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What is proof?

Some teachers might think that proof has no place in the primary mathematics classroom but we would argue that it has. At NRICH we believe that the ideas underlying mathematical proof are central to the understanding of mathematics as a whole. Generalising, justifying and convincing are key elements of mathematical thinking and they form the building blocks of proof. As Haylock (2006) says:
proof is 'a complete and convincing argument to support the truth of an assertion in mathematics which proceeds logically from the assumptions to the conclusion'. Another significant article about proof in primary mathematics classrooms is quite an old one by Lesley Jones (1994) but is is nonetheless worth reading and still relevant.

This idea of proof involves notions of conviction as John Mason (1982) has pointed out. He characterises these levels as being equivalent to convincing oneself, convincing a friend and convincing a sceptical 'enemy'. In our experience of working with young children on proof we find that they are responsive to the idea, easily start looking for opportunities to prove whether things are true, and
offer reasons for their arguments. These reasoned arguments can then be developed into fully fledged proofs.

Generic proof is a special kind of proof that works in rather a different way from the usual idea of logical steps progressing through to a reasoned conclusion, although that is a part of the process. However this is not generalising from a sequence or pattern but it is about situations when the structure of a specific example shows how it all works. This is a great idea for young children to come to terms with and offers lots of opportunities to examine the underlying reasons about why numbers work as they do.

Tim Rowland defines generic proof as 'carefully selecting a particular example that enables one to see, in that example, the general structure'. So what might this mean for us in primary mathematics classrooms? This month we have created a set of primary problems exploring some of the structures in our number system, particularly the properties of odd and even numbers, but also looking at the external angles of polygons so we are presenting ideas about generic proof in the sphere of geometry as well as number theory.

As an example let us look at this month's problem Take Three Numbers. The idea underlying this is of taking a run of three consecutive counting numbers. The children can choose a run for themselves such as 4, 5, 6 and look at the sum of the numbers. They can choose any sequence like this that they like. It doesn't matter where they start as long as the three numbers are all 'next door neighbours'. We are asking them to explore what happens when we add these three numbers together. Is there anything special about the result? Is this special result always true? If it is, can you convince me just through examining your one example? This leads us into a 'generic proof' by careful reasoning with our chosen example.

So how might we support children to do this? We will illustrate this by showing you how it might work for 'Take Three Numbers' and leave you to apply a similar approach to the other problems for this month. First of all ask them to explore their three consecutive numbers and find their sum. Encourage them to use diagrams and 'stuff' like multilink cubes to create models of the numbers and their sum. Make sure you actually do this yourself before you let them loose on it so that you are aware of all the obstacles and opportunities this modelling offers to us in finding a result.

Questions and prompts to use:

Can you tell me three consecutive numbers?

Make multilink towers, use Numicon or draw a picture to represent each.

What is their total?

Why?

How can you re-organise your numbers and their representations to show this?

Can you carry out this exploration yourself? Can you prove that your result holds for any three consecutive numbers by unpacking this one?

Don't click on this until you have convinced yourself of your result!

The key point to get across is that the proof lies in the structure of just this one example. How good is that?

On the website this month we provide some more opportunities to explore this idea of the structure of one example giving us generic proof - another opportunity to embed proof in primary classrooms. In addition to the context of number, there are also some geometric problems about polygons in which the generic proof relies on the idea of closing up the shape. See whether you can see, and help your learners to see, how the notion of 'Walking round a square' links with 'How much to turn' in giving a generic proof for the external angle sum of a polygon. As Tim, whose suggestions have formed the basis of this month's problems has said: 'It would be nice to have some more examples of this kind and we would be delighted to hear from any of you who find something similar'. Just email us at NRICH.

Haylock, D. (2006) Maths explained for primary teachers. London, Sage.

Mason, J., L. Burton & K. Stacey. (1982) Thinking mathematically. Addison-Wesley Publishers Ltd.

Jones, L. (1994) Reasoning, Logic and Proof at Key Stage 2. *Mathematics in School.* Leicester: Mathematical Association