It’s nice to be able to reflect on some of the people I’ve met who have most influenced me. I don’t think it’s going too far to say that most of these heroes have changed my life, but Rex Walford is the only one has changed it twice.
I actually wrote this piece at least five years ago about a man who was not just a hero of mine, but a friend. It’s always been too painful to publish it, but it’s seven years ago today since Rex drowned. He was 76 – the same age I am now – and was still contributing massively in so many ways.
I can’t remember how I came across it, but discovering Rex Walford’s Games In Geography was a real Road to Damascus moment for me. At the end of one summer term our middle school (Y5-Y8) had decided to set up a week with every teacher offering a special programme for children across the school. I decided to try the Railroad Pioneers game, in which teams representing railroad builders compete to construct a transcontinental railroad.
It’s pretty much accurate to say it was an astounding week. For five afternoons mixed-aged, mixed-ability teams researched, discussed, planned, evaluated, schemed and plotted, and in doing so they covered an immense amount of knowledge about American history, geography, economics, politics and probably more besides. It wasn’t just that they were committed and were enjoying it immensely, what really amazed me was the amount and depth of their learning.
And this was in geography, the most tedious and uninspiring curriculum area of all. In my entire career as a pupil I don’t think I ever once gained a pass mark in a weekly geography test, let alone in an exam; yet here I was generating exceptional learning across a geographical topic. So the next step had to be to find out more about Rex Walford. I discovered that he was a lecturer in the Education Department in Cambridge and the leading figure in a revolution that was sweeping his subject – indeed, not only did he become President of the Geographical Association but his was a major voice in devising the geography programme in the National Curriculum.
Of course, I wouldn’t have been much of a teacher if I hadn’t sought to benefit more widely from that week, and attempt to translate as much as possible of the experience into my own subject. I was able to adapt some of the games to emphasise the mathematical aspects, and devised others of my own. I further discovered that there was a network of people playing railway and other games by post; I created several sports games, and a football game called United which became mildly famous. I devised other games to model the processes involved in scientific enquiry – indeed, one of my regrets is that I never managed to launch a group focusing upon the use of simulations in primary and middle schools.
So simulation games became a huge influence on my professional life and the United game and many others became my major hobby commitment as well. None of these huge influences would have happened if I hadn’t discovered Rex Walford, and when I launched an amateur games magazine (which is still going today after 300 issues) I invited him to become involved. Rex was delighted to join us. He set up a team in the football game, enjoying the whimsicality of naming the goalkeeper after the milkman, and using the vicar as a ruthless defender. He joined lots more games and set to work devising a postal game about the aviatrix Amy Johnson.
For some years several of us would meet up for an evening in the pub in Cambridge, though Rex had plenty more to occupy him alongside education and games. He wrote and directed plays (we went to see one he wrote about Amy Johnson), played and recorded in revivals of English pre-war musical theatre songs, and was a Mastermind finalist on TV – indeed, he became a setter of specialist questions. Uniquely, he marked retirement by acquiring both a Harley–Davidson and a doctorate in theology.
So I had the good fortune therefore to know Rex via two routes, professionally and as a hobby friend. But it didn’t stop there, for there was a third connection – Rex’s wife is the sister of one of my wife’s best friends, so we knew him socially as well and we’d often meet at their family occasions. On one splendid evening our three families formed a team of eight to take part in a school general knowledge quiz. The scores were very tight, and at a critical moment we were asked which Commonwealth country used to be known as Pig Island. Seven of us looked at Rex expectantly – and he gave the wrong answer! He apologised profusely for his lapse. “Of course I should have known it’s New Zealand”, he said, “I’ve just been setting the Mastermind questions for a contestant whose special subject is New Zealand!”.
In his mid-seventies Rex showed no signs of reducing his commitments. I have his last Christmas greeting in front of me – “We’re still busy in Cambridge – church, music, drama, and plenty of committees! All the best for 2011 !” Just a handful of days later I opened the daily paper; I saw Rex’s photograph on page 2 and knew something dreadful had happened. Rex and a friend had been lost in an accident for which the word “tragic” was a terrible understatement.
Some weeks later we attended his memorial service. Ely is a small place, and by the middle of the day it was full of visitors and it was apparent that almost everyone we saw was going to the service. Though the town may be small, Ely Cathedral is a huge building – and it was full. There were so many aspects to Rex’s life that we were asked to introduce ourselves to those next to us and explain our connection to Rex. Knowing that there were colleagues from his religious, educational, dramatic, and musical lives present, I diffidently introduced myself as the inventor of a postal football game Rex had enjoyed. The man in the next seat exploded with excitement; “Conington Thursday!”. He called to his wife a couple of seats away, “This is the chap who invented United!”. He was a member of Rex’s village dramatic society, and told me in great glee how every week Rex would recount how the team had performed in the latest postal games and how he’d been forced to drop the butcher for lack of form and introduce the window-cleaner from the next street in his place. You can have no idea how overwhelming it was to have such proof of how much Rex enjoyed being part of something I’d created.
I said Rex changed my life a second time. I was up for interview for the biggest promotion of my life, for the job as maths adviser for the 400+ primary schools in Hertfordshire. The main section was a meeting with the Senior Adviser. His first words were “I’m a geographer. Can you think of any links between geography and mathematics?”. If I’d been asked to choose the first question for myself I couldn’t have come up with anything better. For the next few minutes I talked enthusiastically about Railroad Pioneers and a host of other games such as Rex’s Caribbean Fisherman. I talked about how they were underpinned by mathematics and gave an impetus for developing mathematical exploration, and about how I’d used railway and airline games of my own in my maths teaching. I’d probably have gone on for the rest of the afternoon, but I figured it would be only polite to let him get a word in, and I managed to stop before his eyes began to glaze over. I seriously doubt whether any candidate has ever enjoyed being interviewed for a senior post quite so much – certainly I know I was genuinely sorry when the interview came to an end; and, yes, I did get the job.
I tried and tried, but I really don’t think I ever got it through to Rex how much I owed him. Indeed, the very last words he said to me were to imply that I was the one with remarkable achievements. “You should write your autobiography Alan, you’ve done so many interesting things.” Well, yes, there are some things that have gone well and I’m proud of, but all the same it was probably the biggest mis-statement he ever made. I’m not the one who wrote a pantomime every year, or was a decent middle-distance runner, or who was a sports reporter for the local paper (which led to Rex playing semi-pro football – albeit only for 45 minutes until the missing player turned up). I’m not the one whose books went far beyond geography to include church history and guides to writing one-act plays, or who recorded CDs of 1930 English popular songs, or wrote plays for radio. Most of all, there won’t be over a hundred tributes posted on my professional association’s website, and I certainly won’t be the one with a memorial service bringing well over a thousand people to Ely Cathedral.
This week there was a Twitter chat in which people said they enjoyed using a game I recognised as a version of an activity originally called Caribbean Fisherman. These were secondary maths teachers but actually the game was devised for use in geography classes. The inventor was Rex Walford, and for more than five years now I’ve had a piece about Rex I’ve never yet been able to bring myself to publish.
Even though it was a game for secondary geography students, I did, a quarter of a century ago, manage to convince my team of its potential in primary maths classrooms. In fact we devised a version called Fruit Picking and used it with five yearolds. I enjoyed the Fruit Picking version well enough, but for me the parent version of Caribbean Fisherman was always my favourite. A single 1-6 die was the only equipment needed to work with any group, be it a class of thirty pupils or 150 teachers at a conference.
Each player acts as a fisherman living on a Caribbean island. Every day you go out in your boat to position your lobster pot so you can sell your catch to the tourist hotels. Each day you’ve a decision to make. Place the pot in the waters close to the shore and you get a catch that’ll be worth 2 dollars. And your catch is pretty well guaranteed, because there are plenty of fish and whatever the weather, it’s always safe to fish so close to the shore.
But with a bit more effort you can put the pot in waters further out where the fish are better and your catch will be worth 6 dollars. However, there’s a snag. Sometimes the wind blows hard, and these offshore pots get lost – and so you make no money at all.
So there’s a table like this:
|Good day||Bad day|
One of the great things about even a simple simulation like this is that you can add in more and more as the game goes on. Not only does this add to the fun, it gives the chance to make some realistic points.
So if there’s no offshore catch on a bad-weather day, then the only successful pots are the inshore pots, so their catch has an extra scarcity value. Shouldn’t this make the inshore catch worth more? So the table now looks like this:
|Good day||Bad day|
So for each day every fisherman has to make their choice. Does s/he go inshore or offshore? Each day the pots are placed and then I find out what’s happening with the weather. I roll the die and the resulting weather applies to everyone: 1 to 5 means the weather is fine; but a 6 means the weather is bad.
As soon as everyone knows how to play we can move on a bit. Of course, in practice no fisherman has just a single pot; it’s more sensible for everyone to have six pots, so the figures get a bit more complicated.
I said you can add in more and more features and hence more realism. Your weekly target income might be 80 dollars. Once you’ve done that, what decisions do you make? Do you take the rest of the week off? Or do you keep fishing to build up a cash reserve? And if you don’t make your 80 dollars then how do you pay the bills?
Perhaps you should you split your pots and put some inshore and some further out.
When bad weather means you lose a pot, shouldn’t you need to replace it (at a cost of 5 dollars)?
Bad weather tends to come in spells, so when there’s a bad day, there’s a greater chance it will be bad again the next. So after a bad day, a roll of either 5 or 6 on the die means bad weather the following day.
Depending on how the game goes, you can pile on more and more. Sometimes pupils suggest forming co-operatives or perhaps friendly societies which can offer help to someone who finds themselves in difficulties. On the other hand, if you build up lots of cash, then what do you do with it? Or perhaps there are so many people fishing that the market becomes over-supplied, prices drop, and people look for alternatives – pot-making, money-lending, working in the tourist industry, ….
The beauty of Caribbean Fisherman is that such a simple idea can introduce so much, not just in mathematics and Geography, and in social issues as well. Rex Walford was my friend and pretty well close to my idol; in a few days from now it will be the seventh anniversary of his death in a tragic accident. This time, at last, I really do hope to force myself to say more about him.
There’s a story I tell whenever I get the opportunity. You must know it too. It’s the story of how the young Carl Friedrich Gauss, who in 1785 or so was aged about eight, was set the task of adding the whole numbers from 1 to 100. Rather than adding each number in turn, he promptly wrote the answer on his slate and placed it on the teacher’s desk.
It’s a great story, and it offers probably the only piece of genius mathematics which we can all grasp. I’ll invite children – and indeed teachers – to consider how he might have been able to give the answer so quickly. He never did explain his method, but presumably recognised that you can take the highest and lowest numbers, 1 and 100, and add them to make 101. Then the next highest and the next lowest, 99 and 2, making 101 again, and so on. Then all he had to do was notice that there will be 50 pairs totalling 101, so giving a total of 101×50, equalling 5050.
One of the things I love about this is the immense power it gives us. We’re not restricted to adding the integers from 1 to 100; adding the whole numbers from 1 to 1000 is little more work. Your set of numbers doesn’t have to start with 1, and as long as they increase by the same amount each time they don’t have to be whole numbers either. Once you’ve understood the method you can find the total of sets which include fractions, decimals, and negatives – there’s a formula you can use for summing such series, but learning it becomes wholly redundant.
Another reason the story’s so popular is its great human interest and it’s been told time and time again; there’s a website with well over a hundred versions (http://bit-player.org/wp-content/extras/gaussfiles/gauss-snippets.html ). Many of them are very fanciful, but it’s easy to pull out the basis – the task itself, the little boy, and the school-master Johann Georg Büttner.
Many of the versions have incorporated details which are distinctly fanciful – that Büttner was idle, or a sadistic bully, who was scornful and disbelieving of his young pupil. Often there’s a David and Goliath slant – the ingenious pupil defeating the hulking teacher. Now in the last couple of years I’ve done a large amount of reading about mathematics teaching and I’d like to offer a different interpretation which I think is far more accurate.
It’s lucky Gauss was born in Germany. If he’d been English it’s likely the world would never have heard of him. It’s frequently said England was the worst educated country in Europe; in England it’s unlikely there would have been a school for him to go to, and there was no great desire from anyone to do much about it. The church and the gentry didn’t want their peasants to be too well educated, and parents were happy to put their children out to work – most English eight-yearolds would already have been working and earning for a couple of years.
And where there was provision it was often scarcely deserving of being called a school, with the teacher someone looking to top up his main income, or an older person no longer able to earn a living in other ways. England was so slow developing an educational system that Gauss was middle-aged by the time the first tentative steps towards a national English system of schools were taken, and the first generation who’d studied and trained to be teachers didn’t emerge until he was an old man. Indeed, it’s scarcely believable, but when Gauss died in 1855 there were hundreds of English teachers who were illiterate and couldn’t sign their name to documents.
So Carl was indeed fortunate to have been born in one of the German states. Prussia, for example, had established teacher training programmes before 1750 (virtually a century before England), and had compulsory state education to 13 before 1800. In England attendance didn’t become compulsory until 1880 and it was only at the very end of the century that the leaving age was raised even to 11, and then 12. But even in 1898 attendance was still nowhere near 100% and there were still cases of 5 and 6-yearolds working 12 or 15 hours a week.
Far from being an ignorant oaf Büttner was a trained professional. Rather than ridicule Carl’s achievement, he created an individual programme specially for him. His assistant Johann Martin Bartels lived on the same street as Carl, and Büttner arranged for him to give Gauss individual tuition. Bartels may well have been the most remarkable teaching assistant of all time – indeed, he became a university mathematics professor himself, numbering Lobachevsky among his students. His relationship with Gauss was so productive that they were still corresponding forty years later. What an amazing piece of good fortune that a tiny school should have such a tutor available!
The help Büttner and Bartels gave Carl didn’t end there. From his own purse Büttner bought Carl the best mathematics texts available, and he had the contacts to ensure that Carl’s education didn’t end at the elementary stage but continued into secondary school; from there he and Bartels arranged for the Duke of Brunswick to provide for a university fellowship which set him on the path to become the “Prince of mathematicians”.
Few of us will have the good fortune to number a genius among our pupils – the closest I’ve got is to have known Dick Tahta, who Stephen Hawking has always acknowledged as his inspiration. Johann Georg Büttner appreciated a pupil with exceptional ability, and deserves a far better reputation than he’s been given. He recognised and nurtured one of the greatest mathematical geniuses of all time and rather than traduce his memory all teachers should be proud of the example he set us nearly 250 years ago.
I’ve been lucky enough to meet quite a lot of people on my personal list of heroes. In almost every case my heroes have not just been inspiring, but they’ve been prepared to give me their time and encouragement and they’ve always given a good impression of being pleased to know me.
But there is a saying that you should never meet your heroes, and on just a couple of occasions I found myself rather wishing the meeting hadn’t taken place at all. One of those disappointments was EB.
Early on in my career I had a class who with deliberate malice would make their teachers’ lives miserable, and they were old enough and clever enough to make a pretty decent job of it. I was unlucky enough to have to teach them three years running, and it wasn’t until the final year that they decided I was good enough to be allowed to teach them in an agreeable manner.
For most of those first couple of years they’d have me in near despair. I didn’t have many weapons in my armoury. Just two really; one was a dogged refusal to be beaten by a gang of kids when my self-respect was on the line; the other was a couple of books by EB. The books presented an unglamorised picture of a young teacher in the toughest of London secondary schools, inch-by-inch moving from regular humiliation to something like comfort. If he could do it in the most demanding of environments, then there ought to be some hope for me; I probably read the books half a dozen times, not really looking for tips, but using them like a comfort blanket.
Fifteen years later we’d both moved on. I had indeed become soundly established, and EB had long since left teaching to become a full-time writer and broadcaster. Our English department invited him to speak to our top year and I couldn’t have been more excited at the thought of meeting the person who’d meant so much to me, and whose books are still on my shelves today.
And what a disappointment it was! I couldn’t blame him for being an old-age pensioner rather than a dynamic young teacher, but I could resent that there were no signs of the spark that had won over his pupils a generation earlier. What was left was just another visiting speaker with nothing of interest to say to young people, and – worst of all – not for one moment did he sound like anyone who’d ever met a group of schoolchildren before. Afterwards I introduced myself and attempted to say how much he’d helped me, but he clearly felt he’d left those days far behind, and the conversation didn’t last long.
EB died nearly twenty years ago and an obituary said of his classroom books “EB portrays himself as a martyr rather than the boastful messiah of other autobiographical classroom accounts published around that time. But behind the initial panic he never lost sight of the essential good-humour of the young tearaways he was in charge of. Gradually teacher and taught came to an accommodation satisfying to both. His account of those years is still the best book ever about life in the classroom. Lessons that did not work are described with a rueful honesty that makes descriptions of the more successful times to come all the more convincing.”
I suspect that he and his books are no longer particularly well-known, though anyone interested will be able to identify him easily enough. He was a decent, civilised man and I’ll never forget how much his books meant to me – but I really do wish I’d never met him.
Everyone’s heard of Zoltan Dienes, or at least the multibase blocks he invented for teaching place value and which bear his name. “How could children learn what base ten is if they are not familiar with other number systems?”, he said.
He was born in Hungary – he’d tell people he was actually born in the time of the Austro-Hungarian empire – but grew up and spent much of his working life in England. He travelled widely and subsequently worked in many other countries as well; indeed, he had honorary degrees from several different countries.
Last year I came across his website http://www.zoltandienes.com/ I was pretty well bowled over at what I discovered. Firstly by the fact that, well into his tenth decade, Dienes was still alive and productive. Also by the wealth of materials he’s produced which I’ve downloaded for exploration when I get around to them – lots of articles which start off using very simple and concrete situations to develop some complex mathematics.
Part of my interest was down to the fact that as an undergraduate I was actually a student of his for a term or so. The word had reached us that he was doing remarkable things working with young children, though I guess he wouldn’t claim that teaching differential equations to chemists and physicists was the greatest of his achievements. It’s overdoing things perhaps to put him in the category of my heroes, but he was a remarkable man and few of us can hope to be known throughout one’s profession, fluent in five languages, married for 68 years, have seventeen great-grandchildren, and be active into our nineties.
Dienes lived in Nova Scotia and died last month at the age of 97.
Peter Reynolds played quite a rôle in my career. He invited me to join the Mathematical Association’s Diploma Board, he gave me my first speaking engagement outside my own county, and he was the first to suggest I might have something to offer schools in general rather than just my own. One day a letter arrived: “Dear Alan – have you ever thought of becoming an advisory teacher? I think you’d be excellent in that rôle. Come along for interview ….”
(I turned up, clutching my hand-written letter – and found seven others all with their individual letters! So in fact I never got to work in Peter’s team, but it was this experience that started me thinking and a couple of years later I did make the AT step in my own authority.)
Peter never sought a high profile, but he contributed enormously to mathematical education. Much of his work was done for the Mathematical Association ( http://www.m-a.org.uk ); he was the first editor of Mathematics In School, and the Diploma Board was a leading influence in the development of maths teachers. He also served on the Cockcroft committee which resulted in the hugely influential report “Mathematics Counts”.
I suspect that his image was responsible for much of Peter’s effectiveness. He was always well turned-out, and quietly well-spoken. He looked in fact like the typical grammar school teacher of my own schooldays, and was comfortable with officials and committee members. They looked at Peter and saw someone they could work with and who wouldn’t rock the boat. What they didn’t realise until it was too late is just how deceptive that image was.
Peter was in fact a deeply subversive individual and it was his influence that saw Suffolk as a hot-bed of curriculum development in mathematics. He assembled a team (sadly not including myself) of iconoclasts. Not all of them shared his impeccable dress sense, but they were all committed to innovation, most particularly in the contribution the electronic calculator could play in the development of children’s understanding on numbers. Peter’s team played a large part in Hilary Shuard’s pioneering CAN Project that we in nearby Hertfordshire followed with interest.
Peter was another who died much too early, in 2000 aged 68. Not long before, we’d worked together again on an MA group, and on the last occasion we found we were both planning to look in at Mole Jazz. Mole Jazz was at Kings Cross and even in that somewhat dilapidated area was a bit of an eyesore. I bet not one of those committee members who imagined Peter was one of themselves had ever heard of it.
Hilary Shuard was a giant of mathematical education. To generations of teachers “Williams and Shuard” was mandatory reading, she was a member of the committee that created the Cockcroft Report of 1982 and an important part of the working group that created the National Curriculum in 1989 – as an obituary said “Indeed, it would be unthinkable to have had a national committee concerned with mathematics education on which she did not sit.”
I’d never claim to have known Hilary well, but we served together on a number of working groups and we did meet on dozens of occasions. I was proud to claim after one meeting that it was the only time in my life when a member of the opposite sex plied me with strong drink, pinned me against the wall, and refused to take no for an answer. The result of this meeting was that in Hertfordshire we introduced a project built on one of Hilary’s most dramatic innovations, the Calculator Aware Number curriculum.
As the 1980s progressed and the electronic calculator became widely available and easily affordable Hilary saw earlier than anyone its potential for helping children learn. “For the first time we have a toy that contains the whole number system”, she said. One 6-year-old told me he “Did experiments with my calculator, like see what happens when I multiply numbers by 99”.
Before Paul – whose teacher felt he was not of exceptional ability – no child in history had the opportunity to play with numbers in this way. Hilary maintained passionately that if children were allowed to use calculators they would understand numbers better. For many people this was controversial, even though the CAN results, and indeed the findings of our own Hertfordshire project, gave dramatic support to her view. A group of our own teachers gave a presentation at a national conference and people flatly refused to believe their findings, and of course even today our new National Curriculum refuses to see the calculator as having anything to offer other than as a short-cut to getting sums right.
Tragically, Hilary was prevented from bringing CAN to full fruition. A horrific road accident, where a dislodged cats-eye hit her in the head, caused her to spend many months in hospital. Amazingly she returned to work, but died in 1992 aged just 64.
I can’t believe we’ve ever had someone with such a breadth of expert knowledge as Hilary Shuard. She was as at home in an Early Years class as she was working with A level students; what she wrote for teachers of both groups was recognised as being the state of the art; she also wrote authoritatively on the use of Logo at a time when computers were just being introduced to classrooms.
When I went to her memorial service at Cambridge I learned she’d been just as prominent in a totally different walk of life, and she’d been active in women’s sport at a high level. I’ve a feeling she played top-class hockey, and I know for sure she played cricket at county level and was good enough to be selected to play against touring teams from Australia and New Zealand.
Like many of the great people, Hilary would be happy to treat you as an equal even when it was patently obvious that this wasn’t remotely true. I remember with great pride the evening the phone went; Hilary said she’d more engagements than she could handle and would I mind deputising for her to speak at the National Association of Head Teachers conference? No of course I didn’t mind, though I doubt the NAHT were half as thrilled as I was. A few years later my wife and I were on holiday in Italy and we shared a table one night with a couple. During our conversation we learned that the woman was a teacher and Hilary’s name came up. I mentioned the NAHT story and she couldn’t have been more impressed if I’d told her it was the Prime Minister himself I’d been asked to deputise for. It seems a pretty good reflection of the love that teachers had for Hilary that ten years after she died our new friend couldn’t wait to get back to school to tell her colleagues she’d met someone who just once was the next best thing to Hilary Shuard.