# Farewell Amy

I’d be grateful if you’d help me in a little experiment.

In the photo the letters A to J stand, in some order, for the digits 0 to 9. (As you’ll no doubt guess, a two-letter item stands for a number such as 23 rather than 2×3.)

The first question I’d ask you is roughly how long you needed to solve the whole set of statements and discover the unique solution. Did you see everything straight away? Five minutes? Ten? Were there any blind alleys, and fresh starts?

Secondly, you’ll want to reflect on the mathematical and reasoning skills you needed to call upon.

Thirdly, who might you see as suitable pupils for the problem?

For what it’s worth, I think it probably took me about 15 minutes, perhaps 20; I found four possibilities and had to explore each of them. It wasn’t we were working through it together that I realised there was a much better approach that avoided multiple possibilities and allows you to home in smoothly on the unambiguous solution.

And, as you’ve realised, “we” means the remarkable Amy and her partner Paddy; I’ve written before a couple of times (March 2017) about her unusually highly developed reasoning abilities. Now 55 years of teaching give me a pretty solid feeling that this isn’t a problem you dish out to your average 11-year-old (actually Amy was ten when we first me, but like me she’s had a birthday since then).

But I’ve had plenty of pupils who’ve been able to tackle this problem, and the biggest reason they can handle it is motivation. It’s one of many challenges they meet in Anita Straker’s “Martello Tower” adventure game, and by the time they’ve invested several weeks of effort they’re not going to let one more problem put them off.

Usually, however, I offer a clue or two, and my contribution with Amy and Paddy was much more limited. I did write out each of the statements onto card so they could sequence them as they wished and write on them to keep track, but otherwise

my contributions were restricted to comments like “What does that tell you?” and “What could you do next?” On completion I congratulated them, and Paddy said something to the effect that he didn’t know what I was making a fuss about, it had all seemed pretty easy!

This was our final session together, and eight weeks of working with them has reminded me yet again just how localised children’s abilities can be. Their performance in some other arithmetical problems was nothing like as advanced. In the adventure they need to identify a four-digit number using ‘more than’ / ‘less than’ clues and neither of them were great at that, and Amy was worse than Paddy. In Martello Tower they repeatedly need to use triangular numbers and neither of them ever really reached the stage where they could find TN16 without working from and earlier one like TN10 or TN12. My other pupils have almost invariably called upon the streamlined method long before the end of the adventure.

It’s not simply that some children are good at number and less so at spatial stuff, and vice versa. Amy and Paddy are able to operate a very high level in some number work, and much more mundanely at other activities even in related areas, and the difference in maturity can be quite dramatic.

I’m reminded that one year I was asked to lead the national evaluation of pupil performance in the Key Stage 2 national tests. One thing jumped out at me: for virtually every one of the hardest (level 5) questions something like 10% of the correct answers were given by children whose overall achievement was graded at below average level 3. And of course it was a different 10% each time; clearly there are a lot of Amys around, with a very jagged profile of skills across different areas of the curriculum.

Potentially this has huge consequences for the way we group and teach children, and I thought it was so important that we should be shouting it from the rooftops. But no-one else seemed at all interested, and rather to my relief the curriculum authority decided to keep the process in-house and never invited me to do the job again.

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# “… Tables Not Well Known”

*This is another posting from my history of schools findings.*

*For more than 150 years there’s been one yardstick that’s been used to give a quick judgment by all and sundry about mathematics learning – “Do they know their tables?” And you’ve only got to look at school logbooks or HMI reports to realise that in spite of 150 years of teachers’ efforts the answer has invariably been “No”.*

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Long after many countries had universal elementary education England was known for having perhaps the worst schools in Europe. No-one had much interest in schooling; there had been too many revolutions in Europe for the landed gentry to want the population to become educated, and parents and employers were keen to have children – even those as young as six or seven – working for a living.

Consequently, it wasn’t until 1832 that the Government made the first tentative grant, putting £20 000 towards the building of elementary schools. Of course, the £20 000 covered the country as a whole, but a new school might cost only £60 or so, so it was a useful start. A few years later a system allowing promising pupils to train in their schools was developed, followed by the emergence of training colleges. At last elementary education in England had taken off, and growth and momentum were rapid – I found a reference as soon as 1850 to a teachers’ magazine which encouraged teachers “to make the learning of tables interesting, instead of mere mechanical routine”.

But within a few years the Government found to its horror that the £20 000 grant had grown to nearly a million pounds every year. As governments invariably do in such cases, it set up a committee, and in 1862 the draconian Revised Code was introduced. The Code soon became known as Payment By Results, for schools would only receive a grant for those children who met nationally decreed standards of attainment and attendance.

Anyone who’s been involved in education in recent times will have little difficulty in believing what happened next. Children were tested annually the “Three Rs”, Reading, Writing, and Arithmetic. Children worked in one of six levels of attainment, known as Standards. Not surprisingly, the examination became the focal point of the school year. The Head’s job security depended upon the results, so the curriculum narrowed down to the three Rs and little else, with children spending the preceding weeks or even months doing nothing but practise for the examination. I found that one school even postponed the Christmas and New Year holidays until after the Inspector’s visit!

In the run-up to the tests even Scripture lessons might be abandoned, a serious matter given how important the church, and in particular the Rector, was to most schools. For example, a week before the inspection in 1865 one Head recorded in her logbook “*Instead of having Scripture Lessons children questioned on the Multiplication Table*”.

The examination was carried out via a visit from one of Her Majesty’s Inspectors. Most HMI were appointed for their Church connections, usually with a university background; they’d see themselves having considerably higher social standing than a mere teacher and often they might have little understanding of children. So both teacher and children might dread the annual visit; at least one Head was so terrified by a coming inspection that she drowned herself.

The actual arithmetic syllabus could hardly have been more narrow. In Standard I, for the youngest children, the requirement was “Form on blackboard or slate, from dictation, figures up to 20. Name at sight figures up to 20. Add and subtract figures up to 10, orally, from examples on blackboard”.

Standard II required “A sum in simple addition or subtraction and the multiplication table”, and Standard III “A sum in any simple rule as far as short division (inclusive)”. For most, schooling would finish well before they reached higher Standards.

School logbooks make it clear that such a limited syllabus and so much at stake meant teachers gave the highest priority to the learning of tables. We see teachers devising the same techniques we use today – “*Find the plan of getting St II to learn their Multiplication Tables at home answers well*.” And “*Encouraged children to get table books of their own, bring them to school and say tables from them*.” Those who like to use rock or rap versions of tables are following the example of the Devon teacher of 150 years ago who encouraged her children to sing their tables from 2.30 to 3pm.

Teachers recognised the benefits of a little and often approach: “*Find the II St know much of their Multiplication Table, as I devote a short time on Tuesdays and Fridays to hearing it having been learnt at home*”. They seized every opportunity for a little practice, even when lining up: “*Examined the children in the Multiplication Table while at the line*”. I even found a Head who devised the Buddy approach used in my own school, observing, as we too find, its value to both parties: “*On Thursday adopted a fresh plan for teaching Arithmetic. For about twenty minutes gave everyone on the three upper classes a child from the lower classes to teach …. Found it beneficial to both the elder and the younger ones.*”

It’s frequently asserted that children used to know their tables perfectly, but it’s clear that this common belief simply isn’t true. Virtually every logbook finds Heads bemoaning their pupils’ inadequate knowledge. One Head writes in three successive months he finds it necessary to keep one class in for not learning their tables. Next year’s equivalent class is just as unsuccessful, and the year after that he finds himself keeping them behind not occasionally but every day for a week. (Declining standards, no doubt!) And this is no ogre, but a Head who joins the children at play, and enjoys snowball fights and playing cricket with them. Children bring him flowers, and worry when he’s ill. He’s constantly looking to find better ways to teach; he’s ambivalent about using the cane, but is forced to admit that other punishments don’t always work – “*Find that threatening children with an extra ½ hr at school is no punishment for some say they would like staying*.”

It was the Payment By Results code that required schools to keep a logbook, so logbooks aren’t actually all that rare. Some have been transcribed and others put onto CD ROM, so they can be both convenient and inexpensive to study. Much of what you read comes across as truly historical – children unable to attend because they have no boots, or absenting themselves at harvest time because they’re working in the fields. There are enormous class sizes – in one case 104 children in a room so small they had to take turns to sit down. Illness and epidemics are frequent and pupil funerals are tragically by no means unusual – one terrible story featured a family who lost each of their five children in a measles outbreak.

In other ways you find yourself thinking that things haven’t changed a bit – demanding pupils “*Oliver G cannot be left a minute without his getting into mischief …*”, daunting workloads and endless paperwork, publishers offering workcards and schemes promising to meet syllabus requirements, and – of course – the never-ending struggle to master the multiplication tables.

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# The Logbook revisited

*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.*

I spent a quite fascinating afternoon looking at an old school logbook. It used to be mandatory for schools to keep a record of events, and that the Headteacher had to make an entry at least once a week. Judging by this particular logbook, the Head would have a lot of discretion about how this requirement would be met.

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”.

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 near Banbury, around halfway between London to 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 other duties I’ve never heard of – “leasing”, and “birdminding”. The authorities were clearly 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 clearly had the authority to use some discretion, and on one occasion 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 visited the 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.

We’re told ad nauseam that in the olden days every child knew their multiplication tables. It’s not true!

(“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 wouldn’t have seen his pupils as being impeccable. In a school of just 75 or so, half a dozen pupils 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”.

One incident shocked me when I read of the attack by Ernest L and Clement W (another relation to Minnie and Reginald!) who stoned 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.

Indeed, and contrary to what one might have expected, in this school at least 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”.

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.

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# More Envelopes, More Amy

I spent most of last week telling everyone I could think of about Amy’s insight into the Envelope puzzle, and I couldn’t wait to throw some of my more difficult puzzles at her and her partner. I gave them a ten-envelope set where each contains two cards from a 1 to 20 set, and the displayed products are 10, 24, 26, 45, 55, 63, 136, 168, 320, 342.

They dealt with this quite happily, so I gave them a smaller set, with just four envelopes and a set of 1 to 12 cards – but three cards to an envelope and their products shown:

Once again Amy did something I didn’t expect. Padraig, much as I would have done, targeted the 14 envelope and deduced it contained the 1, 2, and 7. But Amy zoomed straight in on the biggest number, treated it as 96 x 10, recalled that 96 is 12×8, and was home and dry.

What I can’t get my head around is that she’s got brilliant things about how numbers work going on in her head and yet she’s someone who hasn’t found much success in maths. I’m going to have to devise something really special for next week.

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# A Wow! Conversation with Amy

*Every now and then a child says something that really makes you sit up and go Wow! See what you think about this Wow! moment. *

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I’ve borrowed a vast number of ideas from other people, but I have had one or two good ones of my own, and Envelope puzzles are up there with the best of them. I’ve written about them before (April 2015) but I’ve no hesitation in doing so again. They do give a hugely accessible way for children to develop a chain of rigorously justified reasoning.

I gave Amy and her partner this set of envelopes. They knew each envelope contained two cards from a 0 to 9 set of digits and that the product of the two digits was displayed on each envelope. Their job of course was to identify the cards in each envelope.

Amy’s partner and I agreed it would be sensible to leave the 0 envelope till last, since though we could be sure it contained the 0 we wouldn’t know which the other digit was until we’d eliminated all the other possibilities.

“No”, said Amy, “you can say immediately that the 0 envelope must have the 0 and the 1”.

“Why’s that?” I said. I rather assumed Amy was a bit unclear about the multiplicative properties of 0 and 1.

“Well”, she said, “if the 1 is in any other envelope then it must have a single-digit number as its partner. That would mean that one envelope would have a single-digit number written on it, but none does. So 1 cannot be in any other envelope, and so it must be in the 0 envelope.”

Wow! indeed. What a terrific and totally water-tight chain of reasoning that had never occurred to me when I devised the set. With a National Curriculum which aims that we focus upon problem solving, reasoning and fluency I reckon Amy’s pretty much on the right lines.

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A footnote: I was almost as flabbergasted at the end of the afternoon when I eagerly buttonholed a couple of teachers. “Can I tell you about Amy?”, I said. “Ah, Amy”, they said ruefully, “she’s always had problems with maths!”

(Don’t get me wrong – I’m not saying this to show how brilliant I am; these are experienced and committed expert teachers who spend every moment every day devoted to thirty pupils, many very challenging. I, on the other hand, merely swan in for the afternoon and have no other responsibility than to work with two or three children on aspects of their mathematics. My point is rather that locked away in Amy’s head was potential and insight and I was lucky enough to find the right key to bring some of this out into the light of day.)

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# 56÷4

Marilyn Burns ( @mburnsmath ) posted an interesting example of how one child divided 56 by 4.

It just so happens that I’ve a whole collection of different ways pupils tackled this very question and got the correct answer. (There’s one incorrect answer, but it’s another interesting method.) ((PS Everyone has been far too polite to point out this is complete nonsense – Marilyn’s example is actually 56÷14 rather than 56÷4, but I should have spotted that some time ago. My apologies.))

# 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 (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.

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# My Christmas Conversation with Katarina

*I can’t believe I’ve never posted this tale before – everyone should have their own Christmas story, and this one’s mine. And I promise you that every word is true.*

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Part 1 It was Christmas Eve five or six years ago. It was a proper Christmas Eve, cold and with three inches of snow. And on every Christmas Eve my son and I are encouraged to get out of the house for the afternoon, so we set off for our traditional trip to see a film. Halfway to the bus stop I saw an object lying in the snow. It was a combined purse and wallet.

The wallet opened to show a student identity card – a rather attractive student, I had to admit. Like you and me, she also had a whole collection of cards – bank cards, store cards, library cards – so I soon knew she was called Katarina and quite a bit more about where she shopped, but what I didn’t know was her address, nor phone number, nor email. The poor girl, losing her wallet on Christmas Eve! She’d be devastated and if I couldn’t do something about it she’d face the most awful Christmas ever.

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Part 2 So we spent most of the afternoon contacting all the organisations we could think of. Banks, libraries, clubs, stores. Many had closed, plenty were suspicious, but eventually I managed to get an address for her, just a mile or so away. So, like Good King Wenceslas and his page, Simon and I trudged through the falling snow, knowing how thrilled and delighted she’d be and how she would after all be able to get every enjoyment from her Christmas. “My hero!”, she’d cry, as she planted a warm kiss upon my frozen cheek. “Come in, come in, sit by the fire and have a mince pie and glass of mulled wine!”.

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Part 3 We knocked on the door, and waited. I knocked again, and waited again. Eventually we heard sounds of movement upstairs, and finally the front door opened. Now I guess any adult male can fabricate plenty of scenarios built on a young woman in her night attire opening her front door. But it’s fair to say that none of my fantasy scenarios had got even close to this one. Yes, she was just about recognisable, but 5.30pm on Christmas Eve was obviously all too early in the day for her and it would take a lot of work on her make-up, hair, complexion, clothing, and above all her demeanour to become the agreeable and attractive student in the photo. It wasn’t quite a snarl, but it wasn’t far off: “Oo’r’yu, ‘n’whaddya wan’?”

“Katarina?”, I enquired mildly, “I think we’ve found your wallet”.

“’Ow ja geddis? Whad’ja doin’ wivvit?”

Slightly bemused, we went through the whole story, and she became more hostile rather than less. She started by denying she’d ever been at our end of Tring and not even knowing her wallet was missing, and was swiftly moving towards us having stolen it in the first place. I suppose it’s possible she’d ingested some chemical which had affected her manner, but it now looked perfectly possible she was going to make a scene and even call the police. She seemed quite capable of accusing me of helping myself to the contents of the wallet or calling upon her neighbours to sort us out.

So we decided we’d completed our Christmas errand, quickly said farewell, and set off down the path. Belatedly she remembered some of the lessons her mummy had taught her as a little girl. “Oh yeah”, she muttered, “I spose – ‘Appy Chrismuss”, and slammed the door hard enough to dislodge the snow from the rooftops.

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# Animated Factorisation

There’s a brilliant animation of number patterns from Stephen Von Worley. You can find it at http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Try it straight away. It displays first a single dot, then two, then three in a triangle, four in a square, five in a pentagon. From 6 onwards, the number is likely to be shown as a pattern, so for 8 you get two squares of four.

As you explore, it becomes clear that the displays aren’t any old pattern, but are based logically upon the factors of each number.

I’ve not seen so much excited discussion in my classroom for ages. My Y6 children were transfixed. Words and descriptions tumbled out, ideas and predictions were offered, challenged, revised, replaced.

What would 9 look like? There were two opinions. One was we’d see a hollow triangle, the other was that we’d get a triangle of three small triangles. What delight to find both were correct, and the two suggestions were offering alternative descriptions of the same pattern.

If your pupils are anything like mine, one snag you often find when they’re solving a problem is the failure to build on evidence. Not here. Several times when trying to predict a number they asked to look at a relevant previous one. When thinking about 15 it was “Can we see 5 again?”, and used this to decide that 15 would show a pentagon with each vertex a triangle of three dots.

A week after the first session they were knocking the door down to take things further. Why were some numbers not in a pattern but arranged in a circle and labelled “Prime”? Why did we never get two of these in succession? Which numbers were made up of block of four dots in a square?

I had plenty of frustrations. It moves quite fast and the display changes every second, so we need to stop it each time to look at the pattern, and talk about it and discuss what the next one will look like. The control buttons are quite small, so I often miss. And I dearly wanted to be able to call up a number of my choice. But if we want to see what 243 looks like (and you probably will) we have to start again from the beginning, and what seemed fast now becomes rather slow. There is a faster speed option, which changes three times a second, but even that’s prohibitive when we want to explore larger numbers. My pupils were delighted to learn it would display up to 10 000, less so when we talked about how long it would take. By the way, to reset we have to tell it to count back all the way to 1.

I got round some of these problems by taking snapshots of the display for all the numbers up to 100. I’ve put them into a Powerpoint that gives me greater control, and made subsets with odd numbers, even numbers, and multiples of 3, 4 ,5, and 6. When we got back to the classroom after half-term I was pleased I’d done this; it worked really well and we spent a whole hour working through the first thirty or so counting numbers. Virtually nothing went on to paper, but a thousand diagrams were drawn in the air, and as the session went on – and the numbers and patterns increased – these were often dispensed with, so one person’s mental image was articulated and received and understood by their partner.

Yes, I do wish it offered a few more options, but make no mistake – I’m 100% sold on the animation. It’s brilliant, it’s free, and though I was using it with 10 yearolds it will entrance and stimulate any group of children and adults. If you haven’t tried it already, you should do so at once.

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# The All I Can Throwers – #3. More sessions with Den and Jenna.

It was a great pleasure to be working with a couple of pupils I knew would accept any challenge I offered, so you won’t be surprised to know that we spent two or three sessions exploring all the ideas around Stars that I wrote about in several recent posts.

I may well have been the only teacher in the country disappointed that the end of the summer term was coming up fast, but there was still time for one further session. I really don’t think there’s any exploration more accessible and productive than the Tower of Hanoi. It’s intensely practical and visual and you need just two simple rules. I was using it with two very bright nine-yearolds, but I’ve used it both with teachers and with much younger children – one teacher used it with her Reception class “Baby Teddy can sit on Mummy Teddy’s lap or Daddy Teddy’s lap ….” and it worked a treat.

There’s so much to find that even now I’m still discovering new aspects, but it won’t take long to start wondering how many moves it takes to move a stack of 3, a stack of 4, a stack of 5, …., or to observe a dazzling array of patterns and movement rules.

If you need refreshing on the rules and background there must be hundreds of websites devoted to the problem, with diagrams, formulae, and animations. Many of them spoil the fun, but you’ll easily find all the information you could possibly want and much more besides.

In the spring I used it with a Masterclass group of Y6 children and we dealt with numbers up to quintillions, and derived a procedure to allow them to solve the puzzle for a stack of any size. We used boxes gleaned from the supermarket, and I was struck that for these children it’s probably rather rare that they get they chance to manipulate apparatus. It seems a little sad, but I suspect that one reason they enjoyed the session so much was that there was a strong element of play involved. There were 30 people in the group and next year the organiser has decided she wants to invite 90. Collecting enough boxes will be a massive task, and we’re hoping we can persuade IKEA to sponsor us with a few dozen sets of their toddlers’ stacking cups at £1.50 a set.

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