Basic Shapes

Mathematics in the Greek school used to be divided into two parts: algebra and geometry, which were taught separately. Algebra in this case also included logarithms, statistics and percentages. I found geometry a very magical subject and always studied it with great enthusiasm. I really liked the way a two dimensional shape drawn on a piece of paper could be transformed into a three dimensional one with the addition of a few lines and a little imagination. I used to think of a circle, a disc for example, and then imagine a hand picking it straight up from its centre. The course the disc "drew" in space in this case would be a cylinder. And then if a disc was to spin in place around its centre, then that movement would create a sphere in space. I liked the way you could actually touch the solid geometric shapes, as that gave you a feel of the relation of geometry to the real world and made the subject easier to understand for me. When a student struggled a bit with the Pythagorean theorem and its equation and proof, the teacher could draw a right angle triangle, draw all the squares created by its sides and then cut them in such a way to show that indeed the area of the square created by the longest side of the triangle was equal to the sum of the areas of the other two, smaller squares. That availability of a "three dimensional" proof in geometry, gave it a great appeal to me, similar to the one physics held.

I felt very differently about algebra. I could see that it was very useful to everyday life, especially regarding the handling of money, but it always held a strange abstraction to me, it felt much more of a theoretic subject than any other.  Everyone insisted on saying that together with language, mathematics (meaning algebra much more than geometry), was an essential subject that should be mastered, one that if you failed to grasp you would be in trouble for the rest of your days. Even when props were used in algebra, I still thought the whole thing was a bit like an abstract theory, regardless of its very "matter of fact" nature. Beads and sticks were used to represent numbers and units, buttons were used to represent money coins, money were used to represent numbers and value, numbers were used to represent time and so on. The whole thing felt quite like a made up system to me that was left open for discussion and potentially manipulation. So I appreciated algebra to the extend that it facilitated the subject of physics and geometry, but I was left completely uninterested in it, as a subject on its own.

This was not always the case. When I was in primary school, I really liked maths and I saw the exercises we were given as little games, mysteries if you like, that needed to be solved. I liked the fact that 2+2 would always make 4, and there was nothing that could change that, you could count on it. When we started to learn the multiplication table I was very excited when we reached the number 9. It was my favourite number as a child anyway, because we had always lived in a number 9 house or flat, my grandmother was one of 9 siblings and now, when we learnt the multiplication table, I saw that number 9 had a special quality. I noticed that the first five numbers created when multiplying the number 9 with one, two, three, four and five were 09, 18, 27, 36, and 45. Then the following five created when 9 was multiplied by six, seven, eight, nine and ten, were the same numbers but arranged backwards: 54, 63, 72, 81 and 90. I was so excited by this discovery, that I told the teacher about it in class with great enthusiasm. The teacher had the most piercing pale blue eyes,  he leaned above my table and said with a profound frown "All right...I can see that...but it does not affect anything. It does not mean anything, you still need to learn the number 9 multiplications, same as with all the other numbers". I wanted to ask if there was a reason that number 9 multiplications did this or if it happened with larger multiplications with other numbers, if it was normal, but I was too scared to ask and slowly slowly after that I lost my enthusiasm for mathematics.

Things got worse for my idea of maths when Greece joined the European Union. Greece had a very large production of oranges, together with other products such as olive oil, lemons and grapes, and was relying for its economy on exporting a large amount of them to countries in Europe, particularly North Europe. When we entered the European Union, a kind of formula had to be put in place, so that all the countries exporting oranges in Europe (France, Italy and Spain were also large orange producers) were regulated fairly. But the formula they used was based on percentages taken from each country's population and not each country's production of oranges. So Greece's export ability to Europe was sliced dramatically as it was based on the population of the country, which at the time was about 10 million only in comparison to Italy's sixty million, France's sixty five million and Spain's forty six. We were given oranges for free at school, the prices dropped, oranges were piling high on the highways to the South of the country rotting. Producers slowly slowly stopped cultivating oranges and the same thing happened to many other agricultural products, turning a heavily agricultural country to a bare land. My idea of 2+2 always making 4 being a fact you could rely on was shaken, as I realised that the application of this fact would always be up for manipulation from whoever had the power of a decision.

I am a bit nervous about ever having to help teach Aretousa maths, as I only feel at ease doing it through shapes and lines and not so much the traditional way. For the time being we have been playing with some cut out basic shapes, which Aretousa has arranged at random on double spreads of a coloured paper book (she looked concentrated when she was arranging them, so maybe not completely at random). I have then merged these shapes in Photoshop to see what kind of new shapes will emerge. Then I made three-dimensional shapes with the basic cut out ones, took a picture and tried out the same thing. The distortions of the shapes no longer sitting flat on the table have created some new shapes again. The original ones are now less recognisable in the final pictures. However it is still the same seven initial basic shapes which have also created these new ones, even if they look so different. I bet there is someone out there that could tell us exactly how many countless possibilities there are to combine these seven shapes and arrange them either flat on the table or in three-dimensional shapes, so that you could keep on producing maybe thousands or millions (?) of new shapes. Of course that is not the point of interest here, but it reflects how I feel about maths a bit. Numbers are not so factual really, just another thing that can be used this way or the other, in the right hands or in the hands of someone with ulterior motives. It always sounds so proper, so serious when someone drops in statistical figures, numbers and graphs, making you prone to believe them that much more. But the distortions are possible here as they are with anything else, probably a bit more serious here, maths are part of everything as the old primary school teacher said, part of money, of politics, making up a world full of numbers.