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.