Proof: A Brief Historical Survey

Stage: 4 and 5
Article by Leo Rogers

Pedagogical Notes and Questions

1. In mathematics, you are never very far from infinity...
a) Think of the Pythagorean diagram for the square root of 2.
Does this visual representation constitute a proof?
If not, what do you need in order to make it convincing?

b) The method of exhaustion.
Imagine a polygon inside a circle. Now increase the number of sides. How far do you have to go to convince yourself (and others) that the 'polygon' eventually becomes a 'circle'?
Now try constructing a circle using LOGO software. Are the results in LOGO 'true'?

c) The square and its diagonal.
Thinking of the diagonal and the side as numbers, this in effect, is finding a common factor for the two numbers.
Example: chose any two numbers, say 27 and 15. Subtract the smaller from the larger, to get 12. Now subtract 12 from 15 to get 3. So 3 is the common factor of 27 and 15. (What happens if you continue subtracting the smaller number from the larger?) Try this with other pairs of numbers. Which kinds of numbers would you choose to investigate? This process is known as the Euclidean Algorithm.

2. Use dynamic geometry to draw an ellipse and put six points round the edge. Join the six points with lines to make a hexagon and label the points. Keep the points in the same order and move them so that you can find the intesrsections of opposite sides. What happens when you change the order of the points?

3. Definitions are important. Good definitions are elusive.
The relation F + V = E + 2 often appears in schoolbooks as something to 'prove' by filling in a table of examples, but are pupils ever encouraged to ask the question, "What is a polyhedron?"
Imagine an open cubical box with very thick edges.
How many faces, vertices and edges does it have? Is it a polyhedron or not?
Think of a shape like a picture frame with a triangular cross-section. Does it fit the formula? Is it a polyhedron? What other possible 'polyhedra' do not fit the formula, and how do you decide what polyhedra are allowed?

4. Discussion in the Classroom - it happened in history too!
The important story told by Lakatos shows that theories do not appear suddenly 'perfect' in the minds of mathematicians, but come about through long periods of discussion, objection, argument, and efforts to clarify ideas and definitions. His account is written in the form of a discussion in a classroom.
For teachers who wish to organise discussion and mathematical thinking in the classroom, see Watson and Mason (1998).

5. Chinese and Indian methods.
Contemplate the diagrams. Look away or turn off the picture. Can you draw the diagram yourself? What properties and relations do you know which can help you see 'more' in the diagrams? What new things you can you discover?

6 . Investigatons.
In 1964 the ATM invited Lakatos to a seminar, which was one of the important factors which inspired the development of 'Investigations'. Lakatos' thesis was in the philosophy of science. It was about the way in which discoveries in mathematics really happened - by challenges to established ideas, discussion and even argument. It provided a theoretical basis for the use of discussion and promotion of individual research by pupils in the classroom.

7. Students' views on Proof.
D'Amore (2005) shows that "... other cultures have produced intellectual mechanisms of 'truth', generalisation and prediction different from Aristotle's logic... " and the kinds of reasons for believing something is true, offered by his Italian teenagers, correspond closely to a particular style of traditional Indian logic. The fundamental belief of this kind of logic, is that we experience truth through our senses, and of course we do this, from a very early age. Are we paying attention to these kinds of cultural differences in pupils' modes of reasoning in our classrooms?