Thinking in Numbers: How Maths Illuminates Our Lives (6 page)

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 prejudice. 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.

From Mr Baxter I learnt at least one profitable lesson; I learnt how not to teach. This lesson would serve me well on numerous occasions. Two years after I left school, when I was unpeeling the newspaper one morning, I came across an agency’s advert, recruiting for tutors. I had taught English in Lithuania during my gap year and discovered that I liked teaching. So I applied. The interview placed me opposite a lady up in years named Grace, in the office that she kept in her living room. Needlepoint cushions filled the small of my back as I sat before her desk. The wallpaper, if I remember correctly, had a pattern of small birds and honeybees.

The meeting was brief.

‘Do you enjoy helping others to learn something new?’

‘Are you mindful of a student’s personal learning style?’

‘Could you work according to a set curriculum?’

Her questions contained their own answers, like the dialogue taught in a foreign language class: ‘Yes, I could work according to a set curriculum’.

After ten minutes of this she said, ‘Excellent, well you would certainly fit in with us here. For English we already have tutors and there is not much demand for foreign languages. What about primary-school level maths?’

What about it? I was a taker.

Grace’s bookings certainly kept me on my toes. My tutoring patch extended to the neighbouring town five miles away, and the bus ride and walk to the farthest homes took as long as the lessons. Nervous, I learned on the job, but the families helped me. I found the children, between the ages of seven and eleven, generally polite and industrious; their parents’ nods and smiles tamed my nerves. After a while I ceased to worry and even began to look forward to my weekly visits.

Should I admit I had a favourite student? He was a brown-haired, freckled boy, eight years old but small for his age, and the first time I came to the house he fairly shivered with shyness. We started out with the textbooks that the agency loaned me, but they were old and smelly and the leprous covers soon came apart in our hands. A brightly coloured book replaced them, one of the boy’s Christmas gifts, but its jargon was poison to his mind. So we abandoned the books and found some better way to pass the hour together. We talked a lot.

It turned out that he had a fondness for collecting football stickers and could recite the names of the players depicted on them by heart. With pride he showed me the accompanying album.

‘Can you tell me how many stickers you have in there?’ I asked him. He admitted to having never totalled them up. The album contained many pages.

‘If we count each sticker one by one it will take quite a long time to reach the last page,’ I said. ‘What if we were to count the stickers two by two instead?’ The boy agreed that would be quicker. Twice as quick, I pointed out, ‘And what if we were to count the stickers in threes? Would we get to the end of the album even faster?’ He nodded. Yes, we would: three times as fast.

The boy interrupted. ‘If we counted the stickers five at a time, we would finish five times faster.’ He smiled at my smile. Then we opened the album and counted the stickers on the first page, I placing my larger palm over every five. There were three palms of stickers: fifteen. The second page had slightly fewer stickers (two palms and three fingers, thirteen) – so we carried the difference over to the next. By the seventh page we had reached twenty palms: one hundred stickers. We continued turning the pages, putting my palm on to each, and counting along. In all, the number of stickers rose to over eighty palms (four hundred).

After making light work of the album, we considered the case of a giant counting its pages. The giant’s palm, we agreed, would easily count the stickers by the score.

What if the same giant wanted to count up to a million? The boy thought for a moment. ‘Perhaps he counts in hundreds: one hundred, two hundred, three hundred . . .’ Did the boy know how many hundreds it would take to reach a million? He shook his head. Ten thousand, I told him. His eyebrows leapt. Finally, he said, ‘He would count in ten thousands then, wouldn’t he?’ I confirmed that he would: it would be like us counting from one to hundred. ‘And if he were really big, he might count in hundred thousands,’ I continued. The giant would reach a million as quickly as we counted from one to ten.

Once, during a lesson solving additions, the boy hit upon a small but clever insight. He was copying down his homework for us to go through together. The sum was 12 + 9, but it came out as 19 + 2. The answer, he realised, did not change. Twelve plus nine, and nineteen plus two, both equal twenty-one. The fortuitous error pleased him; it made him pause and think. I paused as well, not wishing to talk for fear of treading on his thoughts. Later I asked him for the answer to a much larger sum, something like 83 + 8. He closed his eyes and said, ‘Eighty-nine, ninety, ninety-one.’ I knew then that he had understood.

Of the other students, I recall the Singh family whom I taught on Wednesday evenings for two hours back-to-back. I remember I could never get on with the father, a businessman who put on executive airs, although the mother treated me with nothing but kindness. They had three children, two boys and a girl; the children were always waiting for me around the table in the living room, still dressed up in their school uniform. The eldest boy was eleven: a bit of a show-off, he had the confidence of an eldest son. His sister regularly deferred to him. In the middle, the second son made a lot of laughter. He seemed to laugh for all the family.

In the beginning, the trio took the wan bespectacled man before them only half seriously as a teacher. Between them, they had ten years on me. I looked too young and probably sounded it as well, without the smooth patter that comes with experience. All the same, I held my ground. I helped them with their times tables, in which they were far from fluent. They showed surprise when I failed to berate their errors and hesitations. On the contrary, if they were close to the right answer I told them so.

‘What is eight times seven?’

‘Fifty . . .’ The eldest boy wavered.

‘Yes,’ I said, encouraging.

‘Fifty-
four
,’ he ventured.

‘Nearly,’ I said. ‘Fifty-six.’

This hesitation, a habit of many of my students, intrigued me. It suggested not ignorance, but rather indecision. To say that a student has no idea of a solution, I realised, is untrue. Truth is, the learner
does
have ideas, too many, in fact – almost all of them bad. Without the knowledge necessary to eliminate this mental haze, the learner finds himself confronted with an embarrassment of wrong answers to helplessly pick from.

What had the boy been thinking, I enquired, when he selected fifty-four as his answer? He admitted having previously considered fifty-three, fifty-six, fifty-seven, and fifty-five (in that order). He had felt sure that fifty-one or fifty-two would be too small an answer, both fifty-eight and fifty-nine, too high. Then I asked him why he had finally preferred fifty-four to fifty-three. He replied that he had thought of the eight in the question, and the fact that fifty is half of one hundred, and that half of eight is four.