The Development of Algebra - 1

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 $12$

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

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.