Friday, July 30, 2004

Trudeau: Ch 1. Maths, Games, and Education.

I'm currently reading chapter 4 of Trudeau, but lets do this properly and I'll start with chapter 1.

Chapter 1 is the introduction, and like all introductions everywhere, is largely content free --- or more precisely, is the realm of opinion and motivation. But, before I discuss Trudeau's opinion of why I should read this book, Introduction to Graph Theory has what is probably the most excuberant blurb I think I have ever read on a textbook.

This delightfully written introduction to graph theory offers a stimulating excursion into pure mathematics. It is aimed at those whom the author calls "the mathematically traumatized," but it is a treasury of challenging fun ... "The topics are so well motivated, the exposition so lucid and delightful, that the book's appeal should be virtually universal....[sic] Every library should have several copies"
Wow, somebody was certainly in a good mood. You could almost be forgiven for forgetting that you're actually reading about a textbook. I think this gives new meaning to 'rave review'.

Anyway, back on topic, Trudeau would like us to approach pure mathematics as a game.

Basically pure mathmatics is a box of games. At last count it contained more than eighty of them. One of the is called "Euclidean geometry".
He then goes on to approach the closed-world nature of pure maths; something I as an engineer struggle to deal with.
Games have one more feature in common with pure mathematics. It is subtle but important. It is that the objects with which a game is played have no meaning outside the context of the game. ... You may balk at this, since for historical reasons, chessmen have names and shapes that imply an essential correspondence with the external world. The key word is "essential"; there is indeed a correspondence, but it is inessential. ...this interpretation of the gaem, like all others, has nothing to do with chess per se.
So in a similar way, while the words plane, point, and line suggest real-world analogues, it is only an implied interpretation. In Euclidean geometry they are axiomatic, with specific properties, but without definition. one knows what planes, points or lines are, except to say that they are objects which are related to one another in accordance with the axioms. The three words are merely convenient names for the three types of object geometers play with. Any other names would do as well.
Still the most interesting paragraph in the introduction is right at the beginning. Along with a number of my peers, I cruised through High School mathematics. In fact I largely finished read Lord of the Rings, and War and Peace in my maths classes. It seemed that my teachers would take about a week to teach a days work, and yet it was obvious that dispite the slow pace many of my classmates were still being left behind. Naturally as a geek, I have spent a fair amount of time analysing and self analysing to try and understand the difference. While I was still at school, I just assumed I was just more intelligent. However experience has since taught me I wasn't. Which reopens the question, "What was I doing that made me so much more effective at learning High School maths?". I certainly didn't work harder. I believe Trudeau's introduction gives us a clue:
To understand what mathematics is, you need to understand what pure mathematics is. Unfortunately, most people have either seen no pure mathematics at all, or so little that they have no real feeling for it. Consequently most people don't really udnerstand mathematics; I think this is why many people are afraid of mathematics and quick to proclaim themselves mathematically stupid. Of course, since pure mathematics is the foundation of applied mathematics, you can see the pure mathematics beneath the applications if you look hard enough. But what people see, and remember, is a matter of emphasis. People are told about bridges and missiles and computers. Usually they don't hear about the fascinating intellectual game that lies beneath it all.
My current hypothesis is that my gift was the ability to see past the smokescreen of 'applications', and rote-work, to the fundamental patterns and structure of the pure mathematics underneath. This would also explain why I find such a return on time spent studying Computer Science --- and consequently my interest in language semantics and hence why I'm reading a book on graph theory largely for fun. So assuming I am right, can this insight into how a 'naturally gifted' high-school maths student attains their aptitude be used to improve the pedalogy of our existing maths curriculums?

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