A perpetual Student
I am a great believer in “lifelong education”. You might expect this from a former teacher whose livelihood depended on learning, but amongst my favourite clichés are “It’s a wise man who knows what he doesn’t know” and “You are never too old to learn.”
The first time I remember thinking about this was when I was reading a science fiction novel: “The Day of the Triffids” by John Wyndham. I was at school and there weren’t enough copies of the class reader to go around – so the English teacher gave me a copy of Wyndham’s book to read and told me to write a review about it. It was probably one of the first Science Fiction books I had ever read … and it had a great effect on me. (Not as much as his other book, “The Chrysalids” which I sought out and read next … but “Triffids” awoke in me an interest in reading for it’s own sake.) In the book the survivors of the assault of these monstrous man-eating plants gather on the Isle of White off the Southern Coast of England, where they can purge the land of the carnivorous vegetables with annual culls. There is a public meeting where a management committee for the new community introduces itself and one is elected, or appointed, as a Medical Officer. She is a nurse – not highly qualified but the best candidate that the small band can provide, and as the Chairman of the meeting points out: Better a little practical knowledge recently acquired than a mass of theory learned long ago.
At school I never studied chemistry, economics, Victorian history … amongst other things … and all these represent gaps in my knowledge. Even if there were no changes in the world, they still detract from my understanding of the world in which I live. As it is, however, the world is constantly changing: new facts are being discovered; new theories are espoused; we meet new situations, there are new techniques to master and new skills to acquire.
The classis example that I always quote concerns nose bleeds – yes, nose bleeds.
When I studied First Aid we were told that the correct way to treat a nose bleed was pinch the bridge of the nose and hold the head back with the nose in the air. A couple of years later I went for the refresher course and we were told to pinch the bridge of the nose but hold the head forward with the nose pointing down. The reason for the change in the rule was simple, it was explained to us … if the head is held back and the blood continues to flow then it is possible that patient will swallow it and in some cases may even drown in their own blood.
There is a joke that examination questions in economics are always the same: at every level, and every year … it is just that the answers change depending upon whether it is a school examination, a University examination or whatever … and which year – the answers last year were not the same as this year’s and next year’s will be different once again. That seems a bit cynical and simplistic but, as is often the case, there seems to be a kernel of truth in it. Why else would the economist and the television current affairs interviewer say “we are all Keynesians now …”.
Some things change less frequently – you wouldn’t expect, for example, historical facts to change … what happened, happened … but as new facts come to light our understanding of it what happened in history changes … sometimes we learn that what we thought happened, didn’t, or happened differently from the way that we had earlier thought.
Likewise you wouldn’t expect the laws of Mathematics to change … after all: two plus two equals four, always has and probably always will, won’t it? … but they do change. For many centuries Euclid’s fourth axiom – that parallel lines never meet, was an accepted “fact” … it caused mathematicians and philosophers all sorts of problems because, unlike the other axioms which Euclid enunciated, it did not seem to be self evidently true, although it did seem to work in reality. One of the results based upon this axiom is that the angles of a triangle add up to 180 degrees. The trouble is that the angles of a triangle don’t always add up to 180 degrees.
Take, for example the old conundrum: “A man walks one mile south, one mile east, one mile north and, arriving back at his starting point, sees a bear and shoots it … what colour was the bear?”. At first it looks like nonsense … how is it possible to walk one mile south, another East and another north, and yet arrive back at your starting point? In fact it is perfectly possible and there are an infinite number of possible points from which he could commence his three mile walk … perhaps, the most obvious one is the North Pole. Now, the triangle that he walked included two right angles (when he turned from South to East and from East to North), and there was also an angle between the two paths … the path he left on when he started his journey and the one he used when he arrived back … so the triangle that he walked contained more than 180 degrees. The reason for this is simple … Euclid’s system works perfectly well – on flat surfaces … but the man was walking on the concave surface of a globe … and here the rules are different. They are different once again on a concave surface – the angles of a triangle can add up to less than 180 degrees! So, the veracity of the rules of mathematics depends up on the context.
Think back to when you learned “sums” … maybe you remember learning something like: “five minus three equals two, but three minus five you cannot do”. I did, and I was surprised a little while later to learn that in some circumstances it was possible that: “three minus five equals minus two”. Later I learned that it was possible to multiple these “negative numbers” … “a plus times a plus is a plus – a plus times a minus is a minus and a minus times a minus is a plus”. Ok, so far no big problem … but then we learned about square roots … the square root of 4 is two … actually “plus or minus two” … it could be the result of “plus two times plus two” or “minus two times minus two” … but “you can’t get a square root of a negative number” like minus four. Imagine my surprise when I came across the “imaginary number” i, (the square root of minus one), and the whole new realm of complex numbers and Argand Diagrams, and …. Well, I think you probably get the point. Even in the Queen of the Sciences, Mathematics, just when you thought you “knew” something, you discover that, in fact, you don’t …
It was even worse in the realm of computing, with rapid technological advances … I suppose the first “computing device” I ever used, if we ignore counting beads and abacus of my early schooling, was a slide rule … I really enjoyed how to manipulate and interpret the various scales. At college we have special training on mechanical calculators based on levers, rotating handles, shifting registers and the Leibnitz stepped wheel. When I bought my first electronic calculator it had four functions, (yes, four: add, subtract, multiply and divide). I was furious because a week later a friend of mine went out and bought himself a calculator for half the price … it not only had the basic four functions – but it also had a Memory with “Memory Add”, “Memory Delete” and “Memory Recall” buttons. He could so much more with his device than I could with mine – at half the price.
As technology advanced computers quickly became more advanced, faster and more powerful. The first computer to work in the National Westminster Bank apparently had a core memory of just 256 bytes – and the old basic ZX80 which my students purchased and played their games on was four times more powerful than one which used to process account details for one of Britain’s biggest banks. The one I am sitting at now is many times more powerful than both of them … I had to do a lot of learning when my career took me into computing …
There were other things to learn as well. Subjects like Economics which I had never studied at school. I became like a sponge soaking up new information. I attended a wide variety of courses. Some of them were practical and/or work-related, others were more general interest and seemed to have no obvious bearing on my professional life … but they all brought me enjoyment, a sense of achievement and of purpose, and examples which I used to cite, and still do, from time to time.
These courses varied not just in topic but also in depth and length, from evening seminars on a single topic, to week long courses run in the school holidays, and even a two year part-time Post Graduate Diploma. I also enrolled for several distant-learning courses run by the Open University, some related to education, others to Mathematics, Statistics and Computing, and one on the History of Science, which I would highly recommend. My favourite didn’t carry any credits, it was “Looking at Paintings” from the Open University. I don’t know if they still offer it … but it was very enlightening in many different respects. Attending courses is something I miss here … but the internet is a great tool and I feel very fortunate in being able to research almost any topic.
Not everyone shared my interest in courses, or my opinion of their value. I remember that for one course, I had to interview some colleagues about their experience of In-Service training courses. Two respondents, let’s call them Peter and Paul, stick in my memory: two very good Maths teachers in their own right. Peter was what could be termed an Applied Mathematician. He was very practically minded … for him, a good training course was one which would provide him with materials he could use in the first lesson the following day. When asked for an example of good training course he had attended, his response was that there wasn’t one … they were all a waste of time – and he only went on courses to “tick the box”. I asked about courses we had attended together … to which he replied that although they were interesting – they weren’t “useful”. Paul, on the other hand was what might be termed as a ”Pure Mathematician” … he had a very philosophical approach. When asked for examples of “bad” courses he had attended he said there weren’t any. When I remonstrated that: “Paul, I’ve been on some of the same courses as you … what about the one on …”, he replied that although the sessions themselves were not very good, the course gave the opportunity to meet with colleagues and exchange views and experiences – and that in itself was extremely valuable! I was reminded of this when reading about NLP (Neuro Linguistic Programming) which claims that there is no such thing as a worthless experience – there is always something to be gained, something to be learned.
I am still learning – although it is harder for me to enroll on courses here. What I do have, however, is a far greater range of resources thanks to educational channels that I can receive from Satellite TV, e-mail newsletters, podcasts (which can vary considerably in value), to a vast array of information and opinion availble through internet resources such as Wikipedia.
Which brings me to the principle that one my teachers, (who originally told me about the “wise man” who knows what he doesn’t know …), that knowing something isn’t the important thing – it’s knowing how to find out.
By the way … the bear was white.
Bishkek