Pedagogical Notes
The theme for this second part of the story of algebra is
Representation. Together with Visualisation it is a key activity by
which we are able both to invent new diagrams, pictures and
symbols, and to make correspondences between these representations
in order to describe and explain our mathematical ideas. Not only
this, it is a crucial element in seeing links between one area of
mathematics and another.
1. The 'Babylonian Algorithm'
The 'Think of two numbers game' (see item 5 of the Notes to
Part
1 ) is offered as a means of examining and expanding the
significance and power of an algorithm:
"I am thinking of two numbers; their sum is
seven and their product is twelve, what are the
numbers?"
We start with some easy ones  without 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.)
There are many opportunities here to use elementary arithmetic
for example:
 exploring partitions of the sum, and factors of the
product
 the most efficient way of multiplying brackets together
 operating with fractions and converting 'mixed numbers'
A process emerges:
'Take
half the Sum number and try adding and subtracting the same amount
to it (+/ 1, +/ 2 etc.) and multiply the results together to find
the Product number.'
e.g. For Sum 7 and Product 12
Take S/2 
$~~~~~$7/2 
Square it 
$~~~~~$49/4 
Subtract the Product number 
$~~~~~$49/4  48/4 
Find the square root of the result 
$~~~~~$1/2 
This is the number you add to and subtract from 
$~~~~~$7/2 
So we have (7/2 + 1/2) and (7/2  1/2) which are 4 and 3 

To check, we multiply the brackets together


You can
see how this algorithm appears again and again in slightly
different forms in AlKhowarizmi, Fibonacci, Jordanus, Cardano,
Viete and Descartes and is the basis of the 'quadratic formula'we
use today.
Note above, that 7 is the semiperimeter of a rectangle, so
7/2 is an estimate for the side of a square, and 49/4  48/4
represents the difference between the areas of a square and
rectangle.
New questions can arise when we ask what happens in line 3
above, when the result is not a square number. What about Sum 7 and
Product 11? or Sum 7 and Product 13? What about Cardano's problem:
Sum 10 and Product 40?
2. Proportional Triangles
The diagram of the three similar triangles in the semicircle
introduced in (
link
to section 6 of Part 1) is the basis for much of the
proportional relationships that were fundamental to problem solving
in Mediaeval and Renaissance times. The diagram enables us to find
square roots of numbers  the relationship derived from the similar
triangles can be rewritten as $ab = x^2$ so, if $a = 4$ and $b = 1$
then $x$ (the vertical line) is the square root of $4$. The square
roots of $4$, and $9$ can easily be found by drawing on squared
paper. The square roots of $2, 3$, and $5$ can also be found.
The relationships in this diagram
are used by Viete in his solution of a quadratic.
3. The Development of Notation
Pupils are usually told what notation to use. Why? Having some
experience of inventing and using their own notation provides
opportunities for discussions about which is clearer or more
efficient, and motivations for agreeing on conventions. Much
mathematical notation evolved from the language of the people. Some
appreciation of the evolution of notation is further evidence for
them that mathematics was not 'cast in stone' at some stage and
left for us to get on with it.
Jordanus used letters in all his formulas for the proportional
relationships. His text of Proposition 6, "If the ratio of two
numbers and the sum of their squares is known, then each of the
numbers is known." has no symbols for equality or the operations of
arithmetic, and his explanations were very longwinded,
but his principle of using
letters was to convey the generality of the expressions . At
the end of each discussion he gives numerical examples for his
readers.
By the late 15th century Pacioli and others were using
abbreviations of (mostly Latin) words for plus, minus and equal,
but as the equations became more complicated, the problem of
representing higher powers became acute. As printing developed, so
the language of the common people became used more often to spread
the new knowledge. Latin did not have the technical words for the
new concepts like higher powers of the unknown, and so composite
words began to be made up like quadratum de quodrato in Scheubel, and
zenzizenzike in Recorde,
both for $x^4.$
Cardano's Ars Magna
showed that there were general methods that could be applied to
apparently different cubic equations and that the quartic $(x^4)$
was easily solved. The texts from Scheubel and Recorde show that
mathematicians were having great difficulty in expressing higher
powers of the unknown. For the Printer, hand carving these odd
characters was an expensive job, and his Apprentice often made
mistakes. In the end, economics won, and the Printer probably said
to the Mathematician, "why don't you use what is already in my
box?" and so the alphabet was used for the algebaric
representation.
4. The Text and the Diagram
From AlKhowarizmi's conceptual blending of arithmetic with
geometry, to Descartes algebra in the 17th century, there was
always a correspondence made with a geometrical diagram. Euclid's
geometry and the mode of argument was considered to be the way of
making sure of the truth, and so any arithmetical or later symbolic
argument had to be supported by a diagram. There is a sense in
which our belief in the truth of visual evidence is rooted in our
unconscious mental history. Some people believe that visualisation
is at the foundation of all our knowledge. Be that as it may, our
visual education is vitally important for many aspects of
mathematics, and current research in neuroscience confirms its
importance.
Even today, although we have quite sophisticated mathematical
notation representing complex concepts, language is still
necessary. We invent technical words like 'transformation' or
'function' to describe processes, objects, or processes within
objects, and we often expect students to understand what they mean
without either sufficient experience or discussion of their
meaning. History provides us with the evidence that this was not an
easy process.
5. Reading Descartes.
By choosing the text carefully, the Dover edition can be used
by students with only a limited knowledge of French as a project in
translation  and understanding  of Descartes' basic ideas.
It also shows us that Descartes was addressing an audience.
Who were they? Why did they need to be told this message? What
might have been their response? Do pupils write mathematical
explanations? Who do pupils write for? There are potential contexts
where pupils could address an audience and write explanations. This
had been done where, for example, pupils explain mathematical ideas
to each other, who then respond with their own questions thus
setting up a true dialogue.