(Robyn Williams, of the ABC’s Science program, read my piece in Quadrant about how my own world-view came to be formed, and asked could I do an Ockham’s razor broadcast about my father, mathematics and me. This is the outcome. It was broadcast on Sunday March 12th and interested readers can download the audio here.)
If you grew up in New South Wales in the 1940s, 50s and 60s, there’s a good chance that you studied maths with the help of the textbooks written by two high school teachers, A. G. Aitkin and B. N. Farlow. I knew the books well, because I used them too, and Alec Aitkin was my father. They were mostly for the early years of high school, and he would write them, and draw all the diagrams, on the dining-room table outside my bedroom, from about 6 am until breakfast, which was one of his jobs too. In later life I followed him both in writing books and in making breakfast. He was a good role model.
Dad was convinced that everyone had maths in them, and he had stories to tell to illustrate his belief. One was about a lacklustre student at Canterbury Boys High School in the late 1930s, who explained his poor results by saying that maths meant nothing to him. A few years later Dad encountered him one Saturday morning, the young man, as one might have said at the time, ‘dressed up like a pox-doctor’s clerk’.
‘How’s everything?’ asked my father. ’You seem to be doing well!’
‘Great,’ was the reply. ‘I’m a bookie’s penciller’.
‘You?’ Dad was lost for words. ‘But you didn’t like maths.’
‘Oh, that was then,’ he replied. ‘There’s nothing to maths if you really need it. Actually, I like it.’
Dad loved that story, and we heard it many times. I could counter, later in life, with stories of my friends who had become maths teachers, and loved it, though they had not been proficient at the subject at school. I was the eldest of three boys, and all of us turned out to be decently competent at whatever we studied, but I was the cause of much head-shaking when school reports came in: ‘Don should do much better…’ was a common summary. What vexed Dad was my poor performance in mathematics.
‘You’re too quick and too careless,’ he said. ‘I know you can do all this stuff easily, but you just dash it off, and make simple errors. Why don’t you go back when you’ve finished and check everything you’ve done? You’ll do a lot better that way.’
At the end of my sixth class I followed his advice — perhaps he had given it again the night before the exams. The papers were OK, and as usual, I worked quickly. Instead of looking around to see if I were the first to finish, I went back and checked. I certainly had made errors, and fixed each of them up. When the results came out, I had scored a perfect 400 out of 400 for all the mathematics papers. Dad was jubilant, and the experience stayed in my mind thereafter. What is more, I felt at home with numbers, and still do. Today we call that feeling ‘number sense’.
In 1950 Dad went to be head of the mathematics department at Armidale Teachers College, and most of my secondary schooling was in Armidale. At the end of third year I had to make an awkward choice. My best subject was history, but I liked maths too. Alas, they were opposed, and I couldn’t do physics without both maths. My parents didn’t press for one route or the other. It was, finally, up to me. So I went down the humanities path. I have many times wondered what would have happened if I had done what my brothers did — double maths, physics and chemistry.
Meanwhile my father had become involved in teaching teachers how to teach mathematics, both in primary school and high school. He became known to generations of teachers as ‘Tin Tin’, not because of any daring exploits, but because of his Broken Hill pronunciation of ‘ten’, as in ‘tin times tin’. He already believed that the core problem was the way in which children were taught arithmetic in infants and primary school, and he set out to solve it. He developed a wide network of primary teachers who understood what he was about.
In 1957, on long-service leave overseas, he encountered the coloured rods invented by the Belgian primary school teacher Georges Cuisenaire, and fell for them at once. Back in Australia, they became a basic element in his arsenal, because their use as play allowed pupils to see number relationships for themselves. So many primary teachers, most of them women, had not enjoyed mathematics themselves, and used strict rules as the basis for their teaching. If a child asked a question they could not answer, their tendency was to put the child off, which caused the child to lose interest. In Cuisenaire rods he saw a means by which children could learn by themselves, through play rather than through instruction. His enthusiasm and competence meant that he infected young female teachers with the same possibilities, and they wrote to him about their successes. By the time he retired in 1967, there were few places in Sydney or the bush where he did not have a disciple. His greatest disappointment came when it was decided that learning through play did not sit comfortably with things like the curriculum and the syllabus. If Cuisenaire rods were any good, that would be shown through test results — an understandable perspective that entirely missed the point.
Throughout these years I visited my father regularly, and regaled him with what I was doing, where that bore on mathematics. And it regularly did. My honours and masters degree theses involved what today we would call data analysis, pretty simple stuff, from the vantage point of 2017, but unusual in their day for someone trained in history and moving into political science. I had been taught elementary statistics in psychology in my first year at university, but what I was dealing with now were large numbers — entire election results, and at every level, nation, state, region, electorate, sub-division, polling place. I could see patterns there, I said, explaining my interest, and asked him for advice. ‘You’ll work it out for yourself,’ he said, and I did.
One of the things I had noticed in my work was that in politics people often equated ‘most’ with ‘all’. For example, ‘workers’ or ‘the working class’ voted Labor. It couldn’t be absolutely true, because something like three quarters of the workforce were employees. If they had all voted Labor that party would have been in power since Federation. So what did it mean? When I looked at what passed for data, it seemed that at best about two thirds of the working class voted Labor, however you defined the working class. I thought about it, and some of Dad’s common-sense approach to numbers came to me. If half the workers voted Labor, and the other half Liberal, that meant there was no connection between class and party at all. If all the workers voted Labor, that would mean class and party were the same thing. Where did ‘two-thirds’ fit in that scheme? Well, a good deal close to 50 per cent than 100 per cent. Once you saw politics that way, the narrowness of Australian election results in over a century became clearer. All this did not come to me in a flash, though today it just seems obvious. It came as a puzzle that needed to be solved. When I explained the issue at an academic conference I was not loudly applauded, for I was disturbing fondly-held beliefs, and that does not make you a hero. It still doesn’t.
Some years later, dealing now with several million bits of information in two national surveys for some 2000 people who had found themselves talking for an hour and a half or so, I had to learn how to construct a simultaneous equations model of forty years of national electoral activity that suggested strongly how Australia had become more and more ‘national’ in election outcomes.
At first I knew nothing about simultaneous equations, since they had not been part of what I had learned at school. But, like the bookie’s penciller Dad had once taught, I needed to know, and I needed to know quickly. It proved quite straightforward. By now it was routine for me to check my own work. Yes, it is best practice for any academic, for it is much less embarrassing to detect the flaws in your own work before it is published than to have others point them out when your article or book is in print. But some of my caution went back to that early advice from my father: ‘Go back and check! You’re bound to have made errors. Find them and fix them!’
Later again, I read a paper about the use of Markov chains, and saw at once a use for them in my own work. A Markov chain, named after its Russian inventor, is a way of making predictions about a future (or past) state from the information you have at a given moment. My interest was in how long it took for children to acquire the political leanings of their parents. Remember, the modern party system in Australia started in 1910, when almost all the votes and all the seats went to the Labor and Liberal parties. I had to learn about conditional probabilities, but then I could suggest that Australia by the late 1920s was most likely characterised by party loyalties passed on by about three quarters of the electorate. Again, the bookie’s penciller had showed the way.
I did not get to love mathematics in the way my father did, or in the style of my next brother, who is a professor of mathematical statistics. But I learned two great things from my father in this domain. One is the basic truth that mathematics is not hard if you have a good reason for wishing to employ it. The other is that checking your work is always important. The skilled trades have a phrase for it: ‘measure twice, cut once!’
Too many of us fear mathematics, probably because of the way we were taught in primary school. That is a great pity, because it is a most useful tool in everyday life, and it can be fun. Long live the teachers who teach it that way!