Thinking in Numbers: How Maths Illuminates Our Lives Read online

  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 i
ts 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.

  Beside the ones, it is only the ten times table that produces consistent parallel forms in English, of the kind ‘easy come, easy go’: ‘seven tens (are) seventy’.

  Not all proverbs employ parallels. Many use alliteration – the repetition of certain sounds: ‘One swallow does not a summer make’ or ‘All that glitters is not gold’. English times tables alliterate too: ‘four fives (are) twenty’ and (if we extend our times tables to twelve) ‘six twelves (are) seventy-two’.

  Parallels and alliteration are both in evidence when proverbs rhyme: ‘A friend in need is a friend indeed’ or ‘Some are wise and some are otherwise’. By definition, square multiplications (when the number is multiplied by itself) begin in a similar way: ‘two times two . . .’ ‘four fours . . .’ ‘nine times nine . . .’ though only the squares of five and six finish with a flourish: ‘five times five (is) twenty-five’ and ‘six times six (is) thirty-six’.

  For this reason, learners procure these two multiplication facts (after, perhaps ‘two times two is four’) with the greatest ease and pleasure. This pair – ‘five times five is twenty-five’ and ‘six times six is thirty-six’ truly attain the special quality of the proverb. Other multiplication facts, from the same times tables, get close. For example, multiplying five by any odd number inevitably leads to a rhyme: ‘seven fives (are) thirty-five’. Six, when followed by an even number, causes the even number to rhyme: ‘six times four (is) twenty-four’ and ‘six eights (are) forty-eight’.

  Mistakes? Of course they happen. Nobody is above them. No matter how long a person spends immersed in numbers, recollection can sometimes go astray. I have read of world-class mathematicians who blush at ‘nine times seven’.

  The same troubles we encounter with times tables sometimes occur with words, what we call ‘slips of the tongue’, though more often than not the tongue is innocent. It is the memory that is to blame. Someone who says, ‘he is like a bear with a sore thumb’ (mixing up ‘like a bear with a sore head’ and ‘to stick out like a sore thumb’) makes a mistake similar to he who answers ‘seven times eight’ with ‘forty-eight’ (confusing 7 × 8 = 56 with 6 × 8 = 48).

  Such mistakes are mistakes of unfamiliarity. Proverbs, like times tables, can often strike us as strange, their meanings remote. Why do we talk of bears with sore heads? In what way do swallows conjure up summertime better than other birds? The choice of words seems to us as arbitrary and archaic as the numbers in the times tables. But the truths they represent are immemorial.

  ‘Hold fast to the words of ancestors,’ instructs a proverb from India. Hold fast to their times tables, too.

  Classroom Intuitions

  Television journalists, in their weaker moments, will occasionally pull the following stunt on a hapless minister of education. Mugging through his make-up at the attendant cameras, the interviewer strokes his notes, clears his throat and says, ‘One final question, Minister. What is eight times seven?’

  Such episodes never fail to make me sigh. It is a sad thing when mathematics is reduced to the recollection (or, more often, the non-recollection) of a classroom rule.

  In one particular confrontation of this type, the presenter demanded to know the price of fourteen pens when four had a price tag of 2.42 euros. ‘I haven’t the foggiest,’ the minister whimpered, to the audience members’ howls of delight.

  Of course, the questions are patently asked with the expectation of failure in mind. Politicians are always trying to anticipate our expectations and to meet them. Should we then feel such surprise when they judge the situation correctly, and get the sum wrong?

  Properly understood, the study of mathematics has no end: the things we each do not know about it are infinite. We are all of us at sea with some aspect or another. Personally, I must admit to having no affinity with algebra. This discovery I owe to my secondary-school maths teacher, Mr Baxter.

  Twice a week I would sit in Mr Baxter’s class and do my best to keep my head down. I was thirteen, going on fourteen. With his predecessors I had excelled at the subject: number theory, statistics, probability, none of them had given me any trouble. Now I found myself an algebraic zero.

  Things were changing; I was changing. All swelling limbs and sweating brain, suddenly I had more body than I knew what to do with. Arms and legs became the prey of low desktops and narrow corridors, were ambushed by sharp corners. Mr Baxter ignored my plight. Bodies were inimical to mathematics, or so we were led to believe. Bad hair, acrid breath, lumpy skin, all vanished for an hour every Tuesday and Thursday. Young minds in the buff soared into the sphere of pure reason. Pages turned to parallelograms; cities, circumferences; recipes, ratios. Shorn of our bearings, we groped our way around in this rarefied air.

  It was in this atmosphere that I learnt the rudiments of algebra. The word, we were told, was of Arabic origin, culled from the title of a ninth-century treatise by Al-Khwarizmi (‘algorithm’, incidentally, is a Latin corruption of his name). This exotic provenance, I remember, left a deep impression on me. The snaking swirling equations in my textbook made me think of calligraphy. But I did not find them beautiful.

  My textbook pages looked cluttered with lexicographical debris: all those Xs and Ys and Zs. The use of the least familiar letters only served to confirm my preju
dice. I thought these letters ugly, interrupting perfectly good sums.

  Take: x² + 10x = 39, for example. Such concoctions made me wince. I much preferred to word it out: a square number (1, or 4, or 9, etc.) plus a multiple of ten (10, 20, 30, etc.) equals thirty-nine; 9 (3 × 3) + 30 (3 × 10) = 39; three is the common factor; x = 3. Years later I learnt that Al-Khwarizmi had written out all his problems, too.

  Stout and always short of breath, Mr Baxter had us stick to the exercises in the book. He had no patience whatsoever for paraphrasing. Raised hands were cropped with a frown and the admonition to ‘reread the section’. He was a stickler for the textbook’s methods. When I showed him my work he complained that I had not used them. I had not subtracted the same values from either side of the equation. I had not done a thing about the brackets. His red pen flared over the carefully written words of my solutions.

  Let me give a further example of my deviant reasoning: x² = 2x + 15. I word it out like this: a square number (1, 4, 9, etc.) equals fifteen more than a multiple of two (2, 4, 6, etc.). In other words, we are looking for a square number above seventeen (being fifteen more than two). The first candidate is twenty-five (5 × 5) and twenty-five is indeed fifteen more than 10 (a multiple of two); x = 5.

  A few of Mr Baxter’s students acquired his methods; most, like me, never did. Of course I cannot speak for the others, but for my part I found the experience bruising. I was glad when the year was over and I could move on to other maths. But I also felt a certain shame at my failure to comprehend. His classes left me with a permanent suspicion of all equations. Algebra and I have never been fully reconciled.