All entries for November 2005

November 25, 2005

Cranky Maths 3

Writing about web page

Perhaps the crankiest maths of all time is that devised and implemented by The Party of George Orwell's 1984.

This is hinted at in the opening sentence of the book:

It was a bright cold day in April, and the clocks were striking thirteen.

In 1984 the Lottery is used as a tool of oppression. The downtrodden proles are distracted from what is really going on by engaging in their own cranky maths:

'Can't you bleeding well listen to what I say? I tell you no number ending in seven ain't won for over fourteen months!'
'Yes, it 'as, then!'
'No, it 'as not! Back 'ome I got the 'ole lot of 'em for over two years wrote down on a piece of paper. I takes 'em down reg'lar as the clock. An' I tell you, no number ending in seven-'
'Yes, a seven 'as won! I could pretty near tell you the bleeding number. Four oh seven, it ended in. It were in February — second week in February.'
'February your grandmother! I got it all down in black and white. An' I tell you, no number-'
'Oh, pack it in!' said the third man.
They were talking about the Lottery. Winston looked back when he had gone thirty metres. They were still arguing, with vivid, passionate faces. The Lottery, with its weekly pay-out of enormous prizes, was the one public event to which the proles paid serious attention. It was probable that there were some millions of proles for whom the Lottery was the principal if not the only reason for remaining alive. It was their delight, their folly, their anodyne, their intellectual stimulant. Where the Lottery was concerned, even people who could barely read and write seemed capable of intricate calculations and staggering feats of memory. There was a whole tribe of men who made a living simply by selling systems, forecasts, and lucky amulets.

There's no such thing as bad news in the world of The Party. Even a reduction of weekly rations can be spun into a good thing, with a bit of cranky maths. Here, a change from 30 grammes to 20 grammes of chocolate is seen as an increase:

Bad news coming, thought Winston. And sure enough, following on a gory description of the annihilation of a Eurasian army, with stupendous figures of killed and prisoners, came the announcement that, as from next week, the chocolate ration would be reduced from thirty grammes to twenty.

It appeared that there had even been demonstrations to thank Big Brother for raising the chocolate ration to twenty grammes a week. And only yesterday, he reflected, it had been announced that the ration was to be reduced to twenty grammes a week. Was it possible that they could swallow that, after only twenty-four hours? Yes, they swallowed it.

The following bit of The Party's doctrine would make any of the real world maths cranks I have previously discussed in this blog very proud:

In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four?

The hero of the book, Winston Smith, is unconvinced by this philosophy, and rebels against it:

With the feeling that he was setting forth an important axiom, he wrote:
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.

Winston finds vindication when he gets hold of the banned book by the leader of the underground resistance:

Physical facts could not be ignored. In philosophy, or religion, or ethics, or politics, two and two might make five, but when one was designing a gun or an aeroplane they had to make four.

But, when The Party gets hold of Winston, the instructional approach to enlighten him that he was wrong all along is certainly as effective as any teaching experiment I have read about:

'Do you remember,' he went on, ' writing in your diary, "Freedom is the freedom to say that two plus two make four"?'
'Yes,' said Winston.
O'Brien held up his left hand, its back towards Winston, with the thumb hidden and the four fingers extended.
'How many fingers am I holding up, Winston?
'And if the party says that it is not four but five — then how many?'
The word ended in a gasp of pain. The needle of the dial had shot up to fifty-five. The sweat had sprung out all over Winston's body. The air tore into his lungs and issued again in deep groans which even by clenching his teeth he could not stop. O'Brien watched him, the four fingers still extended. He drew back the lever. This time the pain was only slightly eased.
'How many fingers, Winston?'
The needle went up to sixty.
'How many fingers, Winston?'
'Four! Four! What else can I say? Four!'
The needle must have risen again, but he did not look at it. The heavy, stern face and the four fingers filled his vision. The fingers stood up before his eyes like pillars, enormous, blurry, and seeming to vibrate, but unmistakably four.
'How many fingers, Winston?'
'Four! Stop it, stop it! How can you go on? Four! Four!'
'How many fingers, Winston?'
'Five! Five! Five!'
'No, Winston, that is no use. You are lying. You still think there are four. How many fingers, please?'
'Four! five! Four! Anything you like. Only stop it, stop the pain!
Abruptly he was sitting up with O'Brien's arm round his shoulders. He had perhaps lost consciousness for a few seconds. The bonds that had held his body down were loosened. He felt very cold, he was shaking uncontrollably, his teeth were chattering, the tears were rolling down his cheeks. For a moment he clung to O'Brien like a baby, curiously comforted by the heavy arm round his shoulders. He had the feeling that O'Brien was his protector, that the pain was something that came from outside, from some other source, and that it was O'Brien who would save him from it.
'You are a slow learner, Winston,' said O'Brien gently.
'How can I help it?' he blubbered. 'How can I help seeing what is in front of my eyes? Two and two are four.
Sometimes, Winston. Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once. You must try harder. It is not easy to become sane.'
He laid Winston down on the bed. The grip of his limbs tightened again, but the pain had ebbed away and the trembling had stopped, leaving him merely weak and cold. O'Brien motioned with his head to the man in the white coat, who had stood immobile throughout the proceedings. The man in the white coat bent down and looked closely into Winston's eyes, felt his pulse, laid an ear against his chest, tapped here and there, then he nodded to O'Brien.
'Again,' said O'Brien.
The pain flowed into Winston's body. The needle must be at seventy, seventy-five. He had shut his eyes this time. He knew that the fingers were still there, and still four. All that mattered was somehow to stay alive until the spasm was over. He had ceased to notice whether he was crying out or not. The pain lessened again. He opened his eyes. O'Brien had drawn back the lever.
'How many fingers, Winston?'
'Four. I suppose there are four. I would see five if I could. I am trying to see five.'
'Which do you wish: to persuade me that you see five, or really to see them?'
'Really to see them.'
'Again,' said O'Brien.
Perhaps the needle was eighty — ninety. Winston could not intermittently remember why the pain was happening. Behind his screwed-up eyelids a forest of fingers seemed to be moving in a sort of dance, weaving in and out, disappearing behind one another and reappearing again. He was trying to count them, he could not remember why. He knew only that it was impossible to count them, and that this was somehow due to the mysterious identity between five and four. The pain died down again. When he opened his eyes it was to find that he was still seeing the same thing. Innumerable fingers, like moving trees, were still streaming past in either direction, crossing and recrossing. He shut his eyes again.
'How many fingers am I holding up, Winston?'
'I don't know. I don't know. You will kill me if you do that again. Four, five, six — in all honesty I don't know.'
'Better,' said O'Brien.

As a researcher in the field of mathematical educational technology myself, I'd never realised the rack could be such an efficient pedagogic tool!

November 16, 2005

Cranky Maths 2

Writing about web page

Whilst searching for some resources on the Imagine Logo programming language I unexpectedly came across some wonderfully cranky maths from an obscure website which also happens to be called Imagine.

I quote:

" What does Maths tell us, then? Here are a couple of thoughts. First of all it tells us something about truth. 1 + 1 = 2. The equal sign teaches us about absolute truth. One plus one does not equal 2.130. Imagine somebody comes up to you and says, “One plus one equals 2.130.” You say, “Well that’s fine for you if you want to believe that. That’s just great. If it makes you happy…” No! You can’t run a railroad that way, can you? Maths tells us about a God of order, a God of reason, a God of rationality.

Perhaps Maths also gives us insights into concepts like eternity, or even the Trinity. What is 1 + 1 + 1? The answer is 3. That’s a tougher one. And you say, “You can’t do the Trinity that way, can you?” You cannot do it, which is why the Jews had such a problem. You cannot explain the Trinity using addition, but what is 1×1 x 1? One – which shows how three entities can be equal in value and relate to one another, and still be One. Which solves the problem of the Trinity – which may be of some comfort to the faculty here at Union."

Source (pdf)

My own PhD is about the equals sign and so the bolded statement above, "The equal sign teaches us about absolute truth", is certainly enlightening to me.

November 04, 2005

Arithmetical Poetry

One of the first teacher guides for teaching arithmetic was The Tutor's Guide by Charles Vyse, circa 1775. Although I have not (yet) seen a copy myself I have found the following extracts from it. Apparently Vyse felt that an effective way to teach arithmetic is to present worded problems via the medium of smutty and tongue-in-cheek verse.



When first the marriage-knot was tied
Between my wife and me,
My age to her’s we found agreed
As three times three to three;
But when ten years, and half ten years,
We man and wife had been,
Her age came up as near to mine
As eight is to sixteen.
Now, tell me, I pray,
What were our ages on the wedding day?
[Not original.]


Once as I walked upon the banks of the Rye
I, in the Meads, three beauteous nymphs did spy,
Saying “Well met, we’ve business to impart
Which we cannot decide without your Art:
Our Grannum’s dead, and left a Legacy,
Which is to be divided amongst three:
In Pounds it is two hundred twenty-nine,
Also a good mark, being sterling coin.”
Then spake the eldest of the lovely three,
“I’ll tell you how it must divided be;
Likewise our names I unto you will tell,
Mine is Moll, the others Anne and Nell.
As oft as I five and five-ninths do take,
Anne takes four and three-sevenths her Part to make:
As oft Anne four and one-ninth does tell,
Three and two-three must be took up by Nell.


A castle wall there was, whose height was found
To be an hundred feet from th’ Top to th’ ground:
Against the wall a ladder stood upright,
Of the same length the castle was in height.
A waggish youth did the ladder slide;
(The bottom of it) ten feet from the side;
Now I would know how far the top did fall,
By pulling out the ladder from the wall.

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