Mathematics, like language, is a universal experience. But just as there is a rich variety of languages, so too is there a diversity of methods for counting and recording numbersmethods that have developed over centuries to meet the needs of various groups of people. Count Us In explores these cultural links and differences, drawing examples from the author’s personal experiences. Gareth Ffowc Roberts shows that mathematics“maths” in the United Kingdomis something to enjoy, rather than to fear, and his good-natured, accessible stories will encourage readers to let go of their math anxieties and explore alongside him. As a popular book on mathematics and on the personalities behind its creation, there are no prerequisites beyond the reader’s rudimentary and possibly hazy recollection of primary-school mathematics and a curiosity to know more. Far from being the exclusive domain of specialists and number-crunchers, Roberts makes it clear that math belongs to us all.
|Publisher:||University of Wales Press|
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About the Author
Gareth Ffowc Roberts is Emeritus Professor of Education at Bangor University, and is coeditor of Robert Recorde: the Life and Times of a Tudor Mathematician, also published by the University of Wales Press.
Read an Excerpt
Count Us In
How to Make Maths Real for All of Us
By Gareth Ffowc Roberts
University of Wales PressCopyright © 2016 Gareth Ffowc Roberts
All rights reserved.
MORE 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: 'Make sure that you've got a clean sheet of paper. I'll ask the questions slowly. Everybody ready? No cheating! First question ...' A deathly hush, everyone listening intently. And then, 'Write one word, just one, that expresses how you're feeling: what's going through your mind at this precise moment.' 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.
It's no surprise that the overwhelming number of responses were negative. It was rare for anyone to write words such as 'fantastic' or 'confident'. On the contrary, words such as 'nervous', 'anxious' and 'uncertain' were used far more often. But the most common word by far – year after year – was 'panic' or 'PANIC!' I would have had a negative reaction had I threatened to set a test in any subject – science, language, history, geography, music and so on – 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 × 3) are 6, no arguing, no room for debate, the answer can't 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: 'You still can't remember seven eights (7 × 8)?'
This negativity isn't in our genes; we're 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. 'Well', I answered, 'I'm going to the village hall to give a talk on maths.' 'Oh', said Mari, 'I like doing maths', and then, after a pause, 'And will you be dancing there?' 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: 'Dear algebra, please stop asking us to find your x. She's never coming back, and don't ask y.' 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 wasn't something to be understood; rather it was something to be done – and to be suffered.
The answer often given by children to the gentle parental probe, 'What did you do in school today?' is 'Oh, nothing!' In an effort to snooker her four-year-old daughter at the end of a morning at nursery school, one resourceful parent asked her, 'Now, tell me three things that you did at school this morning.' Not to be outmanoeuvred, the child replied, 'Nothing, nothing, nothing!' 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 she's done before reading further:
Llinos was having trouble writing some of her numerals correctly, reversing the 4 and 5 in this case – a common practice at this age. Some gentle probing on our part revealed that today's lesson had been 'doing add-ups' and the teacher had introduced 'carry one' as a new idea. Llinos had learned to write 'd' (degau – the Welsh for tens) and 'u' (unedau – 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 'u' before moving on to add the numbers under the 'd'. Because the lesson's aim had been to introduce 'carry one', Llinos had also used that in her example, although it didn't 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 she'd been doing. She hadn't interpreted the number in the first line as being thirty-three (33) nor the second number as forty-five (45) – quite a challenge for a five-year-old – and didn't 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. 'That's very good!' was our only possible response.
* * *
'It's all very well for you' is what I'm often told. 'You didn't have any difficulty with maths at school. It wasn't like that for me. I never understood what was going on.' Such sentiments are heart-felt and contain more than a grain of truth. Not that I've 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 can't remember how far I got – possibly somewhere in the hundreds – but I do remember the feeling as I gradually realised that I couldn't 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 'for ever'.
At my infants school the sums, as far as I can remember them (there was no talk of 'mathematics' 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. It's significant that it's 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 – 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 'long division', 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, 'What's going on here? I haven't got a clue. What am I supposed to do?' I experienced my own moment of panic that morning, which wasn't 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 – Miss Williams's 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 I'd managed to understand why whenever we were given new sums to do – add-ups, take-aways and multiplications – but understanding why when faced with long division was completely beyond me. It soon became clear that that was the expectation: don't ask why, just get on with it and keep your head down.
Yesterday, as today, arithmetic includes the skills of addition (+), subtraction (-), multiplication (×) and division (÷). 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. 'Arithmetic is the inheritance of civilised nations' 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. 'Absolute nonsense!', answers the first group. 'Getting it right is the only thing that counts.' The two groups disagree fundamentally regarding the nature and purpose of maths. To which group do you belong?
* * *
Learning to repeat instructions parrot-fashion – rote learning – 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 'modern' 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: 'If it was good enough for me, it's good enough for today's children too.' 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 – he knew how but did not understand why. I was dangerously close to undermining his professional self-respect; hadn't 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 it's 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 – weekly mental tests, ineffectual teachers, nasty teachers. Is there any truth in the saying, 'Maths is like cabbage: you love it or you hate it, depending on how it was served up to you at school'?CHAPTER 2
MEETING OF MINDS
QUÉBEC 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 'ethnomathematics', and are welcomed at the outset by a small Maori choir greeting us in their native language. The choir's leader approaches the microphone to address the audience. 'Mes amis', he begins, in a formal secondary-school French, deliberately showing his respect to the conference location and Québec state's 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: 'The mountains of New Zealand greet the mountains of the state of Québec; our valleys greet your valleys; our rivers greet your rivers', pausing before his final greeting: 'And our people greet your people.' 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 world's 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 'beauty' 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 Pythagoras' 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 – 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 country's 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 didn't recognise the values basic to the Maori culture. This led inevitably to a situation where the Maori viewed many school subjects – and mathematics in particular – as being foreign to their natural culture. The phrase 'cultural alienation' has been used to describe the phenomenon. By today, there have been significant developments in the provision of education through the medium of the Maori language, particularly for the early years; teaching materials that are sensitive to the Maori culture have been developed (such as the inclusion of traditional Maori patterns in work on geometry); and the mathematics is set in contexts that reflect the natural experiences of the children. As a consequence, some of the initial concerns have begun to fade.
Similar experiences have been repeated worldwide. In attempts to modernise their methods of education, and under an historical influence as past colonies of european empires, mathematical ideas and resources were imported from the 'civilised' West during the 1960s and 1970s. As a consequence, mathematics was taught through the language of the country of the 'conqueror' and set within that country's cultural framework. It is hardly surprising that children in the villages of Nigeria were unable to identify with the exploits of Janet and John in an english middle-class suburb. The mathematics curriculum in these countries needed to reflect the local culture and to do so in ways that recognised the richness of the mathematical skills that already existed in those cultures.
Excerpted from Count Us In by Gareth Ffowc Roberts. Copyright © 2016 Gareth Ffowc Roberts. Excerpted by permission of University of Wales Press.
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Table of Contents
Figures and Plates Acknowledgements Preface
1. More cabbage, anyone? 2. Meeting of minds 3. ‘Nothing will come of nothing’ 4. Setting the Record straight 5. ‘Neither a borrower nor a lender be’ 6. Amazing Mayans 7. What do you reckon? 8. Prairie power 9. Putting down digital roots 10. Areas of (mis)understanding 11. Cracking the code 12. Does mathematics have a gender? 13. How to make maths real for all of us
Appendix Answers to puzzles Notes on chapters Further reading