Maths … and all that
For some reason, many people are scared of numbers. I’m lucky in that I’ve never really had that problem.
I wasn’t particularly good at maths in Primary School, in fact I tended to get “average” grades, but in my last years that I found a liking and (minor) talent for Mathematics … the crucial moment was when we started algebra … x times y equals xy, a times b equals ab … hey, I could that, no problem – even if I had no real idea at that time what was actually going on, it caught my imagination and I was no longer “afraid” of “sums”. When I transferred to Secondary school I was frequently top of the class, if not top of the year in terms of Maths.
It came fairly easy to me because I was basically “easy going”. Maths was a set of rules that had to be applied to give facts, most of them involving numbers. numbers were still abstract things but the rules were fairly easy to learn and apply. I made mistakes, of course, (we all do), and there were gaps in my knowledge … but the biggest criticism my teachers had seemed to be with my presentation. (My handwriting has never been that good – and even today my presentation can be quite scrappy.)
I remember the sense of achievement when I was learning about integral calculus and “Solids of Revolution” and accidentally discovered the formula for volume of a sphere when solving a problem. I already knew the formula, of course, but had simply learned it “rote fashion” and discovering it once more, demonstrating it for myself.
One of the things that I learned fairly late on was that the rules change depending upon the circumstances of the situation … In many ways, to me, as I was growing up, Maths was like a game … you had to know the rules and how to apply them. If you worked within the rules, you got a result. It was some time before I realised that, sometimes, in order to solve a problem you have to “think outside the box”, go beyond the rules, at least the rules as know and understand them at the time. Go in search of new rules … which explain the situation better. That’s what “the scientific method” is all about.
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 …
Statistics was worse … just when you thought you had learnt a body of knowledge, say about the Normal Distribution, levels of significance and Confidence intervals, you went on to discover that it wasn’t much good at dealing with small populations, so you had to learn a whole new set of theory associated with Student’s t-test … Both of these tend to works with certain types of data but for others you needed different tools such as the Chi-squared test … and when I was studying Psychology I learned a whole new set of statistical tools.
When I enrolled on an Open University Statistics course, I was quite frustrated and annoyed because they insisted on applying what I considered to be an illogical rule for rounding-off. I had been taught that when rounding figures then anything over 5 was rounded up and anything less than 5 was rounded down. So 3.0, 3.1, 3.2, 3.3 and 3.4 would become 3 and 3.5, 3.6, 3.7, 3.8, 3.9 would become 4. Five cases would round up and five would round down … each would go to the nearest value. That seemed logical. However, the tutors for this required that we round even numbers up and odd numbers down – so 3.1, 3.3, 3.5, 3.7 and 39 would round down to 3 while 3.2, 3.4, 3.6 and 3.8 would round up to 4. As I say, to me this seemed illogical – and although it should have resulted in different in different answers to problems, but I have to take my hat off to the tutors, because some of the problems that they set us were constructed so that it didn’t matter which system you used to round off values – you got the same results.
Perhaps that is one of the reasons I tend towards relativism in philosophical and ethical terms.
Anyway, I quite enjoyed the “game” of Mathematics.
Although I was thinking of entering the church, someone gave me some advice which seemed sensible. Our Minister had retired and his replacement was a young man, with a young wife who didn’t really see herself as a wife of the Manse. Most of the people in the church were a lot older than them and, to be fair, she, in particular, didn’t fit in. I don’t think it was anyone’s fault – indeed, it might just have been a mismatch … if he had been assigned to an inner city church, then I think they would both felt much happier. They would have felt that they were making a real difference rather than being tucked away in some backwater. Eventually, he decided that perhaps he had made a mistake and decided to resign … not just from his post as our Minister, but from the Church as a whole. It was a blow. I was shocked, but it was pointed out to me that I was also young and who knows what the future might bring. Instead of deciding to go to Theological College, perhaps I should think like an actor who always has a second string to his bow for when he is “resting” … that is – when he doesn’t have an acting job … a role to perform on stage. I knew that this was good advice – as I have said elsewhere, I don’t believe in closing doors until you have to … so I looked around and chose teaching as a suitable “second” career. As things turned out, it was to end up being my main vocation for most of my life – sometimes, I think, even today when I am running a tour business.
My choice of college was based on a number of things: it was not far from home; a Methodist college; and it offered me the chance to follow the two main interests at that time … Theology and Mathematics. At that time my main interest was Theology, Mathematics was to be the “second string” to my bow.
In my first teaching job I was “Teacher in Charge of Religious Education” but I also taught a few classes of Maths. It was when I moved to North London that I took up a post teaching just Maths. This had some advantages, chiefly the fact that I had fewer groups to get to know, (as an Religious Studies teacher I saw about 300 pupils every week … and it was difficult to learn all their names, but now I was down to about half that, which was a lot easier), and I was teaching a subject which, even if it wasn’t popular with everyone, was thought to be useful and not just a “necessary chore”, a statutory obligation.
Even though Maths was thought of as a useful tool, not everyone was keen. Some were clearly “Matheholcis” with a phobia for the subject. To be fair, I thought, and still think that some of the topics we had to teach were esoteric and a I understood why some thought they were a waste of time. I am not sure, for example, that I have ever used the Shimbell Index in real life – although it has helped me understand the concept of “remoteness” of settlements in high mountain areas. I remember being castigated by one employer because our “graduates” could handle vulgar fractions … and I asked “why do they need to? … why is it important that they can add 11/16ths and 5/27ths?” … I understood that some branches of engineering still used eights, sixteenths, thirty-seconds and sixty-fourths … but this was clearly a minority and as metrication was being adopted as a norm, even they were becoming irrelevant. Perhaps we were still supposed to teach the unitary fractions of Ancient Egypt? As it happens, I could see reasons for teaching fractions … they are important for understanding a variety of techniques and concepts in higher mathematics, and they do help to teach and develop logical thought processes … but I could not see why they were considered so important for school leavers who going to work as shop assistants, policemen and the like.
I was reminded of this argument when, here in Kyrgyzstan, an acquaintance (he was studying tourism and he knew that I had been a Maths teacher) .asked me to help him with his homework. At first I declined … “Ruslan, ” I said, “it will all be in Russian – I won’t understand”, to which he replied, “Ian, it’s Maths, it’s all numbers …”. So I agreed to have a look. Because he was studying tourism, I expected the exercise to be arithmetic, book-keeping, or statistics … something like that. It turned out to be Thermo-dynamics. Apart from the fact I knew virtually nothing about the subject, I couldn’t see for the life of me why he had to learn Thermo-dynamics. If there was a problem with the central heating, wouldn’t they call for an engineer to sort it all out?
I got on well with most classes – although there ups and downs. I tried to make lessons “fun” if I could – with jokes and stories, competitions, puzzles and quizzes, (with Mars Bar prizes for correct answers). There were times when my sense of humour served me well, defusing potentially difficult situations and potentially problematic students.
I was fortunate in that I that there was a Statistics class that needed a teacher … now Statistics was my applied option, (not Mechanics) … so I was asked to take it. A short while later I was asked by the local technical college to teach Statistics to an adult education evening class. I wouldn’t say that I was in “seventh heaven” but I was beginning to feel that I had “landed on my feet”.
One of my “hobbies” was scouring bookshops for copies of old Mathematics textbooks. It was fascinating to see the various topics and how they were handled over the years … I often used examples in lessons. It was an odd “hobby” and I never met anyone else who had such a collection, although I am sure there must have been some. When I eventually left to come to Kyrgyzstan, I had to dispose of many, many possessions and that included a large part of my library … although I had little difficulty in finding takers for some books (e.g. the Kinsey Reports) nobody was interested in the maths textbooks … and I am afraid that they were lost to the tip. I did bring a few books with me – a “Dictionary of Numbers”, a History of Mathematics, and “A mathematician Reads the Newspapers”. I still occasionally dip into them.
Bishkek