This was originally published in Scientific Computing World, in August 2001, but is no longer available on line so I have reproduced it here.
One day, blind rabbit and a blind snake met on the road. The snake reached out and touched the rabbit, then said: “You are fluffy, with long floppy ears; you must be a rabbit.” The rabbit, in its turn, reached out to touch the snake, then said: “You are dry, hairless and scaly; you have a flat triangular head, forked tongue and small bulging eyes with no lids; you must be a maths teacher.”
Why is mathematics so universally and profoundly unpopular? Why do we passively accept that it should be so? And how far can methods used in scientific computing offer help in finding a solution? It is, I suggest, something which should worry us very much indeed, because it represents a fatal waste of resources. We are (as no Scientific Computing World reader needs to be reminded) an irrevocably science based world. Science professionals are our primary resource, and in a wider sense every citizen must be, to some extent, either a scientist or effectively disenfranchised. Science depends upon conceptual models built and expressed mathematics, a language whose acquisition does not come easily to most human beings. Whether the difficulty is down to nature or nurture, it threatens our supply of mathematically competent potential scientists at a time of escalating demand. One professional acquaintance, doing leading-edge work in a well known mathematics department, privately describes how she was branded an unteachable failure in the subject until, in her mid-twenties, an evening class lecturer spotted her talent and unlocked it. How many other potential mathematicians and scientists are never spotted, and are lost to us for ever? We cannot afford to continue in reliance upon recruiting from that small minority who are either “naturals” or chance survivors.
In recent decades, much has changed and teachers are in the vanguard. But our education systems still bear the mark of mediaeval origins, and the most indefatigable teacher only has so much time in an overworked life. I spend a large minority of my time working with education professionals at various levels; I was interested to find out for myself how far “scientific computing” could improve the support of mathematics learning.
I chose a clutch of software products, and tried them for size from roughly age 9 through to PhD. There is an assumption that age dictates topics and methods; I have never accepted this. Rubber bands are an excellent aid in illustrating vectors to undergraduates, and seven-year-olds find aspects of number theory intriguing. There are limits, of course, but they are far more elastic than conventional wisdom admits. Since we haemorrhage potential scientists most prolifically during the school years, they have my emphasis here; but experience gained there is equally relevant in the upper levels which may be more familiar to most readers of this magazine.
There are two main reasons why maths turns, at some stage, from learning to misery: the hierarchical nature of its learning, and its abstraction. My choice of software was made with these in mind, and were all recognisable modifications of scientific computing applications software. Most were predominantly visual; some emphasised the symbolic; and a hybrid pair straddled both camps.
Primary education is the strongest phase, but even here (see box 1) computer-aided methods have the potential to keep future scientists from slipping through the net. Rather than mount one-off extravaganzas, I planned added-value activities with class teachers in support of their existing themes. This sort of work is great fun -- where else can a mathematician work on magic in the morning, the Spanish armada in the afternoon, and haiku before going home?
Omnigraph saw its first outing in the service of magic. Magic is a stage in the developmental history of science -- a history which each of us retraces as we grow to intellectual maturity. Its study as such by eight-year olds was designed to meet criteria in cultural history and imaginative creation, but also as a context for strengthening critical faculties. The ability to rationally assess likely and implausible explanations of phenomena makes great strides at this age; separation of reality from model is central. A link between mathematical models and spells, illustrated by Omnigraph, was well received and opened up a riot of speculative theorising. It also offered a new stage on which to parade the key concept of the algebraic “placeholder”.
Omnigraph is a graph processor, with facilities for investigating a number of mathematical areas up to very basic calculus. Equations or Cartesian coordinates, entered from keyboard or menus, are instantly reflected in curves, lines, points and shapes drawn in the graph window. Or, looked at another way, “spells” in the lower window produce magical results in the upper one -- but rules can be deduced, even at this age, to predict the result of any given spell. The mouse changes scaling, draws tangents, normals, areas, and the rest; curve drawing can be paused or abandoned, and in many cases the equation/spell is displayed as the mouse passes over a line. A quadratic spell produces a passable model of the path followed by Harry Potter’s broomstick as he swoops to aid a Quidditch team-mate before returning to his normal altitude position. We can also play the part of the villainous Quirrel, interfering with the spell to alter Harry’s flight: alter one part of the spell (the “m” coefficient) to induce suicidal recklessness; change another (the constant “c”) to pull him out of the dive earlier -- or cause him to crash!
If it seems that I am getting carried away -- well, perhaps I am: there is nothing more inspiring than watching young minds leap over their fears and years to grasp an idea. By the end of the morning, any member of the class could evaluate the value of y for any x, plotting the results on a graph paper Quidditch field. They could also deal implicitly with negative values for“m” and “c”, expressed as subtractions in a modified “spell”.
A tolerable Cartesian cartoon representation of Nearly Headless Nick was assembled, behind a transparent overlay carrying a Hogwarts map. The pupils derived great amusement and insight from altering transformation matrix-spells to move Nick about the castle, expand him, shrink him, distort him in various ways ... but enough, for now, of magic.
After the primary years, learners are removed from the integrated learning environment into separate academic subjects. Suddenly, maths is no longer an aspect of global learning but a discrete and arcane discipline. One year seven neighbour told me, miserably, that “last year, maths was real ... like designing an Olympic stadium ... but now it’s just A, B and C filling a bath with a teaspoon while D empties it with an egg-cup and nobody tries to stop him ...” At 12-13, Harry Potter (sadly) ceased to be appropriate. With the maths, history and games teachers I designed a set of “kriegspiel” tasks lasting several weeks, starting with the representation of chess moves and moving on through the modelling of a team game to exploration of a battle, a migration, the launch of a pop group. 
Autograph came into its own, here, its data handling aspects providing a suitable environment from beginning to end. With greater control over display, two-click access to analyses, and constructs such as vectors or dependent data points positioned in relation to other objects, it also lends itself to more extensive modelling activities. Everything from the eight by eight chessboard to contoured maps, epidemics to explosions, could be modelled (albeit simplistically) in Autograph. Translations represented movement of teams, armies or populations, other transformations their spatial behaviour and magnitude.
Omnigraph is a simple, no frills program in its interaction with the user -- which makes it very transparent in use. It is also well known; all the teachers with whom I worked had encountered it, if not used it, before. For work with curves and lines, it could be replaced by the more polished Autograph; this would sacrifice instant usability in favour of added options. Both programs work well in conjunction with graphical calculators, for teaching at the levels where those are appropriate. Autograph offers stronger tools (eigenvalues, for instance), enhanced display options and statistical data plotting.
The move from arithmetic to symbolic algebra is the biggest terror of pupils at this age, and many of our future scientists are lost over the edge at this fracture plane. Graphical work is popular but must support symbolic work, not pleasurably obscure it. Pupils were encouraged record and express what they were doing in standard symbolic short-hand, and to share summaries of the results on an intranet web site. They were introduced to MathType, and their maths teacher used this in place of a whiteboard or hand-outs for his explanations of what was happening in Autograph. Only one copy of MathType was available, but once used to it pupils did much of their work in the cut down version bundled with Microsoft Word, returning to the full program when necessary.
MathType appealed to these teenagers. Its quality of output built their pride in their work; it was used to prepare their worksheets, and they had the experience of feeding back work of equivalent production values. Its ability to produce high quality web material gave them a high-status platform for displaying their achievements. (A new version 5, currently in beta, is more effective still.)
Another catastrophic jump (in both mathematical and human senses) which costs us swathes of pupils in this stage is the advent of formalised geometry, which can seem a meaningless set of hurdles. “To do geom” observes Geoffrey Willans’ schoolboy antihero, Nigel Molesworth (Down With Skool, 1958), “you hav to make a lot of things equal to each other when you can see perfectly well that they don’t”. Dynamic geometry software such as Cabrie-Géomètre II (CG2), a program developed in France and (like Derive -- see box 2) powered by the backing of calculator manufacturer Texas Instruments, offers the solution. Using similar interface methods to Autograph, it was an excellent platform for investigating detailed aspects of the Autograph models -- as preliminary learning in advance, as subsequent consideration of observed phenomena, or as both in a refinement loop.
If Omnigraph is a graph processor, CG2 is a geometry processor adding to axiomatic Euclidean geometry the active, participatory element of transformational or analytic geometry. Here is an opportunity to discover for oneself, in a hands-on way, where the axioms came from. It allows fundamental components (points, lines, shapes) to be combined and moved in ways which obey geometric definitions. If a line is defined as a tangent to a circle at a particular point, for example, then the circle, line and point can all be freely moved around, the circle resized, and so on, but the line will remain tangential to the circle at that point. Additional constraints can be used for particular purposes, as can slider controls. A number of ready-made examples are provided, ready for instant classroom use. During the trial, a physics teacher, borrowing it and used two lines, a circle and an ellipse to demonstrate both the inverse square law and the cause of eclipses in a single pass.
At every stage, the software encouraged rapid explorative investigation whilst also pegging the mathematical representation back to a concrete reality comprehensible to the pupils. At half term, when the experiment ended, both teachers expressed satisfaction at the increased levels of motivation and learning; as I write, they are preparing a bid for purchase of Autograph and CG2 in the next budget round.
(For anyone particularly interested in use of dynamic software, word arrived of a new discussion group just as this piece was being finally put to bed; it can be accessed through http://www.jiscmail.ac.uk/lists/dynamic-geometry.html.)
Finally, learners move into full specialisation -- a phase which lasts the rest of our lives and finally separates us, often irrevocably, into those who accept mathematics and those who do not. More than half a century ago, Brenda Colvin explicitly warned (Land and Landscape, 1947) of this separation as “becoming a danger to society”, and C. P. Snow famously addressed the same issue (The Two Cultures, 1959) but there is no sign of the gulf narrowing. From this point on, learners of mathematics fall into one of two types: those for who mathematics is part of the core activity and those for whom it is an unwelcome adjunct. The line between the types is not a sharp one; many science and engineering students secretly inhabit the latter category, for example, which should worry even those who do not share my belief that “we are all scientists now”. For those whose chosen path has mathematics at its core, software must support deeply learned mathematical skills. For the rest, it must remove impediments to easy use of those skills.
I worked with just over twenty different learner groups during the time in which I was exploring this software: from a class of year ten (14-15 years old) school pupils through onward in a range of scientific and not so scientific settings. Autograph, Omnigraph, MathType and CG2 were still in use here, their value undiminished, but in combination with Fathom, the Warwick Spreadsheet System (WSS), and Maple or Mathcad with education packs.
WSS is a structured system operating which completely colonises its Microsoft Excel host to generate an environment specifically designed for science and applied mathematics education. Supported by a comprehensive and impressive range of content, it introduces mathematical modelling through a range of contexts from business studies to biology, taking in astronomy, engineering, geology, physics and population dynamics, amongst others, along the way. Designed to work up through a skill spectrum, it is not meant to be used piecemeal although that is what I did to good effect in the time available. Several teachers and lecturers commented that the content provided by WSS could keep them going throughout a course, without looking elsewhere.
Startup shows only one menu option, “file”, on which several familiar options have disappeared but some new ones appear; “open data logger file as spreadsheet” is a new arrival, for example. As a minor irritation, most normal toolbars are hidden on the next conventional startup and must be redisplayed. The supplied spreadsheets contain not only data but linked systems of data and chart files with their own, relevant menu systems. We lacked the time to build new examples, during the trials, but we did repopulate sheets with student-generated data. We also did some extensive poking around in the works to see how the examples were constructed; teachers and lecturers are, in real life, short of time for creative building work and these well-constructed exemplars will considerably shorten the development loop.
An utterly different implementation of a very similar conceptual approach lies behind Fathom. Essentially a dedicated statistics education environment which mimics a mainstream package tailored to learning needs, this too presents opportunities to explore data handling as a mathematical model for other subject areas.
Favourable impressions of Fathom start with the browser-based help system. Without being patronising, this reassuringly welcomes users without assumptions of existing knowledge. None of the staff or students who used Fathom had previous experience of it, yet none felt intimidated or resistant. It has a visual drag-and-drop approach which goes well beyond even such graphical mainstream packages as Statview, plots change with a click, user coefficients can be represented as sliders for instant investigation of effects on a model, and so it goes on. The only reservation is that supplied sample datasets are very markedly US-centric.
As with the younger learners, software in this top layer had to be engaging, empowering, and amenable to use within game-style scenarios with real world relevance. Geometry, graphing and data handling lend themselves to this; symbolic work, equally essential, is harder to handle in this way. Ray Girvan has discusses some of the possible approaches in relation to Derive (box 2) but for this area I chickened out and used what I knew best: Mathcad and Maple.
All such market leading products spawn “add-ons”, many of which are often available for free download. In this case, good use was made of “Math Class” and “Post Secondary” packs for Maple as well as Mathcad’s “Applied Statistics” and “Real World Math” electronic books. All of these are usable as-is, or can be raided for fragments which serve an umbrella concept.
Most productive of all, however, is the power of the software itself to amplify and facilitate open exploration. Both Maple and Mathcad, like many similar products, interface with Excel to good effect. Spreadsheet arrays can be used as one of several views onto a model and the experimental reality which it reflects, with the symbolic form of the model residing in the algebra package but linked to tables in the sheet. With some ingenuity, it possible to have manipulate a symbolic equation and see the results in (for example) Autograph or Fathom.
I do not pretend that a combination of playfulness with computer-based methods is an instant cure-all for any and every educational ill; it clearly is not. One problem, ever present, is the age-old need to ensure that a tool does not become a substitute for the skill which it seeks to amplify: my parents were understandably suspicious of the slide rule which I was allowed to use, today’s parents are equally leery of the pocket calculator, and the computer is the latest heir to the same scrutiny. They are right to be cautious; these aids are only a benefit if they are used with understanding in support of existing skills to facilitate the acquisition of new ones.
Then again, there are sociopolitical issues around the hazy border between education (whose Latin root means, let us never forget, “to lead out”) and cultural programming. Naomi Klein points out, and my more acute students remind me, that the commercial underpinning of the drive to information technology has “managed, over the course of only one decade, to all but eliminate the barrier between ads and education” (No Logo, 2000). She could also have mentioned that it has all but eliminated the distinction between education and training.
Nevertheless, software tools represent one more set of specialised arrows in the quiver, of potent use against specific and particular educational problems. The ‘black box, white box’ approach described by Ray Girvan (Box 2) is one way to address the “aid versus crutch” argument. Equally valid in its place is the inverse: teaching and assessing skills first, without software help, then allowing software to power the subsequent use of those skills. Both methods, properly applied, use the amplifying power of information technology produce that most fertile of learning environments: one of individual investigation and exploration. And playfulness is, if not sufficient, certainly essential; we will not get significantly expanded recruitment to mathematics and science tomorrow unless we make them significantly more fun today. All of this software can be used to play more effectively.
Kaylie & Matt investigate latent heat of fusion. (By Chandra Sinha)
Encouraged to use open-ended experiment as a teaching method, I asked my class of ten year old year-5 pupils to investigate what happens over time to water placed in the freezer compartment of a refrigerator. Each was given a spike-and-dial thermometer, and there were also five Xemplar PocketBooks (small, relatively inexpensive palmtop computers) available on a first-come, first served basis. The PocketBook offer was taken up on only one machine – by Kaylie and Matt, a friendship-pair living in the same street. Assessed as being close to the bottom of the class ability range, their motivation for volunteering seemed a mixture of laziness and novelty interest. Accustomed to paper and coloured pens myself, I paid little attention to the low IT take-up.
Observation sheets were prepared in class, the plotting of results on graph paper discussed and practised. Matt and Kaylie sought help with design of a spreadsheet and, by the end of the lesson, had a computerised record form with automatic, auto-scaled plotting. Most of the pupils were, at this stage, interested in the experiment and eager to get started. I advised thermometer readings at roughly 15 minute intervals, then sent them home to experiment. When they returned after the weekend, the difference in educational outcomes between paper and spreadsheet was marked.
Data were patchy, invariably oversampled in the first hour or so but increasingly sparse thereafter. Plotting had generally been abandoned early on. It seemed that my foray into experiential learning was a failure. Kaylie and Matt, however, had been drawn by the automatic plot into enthusiastic continuous monitoring of the temperature curve. Matt said, in his write up: "we thot the flat bits was weird, so we looked then to see what it looked like. Then we looked again ever time it moved again."
Conclusions drawn by most of the pupils were limited to a single figure (although it varied considerably from pupil to pupil) for the time taken freeze solid. Kaylie, having watched the data assemble, said that the water "nearly froze in about two hours, but then it stopped and thought about it for a long time." I prompted with questions about what happened before and after the water froze; most of the class said "nothing" but Matt disagreed, saying that it "jiggled about"; Kaylie added that it "kept stopping and starting."
The pair asked permission to import their handheld data into a desktop spreadsheet for examination during their lunch hour. Returning later, I was startled to find that these two supposedly low-ability pupils had entered all of the class data on their own initiative, plotting multiple graphs for comparison with their own. In an impromptu presentation to their class mates, they took over my role in the lesson to showed considerable insight into the probable significance of similarities and differences between the graphs. They had also merged the sheets (inventing x-wise data transformation in the process) and were eager to discuss the implications of model which they perceived in the resulting scatter graph. I had learned a lesson of my own; I now start any similar activity from computerised methods, rather than working up to them.
(Chandra Sinha is a primary school teacher; at the time of this case study, she was in her probationary year at an inner-city school.)
White box, black box (by Ray Girvan)
Derive is now at version 5.02, fully compatible with Windows so that 2D and 3D plots, text regions and OLE objects can be embedded, and saved, in the algebra worksheet. However, the interface hasn't much altered from its MS-DOS days: no spreadsheet-style recalculation, nor typeset-quality algebra, but simple manipulation of formula by command line and pull-down transformations. Derive offers built-in functions at late high school or first year university level, augmented by 'utility files' of saved function definitions for specialist areas such as ordinary differential equations, elliptic integrals, Laplace transforms, and so on.To traditionalists, the obvious objection to a Computer Algebra System (CAS) is that it will teach a student nothing about algebra, since it jumps straight to a solution. This is often true of Derive, but it may also confuse a student by jumping to an unfamiliar solution equivalent to the textbook answer, or even away from a solution if they have chosen an inappropriate simplification option (Collect and Expand set the transformation direction for identities - for instance, exp(a+b) <---> exp(a)*exp(b) - that need learning).Furthermore, Derive exposes students to rigorous mathematics early on. For instance, the Expand command needs to know by what variables, and whether the expansion is trivial, square-free, rational or radical. Similarly, input and output precision, solution branches and variable domains need declaring.This mix of simplicity, rigour, and occasional outright surprises, needs special strategies to fit into traditional schemes for mathematics teaching. By a combination of circumstances, Derive's possibilities in education have been unusually well-explored as its compactness and low price have favoured the educational market. Educationalists can draw on the long experience since its official adoption in the early 1990s by Austrian grammar schools. The Derive User Group (DUG) was founded by Austrian teacher Josef Böhm, and continues to share related user experience with TI-89/92 calculators, whose similar CAS was coded by Derive's authors.While Derive's application in education is still very much an active research area, it's one informed by a decade of practical trials. Many commercial and non-commercial Derive publications give teachers access to work reported at DUG conferences and the annual academies of the Austrian Centre for Didactics of Computer Algebra (ACDCA).Some educationalists, such as Dr Bernhard Kutzler of the University of Linz, argue for a general rethink of algebraic skills: students should learn "a general minimum competence" and skills such as order-of-magnitude estimation, while much routine algebra and special case rules such as (a/b)+(c/d)=(a*d+b*c)/(c*d) could be dropped from the syllabus.There are also more specific new techniques developed for CAS. With Buchberger's 'White Box / Black Box' Method, students learn an algebraic technique by hand, then use Derive to apply it routinely. With 'Black Box / White Box', they use Derive to experiment first, then postulate and prove rigorously a consequent result; for instance, differentiating various functions of x to discover the general form for d/dx(x^n) or d/dx(f(g(x)). Other possibilities are the 'Module Method', working from pre-programmed Derive files created by the teacher to explore a particular problem; and 'Window Shuttle', understanding a function by replotting a graph for various inputs, which would be impossibly tedious with manual graphing.
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