A Novel Number System

You’re very likely to have come across the Australian mathematician and educator James Tanton.  He’s on Twitter, Facebook, there’s a YouTube channel, and several websites, starting with  https://www.jamestanton.com .   And he doesn’t set his sights low, so there’s a Global Math Project as well.  

He’s particularly known for modelling numbers and the number system by his picture of Exploding Dots.  I recently followed a series of sessions he did for children in which he showed – complete with many exuberant sound effects – how Exploding Dots can be used to cover all basic computation methods in the primary curriculum.  There’s an excellent video of him introducing the idea at

https://www.explodingdots.org/station/I1S1

To recap, Tanton’s model has a line of cells, and dots are fed into the cell at the right hand end.  This cell and indeed all the others has a maximum stable capacity which allows it to hold just one dot.  As soon as a second dot is fed in then both the dots explode.  They annihilate each other and generate a single new dot in the next cell to the left. The process can be continued as far as you like. Hence the first few numbers follow this well-known sequence of binary numbers:

1, 10, 11, 100, 101, 110, 111,1000

Of course, Tanton points out that a change in the basic rule so that each cell can hold a maximum of nine dots produces our own number system.

He took us through how his model can be used to handle all the basic computations in primary arithmetic, though I doubt whether even he seriously believes it’s worthwhile to use exploding dots to explain long division. 

However, in the final session there was a real sting in the tail, which took me somewhere I’d never been before.  Here it comes.

Supposing, said Tanton, that your exploding rule is is that the third dot in a cell causes the explosion and produces not one but TWO dots in the next cell to the left.

OK here we go. The first few go like this:

    1  

    2  

  20 [the third dot into the right-hand box causes the three to explode, creating two in the next cell to the left.

  21 [this is 4 in our normal notation]

  22

210 [we’re now up to the sixth dot; this fills the first- i.e. right-hand – box to capacity, so causes an explosion to create two dots in the second box; this now has four dots of which three explode to make two dots in the third box]

Tanton’s simple rule means it’s easy to generate the numbers for yourself.  Here are the first sixteen:

1   2   20   21   22   210   211   212   2100   2101   2102   2120   2121   2122    21010   21011

Well clearly we’re beginning to get a new notation of the counting numbers – but exactly what is this system? 

Well, three is shown as 2 0.  Now whatever rule you use, the penultimate cell is always the base of the counting system, so we must actually be by using a system whose base is 3÷2 , i.e. 1.5   We’re using base   !

 ……  

I was blown away by how easy Tanton had made it to generate numbers in base one-and-a-half, a base that was completely new to me.  Not just me, in fact, because he made some play with the fact that this has opened up a whole field that’s new to everyone, professionals included.

There’s lots to explore.  For example:

*** Divisibility – it’s easy to recognise numbers which are divisible by three, but there doesn’t seem to be any easy way of recognising even numbers.

*** Is it true that all numbers begin with 2?

*** There are many numbers which look as if they ought to appear in the series but which don’t.  11 is a simple example, 2000 is another.  Do these numbers actually exist?

 …… 

Tanton makes it clear that this is a field that’s so new it’s red-hot.  His 25-page guide is fascinating, and ought to be on the bedside table of anyone interested in exploratory or recreational mathematics:

…… 

One final point.  Can anyone honestly say they realised you can write any integer as a sum of members of the series:  (1) , (1½) , (1½)² , (1½)³  , …. – or if you prefer, 1 , 3/2 , 9/4 , 27/8 , …. using not more than two of any one term?

Six Strips

Here’s an activity I like the look of. At the end I’ll mention how it came about.

Our school has problem-solving and enthusiasm to accept challenges among its aims, so even though many of the children I meet are those who need a bit of a legup they’re usually very positive about mathematics. Certainly after a week or two of working together I’ll usually expect to be able to throw something at them and expect them to get stuck in to making sense of an unfamiliar starting point.

So in this case I’ll probably give them the six strips, sit back with a benign smile, and see what happens.

Another view:

I reckon there are a couple of clues that will help. One strip has four equals signs, so it looks likely there are four statements to sort out.

There are two strips with operation signs. Either that means one is to be used and the other discarded, or that both are needed, in which case each statement will involve two operations.

The zero looks useful. Assuming there are no leading zeroes – and if there were, why is there only one – then there must be a 10, or a 20 or a …., involved.

And running throughout this is the fact that for each of the strips (well, five of them anyway) you’ve got to find which way up they need to go.

In a perfect world I could use this tomorrow, get feedback, and devise a couple more, but it looks as if it will be a bit longer before I can find any guinea pigs.

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Note: the stimulus for this was a recent post by the prolific Sarah Carter ( @mathequalslove ), which in turn she credited to Erich Friedman. My feeling was the version she posted, which had eight strips ( https://mathequalslove.net/equation-rotation-puzzle/ ), was a bit too fierce for me to use as a starter, but might have a part to play later on.

Sarah has a very individual style of presenting an activity, so her posts are always instantly recognisable. Do check her out at https://mathequalslove.net

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Some Battles We Really Ought To Win – Round Midnight

One of Thelonious Monk’s most enduring compositions is Round Midnight – there are well over fifty versions in my collection, but even though I’ve heard it so often I’ve never once wondered when midnight might be.  Certainly the government hasn’t; the announcement that lockdown would begin at midnight on Thursday failed to acknowledge that there‘s a midnight at the beginning of Thursday and another midnight at the end of the day.

In this case midnight Thursday meant the start of the day; when I worked for Pearson a deadline of midnight meant the end of the day.  The armed services recognise any confusion could be catastrophic and so refer to 2359 or 0001, so our current lockdown began at 0001 today rather than 2359 this evening.

You’d think that 1200 in the middle of the day would be less of a problem; at least with midnight there’s one at each end of the day, but there’s only one middle of the day and there are helpful words available like “noon” and “midday”.  Huh.  My online shopping retailer tells me orders must be finalised by 12pm; when I rang for clarification the result was “er …” with a quick and inconclusive check with nearby colleagues; you have to find by experiment that in their view 12pm means one minute after 1159.

And for yet more confusion here’s a honey:

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“Some Weird Animal” – Alfred James Swinburne

In the last couple of years I’ve come across more and more evidence that Her (Victorian) Majesty’s Inspectors of schools were often of vast importance in stimulating the growth of progressive practice in elementary schools.  Alfred Swinburne somehow combined a reactionary background with ideas that still feel radical today.

 

An inspection could be an intimidating and even terrifying experience. Joseph Ashby recounted his experiences as a pupil “One year the atmosphere of anxiety so affected the lower standards that, one after another as they were brought to the Inspector, the boys howled and the girls whimpered”. (From Joseph Ashby of Tysoe 1859-1919, by Mabel Ashby; published by CUP, 1961). All the same, there are enough indications both from what inspectors themselves wrote, and from reading log books that not all inspections were like this. D R Fearon’s guide to conducting an inspection, written on official request in 1876, says “managers and teachers ought to look forward to the visit of an inspector … with hope, and an expectation that he will suggest means of overcoming difficulties and amending defects”.   Fearon suggests that the inspector should be able to observe the children working normally and as “cheerily and naturally as possible”.

A surprising number of inspectors wrote personal memoirs – I’ve got at least four and I know of more. Probably the most personal of all is ‘Memories Of A School Inspector’ by Alfred James Swinburne (1846-1915). Personal indeed – I imagine Swinburne was the only member of HMI before or since to feel it necessary to take his revolver when making a visit to an area of aggressive Irish Catholics. He was an inspector for 35 years, initially in Lancashire but for most of the time in Suffolk.

The social gap between inspectors and teachers was always wide, but it wasn’t until reading Swinburne’s book that I began to grasp just how wide it was. To start with, he was a landowner with the means to fund publishing the book as a vanity project, and there is an arrogance and a pervading sense of entitled privilege that makes much of the book rather uncomfortable to read today.   He spends a considerable part of the first chapter recounting his family history back to the fourteenth century, while his time as an Oxford undergraduate was leisured enough to involve a vast amount of sport, including hunting and steeplechasing with others of his ilk.

This was a man who could not only afford to run a chauffeur-driven car but do without insurance, being perfectly happy to buy a new one when it became a write-off. He takes train journeys with Generals, and stays with the Lord Lieutenant. Time after time he writes disdainfully of those beneath him socially – ‘peasants’, servants, and even parents. In every case his belief of his own superiority is plain.

To be fair, he could also be pretty scathing about those whose social standing was closer to his own. Many of those who were foolish enough to offer him hospitality will have regretted their generosity. He recounts not just one or two, but several occasions where the food or the wine or the accommodation or the servants fell below the standards he felt acceptable for someone of his importance.

But once you get going most of his book is easier to read. The anecdotes have been lovingly polished and many would get a laugh today on a chat show. The old woman who tells him ‘This would be a sad world if it were not for the blessed hope of immorality in the next’. The school manager seeking advice on appointing a new female teacher – who needs to be of mature years to avoid the predatory head ‘who’s ‘a bit amorous, like’.   Then the school manager who greets him ‘Such a dreadful year – master’s wife, bronchitis – mother-in-law, cancer – staying with us – and to crown it all – you’ve come’. (Which reminded me of the occasion when I responded to a local school who’d asked for a visit from an advisory teacher – ‘That’s nice, but isn’t there anyone else?’)

And there’s the occasion on a crowded train en route to the Grand National at Liverpool; the outside door flies open, and in a desperate attempt to save himself from being hurled out he flings out his arms – and happens to grab the watch chains of both men either side of him, thus gaining the professional admiration of the many nearby cardsharps and conmen. And there’s an amusing story of when he’s asked whether a pronunciation should be ‘eeder’down or ‘iider’down. He replied neatly ‘eether or iither’, thus pre-echoing exactly what the Gershwins wrote in 1937 ‘Let’s Call The Whole Thing Off’ (‘You say eether, I say iither …’)

He retired, at the age of 63 in 1911, considerably against his will; indeed he returns several times to his lasting resentment at not being granted a year’s extension to his service.

 

Ahead of his time

However, once you get beneath the patrician surface you find someone who not only cares deeply for ordinary pupils, but is well ahead of his time. Indeed, many of his views are such that you have to wonder if any of today’s inspectors who showed such independence would be able to keep their jobs. However right-wing his background and social life, his professional views were such that he could be asked “Are you a liberal?” – to which he would give the magisterial reply, “Madam, I am an Inspector of Schools”.

He railed against lessons which were nothing more than to learning by rote of lists of historical dates, geographical features, and parts of speech and English grammar (he asked an 8-yearold if he knew anything about Nelson – “Please, sir, we haven’t got to verbs.”

His accounts of teacher controlled lessons where the children had to respond with the approved one-word answer – with contributions of their own, however insightful, being rejected – could still be used with students today. A feature of these accounts is the way children keep trying to guess what’s in the teacher’s mind, and continue to do so enthusiastically even when their ideas are rejected brusquely (“No – no – you silly girl…”, “Nonsense, you stupid creatures!”, “Sit down, you naughty child!”, “How can you be so stupid, you silly girls?”, “Dear, dear, what can you be thinking of?”).

Swinburne frequently mentions corporal punishment, and every example he gives shows it in a negative light. In many cases children are punished for not understanding something that the teacher has explained badly in the first place, or because s/he has failed to organise the class appropriately. He claims that corporal punishment has long been practically dead in elementary schools, though in several cases he finds that there are teachers whose actual practice differs from what they claim – where when he claps to applaud a child’s answer pupils cower in terror as soon as he raises his hands. In what clearly refers to a horrific case he writes cuttingly of boys who have the audacity to fall onto spiked railings.

For Swinburne it is not the building and the facilities that define the school, but the children and indeed the teachers within it – “School exists for the child, not the child for the school”. He reflects “Sympathy, gently lifting over difficulties and stimulating to self-help, which is of the essence of true teaching, has its full weight now”. This was in the 1890s – in the middle of his career – when the era of Payment By Results was finally over.

He sees the teacher as a supporter and a guide to pupils who, after all, are the ones who do the actual learning. There are indications that he believes learning should be active and indeed practical – when he needed to measure the school building he got four pupils to do it (after all, measuring is always a problem-solving activity), and he recounts the absurdity of a gardening lesson consisting of children silently reading a gardening manual rather than actually doing anything outdoors.

He recognised that “we do too much for children” and talks of a teacher who “speaks for them, she thinks for them, she almost breathes for them”. On his retirement he advises teachers “for your own sakes, as well as theirs, do not do too much …”.

As the nineteenth century came to an end education entered an unprecedently progressive era, and Swinburne was fully in sympathy. Today’s teachers might welcome the official “Blue Book” advice that prominence should be given to methods, rather than results, and that teachers should go beyond requiring children to give correct answers and no more. “They now”, says Swinburne “ need to demonstrate understanding of how a calculation works, to give examples illustrating a factual statement ….”.

For Swinburne this means the responsibility of assessing children is now the teacher’s – “The teacher can now choose for himself the best forms of assessing children’s progress”. This, he felt, was a significant reason why he found children happy to be in school; he has no doubt that the change is wholly beneficial.   “There is no question as to the difference in the happiness of the child … a real abiding taste for intellectual pursuits”. Parents too report the benefits – they no longer have to complain of arithmetic cards being worked in children’s sleep

All the same, he is realistic enough to be concerned that even though the rigid focus upon basics of the Payment By Results system had weakened considerably, the authorities were in danger of relapsing to what he calls “the cruel rites of the 3R percentage fetish, as they already worship a horrible percentage fetish in the manner of attendance”.

 

The Prize Scheme

Swinburne spent the vast majority of his career in Suffolk, from about 1881 to 1911. This was a period of great change in schools, and he was at the forefront of many of these changes.

He was particularly proud of the East Suffolk Prize Scheme which he initiated and led for 30 years. One aspect of the scheme brought teachers from different schools together to look at pupils’ work reflecting the curricular standards across several schools. (This moderation aspect was unknown in any of the schools I taught in, and didn’t become a general feature until relatively recently.)

He was particularly proud of the work generated for the scheme and its annual exhibition covering the full range of the curriculum and many practical subjects as well. The work could be of such quality that it featured in the Paris exhibition of 1900, and pupils who took part in the scheme featured in stories in the national press and even the Chicago Tribune.

It was a matter of considerable bitterness and resentment that the LEA decided not to take over what had always been a purely voluntary scheme – even though the Secretary of the Education Department in Whitehall applauded it.

Swinburne was not one to let a good grudge go to waste; he was never a fan of local authorities and officialdom, and this lack of support caused lasting resentment. He refers time and again not just to the failure to support his scheme, but also the refusal given to his request for a year’s extension to his contract, even after a petition in his favour garnered 12000 votes.

 

Teachers

By the end of the century the role of the inspector had become much less the formal examiner and far more the supporter of schools and their teachers. Swinburne reflected this change perhaps more than anyone, and in a retirement speech to teachers he tells them “I am able to realise how much of England’s history is in your hands.” In one of the closing sections of his book he referred to “My true friends – the teachers of East Suffolk …. How often have I enjoyed the lessons given by East Suffolk teachers, and how much they have taught me”.

In many of his reports on visits he’ll often try to make his criticisms constructively, and even in the worst lesson he tries to look for good points. He sees a teacher give an awful geography lesson, but his feedback recognises “she was made of the right stuff – teeming with many good points, especially heart – and far better for infants than clever people often are – but scarcely suitable for Standard III”

After a mediocre lesson on the skylark, rather than attacking the teacher for insufficient preparation he commented drily “Rather meagre your account, wasn’t it?”. The teacher felt able to reply “Well, you see, sir, I don’t tell the class everything about it so as to cultivate their observation better.” On another occasion he pointed out that mistakes were going uncorrected and the mistress claimed she “Didn’t want to discourage him, sir.” I draw from such episodes some clear inferences about the nature of his relationship with teachers. The teachers are prepared to offer their side of the case, even when he suspects them of sloppy work; he may be the face of authority, but it’s a human face and not simply an authoritarian one.

On one occasion he must have been so friendly that the young female teacher misinterpreted some kindly questions as flirtation, and when he asked could she give a lesson on Reindeer she simpered “I have one on clouds and mist but I have not one on rain”.

Swinburne reports that teaching is no longer synonymous with lecturing and emphasises that ‘education’ is derived from the Latin ‘educare’ meaning to draw out. Hence he says the role of the teacher is to draw out the mental faculties of children.   He draws an analogy from a painting of a young woman using stepping stones to cross a stream with a small child behind her. She doesn’t pick up or carry the child, but turns and supports and reassures and guides so the child can complete the crossing herself.

In a retirement speech he reflects on driving to an inspection on a day when the weather was terribly stormy. He worried that the weather was so atrocious that no children would be present. In fact, everyone was there “smiling and happy”. He recognises that this was down the work of the two teachers “with their hearts in their work”.

He compares the work of such teachers with those fighting for the emancipation of the black population of America.   Teachers have to work against the prejudice of [a] of magistrates, who frequently fail to enforce pupil attendance, [b] farmers who’d rather children work for them than attend school (he tells how the board of one School discovered that the school had won an excellent report and promptly advised the teacher to find employment elsewhere!).   And [c] teachers also have to work against the prejudice of parents.

Indeed, parents come lower in his rating than their children and teachers. Early in his book he claims that parents are only interested in money even though he still finds it possible to say a good word for innkeepers and publicans. On a visit to an excellent school a parent who complains his child is not being taught Euclid and algebra is disdainfully dismissed “It would be difficult to award higher praise to an elementary school”.   Another objects his child is not learning anything about Canada. Swinburne is equally cutting: “Another feather in the master’s cap – the year’s course being Europe”.

To a grumble about lack of discipline at the same school Swinburne says “Presumably the parent is complaining that the children do not have long faces and are not trussed like fowls for cooking.”   His support for this school is complete – he says it’s one of the few where he always finds the teachers (head and assistants) out in the playground with the children.

Of course, Swinburne was fiercely right-wing, but his respect for teachers meant relations with teachers’ organisations could be better than might have been expected. He instituted a circulating library for teachers so significant that the King’s sister agreed to be the patron, and the NUT suggested the scheme should be introduced in other areas and indeed used this pattern to set up a library of its own.

The East Suffolk County Association of Teachers marked his retirement after thirty years in the county with a collection and presentation to him. This was a rather grand occasion, and called to mind the very personal tribute paid to him by one teacher:

Still it seems to me, that his kindly hand,

With the breath of his high endeavour,

Has passed a blessing on this land,

And aided the children for ever.

Children

The final chapter of Swinburne’s book is devoted to children and once he starts talking about them you can forgive him for everything – for all the snobbery and the arrogance to those he deems his inferiors.   He met thousands of children every year, and of all the groups he writes about, it is children who come out by far the best. Almost without exception, whenever he mentions children it is in a positive situation – you almost feel a warm glow coming off the page “Artless simplicity and an entire absence of anything like bitterness, cynicism, or profanity….”

Or “One of the most salient characteristics of childhood is a certain delicacy and even tenderness of sympathy.”

I can’t think many inspectors made it so plain how much that being with children was enjoyable for both sides – “The kind pity they have for an Inspector …. They view him as they would some weird animal, that needs strange food, and loves to be petted and stroked.” An inspection by Swinburne was unlikely to be one of those where children howled and whimpered even when he was assessing their learning “I always found that, in anything worth doing, children were my best friends. They never found fault with examinations.”

According to Swinburne, children are always keen to talk and answer questions and say how they’re getting on in their work, and he points out that children do not bear grudges even when teachers make disparaging remarks – “no one forgives quicker and bears less malice than children, especially in groups”

The warmth is reciprocated; children like him just as much – “I thought he was a kind gentleman … and I have done all in my power to please him. He has born [sic] with the children very well and I shall try to read well for him”, and “…. When I go home I should tell my parents I liked him very well”

Most impressive of all is the way in which he gets to know children as individuals. He speaks admiringly of a 13-yearold who has to get all meals for her father and her two older brothers; she has to keep house, do the washing, the darning and all the chores and still comes home from school full of pleasure from her lessons to tell her parents all the interesting things she’s learned.

And I was moved by another example of his deeply caring side. A father’s ill-treatment had caused his wife to leave home; there was a rumour that she might be returning. Swinburne spent twelve hours hand-in-hand with the five children on the platform at Ipswich station waiting with mounting disappointment for her possible return. (His concern didn’t end there; he kept in touch and records she eventually did come back – but the man’s behaviour didn’t improve and she left for good.)

I spent much of the first half of ‘Memories Of A School Inspector’ disliking him intensely, and if I’d ever met him in person I’m pretty sure I might have felt just the same. He’s been called ‘idiosyncratic’, and there’s no doubt about that, but he had the respect of the Secretary of the Board of Education and the leading figures of the day, and it’s difficult to feel ill of someone who could close his memoirs with “Adieu! My dear Suffolk children. I never told you … of all the good you and your teachers have done …..”

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Dotty Six and What If Not?

You’ll find it helpful if you read the article in which Lynne McClure and Lucy Rycroft-Smith of Cambridge Mathematics wrote about a game they called Dotty Six (  https://www.cambridgemaths.org/blogs/connecting-the-dots-on-mathematical-home-learning/  )

 

When I first joined the professional associations I looked on admiringly at those people who were able to look at a simple situation and spin off additional ideas from it like a catherine wheel. Of course you get better with experience, and stealing other people’s ideas (known in polite circles as learning from them) helps, but also there is a simple technique invented by Marian Walter and Steven Brown, dating from 1969. They called it ‘What If Not’.

What If Not is very easy to use – basically, all you do is look at the situation, select one of its aspects and ask yourself what would happen if that aspect were different.

 

The rules of Dotty Six are very simple.   It’s given as a two-player game; the gameboard is a simple 3×3 grid as in Noughts and Crosses.

Players take turns to throw a standard 1 to 6 die.   You put all the dots from that throw into one cell; it may already have dots in, but no cell can hold more than six dots.

You win by being the first to complete a line of boxes, each containing exactly six dots (it doesn’t matter who completed the first two boxes in the row, it’s the act of completing the third which wins the game).

 

Dotty 6 strikes me as being full of interest, and I couldn’t resist applying the What If Not technique.   Even with such a simple situation there’s a wealth of aspects you can modify. There are well over a dozen, and I used a spider chart to help me (don’t try to read it, for heavens sake – it’s simply there to give an idea of how I went about things):

 

So here we go.

Let’s start with the board.   What if it’s not a 3×3 grid?   Noughts and Crosses has long been a very fertile ground for exploring different boards – what if you have a 4×4 board, or if you’re something of a masochist, a 3-D version with a 3x3x3 cube?

What if the board is not a square? Try a row of, say, nine cells.

What if the capacity of each cell is not 6, but perhaps 8? (Lots of number bond reinforcement here.)

What if the cells don’t all have the same capacity? Perhaps some might have capacity 6, others more, or less.

What if you don’t have to use the number you throw yourself? You could choose whether you use it or require your opponent to do so.   Or throw two dice and choose which you want, and give her / him the other.

What if completing a row doesn’t win the game, but loses?

What if you don’t have to put all the dots in one cell? Try: you can put all the dots into one cell, or split the throw into equal piles and put each pile into a different cell.   (I like this – we’re into factors, multiples and prime numbers.)

What if you’re not both trying to do the same thing? Perhaps one player wins by completing a row, the other a column. Or A wins if a row or column is completed within eight throws; if no row / column is completed B wins (and What If Not eight throws, …?)

What if you don’t both throw? So if a 2 is rolled, then both players must place two dots.

What if it’s not for two players?

And a failure – I haven’t come up with any useful suggestion for this – what if it’s not six dots?

And there are lots of What If Nots from the use of the die / dice:

What if you don’t throw one die, but two? Or even three? (Throw three dice and choose which you want, and your opponent chooses from the other two.)

What if it’s not a standard D-6? Actually I don’t think using a D-8 or a D-10 sounds all that interesting, but there are plenty of mobile phone apps that let you generate any range of numbers, so you could try 1-4 or 1-5.

And what if it’s not a die / dice at all? William Gibbs did a lot of work in countries where schools had the most minimal of resources, and devised a simple number generator:

 

What if it’s not a die / dice but a spinner? ([a] If you design you own spinner which numbers would you choose, and in which proportions? [b] Deciding what the dot patterns on pie slices might be was a challenge I hadn’t thought of previously.)

rbt

What if you don’t use a die / dice but counters in a bag? You could use green, red and blue counters and draw three each turn; a green counts 0, red counts 1 and a blue 2. (Lots more possibilities here ….)

And finally – at least until something else occurs to me – what if you don’t use any randomiser at all?   This game now becomes one of pure skill, with all luck eliminated: each turn you may choose your own number from 1 to 6 – but you can’t use any number twice until you’ve used all six numbers (when you start again with another set of 1 to 6). (This strikes me as being distinctly challenging ….)

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Some Battles You Just Don’t Win – A Quantum Change

Lots of words – like data, agenda, criteria, media, graffiti – seem to have lost their singular forms.  These days few people talk of ‘agendum’, or ‘graffito’.

Interestingly there doesn’t seem to be any sign of this happening with ‘quantum’. Perhaps its scientific heritage preserves a slight air of mystery.

Those without such a background need to use words like ‘quantum’ at their own risk. I was at a conference where I found the speaker to be rather patronising. He glibly talked of his project which was certain to introduce a quantum change in the teaching and learning of mathematics. I gave him a slight dose of embarrassment and many of the audience considerable amusement by pointing out that a quantum change was defined as the smallest possible amount of change and that no smaller change was conceivable or possible. Not quite what he had in mind.

I particularly enjoyed an even better example of puncturing an attempt to use pseudo-scientific jargon.   This time someone was foolish enough to attempt to use terms describing his pupils such as ‘cutting edge’ and ‘high-performance’. One of our group was a marvelously droll representative from the aircraft industry.  “Of course you realise”, he said, “that a high performance aircraft is inherently unstable”. I think many of us were still laughing at the end of the afternoon.

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Some Battles You Just Don’t Win – The Agenda Are Dice: Time to Face Up and Take Sides

OK, convincing people that a square is a particular kind of rectangle (see xxxx ) is an uphill battle, but at least that battle is still ongoing. Others have been comprehensively lost.

I used to try to insist on using ‘die’ to mean just one item and ‘dice’ when there are two or more, but these days I’m not doctrinaire about it. I guess I decided it wasn’t worth the effort when I ran a course and a significant number of teachers went away having inadvertently learned that yes, there was a difference – but got it the wrong way round, so that they still talked about one dice but now went home talking about two die! Obviously it would have been better not to bother in the first place.

And NRICH (e.g. https://nrich.maths.org/8303 ) is just one authority to talk about ‘an ordinary dice’, so I’ll give up the die / dice argument. I’ll do this gracefully enough; I don’t really think the language is seriously impoverished by losing a fairly insignificant singular / plural distinction.

(But I definitely do not like it when people – many of whom should know better – talk about 6-sided dice, 10-sided dice etc. 2D shapes have sides, but 3D shapes have faces – so what’s wrong with talking about faces?)

But dice are just one example of words which have lost their singular form – data, agenda, criteria, media, graffiti have all gone exactly the same way, with the plurals being used irrespective of the situation. No-one seems to get too hot under the collar about the disappearance of datum, agendum, criterion, medium, graffito.

Interestingly, this hasn’t happened with quantum – and that’s for another post.

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Some Battles You Just Don’t Win – Squares and Rectangles

Headnote: some people are uncomfortable with my girl / boy analogy and have pointed out that in today’s world we need to acknowledge that gender issues are sensitive, and not always clearcut.  I accept the point that’s been made, and would not wish to give offence.  

If you feel this post is best removed, then please let me know.  (And – of course – if you can give me a better analogy I’ll be delighted to use it.)

 

One battle you never win is trying to convince people that yes, a square actually is a rectangle – a special rectangle but a rectangle nonetheless.

Even dictionaries like to hedge their bets. When I asked Google for a definition of a rectangle these three all popped up immediately:

Google says: a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square.

Merriam-Webster: a parallelogram all of whose angles are right angles; especially one with adjacent sides of unequal length.

YourDictionary.com: a rectangle is a four-sided figure or shape with four right angles that isn’t a square.

So it’s really not surprising that children also tend to agree that a rectangle is a rectangle, a square is a square, and the two are totally separate. The National Curriculum doesn’t help, and has always avoided the issue. Key Stage 2 SAT questions always skirt around the problem.

Life might be a little bit easier if there were more encouragement to use the helpful and apposite term ‘oblong’. The name gives a clue – an oblong is a rectangle with one pair of sides longer than the other, i.e. it’s a rectangle that’s not square.

Now actually you can demonstrate the relationships very simply in this simple diagram. All rectangles are either squares or oblongs. A square is a rectangle and an oblong is a rectangle, but a square is not the same as an oblong.

 

That’s simple enough, but we can draw a parallel that everyone understands.

For rectangles read children, and replace squares and oblongs by boys and girls. No one has any difficulty now! All children are either girls or boys. All girls are children, all boys are children but girls are not the same as boys

This is indeed a very simple and helpful parallel though I can’t claim I’ve had all the success with it I would have hoped. As those dictionary definitions show, the idea that a rectangle is a rather defective square is so well entrenched that it’s one of those battles that’s always going to be an uphill challenge.

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An Index

If I find it a major challenge to locate an item I know is somewhere on the site, then I’m pretty sure you must as well.  Furthermore, anyone who just drops in can’t have any idea of what else might interest them if they only knew it was there.

So I’ve built an index of all postings since 2013.  I’ve built it as what WordPress calls a Page, which means that no matter what you’re looking at there’s a button at the top of the page marked Index which calls up the comprehensive index.

WordPress seems to want to make it as difficult as possible, so every time I tried to plonk the table in I ended up with an impossible mess of different font sizes and types and missing columns, even when I tried to use a PDF version.

I’ve managed to produce something which is readable and undistorted, but I’m afraid I’m afraid the entries aren’t live and are unclickable.  Sorry, but it’s a convincing victory for WordPress, albeit on points rather than a knockout.

By the way, when you do call up the index there are several entries in blue.  These all relate to my recent interest in the history of elementary schools in the nineteenth century.  I’ve eventual plans to move these into a separate blog, but for the moment they’re still here.

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My Mathematical Adventure Games

In my previous post I wrote about a terrific lesson using one of my adventure games – here’s more about what they involve.

 

It was in the 1970s that I first came across a new type of game called Dungeons and Dragons in which the players explore an unknown environment, encountering various puzzles and problems on the way.

Not long after, the first popular computers appeared and soon entered the classroom. There was a wave of brilliant software produced (and, of course, there was a hundred times as much garbage) – to my mind much of the best is still valid today. And some authors realised they could utilise the D&D format in an educational context – and in primary mathematics Anita Straker was amazingly prolific, despite having to teach herself programming in a hospital bed from a textbook. I introduced school after school to her Martello Tower programme. Indeed I may be the last person in the universe still using it today, and it still grips and challenges children nearly forty years later.

I can’t imagine Hertfordshire’s education department ever spent £70 better than buying the licence to use Martello Tower in its 500 primary schools, but unfortunately many of the 500 couldn’t manage to make successful use of Martello Tower. Most teachers were still very nervous of computers, and in any case schools might have just a single computer – i.e. not even one per class. In the most successful schools talented teachers were able to devote half a term to exploring the Tower and all the mathematical challenges it presented, but for most teachers it was too big a step.

I promised myself that when I got the opportunity I would attempt to devise a format that utilised the motivation and challenge of the adventure format while being easy to use even in the classroom with no computer, and when HCC decided it could do without me I set to work.

I decided my adventures had to maintain the idea of a motivating story-based setting. Also essential was that children should meet a variety of problems (about eight or nine) and these should be related to the theme. The whole adventure might be completed in just two or three lessons.   The only equipment would be what you’d find in any classroom.

I wanted an adventure to be usable by any teacher, and furthermore for there to be no need to mark each task the pupils completed – I wanted the teacher to be free to watch and listen to their children (usually working in pairs) as they tackled the problems they met. So I needed to free the teacher from marking, so she could concentrate upon observing the children’s learning. Hence each time they completed a task the children might collect a codeword and at the end all that was necessary was to check that they had the eight correct codewords.

For this to work I decided each activity needed to have half a dozen plausible answers each with its own codeword. (There is, incidentally, a real dichotomy here. An adventure actually feels open-ended and exploratory, but to make this possible the activities themselves need to be closed, with a strictly limited set of possible answers. I’ve never resolved this problem though to be honest in practice it’s never seemed to matter too much.)

Another constraint was that I wanted to avoid imposing an extra photocopying bill upon schools. So I ensured the activities could be done in any order and just two copies of each task were adequate – which incidentally made it easy to store everything – teachers’ notes and the activity sheets should easily into an A4 folder.

Eventually I produced about seven or eight adventures. A teachers’ magazine published several of them and I produced photocopies for interested schools and teachers; a few professionally produced copies are still sold today.   Basically, however, schools became more and more squeezed by the demands of National Curriculum assessments, and for teachers the opportunity to use the adventures largely disappeared.

(Intriguingly, there was a moment when the National Curriculum itself could have led to the adventures being much more widely known. Twenty years or so ago the curriculum body of the time was looking to highlight problem-solving in maths, and commissioned an adventure from me which would be given to every school. They were even kind enough to pay me, and I devised ‘The Fantastic Fairground’. But the next stage never happened; a reorganisation saw a new body appointed and the initiative got forgotten about. Indeed I’d forgotten it myself, and when Dilly and Lemmy (https://established1962.wordpress.com/2020/03/25/a-lesson-that-will-stay-with-me/ ) had a go at it was the first time I’d pulled it off the shelf this century.

Probably ‘Blackbeard’s Treasure’ (an adventure about pirate gold), and ‘S.A.-N.T.A. C.L.A.U.S (where children are working in Father Christmas’ toy distribution centre) are the best known of these. Perhaps I like using ‘The Haunted School’ best – it translates most directly to a year 6 class preparing for Key Stage 2 SATs

It came as some surprise to me a few years back when I realised when I discovered that Questions Publishing group had actually taken the step of publishing five of these commercially. But it does allow me to say that I’ve written seven books, even if in two I was the co-author and the other person did all the work, and I didn’t know I’d written the five in the first place!

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