Published April 2009,February 2011.

The role of visualisation in
the learning and teaching of mathematics. Educational Studies in
Mathematics 52 (3) 2003 (214' 241)

If you can find this in a library, it is certainly worth
looking at.

a) 6x6 Squares of 'dotty paper' have been used here to
indicate the potential for pattern forming and recognition that may
have gone on in early times, and can certainly be used in the
classroom with good effect. There are many possible
explorations:

Odd and even squares and rectangles can be divided with
straight lines to form squares, rectangles and triangles, and
pupils can be asked about the relationships they see. Two
triangular numbers make a square number; what about other figurate
numbers and how they can be built up, etc., ?

b) Building squares on a square dot grid. At some time or
other, pupils meet the diagonal of a square and its irrational
properties. The two examples in this section show how, even with
primary pupils, we can get into discussions about why the square
has a 'whole number' area but we cannot find an exact number or
fraction for the sides. Building sequences of 'skewed' squares
where there are a dot at each corner but no dots on the edges, can
lead to other interesting explorations. See the problem Tilted
Squares .

a) The Method of False Position is an ancient way of solving
proportional problems and is clearly related to the 'unitary
method' taught in school today. The general version of the Egyptian
problem can be posed as: A number '$x$' and its fraction makes
'$N$' or, $x + $(a fraction of $x$) makes $N$

For example:

$$x+\frac{1}{2}x=9,12,15,...
~~~~~~~x+\frac{1}{3}x=8,12,16,...~~~~~~~x+\frac{2}{3}x=5,10,15,...$$

Problem 24 of the Rhind Papyrus states: '$h$' and one seventh
of '$h$' make $19$.

This method was modified as 'Double False Position' in order
to solve more complicated problems like finding the gold value of
particular coins after they had been changed into a different
currency.

b) The properties of the right-angled triangle are one of the
most fundamental pieces of mathematical knowledge. All the ideas of
ratio and proportion can be illustrated with this simple diagram.
The concept of ratio is fundamental to mathematics, and it is the
way in which we are able to compare one thing to another, and set
up relationships between different properties we discover in the
world. Experimenting to see whether one phenomenon is related to
another is the basis of scientific method.

Ratio

The Oxford English Dictionary's definition of Ratio
says:

"Ratio is 'the relation between two similar magnitudes in respect of
quantity, determined by the number of times one contains the other
(integrally or fractionally)".

This is almost exactly the same as can be found in Euclid Book
V Definition 3.

Equal Ratios:

If quantities of the same kind, $A, B, C, D$, can be put in
the same ratio so
that

$\frac{A}{B}=\frac{C}{D}$ then $\frac{nA}{nB}=\frac{C}{D}$ and
ratios of numbers, for example $(
\frac{1}{2},\frac{2}{4},\frac{3}{6}, ...)$ define the equivalence
class 'one half', etc.

Proportion
:

Quantities that have the same ratio are called proportional,
so the fact that$\frac{A}{B}$ is proportional to $\frac{C}{D}$ can
be used to set up relationships between unlike quantities, if there
is sufficient evidence to show a relationship exits.

Until the seventeenth century, all proportions were expressed
in terms of like
quantities. Galileo proved that falling bodies increase their speed
uniformly, and showed that the increase in speed was proportional
to the square of the time. His practical experiments established
proportional relationships between different quantities , speed, distance
and time. This was one of the great breakthroughs in the
application of mathematics in experimental science.

Galileo Galilei (1638) Two New Sciences. New York. Dover
Reprint. (pages 174/79)

(a) 'Peg and Cord' geometry of the Sulbasutras remained the
fundamental practice for setting out the foundations of buildings
and engineering works for centuries. All the 'ruler and compass'
constructions in Euclid Book I are derived from these ancient
methods.

Constructing two straight lines at right angles requires a
knowledge of the fundamental properties of intersecting circles.
There are many constructions; squares of different areas, triangles
of various kinds, parallelograms, etc., using 'peg and cord' that
pupils can make outside on the field or in the playground.

(b) Sanscrit text often replaced numbers with object words
like 'eyes' for two. Pupils could replace number words with words
for objects in writing instructions for a mathematical calculation.
Having made the 'sentence', pass it to a neighbour to translate and
perform the calculation. This could also work for geometrical
constructions.

You may know of the 1965 book on Vedic Mathematics (with
various reprints up to 2004) that claims to be based on an ancient
mathematical system 'rediscovered' from Vedic scriptures. While the
exercises themselves might be useful, there is no historical record
of these arithmetic rules in any traditional study of the
Vedas.

Start with a game: "I am thinking of two numbers; their sum is
seven and their product is twelve, what are the numbers?"

Start with some easy ones - avoid the use of
calculators!

Sum 9 and Product 20, or Sum 10 and
Product 21'...

The numbers get bigger and bigger,
proceeding to e.g.:

"I am thinking of two numbers; their sum is 53 and their
product is 696, what are the numbers?" (Calculators might be
allowed as the numbers get bigger.)

By experimenting with these numbers, pupils would be looking
at partitions of the sum, and factors of the product. A possible
rule from these experiences would be:

'Take half the Sum number and
try adding and subtracting the same amount (+/- $1$, +/- $2$ etc.)
and multiply the results together to find the Product
number.'

e.g. For Sum $7$ and Product $12$

Take $\frac{S}{2} $ | $\frac{7}{2}$ | |||

Square it | $\frac{49}{4}$ | |||

Subtract the Product number | $\frac{49}{4}-\frac{48}{4}$ | |||

Find the square root of the result | $\frac{1}{2}$ | |||

This is the number you add to and subtract from | $\frac{7}{2}$ | |||

So we have (7/2 + 1/2) and (7/2 - 1/2) which are 4 and 3 | ||||

$~~~$ |

Does it work with fractions? Try
$$\mbox{Sum}~6\frac{1}{2}~\mbox{and Product}~7\frac{1}{2}$$

Pupils can make up pairs of numbers and offer their Sum and
Product to a partner to see if they can solve the problem. Clearly,
there will be difficulties to be discussed when the Product is less
than (Half the Sum)$^2$ and when the result of the subtraction is
not a square number.

The simple model 'Sum 7 and Product 12' will appear again in
part 2 to show how non-rational and 'imaginary' numbers
appeared

This algorithm appears again
and again in slightly different forms in different cultures and is
the basis of the 'quadratic formula'.

The comparison of lines and areas by Greek mathematicians was
their fundamental way of 'measuring' lengths and surfaces.
Comparing lines with lines works when the lines are in simple
proportion; for example 2:3 or 17:57. Difficulties were found when
this idea was applied to the diagonal and side of a square, or the
diameter and circumference of a circle. However, geometrical
constructions were found to overcome this problem in special cases.
The construction for finding a square equal in area to a rectangle
is completely independent of the lengths of the sides of either
figure. The three right-angled triangles in the semi-circle are an
example of a proportional relationship which produces a quadratic
relation. These ideas are good material for a classroom
discussion.

By the 10th century CE the Arab mathematicians had translated
and developed most of the scientific knowledge of the ancient
world. Medicine, Astronomy, Optics, Physical Science, Alchemy and
above all, Mathematics were far more advanced than in other places.
In terms of the history of mathematics, the Arab scholars were in a
key cultural position to unify the different problem solving
techniques into a body of knowledge that formed the fundamental
ideas of 'school algebra'. Al-Khowarizmi was probably first among a
number of scholars who showed how the geometrical constructions of
Euclid Book II and the arithmetical heritage from Diophantus'
Arithmetica, and the ideas from the Middle East, Chinese, and
Indian scholars could be brought together. What is special about
Al-Khowarismi's conceptual blending is that for the first time we
find the objects of study
(the unknowns) and the equations that define clearly the
classes of
problems to be solved. Later, Thabit ibn Qurra finally
provided proofs for al-Khowarizmi's six cases showing they were
conceptually equivalent to a series of Euclidean theorems.
Geometry was
used as a means of visualising the processes.