Collatz in the Classroom

Reading Siobhan Roberts’ biography of John Horton Conway “Genius At Play” reminded me of one the problems Conway has worked on, and I posed a piece earlier introducing the Collatz Conjecture, which is easy to present and easy to play with:

Start with any whole number. If the number is odd, triple it and add 1; or if the number is even then halve it.   Treat whatever you get as the second number in the same way, and treat the next number in the same way, and so on.

As I said in the previous piece, it may be easy to play with, but it’s pretty fruitless in school. It’s known that every whole number generates a sequence that eventually ends up with 4, 2, 1, but there’s no way pupils or anyone else can prove this, and though there’s a possibility that a really huge number doesn’t fit the pattern no-one in school is ever going to locate it. Furthermore, the arithmetic is of limited interest and repetitive, and it’s all rather frustrating. Even NRICH, the home of everything to do with exploratory mathematics, can’t come up with anything very interesting.

So here are some thoughts on how you could open things up to do some genuinely intriguing work that I suspect is actually likely to be original.

There are four different bits in the Conjecture:

if the number is odd [this is bit A]

triple it and add 1 [B]

if the number is even [C]

halve it [D]

If you modify any one of these you get a neo-Collatz Conjecture which quite possibly has never been explored by anyone, and certainly will have much less known about it.

Try, for example, replacing [B] with multiply by 5 and add 1.   Does every number reduce to 4, 2, 1 again? Certainly many do (for example, 15 goes to 76, 38, 19, 96, 48, 24, 12, 6, 3, 16, 8, 4, 2, 1). There will plenty of pupils who can check that 3276 does as well, and many of them will be able to discover similar numbers for themselves.

But 17 does not. Try it for yourself – it only takes ten stages to do something different.

And 7 will involve a bit of effort. After 36 stages it’s into six figures and showing every sign it’s going to get larger and larger, and will never reduce to 4, 2, 1. I may be wrong, and I’ve found to my cost it’s very easy to make a simple mistake at any time, so you’re welcome to check.

It certainly wouldn’t be surprising if the multiplying by 5 and adding 1 for part [B] sometimes generates a sequence which goes on increasing.   As an alternative you could use multiply by 3 and add 3 as your new [B]. Here you’ll find that many fewer numbers finish up at 4, 2, 1, but that lots get into a 6, 3, 12, 6, 3, 12, …. loop. Again, I haven’t come across anyone who can tell us what happens.

Messing around with [A], [C], or [D] is possible, but I’m happy to stick with [B]. There are plenty enough opportunities and children can choose their own version of [B] – I’ve just spent 10 minutes with multiply by 3 and subtract 1. Intriguingly, 17 is again interesting.

Some modified versions of [B] will be fruitless, but children will get something out of them even so, and it’s worth letting someone choose multiply by 2 and add 1 and see why it’s not productive.

There’s more Conway to come, including why I got stuck on page 2 of Roberts’ book.

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Conway’s Curious Mind – The Collatz Conjecture

I spent quite a bit of time over the summer reading about John Conway in Siobhan Roberts’ biography “Genius At Play”, and back in August I posted a piece about an original version of Conway’s Life Game.   The subtitle of the book is “The Curious Mind Of John Horton Conway” – so you’ve immediately got a good idea that the book concentrates on some of the lighter aspects of the man and his work.   Roberts had extensive access to Conway (it may be unfair to imagine this was helped by his famous partiality for attractive women) so we learn a lot about the private life of a particularly unconventional character. I suspect Conway’s concept of a cheese sandwich – two slices of cheese spread with butter – will stay with me for all time.

Genius At Play

I’m pretty sure there are plenty of mathematicians for whom much of their work is rooted in play, but I doubt there are any whose frivolous side is quite so important. His Life Game is certainly the best known, and the publicity given to him by Martin Gardner in his Scientific American column has ensured his playful side is more widely known than any other comparable figure. One game that’s reputedly taken up a lot of his time is his Phutball (“Philosopher’s Football”), a game with similarities or resemblances to Go and Chinese Checkers – you can easily find out more about it in the usual places.

Of course, Conway is a mathematician whose professional work into topics such as the Monster Group, and his Surreal Numbers I can’t begin to understand, but his willingness to play with the most basic situations means there are several topics that I can usefully write about.

There’s one that’s cropped up, even potentially at primary school level, for many years. This is The Collatz Conjecture, and it’s very easy to explain and to start exploring.

Start with any whole number.  If the number is odd, triple it and add 1; or if the number is even then halve it.  Treat whatever you get as the second number in the same way, and treat the next number in the same way, and so on.

So if for example you start with 20, this gets halved to 10; 10 is also even, so is halved to 5. 5 is odd, so you multiply by 3 and add 1, making 16. 16 is halved repeatedly to 8, to 4, to 2, to 1.

Pretty obviously if you use 40 as your starting number it immediately enters the same sequence and also ends up as 4, 2, 1 But if you start with 60 things proceed slightly differently, and the sequence goes 30, 15, 46, 23, 70, 35, 106, 53, 160, 80, 40 – and from there down to 4, 2, 1. So 60 too ends up at 4, 2, 1, but there’s a point when you’re into three-digit numbers when you wonder if the tripling effect is outweighing the halving – and perhaps the numbers in the sequence will go on getting larger and larger ….

If you choose 27 you’re in for a bit of a slog; it takes 111 steps, but it too ends up with 4, 2, 1. The Collatz Conjecture suggests that all whole numbers will eventually end up at 4, 2, 1, and certainly no-one has ever found a number for which this isn’t true.

The Conjecture may be easy to explore in the classroom, but it’s really rather disappointing. The arithmetic is simple and repetitive and it doesn’t progress in difficulty or interest, and since the Conjecture is known to hold for all numbers up to 15 digits and beyond none of us is going to find a counter-example.

Furthermore, it’s all rather messy for our conceptions of proof within the school curriculum. There seems to be a general consensus that the Conjecture is true, but no-one has yet succeeded in either proving or disproving it, and it’s suggested that no-one is anywhere near resolving the problem. Indeed, Conway and others believe that no proof either way is possible and that the problem is undecidable.

There’s more on Conway to come, and I’ll post some more Collatz thoughts straight away.

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European Day Of Languages

At the start of September the Head said perhaps we’d like to know the European Day of Languages was coming up.  Perhaps I might do something appropriate?  To my wife’s discomfort – she can’t bear the squeal it makes – it meant some work with the paper trimmer and the sacrifice of a substantial number of cereal packets, but the effort proved well worth while and we ended up with an impressive array.

You can see how things went.

I gave the children a set of cards bearing the numerals 1 to 10, and another set showing the number names.  Naturally enough, they decided to put the cards in order and match names to numerals.

I actually studied Latin at school, and though no-one claimed to know any Latin they were rather gratified to find they could work out how to match Latin names to English ones, and since they’d looked at Roman numbers they were able to add a further row to the table.

And though their French is pretty emergent, we were pleased to see another row could be placed with little difficulty.  By now they were really into things and looking forward to the next challenge.  Italian, Spanish, even Portuguese, all were placed easily.

And what really delighted me was the fact that I didn’t have to say one word about the families that emerged.  It was the children who pointed out the way all these languages showed their links to Latin, and that English has closer links with German.

20181002_145700

I’m not sure the table surface could have held much more, but I went home rather regretting I hadn’t got around to doing cards for Romanian, or Dutch, or Polish.  And we had only 30 minutes for the whole thing, so I left out the cards I’d prepared for other numbers (I’d done 11, 12, 20, and 100 in each language), but I was very pleased I’d made the effort – and though I was working with Y6 pupils I was happy you could use the activity with plenty of other groups.

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Mr Smith of Nash

I realised I hadn’t posted any of my pieces on the history of elementary schools for a while, but I see it’s six months since the last one.

I didn’t even know Nash existed, but its school in the late nineteenth century saw several interesting characters, and one of them deserves to be much better known.

 

We’ve lived in Tring for nearly 50 years, and though Nash is less than 25 miles from us I’d never heard of it and had to look it up in the road atlas. In the 19th century Nash was an isolated village with a very small school; once or twice numbers reached 100, but somewhere around 70 was more common, with about one-third being infants.

Small size means small income, and clearly the managers often found it difficult to attract a suitable person to take charge. Between 1864 and 1900 nearly twenty people became Mistress or Master, two or three of them a couple of times – apparently on the basis that they lived locally, were available, and could take over for a short period. In most cases it was women who were appointed, not for reasons of equality, but because the managers could pay them less.

At times the staffing position was little short of chaotic. Joseph Bate was appointed at the start of 1885. This was the first occasion the Managers had appointed a man, and clearly he was not somebody burdened with a lack of confidence. His very writing seems to shout at you in its bold, assertive floridity. It was common for Heads to play down the state of the school they’d inherited; but Bate may have set something of a new record by listing no fewer than eleven points of complaint. In one of these his account of an incident involving the Rector was so fanciful that the Rector felt it necessary to insert his own version in the logbook.

It’s easy to imagine Bate upsetting people, not excluding the Rector or one of the Managers, and for whatever reason he resigned and within a month of his appointment he was gone.   A temporary replacement took over for just one week and a new ‘permanent’ appointment arrived; she lasted for less than a fortnight. Both she and Bate were described in the inspector’s report a few weeks later as “useless”.

Two or three more people of varying effectiveness came and went, until at the end of 1890 a new Master appears. The complete opposite of Joseph Bate, he sidles modestly upon the scene – so modestly that never throughout his tenure does he give his name. I may be reading too much into things, but unlike Bate his handwriting is neat and understated and would fit comfortably into a twentieth-century style.   We learn from subsequent reports that his name is A Smith, and that as was common, his wife will teach Needlework and take the Infants. Unlike Bate, he makes no complaint about the children he’s inherited, saying their attainments in their various Standards are “in a fair state of efficiency”.

Not one of his dozen predecessors has ever shown any awareness of the world outside, whereas Smith comments that the passing of the Free Education Act should do something to help improve attendance figures. When in January the school cannot open through lack of coal he reflects “I cannot see how the school can be carried on voluntarily there seems to be no means of obtaining funds beyond the base pence & grant.” His analysis no doubt contributes to a meeting in the school next month when it is decided a School Board needs to be set up.

As soon as he’s settled in he begins trying new ideas.   A common theme is broadening children’s experiences – with some passion he writes: “The ignorance of things in common use is awful, many of the children, in fact the majority, have never seen such things as a railway, telegraph post, or wire, a town, river, boat etc.”

The inspectors mention that spoken English and Dictation need attention, and he attributes these to limited vocabulary, and has various strategies to employ: “Tried ‘Novel Reading Lesson’ on Thursday afternoons, viz the upper classes select their own reading from books & papers of their own & read to the whole school”, and “I have been trying the effect of some conversational lessons in the upper School.”

He acquires posters from advertisers to brighten up the classroom walls. He subscribes for a monthly pupils’ magazine called ‘Scholar’s Own’ and uses it to set up competitions for the children to take part in; the magazine is successful enough that the cash-strapped Board continues to subscribe even after he has left the school. There’s a School Savings Bank, and he collects items together to set up a school museum.

In a school the size of Nash Smith’s only support would be a monitor, but both his intellectual and his physical energy are remarkable. In his first autumn term he introduces what’s to become something of an annual event, and launches into an “Operetta” based on Red Riding Hood. This involves evening rehearsals, and a school which has always had a problem with low attendance is suddenly touching 95%   (while other schools regularly lambast the Attendance Officer as ineffective Mr Smith frequently tells him there’s no need for his services). A subsequent Operetta on The Old Woman Who Lived In A Shoe is so successful that the takings provide for a new stove for the classroom.

The inspectors note that it’s not just attendance which improves, but discipline and attainment.   Between 1893 and 1896 their comments include remarks like: “General condition now, both as regards order & attainments is distinctly good”. “… having regard to the circumstances …. the results of the Examination in the elementary subjects are very fair indeed, the Arithc being good. …. The Infants division is being taught with some success by Mrs Smith.”   “…. attainment in the elementary and class subjects, the intelligence, and the order is now satisfactory, and it may be said to have reached a good educational standard.”

Diocesan inspections are also favourable, and the Science and Art Department more than once awards the school an Excellent grade for Drawing.

And when HMI mentions difficult circumstances, they were indeed difficult. Smith takes a full page of the log to explain: “During the past two years the work has been sadly interfered with on account of two severe epidemics; in 92, one of measles occurred, during which, every child attending school was attacked. The school was in consequence closed by order of the Sanitary Authority for nearly 3 months. During the latter part of July 1893, a case of scarlet fever was brought into the village, &, after the school had been opened for only a fortnight, with a very meagre attendance, it was again closed by order of the Sanitary Authority and remained closed until January 1st 1894. During the epidemic as many as 30 cases were at one time reported. I am happy to say that only one death occurred, but many were unable to attend school owing to the after effects till March & the beginning of April; one girl has still been unable to attend. In spite of these severe drawbacks progress has been very satisfactory.”

There are perhaps just a couple of clouds on the horizon. Smith’s burst of new ideas all seem to happen in the first year or two, after which the log has few to offer. Did he run out of steam? Or did the Managers, or the Inspectors, think he was being dangerously progressive and have a quiet word in his ear?

I’ve some regard for the local HMI team of E.M.Kenney-Herbert and Harry Martin. I know from other log books that they were sympathetic and progressive. When Kenney-Herbert retired the log at Wendover says “…. he has been a true friend to both scholars & teachers & has been most helpful & sympathetic to all”, and the Education Committee closes schools so teachers can attend the lunch at Aylesbury in his honour. But this was a dozen years later in a more liberal climate, and their inspection reports for Nash never mention any of Smith’s innovations.

The little clouds foreshadowed storms to come. Some were the responsibility of the Managers. In 1893 HMI had complained about inadequate toilets for the girls, and other building faults; they complained again next year, and things still hadn’t been corrected by 1897.

Mr Martin has already pointed out to Smith defective paperwork, with inadequate records being kept, and he itemises these in a log book entry for April 1897. Three months later, his visit finds matters still haven’t improved, but even then Mr Martin is scrupulous about acknowledging the work of Mr Smith and indeed his wife: “Mr Smith is teaching his school with great care, and the appearance of his scholars and their attentiveness show that he is succeeding. Mrs Smith can show a thriving class of Infants.”

My first reading of the log suggested the inspectors were being nit-picking and pernickety, but I now believe they were bending over backwards to give the Managers and the Smiths chance after chance. A further visit six months later in January 1898 finds that yet again both the record-keeping in the log and the building works are still not being addressed.   Yet even now they offer carrot rather than stick, and a special grant for of £40 is made next month for staffing, repairs and equipment.

Whether through pressure of work or sheer complacency, none of the warnings about record-keeping have been heeded. Mr Martin makes a no-notice visit in February 1898 and finds none of his points – log book, record books, notebooks, time tables – have been taken, and lessons are not being properly prepared. “I am not at all satisfied with the way in which this school is being conducted” probably barely hints at what he had to say.

That wasn’t all Mr Martin had to complain about in his visit. Worse still was his experience with the infants: “On arrival, at 9,30, I found the 22 children in the charge of a little girl aged 12, a daughter of the Master. To my inquiry for her mother, she replied to her mother only came in the afternoons. As her ‘afternoons’ are devoted to Needlework in the upper school the infants can have but very little of Mrs. Smith’s time.”

Given that the teaching of the infants has frequently been praised it must have been embarrassing to discover that Mrs Smith was making negligible input. With remarkable restraint he satisfies himself with the mildest of warnings to the Smiths “that unless the methods of instructions change it may eventually become necessary to warn them under Article 86”.

But even now yet another olive branch is offered. Quite unbelievably, a few months later Mr Martin approves another grant, and almost half of it is for the salary of the negligent Mrs Smith! Clearly Mr Martin has cooled down and is much happier. He acknowledges the buildings are at last being improved, and records only mild warnings about the teaching of the infants and the need to improve the general record-keeping. Kenney-Herbert backs this up in the spring of 1899, but the sense of crisis has quite died down, and he’s reduced to making just a few minor points such as making sure the Drawing books are dated, and that books should be stored neatly in the cupboard.

It feels as if the stalemate could go on for ever. So it comes pretty much out of the blue when in mid- October Smith writes “Resigned position as Master”. We don’t at this stage get further details, but in fact his hand has been forced; we find HMI have recommended his continued recognition should be withdrawn.   It’s not immediately clear why such drastic action needs taking after so long, but a couple of years later a subsequent HMI visit records “The Registration appears to have been neglected in some respects by Mr. Smith who lost or destroyed the admission register.”

They say you can’t beat City Hall, and just as Al Capone was brought down by tax evasion Smith’s downfall was caused by the school registers. A large proportion of a school’s grant was based on the attendance figures, which had to be audited regularly, so keeping the registers correctly was no small matter – one teacher is known to have drowned herself as the inspection approached and she couldn’t get the registers correct. The Smiths had already defrauded the Exchequer in the matter of Mrs Smith’s salary, so they could expect no mercy this time.

Reading and re-reading has quite changed my first impression of the role the Inspectors played. It becomes clear that they were no longer the terrifying authority figures described in books like “From Lark Rise To Candleford”. Rather, the move was well under way towards inspectors becoming facilitators of school development who offered advice (which was not necessarily welcomed – one Head wrote sourly “He made various remarks & recommended what in his idea were improved methods of teaching some subjects”) on curriculum and teaching. Between them Kenney-Herbert and Martin made no fewer than seven visits over three years, and – with the understandable exception of Mr Martin’s tempestuous visit of February 1898 – at every stage they bent over backwards to offer support, even to the extent of approving two special grants to the school.

The other thing Nash School has brought out for me is the sheer pace of change. The undistinguished school Mr Smith joined in 1890 was effectively stuck in the mid-Victorian era. The one he left was a stable school with high standards of attainment, behaviour, and attendance, on the verge of the twentieth century.   Within just a couple of years it was under the aegis of a county council, and led by a “Headmaster”. It was generously resourced, had a broad curriculum, and was highly regarded for its “Elementary Science” and Nature Study.

Clearly Smith never lost the support of the school’s Managers, who could easily have jettisoned him on several occasions if they so wished. But obviously they never chose to do so, and when he does go he leaves a school in a good state. His replacement pays a graceful tribute: “I find the children good & manageable – perhaps a little superior for a country village. I assess their attainments thus: Writing – good & neat; Reading – very fair, & some of it good”. I’ve not come across any other new Master or Mistress so prepared to acknowledge they’d been given a school in such good health.

 

 

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It’s Life, But Not As We Know It ….

Like many of us, I discovered John Horton Conway, one of the great names in recreational mathematics, via Martin Gardner’s column in Scientific American, and for reasons I expect I’ll get around to mentioning another time I plan a few posts about Conway.  Somewhat to his chagrin, although he has plenty more claims to fame, Conway is best known for his Game Of Life.  If by any chance you’ve not come across this, Googling “Conway Life” notches up 55 million references.  True, 55m is only about 10% of the hits you get from enquiring about “Cute Kitten videos”, but it’s a whopping great number and a great deal more than you’ll get from asking about my own best-known game “Alan Parr United”.

Almost any one of the 55 million hits will explain how Conway’s Life is built around applying some very simple rules to growing counter patterns on a grid of squares.  I checked out a YouTube video of Conway talking about Life recently, and in one of the comments someone asked if anyone had ever thought of trying Life on a grid of hexagons rather than squares.  Well, yes they did.  To my horror I find it’s more than 25 years since I wrote an article for Mathematics In School, which to the best of my knowledge has never aroused any interest whatsoever.

Here’s what I called Life91, which actually manages to have rules which are somewhat simpler than Conway’s:

Life91 rules

 

Life91 1

Life91 2

 

Of course, Conway’s Life has been explored to great depths beyond anything you or I could hope to discover, but as far as I know Life91 is completely virgin territory, and all you need is a sheet of hexagon paper (try, for example  https://www.printablepaper.net/preview/hexagon-portrait-letter-1 )  and a supply of counters in two colours.

Have fun!

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Complex Numbers, Pythagoras, and Protractors

Honestly, you could hardly make it up. In the last week or ten days you may have spotted some or all of these:

***   Simon Jenkins in The Guardian pontificates about maths teaching.   Among his assertions is that a primary school complains that a child he knows hasn’t mastered complex numbers.   It’s not clear how much Jenkins knows about complex numbers but it doesn’t seem to extend to knowing the difference between complex numbers and fractions, or possibly decimals.

***   Then there’s a YouTube clip of Piers Morgan smugly telling us that he understands Pythagoras Theorem and can recite it as 3.147….

***   And yesterday I read that a golfer is being investigated for using his protractor – creatively, he’s apparently invented a way of using his protractor to measure distances rather than the usual angles. We then get a helpful clue that his protractor is also called a “compass” – we also learn that sailors have been using them a long time. And in further clarification we learn that the device has a third manifestation as a pair of “split dividers”.   Actually, it was pretty obvious to all of us that the protractor / compass was in fact a pair of dividers all along – but not a single member in the newspaper’s editorial and production team knows what a protractor looks like, let alone what one does.

Protractor

………

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A Year Of Non-French

At the middle school I taught in, the top year (Y8) timetable had a unique subject called non-French.  Pupils were selected for this elite group using the sole criterion that the French department didn’t want to be bothered with them.  As you’ll imagine, this didn’t exactly enhance the children’s self-image; things were made worse still by the fact that nobody had the faintest idea of what the lessons should involve and that the group was taken reluctantly by whichever teacher had a free slot on their timetable that year.

I have to confess that the French department and the Head weren’t far-sighted educational thinkers, and neither of them saw anything wrong with putting a subject on the timetable for which the only guidance was that it wasn’t French.  So one year I decided I’d have a go with the group, and I must say we had a lot of fun.

Rule 1 was no text-books, and above all no worksheets.  Rule 2 was that we’d actually do something positively rather than because it would fill up a lesson or two.  So we read and performed comedy scripts such as the classic Tony Hancock Blood Donor.  Then we spent a month of lessons exploring some number magic tricks, leading up to a nerve-wracking and rather brilliant performance by the group in which they stunned the audience (i.e. the non-non-French remainder of the year group) by a variety of magical effects and amazing predictions.

I thought of them when I spotted a tweet forwarded by Simon Gregg about measurement using a trundle wheel.  One fine morning in early summer we decided to help the groundsman by measuring out the 100-metre track.  They organised themselves in half a dozen groups and remarkably, every group chose a different method.  One group used strides, one pigeon steps, one a metre stick, another made themselves a 10-metre length of string, ….  Even more remarkably, I was flabbergasted that the various methods all agreed within two or three metres – and were in close agreement with the official measurement determined by the school’s trundle wheel.

(Which is why the tweet intrigued me.  Simon’s correspondent reported that his trundle wheel measured differently from A to B than from B to A – and that their groundsman says that’s invariably the case!)

Another big topic we did was to make a simulation game of the development of railways across our local Chilterns area.  The pupils worked in teams, building routes which avoided hills and rivers to connect revenue-generating towns.  We looked at costs, and scheduling, and created a wonderful map of hexagons; the map filled the entire wall of the classroom.  One day a visitor came and was a little snooty that we’d played a game to model the process.  “Wouldn’t it be better to study what really happened?”, she said.  Which was what we’d done the very day before, so the non-Frenchers proudly demonstrated how one of their routes had followed the GWR track out to Reading and the West, while another had followed the Great Central route to Aylesbury and beyond, and another had taken the route out through Luton and Stevenage.  I did enjoy that moment!

As the climax of the year the group created and published a comic magazine called Creeps!, sales of which (enhanced by the donation of a huge box of Monster Munch by Walkers Crisps) were large enough to necessitate a reprint.

There was one spin-off that was nicely gratifying.  Upsetting the French department was always fun, and they were quite aggrieved by the whole business.  They’d been quite happy to slough off their discards, but didn’t much enjoy it when they faced a queue of their students asking if they could drop French and do non-French instead.

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Problems with Gabriel’s Problem

Here’s an activity I’d like some help with. It seemed it would chime pretty well with my Envelope puzzles (see for example, https://established1962.wordpress.com/2015/04/   and also  https://established1962.wordpress.com/2017/03/03/a-wow-conversation-with-amy/  )  Things started well enough, but as the session went on I couldn’t find the right way to take it further.

We started with me sketching out a 2×2 grid. I asked Chaz and Jo to put the digit cards 1 to 4 into the squares. I then wrote in the products of the rows and columns.

A

The next stage was to give them another grid, this time showing the products only. Could they place the digit cards?

B

Of course they were happy to do this, and we were quickly ready to move on.

I think I missed a good line of development here. Chaz and Jo have good knowledge of the multiplication tables, and could have handled a situation where I used other combinations of digit cards. They would have been able to work out the cards used to make this grid:

C

Instead, I moved to a 2×3 grid, with the numbers 1 to 6. This worked fine. There were lots of possibilities, some decent practice in handling two- and three-factor multiplications, and plenty of decisions to make.

D

So, after several of these, and with twenty minutes to go, we simply had to take things a bit further forward. It was pretty obvious that we had to tackle 3×3 grids, using cards from 1 to 9. They were very happy to kick me out of the classroom while they composed puzzles for me to solve. Well, I’m glad I tried it this way round instead of me setting them problems, because the 3×3 grid turned out to be a different beast entirely.

Chaz’ problem made me think for a while, but would have been too much of a challenge for my Y6s, though I’m pretty sure some of their more confident classmates could have wrestled it out.

Chaz

But I found Jo’s to be a complete brute. I gave it 15 minutes of effort, but there were too many Sudoku-type multiplicities and I had no clue which one to follow. Later I did have another look and found a more successful line of attack, but it was definitely more difficult than Chaz’ puzzle.

Jo

I was really disappointed at the way this had developed. It had begun with a nice gentle starting point accessible to all, and when I’d thought I’d turn up the challenge dial one just more notch I found it had jumped from strength 3 to about strength 10.

Afterwards I spent some time trying to figure out how I could move from the rather successful 2×3 grid and introduce a new challenge without making it so intimidating. In fact I spent most of the weekend trying, and rejecting, ideas. I’ll offer a couple of the better ones next time, but in the meantime I’d welcome any suggestions you might have.

And who’s Gabriel?   This was my attempt at using a simpler version of the very similar NRICH puzzle called Gabriel’s Problem. The NRICH version is targeted at secondary pupils, and is located at   https://nrich.maths.org/11750   The challenges set by Jo and Chaz fit very well with the NRICH puzzle.

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One Week’s Photographs

When I was about ten I’d be given a roll of film to take on holiday.  Deciding which scenes to use my exposures on was a tricky business.  Using them too early meant I lost any opportunity to take a special souvenir late in the week, but what if it rained solidly in the last few days?  In practice it didn’t matter much, because the resulting pictures were always pathetic – blurred, dull grey, with the carefully composed subject barely visible in the middle of the landscape.

Now of course, we carry with us the chance to capture almost anything, instantly, and of good quality.  Here are some pictures from the last seven days.

I went to a meeting at the Royal Society on Friday, and have the badge to prove it:

Badge

Now don’t imagine the Royal Society and I are closely connected, but once a year they kindly host a meeting of the Joint Primary Group of the Mathematical Association and the Association of Teachers of Mathematics, and give us a splendid buffet lunch.

Considerably more attractive to my mind than that offered to the Fellows themselves – if you’re an FRS you clearly take very seriously indeed the idea of fish as food for the brain:

FRS lunch

A couple of days earlier I was in Wetherspoon’s.  Can anyone possibly explain why an Americano – which Wetherspoon’s defines as black without milk or cream – contains six times as many calories as black coffee, and more than filter coffee with milk?  (The staff certainly couldn’t, and I don’t blame them):

Wetherspoon

Here’s another I don’t understand (particularly as by my count, the bus only has 28 seats):

Capacity 47

On one of the cooler, damper days last week, this one made me smile:

UK spring

And on one of the sunnier, warmer days here’s the canal.  One of the delightful things about canals is that you get these lovely tranquil scenes when you’re in a shopping street with dozens of cars trundling past, and with the supermarket 100m away to your left, and the railway station 100m away on your right.

Canal

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Valid and Reliable?

When it comes to assessment I guess there’s no name bigger than Dylan Wiliam. Dylan’s advice has been sought, and subsequently ignored, by governments for at least thirty years. His website says “Dylan Wiliam is Emeritus Professor of Educational Assessment at University College London …. he has taught in inner-city schools, directed a large-scale testing programme, … and pursued a research programme focused on supporting teachers to develop their use of assessment in support of learning …. Over the last 15 years, his academic work has focused on the use of assessment to support learning (sometimes called formative assessment). He now works with groups of teachers all over the world on developing formative assessment practices.”

Dylan Wiliam

I can’t recall how we met, but I do know that when in 1980 I launched a postal games magazine built around games of interest to mathematics teachers Dylan was an early subscriber. No doubt I used this connection when we invited him to speak to an audience of a couple of hundred 15-yearolds in Hertfordshire. Hatfield is about 25 miles up the A1, and Dylan arrived on an impressive-looking motor-cycle. Even more impressive looking was Dylan himself.

It’s not quite so obvious in today’s photographs, but the Dylan Wiliam of thirty years ago in full motorbike gear with rich black hair and beard was a hugely glamorous figure. Few of us could even dream of making so dramatic an entrance.   Without a moment’s thought, and quite unconsciously, the female half of the audience reacted as one. The hall filled with pheromones; every bosom swelled, and every skirt was hitched a couple of inches. Simultaneously, every one of the boys shrank and instantly became one of a hundred little brothers. You can guess which half of the audience mobbed Dylan at the end of the afternoon.

I was reminded of this by receiving an invitation to a webinar by Dylan entitled “There is no such thing as a valid test …. All those who produce tests claim that their tests are valid and reliable. Unfortunately, such claims are meaningless, because tests cannot be valid, and they cannot be reliable. A failure to understand why such claims are incorrect means that the substantial power of assessment to support learning is often lost. ….

Now clearly I don’t know what Dylan will be saying, but it does give me a nudge to write once again about Edmond Holmes, the remarkable Chief Inspector of Elementary Schools over a hundred years ago. In a blistering denunciation of how the authorities rely upon tests – and this was before the First World War – Holmes pointed out that tests:

  • Assess the recall of information and rote methods rather than understanding,
  • Label children with a number,
  • Dominate the curriculum and force schools to concentrate on pupils attaining their target grades,
  • Demotivate those given lower scores,
  • Fail to identify other abilities,
  • Force all schools to accept a standard approach,
  • By requiring standards to be attainable by all schools consequently lower aspiration of all those which are above average,
  • And, in his own words, “value the results of education for their measureableness, … ignore those results which are too intrinsically valuable to be measured”.

Over the last twenty years or so we’ve had to make these same points again and again, but it’s rather remarkable to think that they were being made by the Chief Inspector at the very beginning of the twentieth century.

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