Graph 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 looks like I’m 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”.

We assembled a tolerable Cartesian cartoon representation of Nearly Headless Nick, behind a transparent acetate screen 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…

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 involved had encountered it, if not used it, before. For more advanced work it could be replaced by 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.

[originally posted on Scientific Computing World's education pages]

Cabri-Géomètre II

Formalised geometry 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 powered by the backing of calculator manufacturer Texas Instruments, offers the solution. It is 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.

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 our trial a physics teacher borrowed 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.

[originally posted on Scientific Computing World's education pages]