Pure maths takes a small number of basic agreed starting points: the axioms. It sets out statements: propositions are basic statements, theorems are big, important statements, and corollaries are statements that follow directly from theorems. It then proves them by showing that they follow on logically from the axioms or earlier propositions and theorems. This presentation has it origins in Euclid's Elements. Geometry is probably the closest that GCSE maths comes to the real pure maths of Euclid - requiring not only that we find the answer to a problem, but that we explain why.
Why do the angles of a triangle add up to the same as a straight line? First, we could demonstrate that they do by drawing any large triangle on a piece of paper, cutting it out, tearing off the three corners (quite big pieces to make it convincing), then fitting them together to make the straight line. This makes quite an impressive demonstration, but it does not explain why it works, only demonstrating that it works for the particular triangle that you drew. How can students be sure that it will also work for any other triangle? They could, of course, draw lots of triangles and test it out, but there still might be one triangle that they have not drawn that it will not work for. They could not draw all of the possible triangles, so they need another approach. The mathematical approach would be to show that it must be true for all triangles, using facts that they already know to be true. One fact that would be known at this stage is that if a pupil draws two parallel lines and a line across them, then the marked angles must be the same (Figure 1).
Now they use this fact by drawing any triangle and a line parallel to the base of the triangle (Figure 2). Because of what they already know, they understand that the angle marked A in the triangle is the same as the angle marked A on the lne. The two Cs are equal for the same reason. Pupils can then see that the three angles inside the triangle fit together to make a straight line. The argument did not depend at all on the particular triangle that they chose, so it must work for any triangle. This is a proof. Euclid would have expressed it more elegantly and modern mathematicians would have relied on symbols, but the essential idea remains the same.
I recently took a lively Year 8 class for geometry, using Circle Scribe disk compasses. These consist of a large fat acrylic plate with a shallow steel point in the centre and a huge range of holes into which a pencil can be placed to produce circles of varying radius. It takes the place of a normal pair of compasses for many tasks and makes it easier to draw accurate circles. The task was the construction of a hexagon inside a circle. Pupils drew the dotted circle first, then drew a circle the same size with its centre on the first circle. Finally, they drew five more circles with their centres on the crossing points made by the previous circle and the original circle. They joined up all the crossing points to make the hexagon. If they carried on drawing circles on all of the new crossing points then they would create a tessellation of regular hexagons. Pupils produced satisfying results on a task where accuracy is important (Figure 3).
In the Elements, Euclid uses "construction" as a powerful tool in proving his propositions. How do we know that the hexagon we have drawn is a regular hexagon? In Figure 4 there are six triangles. Looking at the sides of one of these triangles in the original drawing, pupils will see that each side is the radius of one of the circles. All their circles were the same size, so the triangle must be an equilateral triangle. So all of the sides are equal. Hence, the hexagon is a regular hexagon. Again, they have proved their proposition.
The Circle Scribe can be used to generate more questions. Could you construct a pentagon? Would it make a tessellation?
Welcome to the world of proof and the power and beauty of Euclid.
* The Circle Scribe costs pound;4.50 with discounts for larger sets. Accompanying book Fun, Art and Geometry by Bill Harper, pound;5.50, from Circle Scribe. Tel: 0151 289 5681. Web: www.circlescribe.com Chris Olley is a maths education consultant. E-mail: firstname.lastname@example.org
* EUCLID'S ELEMENTS
Written around 300 BC the 'Elements' is probably the most famous of all mathematics texts. It runs to 13 volumes covering most of the maths known to the ancient Greeks at the time. The sections dealing with geometry are the best known, covering issues of angle and length in triangles, other polygons and circles. Euclid laid down a small number of essential principles that he held to be true and then devised propositions that could be shown to be logically consistent with the principles. This art of proof has been at the heart of all subsequent maths. Web Link: http:aleph0.clarku.edudjoycejavaelementselements.html