1MORE CABBAGE, ANYONE?
ARE we all guilty of having a certain streak of sadism? When I taught students who were being trained as primary-school teachers, I was often tempted to play an underhand trick on the new intake of freshers. At their first lecture I explained that it was important for me to know something about their current skills in mathematics. In order to do that, I intended to set a short maths test: âMake sure that youâve got a clean sheet of paper. Iâll ask the questions slowly. Everybody ready? No cheating! First question âŠâ A deathly hush, everyone listening intently. And then, âWrite one word, just one, that expresses how youâre feeling: whatâs going through your mind at this precise moment.â A sigh of relief as everyone gradually realised that my threat to set a test was just a cruel joke, and everyone was more than ready to summarise their feelings in one word.
Itâs no surprise that the overwhelming number of responses were negative. It was rare for anyone to write words such as âfantasticâ or âconfidentâ. On the contrary, words such as ânervousâ, âanxiousâ and âuncertainâ were used far more often. But the most common word by far â year after year â was âpanicâ or âPANIC!â I would have had a negative reaction had I threatened to set a test in any subject â science, language, history, geography, music and so on â but the reaction is more extreme, and the cries of anguish more heartfelt, in mathematics than in any other subject.
Why does mathematics provoke such an extreme reaction? Is it because the answers to questions in mathematics tend to be either right or wrong? Two threes (2 Ă 3) are 6, no arguing, no room for debate, the answer canât be 5 or 7. We are conditioned from a young age to think of mathematics as a subject in which there can be no discussion, no possibility of an alternative opinion. The subject has its own inbuilt authority that transfers to teachers who mark answers with a tick or a cross, or to parents who pass on their own uncertainty and worry to the next generation: âYou still canât remember seven eights (7 Ă 8)?â
This negativity isnât in our genes; weâre not born hating mathematics. On the contrary, young children commonly express how much they like doing mathematics. I had been invited to give a talk to a literary society in south Wales and was staying overnight with my son and his family. Before leaving the house, Mari, my five-year-old granddaughter, asked me where I was going and why. âWellâ, I answered, âIâm going to the village hall to give a talk on maths.â âOhâ, said Mari, âI like doing mathsâ, and then, after a pause, âAnd will you be dancing there?â Quite how she made an association between mathematics and dancing remains a mystery but Mari clearly regarded both activities as belonging to the set of things that she liked doing. The trick is to maintain that natural liking and to foster it with care.
For many people their first encounter with algebra was that point in their education when mathematics literally became too abstract and they were not assisted to come to terms with the change of gear. In the words of a tale that has done its rounds on social media: âDear algebra, please stop asking us to find your x. Sheâs never coming back, and donât ask y.â For many, the x remained a perpetual mystery that was never demystified for us during the early years of our secondary education. This only served to reinforce the belief built up during our time at primary school that mathematics wasnât something to be understood; rather it was something to be done â and to be suffered.
The answer often given by children to the gentle parental probe, âWhat did you do in school today?â is âOh, nothing!â In an effort to snooker her four-year-old daughter at the end of a morning at nursery school, one resourceful parent asked her, âNow, tell me three things that you did at school this morning.â Not to be outmanoeuvred, the child replied, âNothing, nothing, nothing!â However, responses can occasionally be more forthcoming and revealing. This was what our then five-year-old daughter, Llinos, wrote down as her response to this evergreen question. Try to work out what sheâs done before reading further:
Llinos was having trouble writing some of her numerals correctly, reversing the 4 and 5 in this case â a common practice at this age. Some gentle probing on our part revealed that todayâs lesson had been âdoing add-upsâ and the teacher had introduced âcarry oneâ as a new idea. Llinos had learned to write âdâ (degau â the Welsh for tens) and âuâ (unedau â the Welsh for units) at the top of the sum and had picked up that it was important to begin by adding the numbers under the âuâ before moving on to add the numbers under the âdâ. Because the lessonâs aim had been to introduce âcarry oneâ, Llinos had also used that in her example, although it didnât apply in this case. She finished off her work with a flourish by adding the tick, and, smiling broadly, turned to her parents for their approval. She had no understanding at all of what sheâd been doing. She hadnât interpreted the number in the first line as being thirty-three (33) nor the second number as forty-five (45) â quite a challenge for a five-year-old â and didnât understand that her answer was eighty-eight (88). She was happy and content that she had accepted the authority of the teacher, her own tick providing the crowning seal of approval. âThatâs very good!â was our only possible response.
âItâs all very well for youâ is what Iâm often told. âYou didnât have any difficulty with maths at school. It wasnât like that for me. I never understood what was going on.â Such sentiments are heartfelt and contain more than a grain of truth. Not that Iâve never had difficulty understanding mathematics but, somehow, I seem to have had sufficient confidence to plough on. When I must have been about six years old, I sat at a table in our back kitchen on a wet Saturday morning and set myself the task of writing out the numbers, beginning with 1, 2, 3 and so on. My aim was to write down every possible number! I canât remember how far I got â possibly somewhere in the hundreds â but I do remember the feeling as I gradually realised that I couldnât possibly finish the task and that numbers simply never come to an end. In that split second I got a fleeting glimpse of infinity, that something could go on and on âfor everâ.
At my infants school the sums, as far as I can remember them (there was no talk of âmathematicsâ in those days), were pretty straightforward. I do, however, have a vivid memory of being distraught at seeing a big red cross alongside every sum in my copybook following one particular lesson. I must have misunderstood something pretty basic. Itâs significant that itâs the memory of that particular morning that has remained.
In my next class, having moved from the Infants to Standard 1, the teaching was much more formal â we all sat in rows and, dipping our steel pens into the inkwells on our desks, we copied into our books whatever Miss Williams wrote in chalk on the blackboard. Here again, one particular lesson stands out: a lesson on âlong divisionâ, a pet hate for many. After going through one example on the board, Miss Williams told us to carry on by ourselves to do a dozen or so similar sums. I remember thinking, âWhatâs going on here? I havenât got a clue. What am I supposed to do?â I experienced my own moment of panic that morning, which wasnât helped by my noticing that the other children in the class were hard at work, apparently unconcerned by the challenge. This was my first experience of not understanding something in mathematics â Miss Williamsâs instructions had made no sense at all. The teacher expected us to be content with knowing how, no more than that, whereas I wanted to understand why. Up to that point Iâd managed to understand why whenever we were given new sums to do â add-ups, take-aways and multiplications â but understanding why when faced with long division was completely beyond me. It soon became clear that that was the expectation: donât ask why, just get on with it and keep your head down.
Yesterday, as today, arithmetic includes the skills of addition (+), subtraction (â), multiplication (Ă) and division (Ă·). Yesterday, as today, arithmetic (and the more broadly based idea of numeracy) is vital to the development of full citizenship and includes the application of number in everyday life as well as across other parts of the school curriculum. âArithmetic is the inheritance of civilised nationsâ was the opinion expressed by the winner of an essay competition at the 1859 Wrexham National Eisteddfod, and that sentiment has been repeated consistently over the years.
If there is broad agreement regarding the importance of numeracy, there have been deep disagreements about the means to develop it. We all experience a constant tension between knowing how and understanding why. For some, knowing how is quite sufficient: the only aim of a maths lesson is to get to know how to get the answer. For others, understanding why is just as important, if not more so. âAbsolute nonsense!â, answers the first group. âGetting it right is the only thing that counts.â The two groups disagree fundamentally regarding the nature and purpose of maths. To which group do you belong?
Learning to repeat instructions parrot-fashion â rote learning â was how generations of children experienced sums. Is it at all surprising that so many people who were at primary school before, say, the 1960s have negative attitudes to the subject? Things improved greatly from the 1960s onwards with more emphasis on giving pupils practical experiences in the classroom and on encouraging the use of language in mathematics. But progress has been gradual and change from one generation to the next is necessarily slow.
As a mathematics adviser during the 1980s, I was invited by a primary-school head teacher to call in to talk with one of the teachers who was refusing point blank to adopt âmodernâ methods. I had an interesting conversation with the teacher, a man in his mid-fifties, who had been brought up on the traditional methods and saw no good reason to change: âIf it was good enough for me, itâs good enough for todayâs children too.â As I probed further, with a certain amount of care and diplomacy, it became clear that the teacher himself did not understand the methods that he was passing on to his pupils â he knew how but did not understand why. I was dangerously close to undermining his professional selfrespect; hadnât he been using these methods for decades without a single complaint? We did, however, manage to prise open some new windows in order to expand his perception but itâs doubtful if I managed to convince him completely. Many of the children under his care are themselves parents by now and can see the very different experiences in mathematics that their children are enjoying compared with what was offered to them by this particular teacher.
Is it, therefore, any wonder that the attitudes displayed by adults towards mathematics tend to be polarised? A relatively small number are fascinated by the subject, delighting in its patterns and its insights. Others loathe it completely and are prepared to boast about their incompetence openly and publicly, often referring to unfortunate experiences at school â weekly mental tests, ineffectual teachers, nasty teachers. Is there any truth in the saying, âMaths is like cabbage: you love it or you hate it, depending on how it was served up to you at schoolâ?
2MEETING OF MINDS
QUĂBEC CITY bathes in sunlight as it welcomes an international conference to the green and lush campus of Laval University. In one of the modern lecture theatres delegates have gathered for a morning on the topic of âethnomathematicsâ, and are welcomed at the outset by a small Maori choir greeting us in their native language. The choirâs leader approaches the microphone to address the audience. âMes amisâ, he begins, in a formal secondary-school French, deliberately showing his respect to the conference location and QuĂ©bec stateâs main language. Only a few in the audience are fluent in French but we all understand his simple phrases and are moved by the sincerity of his message: âThe mountains of New Zealand greet the mountains of the state of QuĂ©bec; our valleys greet your valleys; our rivers greet your riversâ, pausing before his final greeting: âAnd our people greet your people.â Another song from the choir and then, a good quarter of an hour into the meeting, one of the other members of the choir comes forward to present his paper.
By now the audience has been completely captivated by the simplicity and force of the presentation and is eager to hear more. This session is one of scores of others that form a conference of some 3,000 mathematics educators that are held once every four years in different cities across the world. One of the sub-themes of every conference is the link between mathematics and culture or rather the links between mathematics and the worldâs diverse cultures, as it becomes clear that mathematics is not a single body of unchallengeable knowledge. Rather, it is interpreted in differing ways by differing cultures, each through its own cultural prism. That day, it was the turn of the Maori to present their particular prism and to show how it influenced the teaching of mathematics in the schools of New Zealand.
There are (at least) two aspects to the relationship between mathematics and culture. On the one hand, the sparse âbeautyâ of mathematics can spark within us a cultural response having the same quality as our response to, say, a Shakespearean sonnet, a sonata by Schubert or a sculpture by Michelangelo. The perfection of Pythagorasâ theorem, for example, can satisfy us at our deepest levels of emotional understanding.
On the other hand, and at a different level, we use practical mathematical ideas on a daily basis â in our homes, with other people, on the high street and at work. We are taught these ideas at school but our interpretation of them is also influenced by our everyday experiences at home and in the community: they are mediated by our culture, and through the language or languages of that culture.
Mathematics is therefore not only a universal cultural phenomenon but also a product of our daily activities. In this latter respect local cultures and their languages are core factors as children grapple with basic mathematical ideas and as adults use those ideas in their everyday lives.
A specific example can help us reflect on the tensions between these two perspectives. One of the long-term concerns of the Maori has been that their children appear to have underachieved in their school mathematics by comparison with other New Zealanders, using the countryâs standard tests as a yardstick. For over a hundred and fifty years mathematics in New Zealand had been taught only through the medium of English, using curricula and textbooks that were rooted in the majority non-Maori culture, and taught in schools that didnât recognise the values basic to the Maori culture. This led inevitably to a situation where the Maori viewed many school subjects â and mathematics in particular â as being foreign to their natur...