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?