Six and counting... maths, Rishi Sunak, and fairy tales (Anne Rooney)

A prime number of dwarfs, Arthur Rackham

 This week, the British PM suggested the poor numeracy skills of Britain's school leavers could be improved by forcing them to learn maths up to 18 years of age. (Of course, the obvious answer is that their poor numeracy skills could be improved by teaching them maths properly to start with, but never mind that for now.) And today I heard from an American friend and college professor that her students (doing humanities degrees) don't know any fairy tales except any they have seen as Disney movies. That is, 59 out of 60 don't and the 60th is German. So I thought I'd put these two together, because if you don't know any maths, you can't understand fairy stories anyway, because numbers are really important in fairy stories.

There are three bears. There are seven brothers/dwarves etc. There are never seven princesses (someone correct me if I'm wrong).* There are two sisters if they have equal parts to play (Rose Red and Snow White are essentially one). Dancing princesses come by the dozen. But bad step-relatives come in threes (mother and two sisters). Genies give you three wishes. You usually get three tries at something before it all goes to pot — three nights to spin straw into gold, three chances at blowing down a house full of pigs, three chances at frog-kissing, three chances at abandoning your kids in the wood. There are seven corpses in Bluebeard's cold room. What do we notice? Prime numbers if you're male or bad. Even numbers if you're female and good. You can predict that Bluebeard's eighth wife won't die because he already has seven dead wives and no baddie can end up with an even number of dead wives. (I know, there must have been a time when he had four or six, but the story wasn't ending then.) You know the bad fairy is going to turn up in Sleeping Beauty because there are never 11 fairies. And you know she will be defeated in the end because there must be a twelfth wish and as it's an even number it's going to triumph.

I know people have related number in fairy tales to religion and other things, but that's just pushing the issue backwards. It's no use to say there are 12 dancing princesses/disciples and think that's an answer, because why are there 12 disciples? Probably for the same reason there are 12 hours of daylight — 12 is a useful number as it's the first number you get to that has more than two factors (2x6, 3x4). And 3 and 7 go the other way becaus they're prime; you can't do anything with them. You can't pair off the dwarfs, or Billy Goats, or bears - they each need a distinguishing feature.

Of course, thinking about numbers in fairy stories won't help you work out compound interest or calculate your tax. Though... if Goldilocks takes one spoon of porridge from each bowl, is she using the same spoon for each or a spoon of corresponding size? If the first, she takes the same volume from each of the first two bowls but a larger percentage from the middle bowl. If the second, she takes the same percentage but a smaller amount from the middle bowl... Given that her mass is too great for Baby Bear's chair, but not his bed, we can conclude that his parents made/bought a bed he could grow into. How much money did the bears save by cutting down the number of beds they will have to buy over his cubhood? See, we can kill two birds with one stone here. (The two birds are, essentially, a single killing as they don't need to be individually distinguished).

*I am not claiming this is thorough or scholarly. It's an idle muse, riffing on apparently entirely separate stories, because I like seeing what can come out of juxtapositions.

Anne Rooney

Out now: Baby Polar Bear, Oxford University Press

Off limits? By Steve Way

 

I was speaking with some good friends the other day, including someone I consider to be a talented (though as yet unpublished) writer. I had mentioned writing pieces, particularly for children, incorporating maths topics. My friend (and several others) energetically spoke about how much they hated maths and couldn’t do it etc. and said friend said something along the lines of, ‘I was never interested in maths so I when I went to university I studied English to help me to become a writer.’ This seemed to suggest that mathematics was a topic about which one could not, or maybe even should not write about.

Recently I saw a repeat of an interview given by Billy Connolly at one point railing against comedy purists who insisted that some subjects should be taboo, whereas the sainted Billy believed everything should be on the table. I seem to side with Billy, in my case when it comes to considering mathematics a possible topic for creative writing. Perhaps when you’ve read the couple of examples of my poems below you’ll side with Billy’s detractors!

A while ago one of my grandchildren had to learn about ‘platonic solids’, solids formed from perfect ‘regular’ flat shapes, so in order to help I started writing a poem. My wife made a useful point about cubes and so she gains joint authorship for this poem. I hope you like it and/or find it useful.

 

Platonic Solids.  By Steve and Jan Way

This poem is about each ‘Platonic solid’,

I promise you it won’t be horrid.

The most well-known has six square sides,

It’s the good old cube that this describes.

A square of course is a perfect shape,

Perfection others only ape.

It’s got four sides of equal length,

And right-angle corners reveal its strength.

This explains this shape platonic

(Plus, you can balance anything on it!)

A tetrahedron you will adore,

It’s made of triangles numbering four.

A pyramid shape it thus does make,

But please don’t make this big mistake,

It’s not for a pharaoh, (an Egyptian a king)

For that with a square base does begin.

Equilateral triangles form the faces

As they do in two more cases…

One by us is octahedron named,

For having eight sides this is famed.

(As an ‘octopus’ would tell you straight,

The prefix Octo- it means eight!)

An icosahedron’s formed from triangles twenty!

Just like me you may think that’s plenty!

The dodecahedron has a different form,

That certainly varies from the norm.

Perfect pentagons form its faces,

So Tents this shape make useful spaces. *

 

I said these shapes wouldn’t horrid be,

And now we’ve reached the end you see.

 

*Actually they’re usually ‘pentagonal rotunda’ which are made of pentagons combined with triangles (so are technically half an icosidodecahedron) but they’re called ‘dodecahedron tents’. This what is known as artistic licence (i.e. A creative excuse for getting things wrong!) either on the part of the tent makers or me – or both!

~~~~~~~~~~

While I was hunting down this poem, having written it a while ago, I stumbled on another poem about 3D shapes, which I’d written even longer ago and was so pleased to find it, I wanted to share it with you. This poem covers the solids children more usually have to learn about. As I imagine most teachers could tell you, children find it very difficult to pronounce the word ‘sphere’ and I have often jokingly had to dodge an imaginary spear having asked a group of children what maths name was given to the tennis ball I was showing them.

3D shapes poem.

A cube is a 3D shape called solid,

I promise you it is not horrid!

Six flat square faces it does show,

(A square is 2D you should know.)

Cubes do their very best to please,

By having eight sharp vertices.

(That’s posh for corners I should say,

Or else we could be here all day!)

Twelve edges it does proudly boast –

A cube it really has the most!

 

However, you’ll be overjoyed,

To understand the shape cuboid!

Rectangles shape its 3D form,

Proud six of them it does adorn.

Eight vertices cuboids do sport,

“Impressive” You have surely thought!

Twelve edges they are present too –

Surely that impresses you?

I bet if you are really keen,

A pattern you have no doubt seen,

Cuboids and cubes are much the same,

That’s why they have a similar name!

 

Much different is the pyramid,

About this shape let’s lift the lid,

Of pyramids there are several kinds,

It’s all to do with their… behinds.

From triangles they are mainly made,

And on a base are firmly laid.

A base triangle there can be,

To make a tetrahedron – d’ya see?

Egyptian pyramids have a base that’s square,

That’s why they are still standing there.

But it could be nearly any flat shape,

(I know you may have mouth agape!)

 

An ice-cream cornet is a cone,

A shape that stands up on its own!

A circle it does form its base

(A cone you see is really ace!)

Above the circle there does stand,

Something you should understand,

Made from a circle less a big slice,

(That’s how you make a cone that’s nice.)

 

A shape that’s like a tube with ends,

It is a cylinder my friends!

Two circles this time fit,

At the two far ends of it!

The central tube unfolded can,

Be flattened to a rectangle man!

 

A shape that ends in a triangle,

Is special looked at any angle,

The chocolate makers “Toblerone”,

Have really made this shape their own.

Along the shape you can cut a slice,

Of triangle tasting very nice!

A shape the same along its length,

Has a certain depth and strength,

It has the special name of “prism”,

So watch for them, no do not miss ‘em,

Many shapes mentioned here above,

Are prisms too, so deserve love.*

*Well respect.

Cubes and Cuboids they are prisms too,

Along with cylinders, that’s quite a few.

The triangle gives this prism’s name,

To show it is not quite the same,

Triangular prism it is called,

So now you know, please don’t be fooled

 

A special shape it is the sphere,

So listen, don’t yet disappear!

In games it would be called a “ball”,

But for us that will not do at all.

It’s curved whichever way you go,

Nothing’s straight (just so you know!)

If sliced from one side to the other,

By a chef, (such as my brother),

Circles small then large would be created,

(This shape should not be underrated!)

A warning for those who speak its name,

A lot of words are much the same,

But “Sophia” it does name a girl,

And “spear” is a thing you hurl!

So careful when you call out “Sphere”!

It’s easy to get it wrong my dear.

 

Now here my 3D poem’s done,

I hope enjoyed by everyone!

~~~~~~~~~~

If by some miracle you enjoyed these poems you might enjoy my poem about 2D shapes available on YouTube.

https://www.youtube.com/watch?v=Koo5U4eLDss&t=27s

Unearthing a maths poem By Steve Way

Amongst other things – including writing! – I currently teach Spanish adults English via the internet. When we’ve been talking about their children preparing for university, they have been surprised to learn that the UK children begin to specialise from the age of 16 (indeed in reality often sooner) whereas in Spain all subjects are studied up until they do their entrance exam and then specialise when they get to university.

Given that there isn’t an obvious deficit either way in the end result, with Spanish graduates seemingly doing as well as UK graduates, I wonder whether there would be more options for some UK students to gain an broader rather than a deeper education up until the age of 18? More ‘Jack of All Trades’ than ‘Master of a Few’.

I say this as a bit of a rule breaker myself. Although maths and science subjects seemed the most obvious choice for me at 16, I didn’t want to give up studying English Literature - I dreamed of being a writer after all! – and so ended up doing so alongside maths and two sciences. To their credit the staff at the comprehensive I went to in Swindon tried to partly organise the sixth form timetable around me – as you can imagine science subjects were generally taught at the same time as arts/humanities subjects. They managed it perfectly in the first year but in a fascinating duality – that might have interested both quantum physicists and fantasy writers – I was supposed to be in a physics lesson at the same time as in an English lesson. (What was it that Hermione had in Harry Potter? I could have done with that!)

The rapid switching from studying physics to English was pretty mind-blowing (when this overlap occurred, I was always half way through a physics lesson and figured that zipping over to the English class was the best compromise) and emphasised something I noticed even at that relatively young age. It was the marked degree to which the thinking and outlook of my classmates studying science subjects was rapidly diverging from that of those studying arts subjects. I did often feel that I was existing in parallel worlds – none of my classmates in the English class were the same as those in my science classes.

I experienced a similar pronounced dichotomy in my first year at university. I’d registered to study biology but in our first year we were obliged to study two additional subjects. Whilst most of my fellow biologists studied other science subjects, I think I became the first – and as far as I know un until I left – only student to pick the combination of biology, psychology and theatre studies.

I loved it! Of course, there was a similar challenge with timetabling – after a two-hour theatre studies workshop I had to run down the length of the campus to get to a biology lecture! I was heartbroken that I couldn’t continue with a similar combination in my second and third years. I was forced to specialise.

Not surprisingly in that first year I once again mixed with two completely different groups of students, who thought in completely different ways. I also became aware as I completed my biology degree that if I wanted to continue my studies I would have to become even more of a specialist, which you can probably imagine didn’t appeal, and that there was no room for taking a broad approach and attempting to forge links and connections between the different branches of biology. I’d noticed for example that many of our lecturers – who’s fields differed greatly – often (briefly) noted intriguing effects of particular wavelengths of blue light on the diverse life-forms or functions they studied but there seemed no place for someone who might want to draw these observations together.

The catalyst that let to me wanting to explore my experience of this apparently British focus on specialism was rediscovering some maths-poems I’d written. Just before Covid struck my main computer broke down and I’ve only just been able to get it repaired. When I dug down into my reopened files I discovered ideas, poems and parts of poems that I’d completely forgotten about.

The enforced passage of time in which these pieces were buried enabled me to objectively realise which piece needed to stay buried – but also allowed me to appreciate some others that I’m really pleased with. It’s odd to think that I once had an idea that I now think might have been worth having – and which I might not ever have had again, since I’m so surprised by it now! It’s a bit like reading something you wrote a very long time ago and barely remember and feeling like you didn’t do such a bad a job after all!

I want to share a maths poem that was unearthed along with the others. In this case I knew about the first part of the poem, it’s even been published, but I had totally forgotten that I’d added two extra verses.

Now if you’re reading this on the ABBA blog, I know you’re more like to be in (to have been steered towards at an early age?) the arts/humanities camp but I hope you will enjoy it… could it even be a small rebellious step away from specialism! Even if you don’t it may be useful for any teenagers in your orbit who may be on a trajectory towards maths exams and battling with different graphs and would find an alternative way of considering them useful.

PS knowing that we either have a love or hate relationship with maths*, I’ve come across a few people who’ve found the idea of writing about maths or science creatively surprising. I invite you to consider, given my modest attempt to do so, what you might achieve in bringing these subjects to life in an original way!

* I used to privately tutor children and adults in maths. Without exception when first arriving to teach an adult they would declare, ‘I can’t do maths’. Two of them were accounts for large companies! (‘Well, what I do doesn’t count’) Another lovely lady became my wife – so I can easily recall her first words when we met!

~~~~~

Y equals mx plus C.

Y equals mx plus C,

How great can an equation be?

 

m is the gradient, you see,

I don’t know why, don’t ask me!

There’s no “m” in gradient I know…

But this was agreed years ago!

(You may think I’m old, grumpy and weak

But this maths was made up by a Greek!)

 

A gradient of one-in-four,

Means one up and four ‘long the floor.

The line rises from the left to the right,

Just as in the direction we write.

Minus gradients fall from the left side,

As downwardly negative they slide.

A gradient of minus three,

Is one along, three down – do you see?

 

The “intercept” it’s known as “C”,

Describing where on the y-axis it be.

So as I hope you will agree,

It’s located at zero, then C.*     

 

Y equals mx plus C,

How great can an equation be?

All straight lines fall into its realm,

So now you can plot all of them!

 

* (0,C)

 

Square graphs are different as you’ll see,

By rising exponentially.

If a number’s squared, (as you may know…)

Its value it does quickly grow.

As x goes up a little bit,

Y shoots up like a rocket!

Since negatives all square as plus,

Symmetrical graphs are made for us.

 

Cube graphs go quickly up and down,

As if wanting to both fly, then drown!

Cubes change much faster than do squares,

Like changing to a lift from stairs!

Three minuses in a cube do give,

A result that comes out negative.

 ~~~~~

‘Y equals mx plus C’ in an earlier form was published in Using stories to teach Maths Ages 9 to 11 (High Achievers Supplement) by Hopscotch Educational publishing ISBN 978-1-909860-00-1

If you enjoyed my poem you might enjoy others that I have recorded on YouTube under the title Steve Way Writer such as my poem about flat shapes.

 

The link is; https://www.youtube.com/watch?v=Koo5U4eLDss

Flexible Thinking by Steve Way

It’s an interesting responsibility we share as adults writing and working for children. Naturally, we want them to enjoy and benefit from the work we produce and the lessons we teach them. I don’t always believe that arbiters of the decisions, such as the educational authorities, in deciding what and how material should be presented to children always get this right. The knock-on effect I’ve observed is that this then leads to some publishers, teachers and others being misled but influenced by these edicts in their actions and thinking. Despite good intentions these ideas appear to be as fickle as changing fashions and sometimes as disastrous. Most importantly they seem to me to be of no great benefit to the children we are all trying to support.

Several years ago, an educational publisher was showing interest in some of the pieces I’d written incorporating ideas about mathematics. I’d created a character called The Wise Wizz of Woo, who amongst other achievements (allegedly) determined the shape of the numbers we use. The publishers were vehemently opposed to my Wizz. According to them it was completely unacceptable to use witches or wizards, or references to magic, however oblique (he was only a ‘wizz’ after all) in modern publications as apparently this would be unnatural and unrealistic and confuse the children. The theory seemed to be that the children would have difficultly recognising that magic of an imaginary kind doesn’t exist and that it might confuse their understanding and appreciation of their own or others religious beliefs or sensibilities. Wizards and witches were definitely a no no. For me this was doubly a shame as I’d also been writing a couple of stories about a couple of gormless wizards who were the least able students at their school of wizardry.

Unbeknown to me and the staunch anti-magic brigade at this particular publishing house, elsewhere in the country the first edition of a series of stories about a young orphan with a distinctive scar on his forehead was being discovered by an enthusiastic readership. Curiously, even though a few religious zealots made a fuss about it for a while, most children seemed to cope emotionally and spiritually.

Of course I wouldn’t claim that my scribblings concerning The Wise Wizz of Woo and his associates are in the same league as Joanne Rowling’s stories about Harry P but as the absurdity and irony the situation became clear it did seem irksome that my publisher’s thinking about what was acceptable fare for children’s consumption seemed unnecessarily limited. As well as returning The Wizz to the drawing board!

I was amused once, shortly after the Literacy Hour and Numeracy Hour were strictly imposed as each morning’s fare for children – no cross curricular activity allowed! – when after being invited to do a school visit to present my maths stories that this seemed to pose a worrying dilemma for the class teacher. ‘I don’t know whether to have you in the Literacy Hour or the Numeracy Hour,’ she wailed. Goodness, how confusing for the children! It was certain they wouldn’t be able to cope!

I was less amused on another occasion when I was presenting a story about rounding numbers to a class of eight-year-olds. The idea is that Kings Hundreds, Tens and Units go to war and due to rounding numbers differently the army with the least men is mistakenly thought the largest causing the others to retreat. In order to show how this worked I needed to use the board to calculate how many soldiers each of the kings had in their army. There were only three numbers in each case, for the cavalry, archers and infantry. I began showing the first calculation in a column when the story was interrupted by a scream from the back of the classroom. Seconds later the class teacher had run to the board and practically wrenched the pen out of my hand. ‘We don’t add up that way!’ she cried hysterically as seemingly alarmed as if I’d just inappropriately exposed myself to the class. This was when the fashion was for the children to learn to add up sideways rather than in columns. (And at the time in no other way So Help Us God!)

This dramatic mathematical event seems to me to highlight two different schools of thought when it comes to children. One insisting that children can only be taught in certain ways or exposed to certain ideas and characters, the other camp, including me, that believes that children are infinitely more flexible in their thinking than we usually give them credit for. There are many classic children’s books in which the language is highly complex. Is it likely that the children understand the meaning of every word, reference, idea? No. Has that prevented them from gaining an infinite variety of benefits and joy from the novel? Also no. Will they later gain different insights when they read these books again – perhaps as adults to their children – definitely yes.

Instinctively and intellectually I refute the idea that children should be taught in only one way. I’m glad to hear that these days in many schools children are shown a variety of ways in which to carry out calculations and then allowed to use the method which they are most comfortable with. I’m sure this encourages intellectual flexibility, particularly most recently after teaching a couple of eleven-year-old girls English here where we live in France. The girls couldn’t be more normal for their age but like all students in France they are taught only one way to carry out calculations and that is very definitely it. When I started teaching them language related to maths I was astounded at how poor their mathematical skills were, far behind those of similarly able children their age in the UK.

Another educational fad that fazed me was the ‘only teach from the children’s experience’ idea. I was in an international school in Lyon and during the lunchbreak described the science-based sketch I was aiming to present to the children, covering the concept of friction. The idea is that Professor Crackpot visits the bank to ask Mrs Friction, the manager, for funds to finance his inventions that contradict the law of friction, such as bumpy slides and smooth tyres. The teacher was horrified, certain that the children wouldn’t appreciate the piece since it broke the golden rule of only teaching from the children’s experience.

I was very tempted to suggest that if we were only supposed to expose the children to learning based on their own experience that we should immediately send them all home, since the home environment was presumably what they had experienced most up until now. Since the children were around eight or nine-years-old I strongly suspected that they knew what money was, were aware there were places called banks, which some of them may even have been inside in the company of the adults who sometimes have to visit them in the course of their daily lives. I was even more strongly tempted to warn the teacher away from exposing them to any popular children’s books on the off chance that they didn’t actually teach magic at this particular school, or that the children didn’t have bedroom furniture without a back that led into Narnia or… (obviously here we could all have a field day so I’ll leave it at that.)

I had incidentally already shared that sketch with hundreds of children all over the UK and elsewhere and they all seemed to come out of the experience unscathed… as did the children in Lyon.

~~~~~

You may be interested to know that The Wise Wizz of Woo did eventually see light of day with different publishers. A cartoon poem can be seen at http://www.counton.org/explorer/wise-wizz-of-woo/sound/  while this and further exploits can be read in Using Stories to teach Maths Ages 4 to 7 ISBN 978-1-90986-002-5

However the inept wizard students are still languishing in a bottom drawer…