Category Archives: Scrapbook

Give A Dog A Bad Name – Johann Georg Büttner

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 ( ). 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.




Farewell, August

August is traditionally the silly season for newspapers, and there are a couple of educational stories they regularly treat us to. Firstly, there’s the grand announcement of the results, which gives columnists the perfect win-win situation. If the number of successful students goes down, then clearly educational standards are falling. And when the numbers go up – well, it’s obvious the exams are easier and hence standards must be falling.

It’s not just the writers who get busy; the photographers have an even better time. It’s truly remarkable just how many successful students are amazingly attractive young women.   Judging by the photographs, only about 1% of successful candidates are boys or less attractive females.

There’s another regular August story, and this one’s more complicated and more serious as well. Each year the boundary points at which levels are awarded are liable to change slightly. So a mark of 75 may receive a different level this year to what was awarded last year. This can cause immense pain to teachers and pupils, and is liable to be interpreted by columnists as political manipulation, and by teachers as “arbitrary” or “capricious”. Now at the Key Stage 2 / 11-year-old level this is an area I know something about, and I presume the same principles apply elsewhere.

The basic factor is that it’s relatively easy to set examinations that test the syllabus or programme of study in a valid way. However, it’s quite impossible to set two examinations on the same syllabus on which pupils will perform identically, and here’s why.   We’ll simplify our syllabus so that it consists of one element only, that children must master their multiplication facts; so that on a certain date in May every child will be tested and to make it fair they’ll all be tested on the same statement. Let’s say we ask them 6×9. Their papers are sent off for marking, the results are analysed, and in August the results become available and perhaps 85% of children are found to have been successful.

But what do you do next May? You could ask next year’s pupils 6×9 again, and perhaps 93% of children are successful.   But in the meantime you know parents have been practising their children on 6×9, publishers have been bringing out 6×9 games, worksheets, and practice cards; schools have been putting on 6×9 practice sessions, and a thousand Youtube channels show 6×9 rhymes and raps.

So next May you decide to ask 7×8 instead, and this time 87% pass. But we don’t have much idea why this is. Is it because all the extra practice has meant children know their tables better this year? Teachers will claim the higher pass rate is down to their skill and commitment, while the government will claim a triumph for their enlightened policies. But perhaps the whole cohort is of slightly different ability. And it’s certainly true that 7×8 won’t present exactly the same level of difficulty as 6×9 to every child – some may find it easier to remember, and others harder.

And of course in the real exam there’s not just one, but dozens of questions sampling dozens of syllabus skills, so while we can be pretty sure we’re setting an exam that is fair and valid we really can’t say that a mark of 60% indicates the same level of performance as it did last year.

So just how do we ensure that a grade from last year is comparable to the same grade this year? This is vitally important and it’s a hugely sensitive issue. It’s also fiendishly difficult, and the boards use every method they can think of; many aren’t particularly watertight in themselves, but they do offer pointers. There may be an Anchor test, taken by a random selection of the age-group; the Anchor test stays the same year after year, so that gives an indication how each cohort compares to the last. Another process is that some of last year’s candidates sat this year’s paper immediately before they sat their own test, so we can reason their performance on the two tests will be similar. Of course the statisticians will be at work as well. There are likely to be other processes involved I don’t know about, but one I have experienced is where a panel of the most expert authorities sit down and examine a selection of papers at this year’s borderlines and compare them with borderline papers from previous years.

All of these are partial indicators only, and they all have disadvantages, but when they’ve all been taken into account it may be necessary to take the decision that one or more of last year’s boundaries may need to be adjusted by a point or two.  I’m as sceptical about politicians as you are, but I’ve been given every assurance that this judgment is made on educational grounds and nothing else whatsoever.  You can at least be certain that no examination board ever adjusts boundaries without a huge amount of thought and effort, and you can be 100% sure this is never done in an “arbitrary” or “capricious” manner.



The New Sink

We’ve recently had some kitchen improvements, so we now have a nice shiny stainless steel sink.   We’re rather proud of it, but what no-one told us is that even the top-range models are so sensitive that they take offence if you don’t speak nicely, or if you forget to wish them good morning in an agreeable manner.

Consequently, within five minutes the first scratch appeared. True it was invisible to all but high-powered lighting and advanced microscopy, but I was at the hardware store buying a sink mat moments later.   The sink mat worked very well but it really didn’t look state-of-the-art, and a couple of weeks later we were in a kitchen shop and found something rather more interesting.

We’d discovered a device called a SinkSaverTM. The basis of this is that the designer has said that the business area of your sink can be thought of as a 3×3 square. One of the nine cells will be the drain area, so there are eight cells which the Sinksaver needs to protect.

The problem for the designer is that different models of sink have the drain in different positions.   Sometimes the drain is at a corner, sometimes in the centre, and others – like ours – have the drain in the middle of one side. So the designer has produce a device which is made of two separate sections which can be clipped securely together to accommodate the drain in any one of the three possible positions.

You can find out more at  – but before you do that, can you work out the shapes of the two sections?























Sink 1


Sink 3


Sink 4



The Logbook

I first published this piece two years ago following a chance encounter with a school logbook.  By the end of the week the owners of the logbook had invited me to talk to their history group about the history of mathematics teaching in elementary schools.  Only after agreeing did it sink in that I actually didn’t know very much at all about the subject, and I’ve spent the last couple of years trying to find out.  

It’s become a major interest; I’ve explored texts, archives, reports, lots more logbooks; I’ve picked the brains of everyone I can think of and I seem to have run out of people who know more than I do.  I’ve found out a lot of interesting things along the way, so I plan to make regular postings on the topic.  To start the ball rolling, this is my original piece.


logbook cover

Recently I spent a quite fascinating afternoon looking at an old school logbook.  It wasn’t until the 1870 Education Act that education became universal, and it used to be mandatory for schools to keep a record of events.  The Headteacher would make an entry at least once a week; judging by this particular logbook, dating from the latter years of the nineteenth century, the Head would have a lot of discretion about how this requirement would be met.

logbook (1)

Over a period of thirty years or so the job changed hands a few times, and some incumbents wrote just a single line – sometimes simply “Nothing important happened this week”.

logbook (3a)

Later Heads wrote more, and as the book filled up over the years they would regularly be writing a page or more.

The school was in a village  with a population of 500 or so near Banbury, around halfway between London and Birmingham, and the book covers the period from the mid-1880s to 1906.  Typically, roll numbers were around 75 with an infants class and another class for older children.

It was a rural community and children were often away from school helping with duties like potato-picking and harvesting, and others I’ve never heard of – “leasing”, and “birdminding”.   A consequence of the 1870 Act was that the authorities were pretty strict about attendance, with visits from the attendance officer and the attendance registers being audited frequently.  Later Heads would state the percentage attendance for both classes every week, but even so clearly had the authority to use some discretion (on one occasion the Head decided not to open school on the day Barnum and Bailey’s circus came to town).

It wasn’t just the attendance officer that the Head had to worry about.  He himself taught fulltime and needed to supervise other classes to check on progress; the Rector visited regularly, and the Government Inspector came as well, perhaps once a year.   I was a little surprised to note that often the reports of the Head and the Inspector would often give mathematics (more precisely, arithmetic) a low profile, being subsumed within “Basic” studies. Greater priority might be given, particularly in the Infants, to Handwriting, Singing, Needlework, or Recitation.

logbook (4)

(“…. The knowledge of the elementary subjects is good on the whole, but Arithmetic is weak in the fifth and seventh standards.  Geography is good, History fair, and Needlework is well done.”

Ever since I began teaching we’ve been told that in the olden days every child knew their multiplication tables.   It’s not true!logbook (5a)

(“Standard III want great attention in their arithmetic tables not well known.”)

There’s another widespread belief – that children in the past were impeccably behaved, and that today’s society, and teachers in particular, have allowed standards of behaviour to plummet. The 1890s Head had plenty of non-impeccable pupils. In a school of just 75 or so, half a dozen are named week after week and several others less frequently.  Not all of them were boys – Minnie W seems to have been a real problem, being excluded from class time after time.  Her brother? / cousin? Reginald is pretty well as bad, while Oliver G “can’t be left for a moment without getting into mischief”. One senses a grim smirk on the next page when Oliver falls off a prohibited wall and breaks his leg – but a year later he “is just as bad as before he broke his leg”.

John J was another regular offender, with a particular habit of “molesting the girls on their way to school”.

I was shocked by one incident, when I read that Ernest L and Clement W (another relation to Minnie and Reginald!) attacked their teacher on her way home. I’ve never heard of such an incident, and I hope the teacher was satisfied that sending offenders home and making them apologise dealt adequately with the matter.

logbook (13b)

(“Two boys, Ernest L and Clement W, waylaid their teacher on her way home and stoned her – troublesome boys but the first is an imbecile and dangerous.  The correspondent asked that he might be sent home and the other to apologise.”)

Indeed, and contrary to what one might have expected, corporal punishment seems to have been rare. In 300 pages I found only one direct mention, when John J “an excessively bad boy … at last had a stripe this Friday afternoon”.  From another source I find that boys in their early teens would routinely receive fierce punishment (birching, or hard labour) for stealing items worth just a few pence, so if physical punishment at the school was indeed as rare as it seems then that does indeed surprise me.

No doubt the teachers breathed sighs of relief when Oliver and John and Reginald left school for the last time, probably at the age of 13.  Little did anyone know that several of those happy, carefree, mischievous boys had fewer than fifteen years left to look forward to.  This tiny village of just a few hundred sent 86 men to fight in the Great War, and no fewer than 25 never returned.  Reginald and Clement died on the Somme within a few months of each other; to the unimaginable grief of their parents both lost an elder brother as well.



Some Limericks

Chris Smith ( ) sends me his weekly Newsletter.  It’s really for staff and pupils at his school, but it always contains a few jokes and puzzles and he’s happy to distribute it more widely.


This week’s issue contains some limericks, and I enjoyed this one:

There was a young man from Lahore
Whose limericks stopped at line four.
When asked why this was,
He responded, “Because”


I decided we really ought not to stop there, so I offered Chris four more, of which I think the last one, at least, deserves to be better known:

An inventive person from Dundee
Wrote limericks to end at line three
“That’s all I have to say, you see”

There was a young man from Peru
Whose limericks stopped at line two

There was a young girl from Verdun

And, finally, the piece de resistance:


Chris then challenged me to produce a limerick for negative one. I couldn’t manage that, but I did offer him two more:

My friend Metcalfe




Further contributions welcome.

Footnote: I came across this limerick for pi.  I couldn’t decide whether it fits into the piece on limericks, or the one about pi.  Easy; just put it in both:

It’s a favourite hobby of mine
a new value for pi to assign
I would fix it at three
because it’s easier, you see
than three point one four one five nine



Any blogger who doesn’t realise that 14th March can’t be paying attention.  We’ll need to forget that in the UK we write 14/3 rather than 3/14, because this year is of course special, since Sunday will be 3/14/15.

We all know the usual approximations of       22/7    3.14      3.142

All these are good enough for most of us; I can’t think I’ve ever need to know that the first eight digits are 3.1415926, but one of my old pupils Johanna kindly created the mnemonic I’ve used for 25 years:

Dad, I want a scaly armadillo to cuddle


And that’s all I have to offer as a contribution to Pi-Day.  In any case, the 15th is Mothering Sunday and I’ll have other commitments.


Footnote: I came across this limerick for pi.  I couldn’t decide whether it fits into the piece on limericks, or the one about pi.  Easy; just put it in both:

It’s a favourite hobby of mine
a new value for pi to assign
I would fix it at three
because it’s easier, you see
than three point one four one five nine




A Promise Broken

I’m pretty sure I promised I’d try and avoid writing about either Brighton and Hove Albion or jazz, but I’m going to recant on one of those.  This week we went to the Sheldonian Theatre in Oxford, designed by Christopher Wren and built in the 1660s.  It’s a notoriously uncomfortable building and sufficiently vertiginous that you get given a card with a safety warning, but it’s very significant for us, because it was there that our son Simon received his MSc in eLearning from Oxford University a couple of years back.


The three of us went to hear the Oxford University Jazz Orchestra and the choir Schola Cantorum of Oxford give a performance of the Duke Ellington Sacred Concert music.  It’s good to know that a university can still raise a big band and indeed one of near-professional standard; better still, that several hundred people can turn out on a wet Sunday evening to fill the Sheldonian.  I imagine that any university student has more sense than to try and make a living from playing jazz, but the bass player Lila Chrisp and particularly the terrific drummer Ben Varnam are names I’ll be keeping an eye open for just in case.

Sheldonian 150221

The band was supported by a couple of professionals: vocalist Tina May and the master saxophonist Nigel Hitchcock.  Annette Walker tap-danced a couple of selections, naturally enough including “David Danced Before The Lord”, and James Burton conducted choir and orchestra with immense enthusiasm.  I’ve never heard the music played better or with more emotional power.  Duke Ellington’s sacred music was very important to him but received some pretty poor reviews from critics at the time, but I doubt whether anyone in a building which holds up to 1000 people had a negative word to say about the music or the performance.

Sheldonian 150222 (6)

And, yes, I do have just the most tenuous justification for mentioning Duke Ellington.  Iva Sallay in her blog at  recently mentioned a quote attributed to Duke:

“A problem is a chance for you to do your best.”

You can find it in his autobiography ‘Music Is My Mistress’.  There are a few books about Duke on my shelves – one of them I came across a few years back, very cheaply in a secondhand book shop.  I’d never seen it before, and it looked interesting, so I happily coughed up a few pence.  When I got home I looked more carefully and found proof I must indeed have seen it before – I’d written my name inside the front cover when I’d purchased that very copy the first time!






My School – and Leapfrogs

I feel a bit of a fraud sometimes, strolling into school to do my one or two sessions a week.  Clearly, I’m not more than the smallest cog in the machine.   However, I get the staff emails, I have the security code for the front door and the photocopier as well, and my photo’s on the board in the entrance and on my staff badge, all of which go to confirm that I am indeed officially part of the staff. I find I’ve been there for five years, which of course is longer than all the pupils and many of the teachers.

But what makes it really special for me is that it’s the same school where I was Deputy Head not far short of 40 years ago, which means that my first pupils there are now fast closing in on their 50th birthdays. Sometimes I’ll stroll through the building calling up memories – I taught there for nearly ten years in the 70s and 80s, so there are plenty of those in every corner and every classroom.

Of course, much is completely different these days, from the moment I drive into the car park and see the new buildings and lots of girls playing football. Perhaps the biggest change is the place of the computer. 35 years ago I had to raise funds to buy our first computer, which was quite likely one of only a few dozen in the whole town, but I’m afraid left most of our staff underwhelmed. Today we have a room full of computers and every child can use a variety of programs and sees nothing remarkable about it.

And to be strictly accurate, it’s not the same school; it’s been reorganised into a different school, with a new organisation and new set of governors, and is now a primary school for 7s to 11s rather than a middle school for 9s to 13s – but they still feel like much the same pupils coming from the same streets as forty years ago. I don’t think the principles guiding what I do, or many of the activities, have changed much over the years, though I was amused when I loaded up my favourite computer adventure game for the first time in a quarter of a century. Its introductory screen says “It is a warm sunny day in 1993”, so what offered children an exploration set ten years in the future has become a trip into the past – but who cares, the program still enthuses today’s children just like those of thirty years ago.

There’s one anecdote no-one will believe. I first came to Berkhamsted in 1971. It was an exciting time and I was delighted to be the first head of mathematics and science in a brand-new middle school. The County Treasurer came to a welcome party for the staff. “We are the richest county in England”, he said.   “Anything you ask for you can have!”   It was true. Lorries drew up, delivering photographic enlargers and equipment we hadn’t asked for, and woodworking machines that no-one on the staff knew how to use.

Counties, and their advisory staff – and I was later lucky enough to lead the Hertfordshire advisory team for primary maths – were major sources of curriculum development on a scale that’s already just a fading memory. In-service courses, residential centres, and advisory teams all developed innovation and advice. And it wasn’t just the counties, either. The professional associations of the Mathematical Association and the Association of Teachers of Mathematics had a profile that ensured that governments willingly sought their views.

But you’ll be wondering what on earth Leapfrogs have to do with this post.  It was during my first period at the school in the late 1970s that I first met some of those at the cutting edge of innovation in mathematics teaching.  I joined an ATM spinoff called Leapfrogs and for three years we went to their summer conference.  I say “we” because a Leapfrogs summer conference was a combination of workshop, blue-sky thinking, and family holiday for the three of us – at £10 the only one we could afford at the time.

Leapfrogs weeks were probably the most exciting development sessions of my life.   Teachers, partners, young children were all involved in informal and exciting sessions which were important enough to attract university lecturers and participants from Europe and the USA.  Somebody must have secured some significant funding, because several booklets which were the very antithesis of textbooks were published – you can find the Leapfrogs Link and Action books at

I remember talking a lot of nonsense at Leapfrogs sessions, but it was an atmosphere where you could say anything and have your opinions respected, and it was a wonderful opportunity for a classroom teacher to mingle with experts and innovators from around the world.

img415 Monkton Wyld c. August 1977


img350 Monkton Wyld c. August 1977

(That’s Jill and myself at the base of the steps, vintage 1977.  I can’t recall the woman at the top of the steps, and I’ve no idea who actually took the photograph.)




A Coincidence

I guess that by the time you’ve notched up several dozen years it’s pretty likely that some distinctly unlikely things have happened to you.

I went to a lecture at Logan Hall in London.  The Logan Hall is a huge lecture theatre and holds nearly a thousand people.  Speaking in public has never been much of a problem for me, and as I looked around I wondered what speaking to such a large audience might feel like.  The only venue I could think of that was more prestigious was the Lecture Theatre at the Royal Institution.

When I got home I switched on the computer to check my emails.  At the top of the list – and I still have it today – I read:

“[We] were wondering whether you would be able to do a Primary
Maths lecture for us at the Royal Institution in London …. 

“We would be very keen for you to do something based on your “Take Ten
Cards” activities which you introduced to Jenni during the ATM conference
(although Jenni has already used the envelope problem at a lecture last

I really don’t think anything has topped the excitement and pride of speaking to 300 children at the Royal Institution, standing at the very desk where Michael Faraday and some of the world’s most famous scientists stood.  Evening discourses at the RI were one of the great events in the London calendar, so much so that the resultant congestion meant that Albemarle Street became London’s first-ever one-way street.

RI b



I did indeed include the envelope problem, and that’s what I want to write about next time.






Confessions Of A Reluctant Inspector

Inspections were a part of my job, but the part I enjoyed least.  At least when I was on home ground, inspecting mathematics in primary and middle schools, I felt I had something to offer and didn’t feel too much of a fraud.  Even then, I couldn’t avoid the fact that however nice I tried to be I was still putting enormous stress on the poor teacher.  One lady found herself so frazzled that the lesson quickly became a disaster, and in desperation she asked me to take over!  Fortunately, the problem was simply that the strain had caused her to overlook a single step in the lesson and I could do so easily, but she was an experienced and capable teacher and it would never have happened if I hadn’t been there in the first place.

Another teacher demonstrated the stress even more dramatically.  In a debrief she suddenly mumbled “I’m sorry, I think I’m going to be sick”, and was.  (I sent her some flowers but decided against asking for a testimonial to my inspecting rigour.)

I was glad the request to take over the lesson only happened on the one occasion, because on many occasions I felt completely out of my depth.  There was the school for teenagers with behavioural problems.  I cried all the way home thinking of the poor young woman, with just a few months of teaching behind her, expected to teach Romeo and Juliet to a class of 15-year-old  boys in an EBD school.  Even more searing was the week I spent in a Special School which took children all the way up the scale to where you couldn’t really be sure the child actually had any significant level of awareness.  That was the first time I realised that we were actually lucky that our first child, who was born with severe brain trauma, died soon after she was born.   I spent just a week in both these schools, and all the others, but those who worked in them had to face their problems every single day.

The story I tell most often is inspecting a secondary school.  I joined a Y9 class at the start of their lesson and immediately noticed a boy abusing his support teacher.  Within a couple of minutes she left the room in tears, and he turned his attention to the teacher and subjected him to a similar load of invective.  The teacher in turn decided he needed to spend a couple of minutes in the stock cupboard.  The pupil looked around for another victim and his eyes lit upon me, sitting inoffensively in the corner.  He strode up to me.  “What I wanna know”, he snarled, “is how long you and your inspectin’ mates are goin’ to be ‘ere”.  “Why’s that, Wayne?”, I smiled politely.  “So I can stop ‘avin’ to be on my best bloody behaviour ALL THE BLOODY TIME!”