Some months ago I wrote a short piece touching on innumeracy and promised to return to the theme later. By one of those strange, but probably predictable, coincidences, I was thinking about this again and low and behold there was an article in the Independent about a Government commissioned report into the state of mathematics in the UK. Again, quite predictably, the article began, “Next month a Government-commissioned inquiry into the state of mathematics in Britain will report that radical measures are needed to save the subject from slipping into terminal decline in schools and universities.” And, as you might expect, Professor Adrian Smith, the author of the report, said, “We need to make the material much more inspirational so that people want to study maths for longer than they do now.”
I would agree that we need to make the material more inspirational, but fear that, in practice, this would mean using more practical examples like changing money into foreign currencies. What might actually be inspirational would be to communicate the fact that mathematics gives us a different way of understanding the world. And, perhaps, as a kicker, that it can offer a means of uncovering some of the dirty little secrets that others use numbers to conceal from us.
I had begun thinking about the question of innumeracy again after coming across the work of Steven Levitt. He has just won The Clark medal – for the most important economist aged under 40. “Levitt won the medal for his genius in crunching data afresh to answer real-world questions. For instance, if drug dealers are so rich, why do so many of them live with their mothers? (Answer: most drug dealers are poor.)”. He has also looked at things as diverse as the relationship between legalised abortion and crime and the most effective tactics for scoring penalties in soccer.
As you will probably have gathered, my interest in Levitt is because he is someone who is using maths to understand the world. My concern is that I, like most people, have to take his work on trust. We can apply some tests as to the plausibility of his conclusions, but those tests are all non-mathematical. The reality is that most of us are in varying degrees mathematically illiterate, including many who use maths in their work. What tends to be missing is an understanding of what we are doing. The process is largely taught as a series of mechanical operations, even at graduate and postgraduate level in subjects like, science, the social science and medicine.
Speaking as someone who is at the higher end of the innumeracy scale, I feel a sense of deprivation that I still have struggle with numbers and that there so many areas where it is hard for me to make informed judgements, because I can’t test the calculations myself.
Up until about the age of eleven or twelve I appear to be quite good at maths – despite some problems with the rote learning of times-tables. But I say appear because all I had learned were some tricks that I was able to carry out mechanically, but with no understanding. After twelve my interest in maths went into freefall as the sense of tedium of carrying out mechanical operations grew and I failed my “O” level – the exam we take at sixteen – several times. Since I could do the simple stuff, like checking my change, this didn’t seem to be too much of a handicap.
Then in my late teens, in my first job, maths came back into my life. I was working in a film lab that specialised in 16mm films, mostly industrial, a bit of TV work, some medical and a few art films. We had got into a real mess with delivery times, which up until then had been given with a few simple rules of thumb. The mess had occurred because we’d got one new printing machine, which when it worked was very fast, but was also very unreliable, breaking down often. So we had a vast backlog of work and a lot of angry customers.
The MD of the company asked me to set up and run a system to control and monitor the flow of work through the company, which I did. It was a very primitive but effective system based on a form for each order that travelled through the plant with the job and was returned to me at the completion of each stage, a white board where I tracked the progress of each job and a very simple adding machine which enabled me to calculate how many orders we could put through the printing machines each day.
What I didn’t realise at the time was that I was constructing a model of the plant based on numbers – the numbers in this case being chunks of time – and that even a very crude numerical model could be more effective and reveal things that rule of thumb models couldn’t. What I did realise using my primitive adding machine, and that came as a staggering insight, was that multiplication was the same thing as adding a number together several times. Now to many of you this may seem blindingly obvious, but it had never occurred to me in all the years I was being formally taught arithmetic.
My next insight came many years later. At that time I was working in a college that was so old fashioned in its governance that perhaps it should have been preserved as a historical monument. The only person who knew what was going on in terms of finances and resources was the Vice-Principal and he and the Principal handed out resources and budgets to departments rather like Royalty rewarding their Barons.
I had come from a College where I had sat on a finance and resources committee and, as a student knew more about the College’s finances than the Heads of Department in the College where I was now working. My Head of Department believed that we were being starved of resources that rightly should be ours, but had no way to prove it and asked me to look into it.
By one of those odd conjunctions, that seem to occur so frequently, my mother had a lover who was a management consultant and was helping her sort out the finances of a shop she ran. He used a technique of indexing to get a clearer picture of what going on and I clicked that I could use this same technique at my College.
So I gathered together figures that were scattered over a variety of reports and documents and indexed them. Very soon a picture began to emerge of two successful departments, ours and another, that were effectively subsidising others. With a bit more digging around it began to look as if what was happening was that one department, in particular, was being artificially boosted so that its Head of Department who due to retire soon would get a bigger pension.
Now we never explicitly used what I had found, but even a few hints that we knew what was going on meant that we did get an increase in resources even though it was not as much as we were entitled to get.
So the insight I gained from that experience was that maths could be a very useful tool for uncovering dirty little secrets, even when the numbers had been presented in a way that was intended to mystify. However, to do this I needed two bits of technology. First, the idea of indexing itself and second an electronic calculator that helped me do the necessary calculations.
The next insight came a long time after. A friend of mine, a journalist, Paul Lashmar, wanted to construct a league table of local government performance, based on Audit Commission performance indicators. He knew I could use spreadsheets so asked me to help him. We constructed a fairly simple method for doing this, which while crude looked reasonably plausible. We checked it with someone who had worked with the Audit Commission in this area, who agreed that providing we put in appropriate caveats the method could provide some useful information.
We ran the League Tables for three years, first in the Observer and for two years in the Independent. Frankly, it was a nightmare of a job, because it generated a huge amount of controversy and I had deal with mass of queries and complaints from local councils who thought they were too low in the table. And, with each complaint there was always the fear that I had made a mistake, which fortunately I hadn’t. Also there was the inherent problem with league tables that they amplify very small differences so that a council that was say twenty places lower than another might actually be pretty much the same. So when the Audit Commission, who had previously said that national league tables were too difficult to do, started to do it themselves I breathed a sigh of relief and was pleased I wouldn’t have to do it again.
What I found most interesting from the exercise was an unexpected insight, which we mentioned, but nobody picked up. This was that the quality of management could make a difference. When we began I thought we would see a clear relationship between social deprivation and performance i.e. that the councils with the most prosperous populations would be at the top and those with the least would be at the bottom. In fact at one point we were seriously thinking of trying to factor in an index of social deprivation, but gave up because I couldn’t see how to do it.
Now as a rule of thumb it was roughly true. But there was a big but. One of the best performing councils South Tyneside, in terms of almost any measure of social deprivation you could think of, was the same as one of the worst performing Kingston-upon-Hull. The only explanation we could come up with was that South Tyneside was better managed than Kingston-upon-Hull. I found this a very encouraging and hopeful lesson, but unfortunately the larger more important picture got lost in squabbling about relatively unimportant details of relative positions in the league tables.
Now what is it I am trying to say here? Making sense of and with numbers is a powerful way of understanding what is going on the world and seems to be one of those things you need to learn young. If you don’t, you remain one way or another handicapped. Technology can help with the mechanics, but doesn’t help with finding the right technique for helping to explore a question or to help with understanding the significance of a mathematical enquiry. In an interview I quoted in an earlier post, “Smart Heuristics”, Gerd Gigerenzer suggested an approach that might help professionals and by inference could be useful in more general education.
But, I fear, the problem lies much deeper than just mathematical education. The central problem would seem to lie in an approach to education which is essentially about presenting a picture of the world as it is and rewarding the learners who can regurgitate this vision. Since this “world as it is” is a fiction or is intrinsically uninteresting its not surprising that so many young people get turned off.
Human beings are curious and creative creatures, who invented machines, but are not machines. Why then do I wonder, do we treat them as if they are? Just imagine what it would be like if we created forms of education that took account of our nature. What would happen if our education was seen as being about how to explore, interrogate and create our worlds? Who knows we might even find that innumeracy became a thing of the past.