Thinking in Numbers: How Maths Illuminates Our Lives (5 page)

Counting people directly is believed by the
Kpelle
to bring bad luck. In Africa, this taboo is both ancient and widespread. There exists as well the sentiment, shared by the authors of the Old Testament, that the counting of human beings is an exercise in poor taste. The simplicity of their numbers is not linguistic, or merely linguistic, but also ethical.

I read with pleasure a book of essays published several years ago by the Nigerian novelist Chinua Achebe. In one, Achebe complained of the Westerners who asked him, ‘How many children do you have?’ Rebuking silence, he suggested, best answered such an impertinent question.

‘But things are changing and changing fast with us . . . and so I have learned to answer questions that my father would not have touched with a bargepole.’

Achebe’s children number
ano
(four). In Iceland, they would say,
fjögur
.

Proverbs and Times Tables

I once had the pleasure of discovering a book wholly dedicated to the art of proverbs. It was in one of the municipal libraries that I frequented as a teenager. The title of the book escapes me now, the name of its author, too, but I still recall the little shiver of excitement I felt as my fingers caressed its quarto pages.

‘Penny wise, pound foolish’

‘Small fishes are better than empty dishes’

‘A speech without proverb is like a stew without salt’

Now that I come to think of it, I doubt this book had any single author. Every proverb is anonymous. Each appears in a society’s mental repertoire by some process akin to immaculate composition. Like the verses in the Muslims’ Koran, proverbs seem pre-written, patiently awaiting a mouth to utter their existence. Some linguists contend that language happens independently of our reason, its origins traceable to a still mysterious and exclusive gene. Perhaps proverbial logic is like language in this respect, its existence as essential to our humanity as the power of speech.

Whoever the author or the editor, this book proved to me that there are only so many proverbs a healthy man can take. A point of surfeit is reached, beyond which the reader can no longer follow: his brain starts to ache, his eyes to water. Taken in excess, proverbs lose all the familiar felicities of their compact structure. They start to read merely as repetitions – which many by then quite probably are. From my experience, I estimate this limit at about one hundred.

One hundred proverbs, give or take, sum up the essence of a culture; one hundred multiplication facts compose the ten times table. Like proverbs, these numerical truths or statements – two times two is four, or seven times six equals forty-two – are always short, fixed and pithy. Why then do they not stick in our heads as proverbs do?

But they did before, some people claim. When? In the good old days, of course. Today’s children, they suggest, are simply too slack-brained to learn correctly. Nothing interests them but sending one another text messages and harassing the teacher. The critics hark back to those days before computers and calculators; to the time when every number was drummed into children’s heads till finding the right answer became second nature.

Except that, no such time has ever really existed. Times tables have always given many schoolchildren trouble, as Charles Dickens knew in the mid-nineteenth century.

Miss Sturch put her head out of the school-room window: and seeing the two gentlemen approaching, beamed on them with her invariable smile. Then, addressing the vicar, said in her softest tones, ‘I regret extremely to trouble you, sir, but I find Robert very intractable, this morning, with his multiplication table.’ ‘Where does he stick now?’ asked Doctor Chennery. ‘At seven times eight, sir,’ replied Miss Sturch. ‘Bob!’ shouted the vicar through the window. ‘Seven times eight?’ ‘Forty-three,’ answered the whimpering voice of the invisible Bob. ‘You shall have one more chance before I get my cane,’ said Doctor Chennery. ‘Now then, look out. Seven times . . .

Only his younger sister’s rapid intervention with the answer – fifty-six – spares the boy the physical pain of another wrong guess.

Centuries old, then, the difficulty that many children face acquiring their multiplication facts is also serious. It is, to borrow a favourite term of politicians, a ‘real problem’. ‘Lack of fluency with multiplication tables,’ reports the UK schools inspectorate, ‘is a significant impediment to fluency with multiplication and division. Many low-attaining secondary school pupils struggle with instant recall of tables. Teachers [consider] fluent recall of multiplication tables as an essential prerequisite to success in multiplication.’

The facts in a multiplication table represent the essence of our knowledge of numbers: the molecules of maths. They tell us how many days make up a fortnight (7 × 2), the number of squares on a chessboard (8 × 8), the quantity of individual surfaces on a trio of boxes (3 × 6). They help us evenly divide fifty-six items between eight people (7 × 8 = 56, therefore 56/8 = 7), or realise that forty-three of something cannot be evenly distributed in the same way (because forty-three, being a prime number, makes no appearance among the facts). They reduce the risk of anxiety in the young learner, and give a vital boost to the child’s confidence.

Patterns are the matter that these molecules, in combination, make. Take, for instance, the consecutive facts 9 × 5 = 45, and 9 × 6 = 54: the digits in both answers are the same, only reversed. Thinking about the other facts in the nine times table, we see that every answer’s digits sum to nine:

9 × 2 = 18 (1 + 8 = 9)

9 × 3 = 27 (2 + 7 = 9)

9 × 4 = 36 (3 + 6 = 9)

Etc . . .

Or, surveying the other tables, we discover that multiplying an even number by five will always produce an answer ending in zero (2 × 5 = 10 . . . 6 × 5 = 30), while multiplying an odd number by five gives answers that always end in itself (3 × 5 = 15 . . . 9 × 5 = 45). Or, we spot that six squared (thirty-six) plus eight squared (sixty-four) equals ten squared (one hundred).

Sevens, the trickiest times table to learn, also offer a beautiful pattern. Picture the seven on a telephone’s keypad, in the bottom left-hand corner. Now simply raise your eye to the key immediately above it (four), and then again to the next key above (one). Do the same starting from the bottom middle key (eight), and so on. Every keypad digit in turn corresponds to the final digit in the answers along the seven times table: 7, 14, 21, 28  . . .

Not all multiplication facts pose problems, of course. Multiplying any number by one or ten is obviously easy enough. Our hands know that two times five, and five times two, both equal ten. Equivalencies abound: two times six, and three times four, both lead to twelve; multiplying three by ten, and six by five, amounts to the same thing.

But others are trickier, less intuitive, and far easier to let slip. A numerate culture will find whatever means at its disposal to pass these obstinate facts down from one generation to the next. It will carve them into rock and scratch them onto parchment. It will condemn every inauspicious student to threats and thrashings. It will select the most succinct form and phrasing for its essential truths: not too heavy for the tongue, nor too lengthy for the ear.

Just like a proverb.

For example, what did our ancestors mean precisely when bequeathing us a truth like ‘An apple a day keeps the doctor away’? Not, of course, that we should read it literally, superstitiously, imagining apples like the cloves of garlic that are supposed to make vampires take to their heels. Rather, the sentence expresses a core relationship between two different things: healthy food (for which the apple plays stand-in), and illness (embodied by the doctor). Consider a few of the alternate ways in which this relationship might also have been summed up:

‘A daily fruit serving is good for you’

‘Eating healthy food prevents illness’

‘To avoid getting sick, eat a balanced diet’

These versions are as short, or even shorter, than our proverb. But none is anywhere near as memorable.

Long before Dickens wrote about the horrors of multiplication tables, our ancestors had decided to sum up fifty-six as ‘seven times eight’, just as they described health (and its absence) in apples and doctors. But as with a concept like ‘health’, understanding the number fifty-six can be achieved via many other routes.

56 = 28 × 2

56 = 14 × 4

56 = 7 × 8

Or even:

56 = 3.5 × 16

56 = 1.75 × 32

56 = 0.875 × 64

It is not difficult to see, though, why tradition would have privileged the succinctness and simplicity of ‘seven times eight’ for most purposes, over rival definitions such as ‘one and three-quarters times thirty-two’ or ‘seven-eighths of sixty-four’ (as useful as they might be in certain contexts).

What is seven times eight? It is the clearest and simplest way to talk about the number fifty-six.

These familiar forms may be simple and succinct, but they are finely wrought, nonetheless, whether in words or figures. The proverbial apple, for example, begins the proverb, though its meaning (as a protector of health) cannot be grasped until the end. ‘Apple’ here is the answer to the question: What keeps the doctor away? Other proverbs also share this structure, where the answer precedes the question. ‘A stitch in time saves nine’ (What saves nine stitches? A stitch in time) or ‘Blind is the bookless man’ (What is the bookless man? Blind).

Placing the answer at the start compels our imagination: we concede more freely the premise that an apple can deter illness, in part because the word ‘apple’ precedes all the others. Using this structure can also arouse our attention, inciting us to picture the rest of the proverb with the opening image in mind: to see the bookless man, for example, more clearly in light of his blind eyes.

When I discussed the ways in which we could think about the number fifty-six I borrowed this feature of proverbs and put the sum’s answer at the start. Saying, ‘Fifty-six equals seven times eight’ lends emphasis where it is needed most: not on the seven or the eight, but on what they produce.

Form is important. A pupil reads 56 = 7 × 8 and hears the whisper of many generations, whilst another child, shown 7 × 8 = 56, finds himself alone. The first child is enriched; the second is disinherited.

Today’s debates over times tables too often neglect questions of form. Not so the schools of nineteenth-century America. The young nation, still younger than its oldest citizens, hosted educational discussions unprecedented in their inquisitive detail. Teachers pondered in marvellous depth the kind of verb to use when multiplying. In
The Grammar of English Grammars
(published in 1858) we read, ‘In multiplying one only, it is evidently best to use a singular verb: “Three times one is three”. And in multiplying any number above one, I judge a plural verb to be necessary: “Three times two are six”.’

The more radical contributors to these debates suggested doing away with excess words like ‘times’ altogether. Instead of learning ‘four times six is twenty-four’, the child would repeat, ‘four sixes are twenty-four.’ These educators urged a return to the way ancient Greek children had chanted their times tables two millennia before: ‘once one is one,’ ‘twice one is two’, et cetera. Others went even further, suggesting that the verb ‘is’ (or ‘are’) also be thrown out: ‘four sixes, twenty-four’, in the manner of the Japanese.

Schools in Japan have long lavished attention on the sounds and rhythms of the times tables. Every syllable counts. Take, say, the multiplication 1 × 6 = 6, among the first facts that any child learns. The standard Japanese word for one is
ichi
; the usual Japanese word for six is
roku
. Put together, they make:
ichi roku roku
(one six, six). But Japanese pupils never say this: the line is clumsy, the sounds cacophonous. Instead, the pupils all say,
in roku ga roku
(one six, six), using an abraded form of
ichi
(
in
) and an insertion (
ga
) for euphony.

The trimming of unnecessary words or sounds shapes both the proverb and the times table. ‘Better late than never,’ says the parent in New York when his son complains of pocket money long overdue. ‘Four fives, twenty’ says the boy when he finally counts his quarters.

In Japanese, the multiplication 6 × 9 = 54 is an extreme example of ellipsis. Being similar in sound, the two words –
roku
(six) and
ku
(nine) – merge to a single
rokku
. This new number would be a little like pronouncing the multiplication 7 × 9 in English as ‘sevine’.

Why is
in roku ga roku
judged more pleasing than
ichi roku roku
? Both phrases contain six syllables; both phrases use the ‘roku’ word twice, yet the first sounds beautiful, while the second seems ugly. The answer is, parallelism.
In roku ga roku
has a parallel structure, which makes it easier on the ear. We hear this balanced structure frequently in our proverbs: ‘fight fire with fire’. One six is six.

It is much harder to make good parallel times tables in English than it is in Japanese. The same is true of many European languages. In Japanese, a child says
roku ni juuni
(six two, ten two) for 6 × 2 = 12, and
san go juugo
(three five, ten five) for 3 × 5 = 15, whereas an English child must say ‘twelve’ and ‘fifteen’, a French child
douze
and
quinze
, and a German child
zwölf
and
fünfzehn
.