networkable package with something approaching top-flight power and an interface that allows the novice to get stuck in. CalculationCenter is already capable of doing what mathematically fluent scientists and engineers ask of it; where it won't reach, Mathematica takes over; and 'incorporating Mathematica Technology' is one of the features expected in Calculation-Center's new version.

CalculationCenter, having sparked off the idea, was central - but it wasn't the only software involved. There were three other mathematical software packages, Mathematica and two from other stables, for comparison. Half of the interviewees were exposed to a competitor before CalculationCenter, half of them afterwards. In each half the three competitor products were rotated, giving six equal subgroups in all.

Intrigued by Ray Girvan's descriptions of MuPad's addins for Microsoft Word, I downloaded them with a view to including them in the quest. Alas; since my copy of Word on the lap top immediately stopped working, and nothing has yet coaxed it back to life, I wasn't able to use them with most of my interviewees. Asking around, nobody else has experienced the same problem so it presumably relates to my particular set-up rather than to the add-ins - but I haven't been brave enough to try another installation on another machine. I found someone with installed copies in a functioning copy of Word, and was able to try them out on a dozen or so volunteers with an encouraging level of success.

Additionally, I took along Publicon, a mathematical and technical publishing product (also from Wolfram) long in development and sharing the conventions of Mathematica (therefore also of CalculationCenter). Emerging from a couple of years in non-disclosure country, it came to market part-way through writing and was demonstrated to about a third of the subjects. It differs from the other products in being, in effect, a word processor or desk top publisher for mathematical text. It can be seen as a dedicated replacement for word processor and equation editor. It appealed to many who have to type up material that contains mathematical material, even when they were not interested in exploring that material. Reactions were even more enthusiastic when preceded by experience of CalculationCenter, but not by any of the other packages (including Mathematica, which it equally resembles) - further suggesting that the supportive interface is all important.

I referred, above, to more than 100 conversations or interviews. Besides Aaron, Parveaz, and Charlotte, they included about 60 subjects in post-compulsory education and 50 in corporate, military or public services employment - none of whom had studied mathematics beyond the minimum level absolutely required by their education systems. To provide the clearest illustration, I'll take you to my youngest interviewee. Let me introduce Laura Broad: 13 years old, highly intelligent, but not mathematically inclined.

'Maths', says Laura, 'is OK - I don't hate it, but it's quite difficult'. She has her own PC, which she uses for many subjects but 'it's no good for maths'. She has a sound understanding of arithmetic; is happy with most types of two-dimensional plotting, graphing and charting; but algebra 'doesn't make any sense'. Expansion of brackets headed her list of troublesome topics. One of CalculationCenter's tricks is a set of solvers and calculators: pre-programmed routines that guide the user through program functions. One option is 'algebra', and below it is 'expand expression', so we started there.

If there is an expression on the screen, in a CalculationCenter notebook, highlighting it and clicking on 'expand expression' will open a calculator with that expression in place. Clicking the 'calculate' button runs it out to its expanded form. Alternatively, and more useful to Laura in her first few minutes of an unfamiliar environment, opening the calculator and clicking 'show an example' enters a simple example of the expression form which can be edited. Finally, clicking 'convert to text input' demonstrates how the calculator can be bypassed by typing in function and equation directly. Laura shed her problems with bracket expansion in five minutes, abandoning the calculator robot and learning some proficiency in text entry.

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  • Using the Expand Brackets helper, Laura Broad moves from problems with basic algebra (top left) to complex numbers (frame centre). The second box ('enter an imaginary number') shows how the helper sits within the program's normal operational context.

She also ran up against a problem. The expansion of (ax+b)(cx-d), for example, is to the form a.x.c.x+b.c.x-a.x.d-b.d, which was useful in demonstrating the mechanics but offered no help in getting beyond that to the acx2-(bc+ad)x-bd form. Looking up the terminology she knows (such as 'collect' and 'like terms') in the help browser yields nothing. Trying 'Collect[a.x.c.x+b.c.x-a.x.d-b.d]' yields an error message showing that such a function exists (but not how to use it) and that a parameter is missing (but not what it is). CalculationCenter's help system seems, curiously, less helpful than Mathematica's. Laura was able to convert the form manually without any trouble, but that rather misses the point; and repeatedly drawing a blank was dispiriting. I had to use prior experience of Mathematica to get her past this hitch.

Despite that hiccup, a lot was accomplished in three quarters of an hour. Moving on from expansion of brackets, for instance, and with CalculationCenter to carry the manipulative load, she was able to explore the idea of complex numbers. Similarly, with fast and easy 2D there was little problem with the extension of number definition space from line to plane. Finally, a quick and superficial look at the extension of bracketed Cartesian pairs to vectors and matrices allowed some play with the idea of groups. I don't pretend that this is firm, long-term learning, nor that Laura could have made this journey without guidance. Nevertheless, in 45 minutes she was able to encompass, without strain, a set of ideas a good five years ahead of her skill level. Her closing judgement was that CalculationCenter was useful, and fun, but within limits. It would be a good way to try out new learning, provided that its methods were easily discovered; where they were hard to find, she would probably do it by hand.

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  • Stepping stones. There is a 'construct equation' solver menu option to generate this system of simultaneous equations with nonreal solutions for x (background, top and left), but there is no need to use it once the web style paging interface is understood. In the foreground, Laura has opened the equation construction pane (top) to make it readily available and then gone straight to the 'solve an equation' option. The equations are then constructed by paging back to the symbolic palette using the left and right arrow buttons (bottom of left hand panel, above the CalculationCenter logo). In doing so, she illustrates how much she has learnt by using the program; the equations are designed to test her own understanding of the way complex numbers relate to bracket expansion.

The pattern played out in much the same way amongst my older subjects. With accessible software and a relevant or interesting idea to use it on, I met only a very few resistant cases. Everyone was seduced into playing with, and learning, maths well beyond what they were prepared to countenance beforehand. I don't claim that all (or even any) of my hundred or so respondents are future mathematicians; nor do I claim that all of them would necessarily extend what they did with me to new problems, if the software had remained after I have gone. Nevertheless, almost all are converts from disinterest at best (and loathing at worst) to somewhere between cautious and enthusiastic to the idea that they might use mathematics to solve real problems. Amongst them are a sculptor and a painter, lured into use by structural strain models and precise pigment colour specification respectively; an administrative assistant interested in what queue theory can do for her spasmodic work load (as a result of her ensuing suggestions, she has since been dramatically promoted); a historian developing experimental economic and political models for aspects of the Roman Empire; a janitor intrigued by the ability of network and transition matrices to model and explain previously mysterious behaviour in ventilation, elevator and other building infrastructure systems; and on it goes.

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  • Family resemblance, but crucial difference. The compact and efficient arrangement of Mathematica (foreground) contrasts with the much simpler, less intimidating though more user intensive, web-like CalculationCenter interface (background) which is so much a part of its encouragement to the wary. Between them, both in this graphic (midground) and in philosophy, is Publicon.

Comparing reactions to CalculationCenter's interface with that to other similar products (Mathematica included) the difference was clear. Only a small percentage was comfortable with any of the others. Those who encountered CalculationCenter before the alternative were more receptive to learning and trying than those who tried the alternative first. In every case, without exception, the interface was given as the reason for preference.

The negative side shouldn't be ignored. Almost always, as with Laura's 'collect terms' problem, the obstacles are a mismatch between what the user knows how to ask for and what the help system knows how to reply to. Every complaint or negative comment about CalculationCenter related to frustration at finding that the help system let down the interface. But that's something which can be worked upon, and developed. Who knows, by the time this is printed, perhaps the new CalculationCenter release will have appeared and swept away some of the problems.