One problem is that the bits of geometry included in the syllabus (a collection of what I assume somebody decided were the more interesting theorems about lines, triangles, polygons and circles) are not presented as part of an edifice, but as a bag of results, all too often just used to solve some fairly contrived problems. The individual theorems may indeed be proven, in the sense that the pupil is presented with an argument that it is hard to disagree with. But there's no real sense of just where it all starts from, and how it is all built from a remarkably small collection of givens (the definitions, postulates and common notions).
Another is when a pupil asks the perfectly reasonable question "What's the point of this?". Brave attempts to find practical everyday uses such this tend not to be convincing: nobody really believes that painters routinely use Pythagoras' theorem to find out what length of ladder they need. The actual practical application of the plane geometry is vanishingly small for almost everybody, and the pupils are quite capable of spotting this. The obvious consequence is that they become less motivated, rather than more. The issue was raised a while ago by David Wees , where he took Pythagoras' theorem in particular as a starting point for a 'Why should we teach this topic?' discussion. Much of what I write below could be read as my answer to that question.
The real reasons for doing plane geometry are quite different from its practical application. But, as I noted above, the context of syllabus and assessment that they crop up in can be a potent distraction from the value of the material. I want to argue for the material, but also for the importance of having it placed in a context (i.e. in the specification and in the corresponding textbooks) that makes the value explicit. The hope is to give the students a different, and I hope a better, reason for working on the material: and maybe at least as important, to give a response to those adults who also have the attitude that the point of school mathematics is to teach material which is of everyday practical use, as exemplified in this blog, by a maths teacher. (That blog, and some of the responses to it, are part of the reason I don't think I'm just preaching to the choir.)
So, what do I claim are the reasons that pupils should study plane geometry?
One is, of course, that there is strong pressure on schools and pupils for everybody to get a passing grade on GCSE maths, and being able to do the geometry makes a useful contribution to this. This is, also of course, a terrible reason, to which the easiest response would be to take it out of the syllabus entirely, so I'll say no more on it.
I'm also not going to argue that a study of Euclidean geometry teaches logical thought. It would be nice if it were true, but I don't think it's any truer of geometry than it is of Latin. Tempting though it is to believe that understanding how to prove a geometric proposition or analyse the grammatical structure of a long Latin sentence will transfer to skills of logical deduction and analysis in other contexts, the evidence is just not there. If you want that, you have to teach that.
So what are the good reasons?
One good reason for it is that it is an opportunity to understand an elegant mathematical structure. But this requires a quite different approach, one of seeing how the results start off with very basic ideas and build up to more advanced ones. Jumping straight to the 'interesting' bits without seeing the journey doesn't give an appreciation of the intellectual journey. Is the alternative demanding? Yes, it is. I don't see that it's more demanding that reading Shakespeare though, and we expect that of our school pupils.
I'm not advocating a return to some kind of nineteenth century public school syllabus of working through and memorizing the entirety of book one of Euclid's elements, culminating in the proof of Pythagoras' theorem. (That said, I wish that schooling afforded the time to work through and understand the material.)
Another good reason is that it lets the pupils in on some of the few eternal truths that a school education can impart. It doesn't matter what culture you are from, or when you study it. From the basic assumptions of Euclidean geometry will always and inevitably flow the conclusions. They aren't a matter of opinion, or preference, or cultural heritage.
And all that isn't to say that there is nothing interesting about the way the different cultures have developed their understanding of plane geometry, and the relationship between those developments and the particular synthesis offered by Euclid. But this is probably to go considerably beyond the scope of what might be possible in the curriculum up to the age of 16, even in my rather Utopian ideal school.
So, bearing in mind the unpleasant fact that there are limits to the amount of time which will be devoted to mathematics in the school curriculum, what might be done to improve the situation? Of course, I do have an opinion about this, which comes in two parts.
- The first part of the solution would be to be more honest about the reasons for studying the material. We don't do it for reasons of everyday practicality, but because it has an intrinsic beauty and elegance, and demonstrates the power of careful logical thought, and in particular how so much can come from a relatively small basis. How the results of this show up in the real world, and why some of them have been important historically then demonstrates that there are real-world connections and consequences; but they aren't the reason for studying geometry, any more than holding up a layer of paint is the reason for the Eiffel Tower.
- The second part would be to present the material in a way which makes the logical structure clearer. Obviously (I hope) I'm not suggesting a complete presentation of the geometric books of the elements, but rather a whistle-stop tour starting with the rules of the game, and outlining what goes into getting to the particular results of interest, emphasising the logical dependencies. It would also be useful to discuss at least briefly why these rules, and how you can do more if you extend the collection of tools (which was, of course, extremely well-known by the time of Euclid).