Let primary students cut to the action with these intriguing 3D shapes, says John Dabell.
As weapons of maths instruction, Sliceforms are models of engineering. They are collapsible 3D mathematical configurations fashioned out of flat sheets of card wafers interleaved together. They are assembled by intersecting multiple parallel planes (slices) within the geometric shape. These intersections act as a pack of hinges. Each model can be constructed to flatten in two different ways and pass through a myriad of distinct but related shapes between these two extreme positions.
Sliceforms are the manifestation of the belief that maths is creative, artistic, colourful and dynamic. These curious and interesting magical marvels have largely passed unnoticed since their mathematical birth.
Mathematicians Olaus Henrici and Alexander von Brill used Sliceform skills more than 100 years ago to make models using cross-sections of quartic surfaces. It is thought that Russian constructivist artist Naum Gabo was inspired by these surface models to make his sculptured heads which are now in the Tate Modern, London. Today, John Sharp (above) is at the cutting edge of mathematical surface modelling and continues to act as the leading light adapting and evolving similar techniques.
Sliceforms have many advantages for the primary maths curriculum. They are highly practical, they teach flexibility and creativity, expand construction skills, encourage investigation and exploration, expand mathematical vocabulary, teach mathematical formulas, definitions and concepts and are cross-curricular (links can be made to art, design and technology, science, history, literacy) and are cheap to make.
There are also disadvantages to consider. Sliceforms are time-consuming, need a lot of individual teaching, require manual dexterity which could demoralise children with poor motor skills, and may be difficult, delicate and complicated to assemble.
But the benefits considerably outweigh the disadvantages. They may be more suited to key stages 2 and 3, but ability ranges within classes always make a mockery of such divisions. Age does come into the equation when teaching Sliceform geometry because the motor skills required to cut and manipulate pieces are tricky. The success of Sliceforming will largely depend on learning preferences and modes of learning.
Sliceforms suit the learning styles of visual thinkers best but will provide food for thought for logical thinkers as well.
Right-brain learners are good with spatial relationships, so that while they may not be good at dividing, they may be extremely good at visualisation and hands-on geometry. Left-brain learners may find Sliceforming harder because they deal with things in a linear way. The left-brain is reality based, verbal, temporal and can deal with abstract concepts. It is ideally suited to the skills of handwriting, symbols, language, reading, phonetics, finding out details and facts, talking and reciting, following directions and listening. Education has mostly centred on left-brain skills so Sliceforming contributes meaningfully to right-brain learning and should allow right-brain learners to thrive for a change. Sliceforms do deserve a slice of the maths action in our classrooms but not without considered planning, patience and plenty of practice.
Sliceforms will always remain a playful example of maths and artistic exploration at its best.
John Sharp's book Sliceforms: Mathematical Models from Paper Sections, pound;3.95 from Tarquin Publications, includes eight models to cut out and construct and details of how to construct your own.Tel: 01379 384 218 www.tarquin-books.demon.co.ukAlso available in June is Surfaces: Explorations with Sliceforms from QED (pound;19.95) which looks at the geometry of surfaces. Tel: 01494 772973 www.mathsite.co.ukOf paramount importance is to have a go yourself and decide the suitability of Sliceforms for your pupils. Why not try making a zonohedron (see box)? (Go to www.counton.org and click on the Explorer icon to access the section on Sliceforms)
A zonohedron is a rhombic prism in which all the faces are rhombuses and all edges have the same length. Slicing in two directions gives two sets of planes that are identical.
A good starting point to this activity is to make a zonohedron from a cube using art straws joined together with string. If the cube is deformed by pulling two opposite corners away from one another so that the sides of the cube remain parallel, a zonohedron is formed.
A zonohedron Sliceform is constructed using six template slices. Each slice has three slots and there are two ways these are cut. To start, cut out all the pieces and cut the slots on each one. To construct, start with the two slices that have opposite slots in the centre as these are the central slices. Now add the two outside pairs of slices in each direction. Ensure that you insert them the same way as the central ones in each direction.
When you have made the model, fold it flat in two directions.
It is important to cut a slot the width of the card and not just a single cut. If you don't, the card will fight for space and the model will be distorted and not fold properly. Make two cuts either side of the slot line and then gently pull out the hair-like piece you have cut out.