The Development of Algebra - 2

Age 11 to 18
Article by Leo Rogers

Published 2011

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 Al-Khowarizmi, Fibonacci, Jordanus, Cardano, Viete and Descartes and is the basis of the 'quadratic formula'we use today.

Note above, that 7 is the semi-perimeter 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 semi-circle 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 long-winded, 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 Al-Khowarizmi'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.