Sunday evening comes and I find myself, as usual, beginning to think about the lessons I am teaching this week in more detail. This week I am looking at some 3D Geometry with the Maths Studies group. Mainly, volumes and surface areas of 3D shapes. It often pays to sit and ponder why we teach things, not only so that you have a half decent answer if, and probably when, students ask and more importantly so we are clear ourselves of our goals. This 3D mensuration is one of those topics that I have struggled more with to find answers about. Over the years I have run off numerous tenuous explanations in supposedly 'real' contexts, all of which have at least some value but none of which completely satisfy me.

Can we really argue that it is relevant to to know the surface area of a cone or a sphere? If we run off a list of people who do need to know for their jobs, who would it include? I am not suggesting that there is no one and am sure that it figures in to the work of architects and engineers, but simply questioning the reasons for including it in a syllabus. 3D shapes can be very appealing visually and we live in a 3D world. Natural shapes and man made buildings etc offer us lots of examples of these shapes, although very rarely isolated and perfect. As such I have been tempted to use these examples as a context for teaching this topic, but increasingly find myself leaning towards less direct or more abstract ideas associated with the counter intuitive nature of this topic.

Let me offer an example. I am lucky enough to run a swimming pool in my back garden, and at the end of each summer its necessary to empty the pool by at least 30 cm before servicing and closing the filter system. (this is to allow enough space for rain water to fill the pool without getting in to the filter pipes and freezing). It can be done easily, if slowly, using the filter before its switched off. One year, however, I forgot to and found my self needing to empty the pool by about 30 cm without the aid of the filter. It was suggested to me that I buy or hire a pump to do so. Always anxious the keep a little money, and (despite) being a practising maths teacher, I set about trying to figure out how many buckets of water I would have to scoop out of the pool myself to achieve this! No surprises then, that I hired a pump later that afternoon! Approximately 8000 buckets of water was my estimation and I did not go on to estimate how long that may take me, considering the likely onset of muscle fatigue. Note, I have avoided giving you enough details to check my estimation, but the real problem here is to work out what the possible dimensions of my pool and size of my bucket could be. Volume is so wonderfully counter intuitive. It might take me weeks to count to 1 million, but there are 1 million cm³ in 1m³. The rate we consume fossil fules is phenomenal, but yet we haven't run out yet? The point of the example is not that I think being able to estimate the number of buckets is a particularly important life skill (although it was of benefit to me in this case), but that volume is a topic that we cannot necessarily rely on our instincts to understand. As such, this becomes one of my reasons for wanting to teach it.

To other ideas; who found out the the curved surface of a cone has area pi x r x l? Who even knows what the net of a cone looks like? This is another wonderfully counter intuitive concept. Discovering that the curved surface area of a cone is actually a part of a 2D circle is wonderful and rarely the first thing people think. Establishing the relationship between the radius of the curved surface area and the radius of the base is a lovely bit of mathematics.

The sphere is another ball game altogether! Here is a curved surface area that cant be laid flat? A workshop participant once told me of an activity she did with an orange sliced in two through the centre, then peeled showing that the bits of orange peel could fill 4 circles drawn around the flat base of one of the orange halves. Is it not, at first, astonishing that the surface area of the sphere, this uniform, smooth, yet apparently unruly area is exactly 4 times the area of the circle that divides the shape in two?

In none of these cases am I suggesting that Maths Studies teachers replace the syllabus items with more complex mathematics, but I am suggesting that there can be much more to teaching this part of the syllabus than meets the eye and that it offers some very rich opportunities to explore some of the wonderful, often counter intuitive mathematics of relations. Any number of topics could have been generated to test students' ability to substitute numbers in to formulae, but I am glad its this one this week!

For anyone still reading, I would like to point you in the direction of the work of John Bryant and Chris Sangwin and their book 'How round is your circle?', details of which can be found at the books website www.howround.com/ and a sneak preview of which you can get from the video below! This should get your curiosity on 3D geometry going nicely! Maybe this is for another blog entry one time!