THINKING ABOUT PROOF

I put before you my two main conjectures:

What is important about geometry is being aware of the fact that there are facts, rather than mastery of some particular few facts.

Geometry takes place in a world of forms and images, entry to which is gained through the power of mental imagery, augmented and extended by dynamic images, drawings on paper and discussion with colleagues.

(Mason, 1989) As mentioned earlier, geometry was not part of the ancient Greek curriculum. It was included in the medieval quadrivium (fourfold curriculum) and in Victorian times a two-column layout of assertions and justifications became a central part of UK gram- mar school education. Geometry was credited with teaching young people how to reason effectively and to educate their awareness about spatial relationships.

In the 20th century, geometry and its reasoning faded from the UK mathematics curriculum, largely because it seemed very difficult to teach learners to prove things, so learners were encouraged to memorise proofs instead. Recent curriculum devel- opment in England has recognised a place for geometry: there are geometrical facts that govern how the material world works.

More recent research in mathematics education has revealed some of the reasons why learners find proof challenging. The van Hiele (1986) levels give teachers a per- spective on adolescents’ experience of moving from truth and rules belonging to adult authority to discovering that mathematical truth comes from the mathematics itself and so is in the purview of learners themselves.Thus reasoning and proof in geometry offer learners an opportunity to become their own authority, within the social conven- tions of how mathematical reasoning proceeds, so that they need not take the word of any authority other than reasoning and mathematical structure.

Most of the difficulties encountered with proof have to do with knowing what you are allowed to assume, and what you are expected to justify.To prove, you need to

appreciate that properties can apply to a variety of objects, and that reasoning pro- ceeds on the basis of established and accepted properties only.

Being aware of how this way of working developed is sometimes helpful to both teachers and pupils.To sketch in a little historical background:

The father of all the Western philosopher mathematicians is claimed by many to be Pythagoras (sixth century BCE) who was described by the historian E. Bell (1953,

p. 20) as ‘mystic, mathematician, ... one-tenth of him genius, nine-tenths sheer fudge’. Bell suggests that Pythagoras was the first European to realise (or to learn from those with whom he came into contact from other cultures) that proof must proceed from agreed properties and explicit assumptions.

In the 200 years following Pythagoras’ teaching, there was continuing, but spo- radic, geometrical development in several Greek city states. It became increasingly understood that from a set of arbitrary postulates new ideas could be reached by close deductive reasoning. Proof Four in section 7.2 is an example of organising such a chain of deductive reasoning.

Socrates (469–399 BCE) and his pupil Plato (427–347 BCE) both worked in Athens.

Euclid (330–275 BCE) worked in Alexandria, a new city at the mouth of the Nile

founded by Alexander and ruled by one of his generals, Ptolemy. Euclid collected geo- metrical work and codified it into 13 books. The first six books are about plane geometry; books 7–9 deal with number theory; book 10 is about Eudoxus’ theory of irrational numbers and books 11–13 concern solid geometry, ending with a discussion of the properties of the five regular classical polyhedra and a proof that there can be no more than these five. Examples of all five polyhedra can be found in museums in Edinburgh and elsewhere, carved in granite balls by Neolithic farmers.

Euclid was a little older than Archimedes but two generations younger than Plato. A contemporary of Euclid was Aristarchos of Samos (who decided that the earth went round the sun and not, as previously thought, the other way about). For more information, see St Andrews (webref).

What Euclid offers to us in books 1–6 is a logical structure for plane geometry developed from a few postulates, or axioms.The axioms are not unique; a different set to the ones proposed by Euclid could be used. Leading up to the time of Euclid, the Greek geometry programme insisted that all geometry had to be made with a straight edge and pair of compasses only. From these two tools, the world of classical geome- try was open. It was also a model of ‘postulates asserted, theorems deduced’.

Ideas were communicated in writing between the various ‘Schools’ (groups of intellectuals who necessarily were landowners and so independently wealthy). What we have is written evidence that these ideas were communicated around the Mediterranean but we cannot know from the written evidence how much thought experiment and visualisation were used by the founders of Euclidean geometry in their creative teaching and thinking.

Fragments of stories have survived about individuals working as teachers. Plato comes over as a superb teacher. Euclid does not come over as a sensitive teacher. He is reported to have dealt with Ptolemy in the following way: Ptolemy attending one of Euclid’s seminars asked for a simpler chain of reasoning. Euclid’s cutting response was ‘There is no royal road to geometry’. On another occasion, a member of the Academy asked ‘What’s the use of this (theorem)?’ Euclid’s response was to address the rest of the group and say ‘He wants to profit from his learning, give the man a penny’. 112 BLOCK 2

A problem in working with ‘geometrical proof ’ at school or university level is that learners find it hard to enter into the rules of the game.What are the givens? What do we need to prove or deduce? So often the ‘fact to be proved’ appears to be obvious. Why are we taking all this effort to ‘prove’ when I know it is the case? Why won’t a few examples suffice, or perhaps a diagram in dynamic geometry software with the opportunity to ‘drag’ in order to test for generality?

Historical records identify only what authors chose to articulate and record: their best thoughts. To try to appreciate what they were thinking and how they reached their conclusions it is necessary to reconstruct objects that might have been manipu- lated, calculations that might have been done, special cases that might have been tried. To teach geometrical thinking requires tasks for learners that engage them in appro- priate manipulations, which give them opportunities to get a sense of relationships and to see these as properties independent of the particular situation, and finally to reason solely on the basis of agreed properties.

This chapter takes the idea of chains of logical reasoning and builds a number of ‘proofs’ from different starting points.The task, each time, is to ask yourself:

● Do I know what the givens (assumptions) are? What properties are used in the

reasoning?

● Is there enough manipulating to get-a-sense-of for me to believe this proof? ● What needs to be added, or telescoped, if I were to use this proof with pupils?